LECTURE NOTES ELECTRICAL CIRCUITS. Prepared By. Dr.D Shobha Rani, Professor, EEE. Ms. S Swathi, Assistant Professor, EEE. B.

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1 LECTURE NOTES on ELECTRICAL CIRCUITS Prepared By Dr.D Shobha Rani, Professor, EEE Ms. S Swathi, Assistant Professor, EEE B.Tech II semester Department of Electrical and Electronics Engineering INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad

2 SYLLABUS Module-I INTRODUCTION TO ELECTRICAL CIRCUITS Circuit concept: Basic definitions, Ohm s law at constant temperature, classifications of elements, R, L, C parameters, independent and dependent sources, voltage and current relationships for passive elements (for different input signals like square, ramp, saw tooth, triangular and complex), temperature dependence of resistance, tolerance, source transformation, Kirchhoff s laws, equivalent resistance of series, parallel and series parallel networks. Module-II ANALYSIS OF ELECTRICAL CIRCUITS Circuit analysis: Star to delta and delta to star transformation, mesh analysis and nodal analysis by Kirchhoff s laws, inspection method, super mesh, super node analysis; Network topology: definitions, incidence matrix, basic tie set and basic cut set matrices for planar networks, duality and dual networks. Module-III SINGLE PHASE AC CIRCUITS AND RESONANCE Single phase AC circuits: Representation of alternating quantities, instantaneous, peak, RMS, average, form factor and peak factor for different periodic wave forms, phase and phase difference, j notation, concept of reactance, impedance, susceptance and admittance, rectangular and polar form, concept of power, real, reactive and complex power, power factor. Steady state analysis: Steady state analysis of RL, RC and RLC circuits (in series, parallel and series parallel combinations) with sinusoidal excitation; Resonance: Series and parallel resonance, concept of band width and Q factor. Module-IV MAGNETIC CIRCUITS AND THREE PHASE CIRCUITS Magnetic circuits: Faraday s laws of electromagnetic induction, concept of self and mutual inductance, dot convention, coefficient of coupling, composite magnetic circuit, analysis of series and parallel magnetic circuits; Three phase circuits: Star and delta connections, phase sequence, relation between line and phase voltages and currents in balanced systems( both Y& ),three phase three wire and three phase four wire systems, analysis of balanced and unbalanced three phase circuits, measurement of active and reactive power Module-V COMPONENTS OF ELECTRICAL SYSTEMS Components of LT Switchgear: Switch Fuse Unit (SFU), MCB, ELCB, MCCB, types of wires and cables, Earthing. Types of batteries, Alkaline battery, zinc carbon battery, dry cell battery, nickel cadmium battery, lead acid battery, lithium ion battery, nickel metal hydride battery, important characteristics for batteries, applications, Elementary calculations for energy consumption. Text Books: 1. A Chakrabarthy, Electric Circuits, DhanipatRai& Sons, 6 th Edition, A Sudhakar, Shyammohan S Palli, Circuits and Networks, Tata McGraw-Hill, 4 th Edition, M E Van Valkenberg, Network Analysis, PHI, 3 rd Edition, D.P. Kothari and I.J. Nagrath, Basic Electrical Engineering, Tata McGraw Hill, 3 rd Edition Reference Books: 1. John Bird, Electrical Circuit Theory and Technology, Newnes, 2 nd Edition, C L Wadhwa, Electrical Circuit Analysis including Passive Network Synthesis, New Age International, 2 nd Edition, David A Bell, Electric circuits, Oxford University Press, 7 th Edition,

3 UNIT I INTRODUCTION TO ELECTRICAL CIRCUITS 3

4 Basic definitions: INTRODUCTION: With increase in population the need for electricity also increases therefore it is necessary to rise number of electrical engineers. The electrical engineering mainly deals with generation, transmission and distribution of electricity. Electrical circuits is the basic and fundamental subject which lays path to understand subjects related to generation, transmission and distribution of electricity. In the first unit we shall deal with what is electrical circuit and formation of electrical circuit. Knowing or unknowing we come across electron in our life daily, studying the properties of these electrons leads to the terms voltage, current and power. Hence the basic definitions of voltage, current, power and energy are studied here. CIRCUIT: The electrical circuit consists of mainly three parts, they are source, connecting wire and load or sink. Source : An source may be battery which forces electrons into the circuit. Connecting wire: This is part which provides path for electrons to flow. Load : The load may be bulb etc. when electrons flow through it an reaches the Source it glows. If the electrons are provided closed path to flow, leads to current is called as electrical circuit. BASIC DEFINITIONS: Voltage (V): The potential difference between force applied to two oppositely charged particles to bring them as near as possible is called as potential difference.( in electrical terminology it s voltage). V = W / Q (v) υ = dw / dq (v) v- volts, units of voltage. - Unit volt is defined as 1C of charge developed when 1 J of energy is applied. Current (I): The flow of electrons develops the current. I = Q/ t (A) i = dq / dt (A) A = Ampere, units of current. - Unit ampere of current is constituted when 1C of charge is flowing in 1S. 4

5 Power(P): It is defined as product of voltage and power in electrical circuits or Rate of change of energy. P = dw/dt = dw/ dq. dq/dt = υ.i (W) W = watts, units of power. Unit watt is the 1J of energy is dissipated in 1S. Energy : It is the capacity to do work or it is defined as power consumed over Given interval of time.(w) W = ʃ P dt.(j) J = Joules, units of energy 1J = 1 watt-sec. Conventional Current Flow Figure 1.1: Conventional current flow Conventionally this is the flow of positive charge around a circuit, being positive to negative. The diagram at the left shows the movement of the positive charge (holes) around a closed circuit flowing from the positive terminal of the battery, through the circuit and returns to the negative terminal of the battery. This flow of current from positive to negative is generally known as conventional current flow. This was the convention chosen during the discovery of electricity in which the direction of electric current was thought to flow in a circuit. To continue with this line of thought, in all circuit diagrams and schematics, the arrows shown on symbols for components such as diodes and transistors point in the direction of conventional current flow. Then Conventional Current Flow gives the flow of electrical current from positive to negative and which is the opposite in direction to the actual flow of electrons. Electron Flow Figure 1.2: Electron flow 5

6 The flow of electrons around the circuit is opposite to the direction of the conventional current flow being negative to positive. The actual current flowing in an electrical circuit is composed of electrons that flow from the negative pole of the battery (the cathode) and return back to the positive pole (the anode) of the battery. This is because the charge on an electron is negative by definition and so is attracted to the positive terminal. This flow of electrons is called Electron Current Flow. Therefore, electrons actually flow around a circuit from the negative terminal to the positive. Both conventional current flow and electron flow are used by many textbooks. In fact, it makes no difference which way the current is flowing around the circuit as long as the direction is used consistently. The direction of current flow does not affect what the current does within the circuit. Generally it is much easier to understand the conventional current flow positive to negative. Ohm s law at constant temperature: Georg Ohm found that, at a constant temperature, the electrical current flowing through a fixed linear resistance is directly proportional to the voltage applied across it, and also inversely proportional to the resistance. This relationship between the Voltage, Current and Resistance forms the basis of Ohms Law and is shown below. Ohms Law Relationship Voltage (v) Current (I) = Resistance (R) by knowing any two values of the Voltage, Current or Resistance quantities we can use Ohms Law to find the third missing value. Ohms Law is used extensively in electronics formulas and calculations so it is very important to understand and accurately remember these formulas. To find the Voltage, ( V ) [ V = I x R ] V (volts) = I (amps) x R (Ω) To find the Current, ( I ) [ I = V R ] I (amps) = V (volts) R (Ω) To find the Resistance, ( R ) [ R = V I ] R (Ω) = V (volts) I (amps) It is sometimes easier to remember this Ohms law relationship by using pictures. Here the three quantities of V, I and R have been superimposed into a triangle (affectionately called the Ohms Law Triangle) giving voltage at the top with current and resistance below. This arrangement represents the actual position of each quantity within the Ohms law formulas. Electrical Power in Circuits Electrical Power, ( P ) in a circuit is the rate at which energy is absorbed or produced within a circuit. A source of energy such as a voltage will produce or deliver power while the connected load absorbs it. Light bulbs and heaters for example, absorb electrical power and convert it into either heat, or light, or both. The higher their value or rating in watts the more electrical power they are likely to consume. The quantity symbol for power is P and is the product of voltage multiplied by the current with the unit of measurement being the Watt ( W ). Prefixes are used to denote the various multiples or sub-multiples of a watt, such as: milliwatts (mw = 10-3 W) or kilowatts (kw = 10 3 W). Then by using Ohm s law and substituting for the values of V, I and R the formula for electrical power can be found as: To find the Power (P) [ P = V x I ] P (watts) = V (volts) x I (amps) 6

7 Also: [ P = V 2 R ] P (watts) = V 2 (volts) R (Ω) Also: [ P = I 2 x R ] P (watts) = I 2 (amps) x R (Ω) Again, the three quantities have been superimposed into a triangle this time called a Power Triangle with power at the top and current and voltage at the bottom. Again, this arrangement represents the actual position of each quantity within the Ohms law power formulas. Electrical Power Rating Electrical components are given a power rating in watts that indicates the maximum rate at which the component converts the electrical power into other forms of energy such as heat, light or motion. For example, a 1/4W resistor, a 100W light bulb etc. Electrical devices convert one form of power into another. So for example, an electrical motor will covert electrical energy into a mechanical force, while an electrical generator converts mechanical force into electrical energy. A light bulb converts electrical energy into both light and heat. Also, we now know that the unit of power is the WATT, but some electrical devices such as electric motors have a power rating in the old measurement of Horsepower or hp. The relationship between horsepower and watts is given as: 1hp = 746W. So for example, a two-horsepower motor has a rating of 1492W, (2 x 746) or 1.5kW. Classifications of elements: Based on the property of the ciruit element are classified as follows 1. Active or passive elements 2. Unilateral or bilateral elements 3. Lumped or distributed elements 4. Linear or non linear elements Active or passive elements Active Circuit Element The circuit elements which supply energy to the circuit are called active circuit element. Examples: Include voltage sources, current sources, and generators such as alternators, DC generators etc. Passive Element The circuit elements that receive energy (or absorb energy) and either convert it into heat or store it in an electric field or a magnetic field are called passive circuit elements. Example: Resistor Unilateral or bilateral elements Unilateral Circuit Element Conduction of current in one direction is termed as unilateral circuit element. It does not offer same resistance to the current of either direction. The resistance of the unilateral circuit element is different for forward current than that of reverse current. Example: diode, transistor etc. Bilateral Circuit Element Conduction of current in both directions in a circuit element with same magnitude is termed as bilateral circuit element. It offers same resistance to the current of either directions. Example: resistor, inductor, capacitor etc. Lumped or distributed elements 7

8 Lumped Circuit Elements When the voltage across and current through the element don't vary with dimension of the element, it is called lumped circuit elements. Example: Resistor connected in any electrical circuit. Distributed Circuit Element When the voltage across and current through the element change with dimensions of the element, it is called distributed circuit element. Example: Resistance of the transmission line. It varies with the length of the line. Linear or non linear elements Linear circuit elements The elements which possess the linear voltage and current characteristics are called linear cicuit elements Example: Resistor Non linear elements The elements which does not possess the linear voltage and current characteristics are called non linear circuit elements Example: diode, transistor etc BASIC PARAMETERS: Any electrical mainly consists of three important elements, they are resistor, inductor and capacitor. Let us deal these parameters in detail. Resistor: resistor is the element which restricts flow of electrons and this property of opposing electrons is called as resistance. OHM s Law : ohm s law states that current flowing through circuit is directly proportional to potential difference applied. ( at constant temperature) I α V, at constant T. I.R = V. R = V / I, hence resistance can be calculated as ratio of Voltage to current in any element or circuit. = OHMS ῼ, units of resistance. R = Inductor: An length of wire twisted forms the basic inductor.(l). when alternating curent is allowed 8

9 through such a element it induces voltage in it. Where, e = emf induced ɸ = flux developed in it for current i. Hence, e = L di/dt (v). L= is the inductance of the coil(h) H = Henry units of inductance. Unit H is the 1v of voltage induced in coil when current Changing at arte of 1A/S. by solving above equation, current flowing through coil is given as, i = (1/L) ʃ v dt + i(o+). Energy stored in indcutor, W = ʃ e.i dt = ʃ L di/dt. i dt. = (1/2) L i 2 Properties: Indcutor doesn t allow sudden changes in current. If DC supply is provided to indcutor it acts as short circuit. Pure inductor is non-dissipative element i.e its internal resistance is zero. Stores energy in the form of magnetic field. Capacitor: Two parallel plates oppositely charged separated by an di-electric medium constitutes an capacitor. v, i when some voltage v is applied, i is the current flowing through capacitor, given as i = c dv/dt c = capacitance of the capacitor.(f) Farad is the unit of capacitance, 1F is the when 1A 9

10 Flow if 1v applied for 1S. Voltage across capacitor is given as, v = (1/c) ʃ i dt + v(0+). Energy stored in capacitor is, w = ʃ v.i dt = ʃ c dv/dt. i dt. = (1/2) cv 2 Properties: Capacitor doesn t allow sudden changes in voltage. If DC supply is provided to capacitor it acts as open circuit. Pure capacitor is non-dissipative element i.e its internal resistance is zero. Stores energy in the form of electric field. Independent and dependent sources: The Voltage Source A voltage source, such as a battery or generator, provides a potential difference (voltage) between two points within an electrical circuit allowing current to flowing around it. Remember that voltage can exist without current. A battery is the most common voltage source for a circuit with the voltage that appears across the positive and negative terminals of the source being called the terminal voltage. Ideal Voltage Source Figure1.4: Ideal voltage source and its characteristics An ideal voltage source is defined as a two terminal active element that is capable of supplying and maintaining the same voltage, (v) across its terminals regardless of the current, (i) flowing through it. In other words, an ideal voltage source will supply a constant voltage at all times regardless of the value of the current being supplied producing an I-V characteristic represented by a straight line. Then an ideal voltage source is known as an Independent Voltage Source as its voltage does not depend on either the value of the current flowing through the source or its direction but is determined solely by the value of the source alone. So for example, an automobile battery has a 12V terminal voltage that remains constant as long as the current through it does not become to high, delivering power to the car in one direction and absorbing power in the other direction as it charges. Practical Voltage Source We have seen that an ideal voltage source can provide a voltage supply that is independent of the current flowing through it, that is, it maintains the same voltage value always. This idea may work well for circuit analysis techniques, but in the real world voltage sources behave a little differently as for a practical voltage source; its terminal voltage will actually decrease with an increase in load current. As the terminal voltage of an ideal voltage source does not vary with increases in the load current, this implies that an ideal voltage source has zero internal resistance, R S = 0. In other words, it is a resistor less 10

11 voltage source. In reality all voltage sources have a very small internal resistance which reduces their terminal voltage as they supply higher load currents. For non-ideal or practical voltage sources such as batteries, their internal resistance (R S ) produces the same effect as a resistance connected in series with an ideal voltage source as these two series connected elements carry the same current as shown. Ideal and Practical Voltage Source Figure1.5: Ideal and practical voltage source Practical Voltage Source Characteristics Figure1.6: practical voltage source and its characteristics Dependent Voltage Source Unlike an ideal voltage source which produces a constant voltage across its terminals regardless of what is connected to it, a controlled or dependent voltage source changes its terminal voltage depending upon the voltage across, or the current through, some other element connected to the circuit, and as such it is sometimes difficult to specify the value of a dependent voltage source, unless you know the actual value of the voltage or current on which it depends. Dependent voltage sources behave similar to the electrical sources we have looked at so far, both practical and ideal (independent) the difference this time is that a dependent voltage source can be controlled by an input current or voltage. A voltage source that depends on a voltage input is generally referred to as a Voltage Controlled Voltage Source or VCVS. A voltage source that depends on a current input is referred to as a Current Controlled Voltage Source or CCVS. 11

12 Ideal dependent sources are commonly used in the analysing the input/output characteristics or the gain of circuit elements such as operational amplifiers, transistors and integrated circuits. Generally, an ideal voltage dependent source, either voltage or current controlled is designated by a diamond-shaped symbol as shown. Dependent Voltage Source Symbols Ideal Current Source Figure1.7: dependent voltage source symbol Figure1.8: Ideal current source and its characteristics Then an ideal current source is called a constant current source as it provides a constant steady state current independent of the load connected to it producing an I-V characteristic represented by a straight line. As with voltage sources, the current source can be either independent (ideal) or dependent (controlled) by a voltage or current elsewhere in the circuit, which itself can be constant or time-varying. Ideal independent current sources are typically used to solve circuit theorems and for circuit analysis techniques for circuits that containing real active elements. The simplest form of a current source is a resistor in series with a voltage source creating currents ranging from a few milli-amperes to many hundreds of amperes. Remember that a zero-value current source is an open circuit as R = 0. The concept of a current source is that of a two-terminal element that allows the flow of current indicated by the direction of the arrow. Then a current source has a value, i, in units of amperes, (A) which are typically abbreviated to amps. The physical relationship between a current source and voltage variables around a network is given by Ohm s law as these voltage and current variables will have specified values. It may be difficult to specify the magnitude and polarity of voltage of an ideal current source as a function of the current especially if there are other voltage or current sources in the connected circuit. Then we may know the current supplied by the current source but not the voltage across it unless the power supplied by the current source is given, as P = V*I. However, if the current source is the only source within the circuit, then the polarity of voltage across the source will be easier to establish. If however there is more than one source, then the terminal voltage will be dependent upon the network in which the source is connected. 12

13 Practical Current Source We have seen that an ideal constant current source can supply the same amount of current indefinitely regardless of the voltage across its terminals, thus making it an independent source. This therefore implies that the current source has an infinite internal resistance, (R = ). This idea works well for circuit analysis techniques, but in the real world current sources behave a little differently as practical current sources always have an internal resistance, no matter how large (usually in the mega-ohms range), causing the generated source to vary somewhat with the load. A practical or non-ideal current source can be represented as an ideal source with an internal resistance connected across it. The internal resistance (R P ) produces the same effect as a resistance connected in parallel (shunt) with the current source as shown. Remember that circuit elements in parallel have exactly the same voltage drop across them. Ideal and Practical Current Source Figure1.9: Ideal and practical current source Practical Current Source Characteristics Figure1.10: practical current source and its characteristics Dependent Current Source We now know that an ideal current source provides a specified amount of current completely independent of the voltage across it and as such will produce whatever voltage is necessary to maintain the required current. This then makes it completely independent of the circuit to which it is connected to resulting in it being called an ideal independent current source. A controlled or dependent current source on the other hand changes its available current depending upon the voltage across, or the current through, some other element connected to the circuit. In other words, the output of a dependent current source is controlled by another voltage or current. 13

14 Dependent current sources behave similar to the current sources we have looked at so far, both ideal (independent) and practical. The difference this time is that a dependent current source can be controlled by an input voltage or current. A current source that depends on a voltage input is generally referred to as a Voltage Controlled Current Source or VCCS. A current source that depends on a current input is generally referred too as a Current Controlled Current Source or CCCS. Generally, an ideal current dependent source, either voltage or current controlled is designated by a diamond-shaped symbol where an arrow indicates the direction of the current, i as shown. Dependent Current Source Symbols Figure1.11: dependent current source Kirchhoff s laws: Kirchhoff s First Law The Current Law, (KCL) Kirchhoff s Current Law or KCL, states that the total current or charge entering a junction or node is exactly equal to the charge leaving the node as it has no other place to go except to leave, as no charge is lost within the node. In other words the algebraic sum of ALL the currents entering and leaving a node must be equal to zero, I (exiting) + I (entering) = 0. This idea by Kirchhoff is commonly known as the Conservation of Charge. Kirchhoff s Current Law Figure1.12: Kirchhoff s current Law Here, the three currents entering the node, I 1, I 2, I 3 are all positive in value and the two currents leaving the node, I 4 and I 5 are negative in value. Then this means we can also rewrite the equation as; I 1 + I 2 + I 3 I 4 I 5 = 0 The term Node in an electrical circuit generally refers to a connection or junction of two or more current carrying paths or elements such as cables and components. Also for current to flow either in or out of a node a closed circuit path must exist. We can use Kirchhoff s current law when analysing parallel circuits. Kirchhoff s Second Law The Voltage Law, (KVL) Kirchhoff s Voltage Law or KVL, states that in any closed loop network, the total voltage around the loop is equal to the sum of all the voltage drops within the same loop which is also equal to zero. In other words the algebraic 14

15 sum of all voltages within the loop must be equal to zero. This idea by Kirchhoff is known as the Conservation of Energy. Kirchhoffs Voltage Law Figure1.13: Kirchoff s voltage law Starting at any point in the loop continue in the same direction noting the direction of all the voltage drops, either positive or negative, and returning back to the same starting point. It is important to maintain the same direction either clockwise or anti-clockwise or the final voltage sum will not be equal to zero. We can use Kirchhoff s voltage law when analysing series circuits. When analysing either DC circuits or AC circuits using Kirchhoffs Circuit Laws a number of definitions and terminologies are used to describe the parts of the circuit being analysed such as: node, paths, branches, loops and meshes. These terms are used frequently in circuit analysis so it is important to understand them. Common DC Circuit Theory Terms: Circuit a circuit is a closed loop conducting path in which an electrical current flows. Path a single line of connecting elements or sources. Node a node is a junction, connection or terminal within a circuit were two or more circuit elements are connected or joined together giving a connection point between two or more branches. A node is indicated by a dot. Branch a branch is a single or group of components such as resistors or a source which are connected between two nodes. Loop a loop is a simple closed path in a circuit in which no circuit element or node is encountered more than once. Mesh a mesh is a single open loop that does not have a closed path. There are no components inside a mesh. Application of Kirchhoff s Circuit Laws These two laws enable the Currents and Voltages in a circuit to be found, ie, the circuit is said to be Analysed, and the basic procedure for using Kirchhoff s Circuit Laws is as follows: Assume all voltages and resistances are given. ( If not label them V1, V2, R1, R2, etc. ) Label each branch with a branch current. ( I1, I2, I3 etc. ) Find Kirchhoff s first law equations for each node. Find Kirchhoff s second law equations for each of the independent loops of the circuit. Use Linear simultaneous equations as required to find the unknown currents. As well as using Kirchhoffs Circuit Law to calculate the various voltages and currents circulating around a linear circuit, we can also use loop analysis to calculate the currents in each independent loop which 15

16 helps to reduce the amount of mathematics required by using just Kirchhoff's laws. In the next tutorial about DC circuits, we will look at Mesh Current Analysis to do just that. A Single Circuit Loop Kirchhoff s voltage law states that the algebraic sum of the potential differences in any loop must be equal to zero as: ΣV = 0. Since the two resistors, R 1 and R 2 are wired together in a series connection, they are both part of the same loop so the same current must flow through each resistor. Thus the voltage drop across resistor, R 1 = I*R 1 and the voltage drop across resistor, R 2 = I*R 2 giving by KVL: We can see that applying Kirchhoff s Voltage Law to this single closed loop produces the formula for the equivalent or total resistance in the series circuit and we can expand on this to find the values of the voltage drops around the loop. 16

17 Kirchhoff s Voltage Law Example No1 Three resistor of values: 10 ohms, 20 ohms and 30 ohms, respectively are connected in series across a 12 volt battery supply. Calculate: a) the total resistance, b) the circuit current, c) the current through each resistor, d) the voltage drop across each resistor, e) verify that Kirchhoff s voltage law, KVL holds true. a) Total Resistance (R T) R T = R 1 + R 2 + R 3 = 10Ω + 20Ω + 30Ω = 60Ω Then the total circuit resistance R T is equal to 60Ω b) Circuit Current (I) Thus the total circuit current I is equal to 0.2 amperes or 200mA c) Current Through Each Resistor The resistors are wired together in series, they are all part of the same loop and therefore each experience the same amount of current. Thus: I R1 = I R2 = I R3 = I SERIES = 0.2 amperes d) Voltage Drop Across Each Resistor V R1 = I x R 1 = 0.2 x 10 = 2 volts V R2 = I x R 2 = 0.2 x 20 = 4 volts V R3 = I x R 3 = 0.2 x 30 = 6 volts e) Verify Kirchhoff s Voltage Law Thus Kirchhoff s voltage law holds true as the individual voltage drops around the closed loop add up to the total. 17

18 Kirchhoff s Circuit Loop We have seen here that Kirchhoff s voltage law, KVL is Kirchhoff s second law and states that the algebraic sum of all the voltage drops, as you go around a closed circuit from some fixed point and return back to the same point, and taking polarity into account, is always zero. That is ΣV = 0 The theory behind Kirchhoff s second law is also known as the law of conservation of voltage, and this is particularly useful for us when dealing with series circuits, as series circuits also act as voltage dividers and the voltage divider circuit is an important application of many series circuits. Current Divider Statement: The electrical current entering the node of a parallel circuit is divided into the branches. Current divider formula is employed to calculate the magnitude of divided current in the circuits. Let's understand the basic definitions: Node: A point where two or more than two components are joined. Parallel circuit: The circuit in which one end of all components share a common node, and the other end of all components share the other common node. You can learn more about parallel circuit configuration here. General formula A parallel circuit with 'n' number of resistors and an input voltage source is illustrated below. We are interested to find the current which is flowing through R x. 18

19 In the above formula: I x : The current through R x. I t : The total current which enters the circuit. R x : The resistance of the component whose current value is to be determined R t : The equivalent resistance of the parallel circuit For two resistors Let's consider a parallel circuit having two resistors R 1 and R 2. The current I t enters the node. We are interested to calculate the current that is flowing through. The general formula and circuit now take the form: We can modify the previous equation to obtain an alternative formula: 19

20 : Resistors in parallel divide up the current. When we have a current flowing through resistors in parallel, we can express the current flowing through a single resistor as ratio of currents and resistances, without ever knowing the voltage. In the circuit above or where i is the current flowing through all the resistors. Note that the numerator on the right is R2, not R1. Remember that a larger resistance will carry a smaller current. We can generalize the equation for N resistors in parallel with the equation: where i k is the current flowing through resistor k and i is the current flowing through all the resistors. 20

21 Example # 1: A 5 kω resistor connects in parallel to a 20 kω resistor. 5 A current enters the node. Find the current across both resistors. Solution: Derivation of Current Divider formula The derivation of CDR formula is very simple. Let's reconsider the general circuit: Apply the Ohm's law on R x. I x = E/R x where E = I t R t. I x = I t R t /R x. or I x = [R t /R x ] * I t 21

22 Equivalent resistance of series and parallel networks: Resistors in Series Resistors are said to be connected in Series, when they are daisy chained together in a single line. Since all the current flowing through the first resistor has no other way to go it must also pass through the second resistor and the third and so on. Then, resistors in series have a Common Current flowing through them as the current that flows through one resistor must also flow through the others as it can only take one path. Then the amount of current that flows through a set of resistors in series will be the same at all points in a series resistor network. For example: Figure1.14: Resistors connected in series R EQ = R 1 + R 2 + R 3 Where four, five or even more resistors are all connected together in a series circuit, the total or equivalent resistance of the circuit, R T would still be the sum of all the individual resistors connected together and the more resistors added to the series, the greater the equivalent resistance (no matter what their value). This total resistance is generally known as the Equivalent Resistance and can be defined as; a single value of resistance that can replace any number of resistors in series without altering the values of the current or the voltage in the circuit. Then the equation given for calculating total resistance of the circuit when connecting together resistors in series is given as: Series Resistor Equation R total = R 1 + R 2 + R R n etc. Resistors in Parallel Since there are multiple paths for the supply current to flow through, the current may not be the same through all the branches in the parallel network. However, the voltage drop across all of the resistors in a parallel resistive network IS the same. Then, Resistors in Parallel have a Common Voltage across them and this is true for all parallel connected elements. So we can define a parallel resistive circuit as one where the resistor are connected to the same two points (or nodes) and is identified by the fact that it has more than one current path connected to a common voltage source. Then in our parallel resistor example below the voltage across resistor R 1 equals the voltage across resistor R 2 which equals the voltage across R 3 and which equals the supply voltage. 22

23 Parallel Resistor Equation Figure1.15: Resistors connected in parallel 1 = Req R1 R2 R3 Rn Series, parallel connection of resistors, inductors and capacitors and their equivalents: Resistors in Series and Parallel Resistor circuits that combine series and parallel resistors networks together are generally known as Resistor Combination or mixed resistor circuits. The method of calculating the circuits equivalent resistance is the same as that for any individual series or parallel circuit and hopefully we now know that resistors in series carry exactly the same current and that resistors in parallel have exactly the same voltage across them. For example, in the following circuit calculate the total current ( I T ) taken from the 12v supply. At first glance this may seem a difficult task, but if we look a little closer we can see that the two resistors, R 2 and R 3 are actually both connected together in a SERIES combination so we can add them together to produce an equivalent resistance the same as we did in the series resistor tutorial. The resultant resistance for this combination would therefore be: 23

24 R 2 + R 3 = 8Ω + 4Ω = 12Ω So we can replace both resistor R 2 and R 3 above with a single resistor of resistance value 12Ω So our circuit now has a single resistor R A in PARALLEL with the resistor R 4. Using our resistors in parallel equation we can reduce this parallel combination to a single equivalent resistor value of R (combination) using the formula for two parallel connected resistors as follows. The resultant resistive circuit now looks something like this: We can see that the two remaining resistances, R 1 and R (comb) are connected together in a SERIES combination and again they can be added together (resistors in series) so that the total circuit resistance between points A and B is therefore given as: R (AB) = R comb + R 1 = 6Ω + 6Ω = 12Ω. and a single resistance of just 12Ω can be used to replace the original four resistors connected together in the original circuit. Now by using Ohm s Law, the value of the circuit current ( I) is simply calculated as: 24

25 So any complicated resistive circuit consisting of several resistors can be reduced to a simple single circuit with only one equivalent resistor by replacing all the resistors connected together in series or in parallel using the steps above. It is sometimes easier with complex resistor combinations and resistive networks to sketch or redraw the new circuit after these changes have been made, as this helps as a visual aid to the maths. Then continue to replace any series or parallel combinations until one equivalent resistance, R EQ is found. Lets try another more complex resistor combination circuit. 25

26 UNIT II ANALYSIS OF ELECTRICAL CIRCUITS 26

27 Introduction: Three branches in an electrical network can be connected in numbers of forms but most common among them is either star or delta form. In delta connection, three branches are so connected, that they form a closed loop. As these three branches are connected nose to tail, they form a triangular closed loop, this configuration is referred as delta connection. On the other hand, when either terminal of three branches is connected to a common point to form a Y like pattern is known as star connection. But these star and delta connections can be transformed from one form to another. For simplifying complex network, delta to star or star to delta transformation is often required. Delta - Star Transformation: The replacement of delta or mesh by equivalent star connection is known as delta - star transformation. The two connections are equivalent or identical to each other if the impedance is measured between any pair of lines. That means, the value of impedance will be the same if it is measured between any pair of lines irrespective of whether the delta is connected between the lines or its equivalent star is connected between that lines. Consider a delta system that's three corner points are A, B and C as shown in the figure. Electrical resistance of the branch between points A and B, B and C and C and A are R 1, R 2 and R 3 respectively. The resistance between the points A and B will be, Now, one star system is connected to these points A, B, and C as shown in the figure. Three arms R A, R B and R C of the star system are connected with A, B and C respectively. Now if we measure the resistance value between points A and B, we will get, Since the two systems are identical, resistance measured between terminals A and B in both systems must be equal. Similarly, resistance between points B and C being equal in the two systems, And resistance between points C and A being equal in the two systems, Adding equations (I), (II) and (III) we get, 27

28 Subtracting equations (I), (II) and (III) from equation (IV) we get, The relation of delta - star transformation can be expressed as follows. The equivalent star resistance connected to a given terminal, is equal to the product of the two delta resistances connected to the same terminal divided by the sum of the delta connected resistances. If the delta connected system has same resistance R at its three sides then equivalent star resistance r will be, Star - Delta Transformation: For star - delta transformation we just multiply equations (v), (VI) and (VI), (VII) and (VII), (V) that is by doing (v) (VI) + (VI) (VII) + (VII) (V) we get, Now dividing equation (VIII) by equations (V), (VI) and equations (VII) separately we get, 28

29 Star Delta Example Convert the following Star Resistive Network into an equivalent Delta Network. Numerical problems on star-delta transformation: 1. Determine the resistance between the terminals A&B and hence find the current through the voltage source. Refer figure 1 Figure: 1 29

30 Sol: See figure 2 Figure: 2 The resistors in between point 1, 2&3 are about to replace by a star connected system. Otherwise is difficult to find the total resistance. So we have to use the delta to star transformation equations. R 1 = R 12 R 31 / (R 12 +R 23 +R 31 ) R 1 = (60*40)/ ( ) R 1 = 12Ω R 2 = R 23 R 12 / (R 12 +R 23 +R 31 ) R 1 = (100*60)/ 200 R 1 = 30Ω R 3 = R 31 R 23 / (R 12 +R 23 +R 31 ) R 3 = (100*40)/ 200 R 3 = 20Ω So we can redraw the network as shown in figure 3 30

31 Figure: 3 Now we can easily find the total resistance between A&B terminals R total = [(80+20)//(88+12)] + 30 R total = R total = 80Ω Applying ohm s law to the total resistance, I = V/R I = 160v/80Ω I = 2A Mesh Analysis: While Kirchhoff s Laws give us the basic method for analyzing any complex electrical circuit, there are different ways of improving upon this method by using Mesh Current Analysis or Nodal Voltage Analysis that results in a lessening of the math s involved and when large networks are involved this reduction in math s can be a big advantage. For example, consider the electrical circuit example from the previous section. Mesh Current Analysis Circuit: Figure: 1 31

32 One simple method of reducing the amount of math s involved is to analyse the circuit using Kirchhoff s Current Law equations to determine the currents, I 1 and I 2 flowing in the two resistors. Then there is no need to calculate the current I 3 as it s just the sum of I 1 and I 2. So Kirchhoff s second voltage law simply becomes: Equation No 1 : 10 = 50I I 2 Equation No 2 : 20 = 40I I 2 Therefore, one line of math s calculation has been saved. Mesh Current Analysis: An easier method of solving the above circuit is by using Mesh Current Analysis or Loop Analysis which is also sometimes called Maxwell s Circulating Currents method. Instead of labeling the branch currents we need to label each closed loop with a circulating current. As a general rule of thumb, only label inside loops in a clockwise direction with circulating currents as the aim is to cover all the elements of the circuit at least once. Any required branch current may be found from the appropriate loop or mesh currents as before using Kirchhoff s method. For example: i 1 = I 1, i 2 = -I 2 and I 3 = I 1 I 2 We now write Kirchhoff s voltage law equation in the same way as before to solve them but the advantage of this method is that it ensures that the information obtained from the circuit equations is the minimum required to solve the circuit as the information is more general and can easily be put into a matrix form. For example, consider the circuit from the previous section. Figure: 2 These equations can be solved quite quickly by using a single mesh impedance matrix Z. Each element ON the principal diagonal will be positive and is the total impedance of each mesh. Whereas, each element OFF the principal diagonal will either be zero or negative and represents the circuit element connecting all the appropriate meshes. First we need to understand that when dealing with matrices, for the division of two matrices it is the same as multiplying one matrix by the inverse of the other as shown. 32

33 Having found the inverse of R, as V/R is the same as V x R -1, we can now use it to find the two circulating currents. Where: [ V ] gives the total battery voltage for loop 1 and then loop 2 [ I ] states the names of the loop currents which we are trying to find [ R ] is the resistance matrix [ R -1 ] is the inverse of the [ R ] matrix And this gives I 1 as Amps and I 2 as Amps As: I 3 = I 1 I 2 The combined current of I 3 is therefore given as : (-0.429) = Amps Which is the same value of amps, we found using Kirchhoff s circuit law in the previous tutorial. Numerical Problems on KVL: 1. Find the current flowing in the 40Ω Resistor, R 3 The circuit has 3 branches, 2 nodes (A and B) and 2 independent loops. Using Kirchhoff s Current Law, KCL the equations are given as: At node A: I 1 + I 2 = I 3 33

34 At node B: I 3 = I 1 + I 2 Using Kirchhoff s Voltage Law, KVL the equations are given as: Loop 1 is given as: 10 = R 1 I 1 + R 3 I 3 = 10I I 3 Loop 2 is given as: 20 = R 2 I 2 + R 3 I 3 = 20I I 3 Loop 3 is given as: = 10I 1 20I 2 As I 3 is the sum of I 1 + I 2 we can rewrite the equations as; Eq. No 1: 10 = 10I (I 1 + I 2 ) = 50I I 2 Eq. No 2: 20 = 20I (I 1 + I 2 ) = 40I I 2 We now have two Simultaneous Equations that can be reduced to give us the values of I 1 and I 2 Substitution of I 1 in terms of I 2 gives us the value of I 1 as Amps Substitution of I 2 in terms of I 1 gives us the value of I 2 as Amps As : I 3 = I 1 + I 2 The current flowing in resistor R 3 is given as: = Amps and the voltage across the resistor R 3 is given as : x 40 = volts The negative sign for I 1 means that the direction of current flow initially chosen was wrong, but never the less still valid. In fact, the 20V battery is charging the 10v battery. Nodal Analysis in Electric Circuits: Definition of Nodal Analysis: Nodal analysis is a method that provides a general procedure for analyzing circuits using node voltages as the circuit variables. Nodal Analysis is also called the Node-Voltage Method. Some Features of Nodal Analysis are as Nodal Analysis is based on the application of the Kirchhoff s Current Law (KCL). Having n nodes there will be n-1 simultaneous equations to solve. Solving n-1 equations all the nodes voltages can be obtained. The number of non reference nodes is equal to the number of Nodal equations that can be obtained. Types of Nodes in Nodal Analysis Non Reference Node - It is a node which has a definite Node Voltage. e.g. Here Node 1 and Node 2 are the Non Reference nodes Reference Node - It is a node which acts a reference point to the other entire node. It is also called the Datum Node. Types of Reference Nodes 1. Chassis Ground - This type of reference node acts a common node for more than one circuits 2. Earth Ground - When earth potential is used as a reference in any circuit then this type of reference node is called Earth Ground. Solving of Circuit Using Nodal Analysis: Basic Steps Used in Nodal Analysis 1. Select a node as the reference node. Assign voltages V 1, V 2... V n-1 to the remaining nodes. The voltages are referenced with respect to the reference node. 34

35 2. Apply KCL to each of the non reference nodes. 3. Use Ohm s law to express the branch currents in terms of node voltages. Node always assumes that current flows from a higher potential to a lower potential in resistor. Hence, current is expressed as follows 4. After the application of Ohm s Law get the n-1 node equations in terms of node voltages and resistances. 5. Solve n-1 node equations for the values of node voltages and get the required node Voltages as result. Nodal analysis with current sources is very easy and it is discussed with a example below. Example: Calculate Node Voltages in following circuit In the following circuit we have 3 nodes from which one is reference node and other two are non reference nodes - Node 1 and Node 2. Step I. Assign the nodes voltages as v 1 and 2 and also mark the directions of branch currents with respect to the reference nodes 35

36 Step II. Apply KCL to Nodes 1 and 2 KCL at Node 1 KCL at Node 2 Step III. Apply Ohm s Law to KCL equations Ohm s law to KCL equation at Node 1 Simplifying the above equation we get, Now, Ohm s Law to KCL equation at Node 2 Simplifying the above equation we get Step IV. Now solve the equations 3 and 4 to get the values of v 1 and v 2 as, Using elimination method And substituting value v 2 = 20 Volts in equation (3) we get- Hence node voltages are as v 1 = Volts and v 2 = 20 Volts. Nodal Analysis with Voltage Sources: Case I. If a voltage source is connected between the reference node and a non reference node, we simply set the voltage at the non-reference node equal to the voltage of the voltage source and its analysis can be done as we done with current sources. v 1 = 10 Volts. Case II. If the voltage source is between the two non reference nodes then it forms a supernode whose analysis is done as following 36

37 Super node Analysis Definition of Super Node Whenever a voltage source (Independent or Dependent) is connected between the two non reference nodes then these two nodes form a generalized node called the Super node. So, Super node can be regarded as a surface enclosing the voltage source and its two nodes. In the above Figure 5V source is connected between two non reference nodes Node - 2 and Node - 3. So here Node - 2 and Node - 3 form the Super node Properties of Super node Always the difference between the voltage of two non reference nodes is known at Super node. A super node has no voltage of its own A super node requires application of both KCL and KVL to solve it. Any element can be connected in parallel with the voltage source forming the super node. A Super node satisfies the KCL as like a simple node. How Solve Any Circuit Containing Super node Let's take a example to understand how to solve circuit containing Super node Here 2V voltage source is connected between Node-1 and Node-2 and it forms a Super node with a 10Ω resistor in parallel. Note - Any element connected in parallel with the voltage source forming Super node doesn t make any difference because v 2 - v 1 = 2V always whatever may be the value of resistor. Thus 10 Ω can be removed and circuit is redrawn and applying KCL to the super node as shown in figure gives, 37

38 Expressing and in terms of the node voltages. From Equation 5 and 6 we can write as Hence, v 1 = V and v 2 = V which is required answer. Super mesh Circuit Analysis: Super mesh or Super mesh Analysis is a better technique instead of using Mesh analysis to analysis such a complex electric circuit or network, where two meshes have a current source as a common element. This is the same where we use Super node circuit analysis instead of Node or Nodal circuit analysis to simplify such a network where the assign super node, fully enclosing the voltage source inside the super node and reducing the number of none reference nodes by one (1) for each voltage source. In Supermesh circuit analysis technique, the current source is in the inner area of the Super mesh. Therefore, we are able to reduce the number of meshes by one (1) for each current source which is present in the circuit. The single mesh can be ignored, if current source (in that mesh) lies on the perimeter of the circuit. Alternatively, KVL (Kirchhoff s Voltage Law) is applied only to those meshes or super meshes in the renewed circuit. By the way, it is difficult to understand by Preamble, so we will first solve a simple circuit by super mesh circuit analyses, and then, we will summarize the whole super mesh analysis (step by step). Summary of Super mesh Analysis: 1. Evaluate if the circuit is a planer circuit. if yes, apply Super mesh. If no, perform nodal analysis instead. 2. Redraw the circuit if necessary and count the number of meshes in the circuit. 3. Label each of mesh currents in the circuit. As a rule of thumb, defining all the mesh currents to flow clockwise result in a simpler circuit analysis. 4. Form a super mesh if the circuit contains current sources by two meshes. So that, the super mesh would enclose both meshes. 5. Write a KVL (Kirchhoff s Voltage Law) around each mesh and super mesh in the circuit. Begin with an easy and will fitted one node. Now proceed in the direction of the mesh current. Take the - sign in the account while writing KVL equations and solving the circuit. No KVL equation is needed if a current source lies on the periphery of a mesh. So, the mesh current is determined and evaluated by inspection. 38

39 6. One KCL (Kirchhoff s Current Law) is needed for each super mesh defined and can be accomplished by simple application of KCL. in simple words, relate the current flowing from each current source to mesh currents. 7. An additional case can be occurred if the circuit contains on further dependent sources. In this case, express any additional unknown values and quantities like currents ir voltages other than the mesh currents in terms of suitable mesh currents. 8. Arrange and organize the system of equations. 9. At last, solve the system of equations for the Nodal voltages such as V1, V2, and V3 etc. there will be Mesh of them. if you find difficulties to solve the system of equations, refer to the above example. Super node Circuit Analysis: Today, we will try to answer the common question that why we use Super node circuit analysis while we can simplify the circuit by simple Node or Nodal Circuit analysis. In previous article, we have discussed that why we use super mesh circuit analysis instead of using simple mesh analysis for circuit simplification. If you got this point, then this is the same case about the discussion. If not satisfied, let me try to explain in the following example. Consider both circuits in the following fig 1. Did you notice something different? Difference between Node / Nodal & Super node Analysis The difference in both circuits is that there is an additional voltage source of 22V instead of 7Ω resistor between node 2 and node 3. And this is the main point. In Node or Nodal analysis, we apply the KCL (Kirchhoff s Current Law) at each non-reference nodes i.e. we apply the simple KCL at once on three nodes in fig 1(a). If we do the same i.e. apply the Nodal analysis instead on Super node circuit analysis on the circuit in fig 1 (b), we face some difficulty at Node1 and Node2, because we don t know that what is the current in the branch with the voltage source? In addition, there is no such a way by which we adjust the situation i.e. we can t express the current as a function of the voltage, where the definition of the voltage source is that the voltage is independent of the current. Due to these difficulties and troubles, we use super node circuit analysis instead of Nodal analysis in the above fig 1 (b). There are two methods to simplify the circuit in the above fig 1 (b). The 1 st one, which is more complex, is that to assign an unknown current value to the branch contains the voltage source. Then apply KCL three times on the 3 Nodes (one KCL equation for each node). At last, apply KVL (Kirchhoff s Voltage Law) which is v 3 v 2 = 22V between Node2 and Node3. In this case, we get four (4) equations for unknown values in the above example, which is little bit complex to simplify. The 2 nd method is easier than the above method which is called Super node analysis. In this method, we treat Node2, Node3 and the voltage source of 22V together as a sort of Super node and apply KCL to both nodes (Nod2 and Node3) at once. The super node is indicated by the region enclosed by the dotted line. This is possible because, if the total current leaving Node2 is zero (0) and the total current leaving Node3 is zero (0), the total current leaving the combination is zero. This concept is shown in the following fig 2 (b) with the super node (the area enclosed by the broken line). 39

40 Expanded view of the region defined as Super node Now, we will solve the circuit below by step by step super node circuit analysis and then, we will summarize the whole super node analysis (step by step). Solved Example of Super node Analysis: Example: Use Super node analysis to find voltage across each current source i.e. v 1 & v 2 in the following fig SUPERNODE Circuit Analysis Step by Step with Solved Example Solution: First, we redraw the circuit as shown in fig 3(b) We begin by writing a KCL equation for Node1. 4 = 0 + 3v 1 + 3v 3. Eq 1. Now, consider the super node (Combination of Node1 and Node2). Moreover, one current source and three resistors are connected. Thus, Apply KCL at Super node (Node1 & Node2) 9 = 2v 2 + 6v 3 + 3v 3 3v = 3 v1 + 2v 2 + 9v 3. Eq 2. Since we have three unknown values, therefore, we need one additional equation. Obliviously, we will go for the 5V voltage source between Nodes 2 and 3, which is; v 2 v 3 = 5. Eq 3. Solving equations 1, 2 and 3 by Cramer s rule or Cramer s rule calculator, Elimination, Gauss Elimination or computer aided program such as MATLAB, we find, v 3 =0.575V or 375mV v 2 =5.375V v 1 = V 40

41 NETWORK TOPOLOGY: Introduction Network topology is a graphical representation of electric circuits. It is useful for analyzing complex electric circuits by converting them into network graphs. Network topology is also called as Graph theorynetwork topology is the one of the technique to solve electrical networks consisting of number of meshes or number of nodes, where it is difficult to apply mesh and nodal analysis. Graph theory is the technique where all the elements of the network are Represented by straight lines irrespective of their behavior. Here matrix methods are used to solve complex networks. Before seeing the actual matrices, the knowledge of some of the definitions is very important. They are--- DEFINITONS: Node : An node is junction where two or more than two elements are connected. Degree of the node: number of elements connected to the node is defined as degree of the node. Branch: An branch is a element(s) connected between pair of nodes. Path: It is traversal of signal between pair of nodes. Loop: It is the path started from an node and ends at the same node. Graph: An graph is formed when all the elements of the network are replaced by straight line irrespective of their behaviour. Oriented and non-oriented graph: If the graph of the network is represented with directions in each and every branch then it is oriented, if at least one branch of graph has no direction then iti is non-oriented graph. Planar and non-planar graph: if an graph can be on plane surface without cross over then system is planar and vise-versa is non-linear. GRAPH, TREE,CO-TREE Graph Network graph is simply called as graph. It consists of a set of nodes connected by branches. In graphs, a node is a common point of two or more branches. Sometimes, only a single branch may connect to the node. A branch is a line segment that connects two nodes. Any electric circuit or network can be converted into its equivalent graph by replacing the passive elements and voltage sources with short circuits and the current sources with open circuits. That means, the line segments in the graph represent the branches corresponding to either passive elements or voltage sources of electric circuit. EXAMPLE 41

42 In the above circuit, there are four principal nodes and those are labelled with 1, 2, 3, and 4. There are seven branches in the above circuit, among which one branch contains a 20 V voltage source, another branch contains a 4 A current source and the remaining five branches contain resistors having resistances of 30 Ω, 5 Ω, 10 Ω, 10 Ω and 20 Ω respectively. THE EQUIVALENT GRAPH CAN BE PLOTTED AS BELOW: In the above graph, there are four nodes and those are labeled with 1, 2, 3 & 4 respectively. These are same as that of principal nodes in the electric circuit. There are six branches in the above graph and those are labelled with a, b, c, d, e & f respectively. In the above graph, there are four nodes and those are labelled with 1, 2, 3 & 4 respectively. These are same as that of principal nodes in the electric circuit. There are six branches in the above graph and those are labelled with a, b, c, d, e & f respectively. In this case, we got one branch less in the graph because the 4 A current source is made as open circuit, while converting the electric circuit into its equivalent graph. The number of nodes present in a graph will be equal to the number of principal nodes present in an electric circuit. The number of branches present in a graph will be less than or equal to the number of branches present in an electric circuit. Types of Graphs Following are the types of graphs Connected Graph Unconnected Graph Directed Graph Undirected Graph Now, let us discuss these graphs one by one. 42

43 Connected Graph If there exists at least one branch between any of the two nodes of a graph, then it is called as a connected graph. That means, each node in the connected graph will be having one or more branches that are connected to it. So, no node will present as isolated or separated. The graph shown in the previous Example is a connected graph. Here, all the nodes are connected by three branches. Unconnected Graph If there exists at least one node in the graph that remains unconnected by even single branch, then it is called as an unconnected graph. So, there will be one or more isolated nodes in an unconnected graph. Consider the graph shown in the following figure. In this graph, the nodes 2, 3, and 4 are connected by two branches each. But, not even a single branch has been connected to the node 1. So, the node 1 becomes an isolated node. Hence, the above graph is an unconnected graph. Directed Graph If all the branches of a graph are represented with arrows, then that graph is called as a directed graph. These arrows indicate the direction of current flow in each branch. Hence, this graph is also called as oriented graph. Consider the graph shown in the following figure. In the above graph, the direction of current flow is represented with an arrow in each branch. Hence, it is a directed graph. Undirected Graph If the branches of a graph are not represented with arrows, then that graph is called as an undirected graph. Since, there are no directions of current flow, this graph is also called as an unoriented graph. The graph that was shown in the first Example of this chapter is an unoriented graph, because there are no arrows on the branches of that graph. Subgraph and its Types A part of the graph is called as a subgraph. We get subgraphs by removing some nodes and/or branches of a given graph. So, the number of branches and/or nodes of a subgraph will be less than that of the original graph. Hence, we can conclude that a subgraph is a subset of a graph. Following are the two types of subgraphs. Tree 43

44 Co-Tree Tree Tree is a connected subgraph of a given graph, which contains all the nodes of a graph. But, there should not be any loop in that subgraph. The branches of a tree are called as twigs. Consider the following connected subgraph of the graph, which is shown in the Example of the beginning of this chapter This connected subgraph contains all the four nodes of the given graph and there is no loop. Hence, it is a Tree. This Tree has only three branches out of six branches of given graph. Because, if we consider even single branch of the remaining branches of the graph, then there will be a loop in the above connected subgraph. Then, the resultant connected subgraph will not be a Tree. From the above Tree, we can conclude that the number of branches that are present in a Tree should be equal to n - 1where n is the number of nodes of the given graph Co-Tree Co-Tree is a subgraph, which is formed with the branches that are removed while forming a Tree. Hence, it is called as Complement of a Tree. For every Tree, there will be a corresponding Co-Tree and its branches are called as links or chords. In general, the links are represented with dotted lines. The Co-Tree corresponding to the above Tree is shown in the following figure. his Co-Tree has only three nodes instead of four nodes of the given graph, because Node 4 is isolated from the above Co-Tree. Therefore, the Co-Tree need not be a connected subgraph. This Co-Tree has three branches and they form a loop. The number of branches that are present in a co-tree will be equal to the difference between the number of branches of a given graph and the number of twigs. Mathematically, it can be written as l=b (n 1)l=b (n 1) l=b n+1 Where, l is the number of links. b is the number of branches present in a given graph. n is the number of nodes present in a given graph. If we combine a Tree and its corresponding Co-Tree, then we will get the original graph as shown below. 44

45 Properties of a Tree- (i) It consists of all the nodes of the graph. (ii) If the graph has N nodes, then the tree has (N-1) branch. (iii) There will be no closed path in a tree MATRICES ASSOCIATED WITH NETWORK GRAPHS Following are the three matrices that are used in Graph theory. Incidence Matrix Fundamental Loop Matrix Fundamental Cut set Matrix INCIDENCE MATRIX(A): An Incidence Matrix represents the graph of a given electric circuit or network. Hence, it is possible to draw the graph of that same electric circuit or network from the incidence matrix. We know that graph consists of a set of nodes and those are connected by some branches. So, the connecting of branches to a node is called as incidence. Incidence matrix is represented with the letter A. It is also called as node to branch incidence matrix or node incidence matrix. If there are n nodes and b branches are present in a directed graph, then the incidence matrix will have n rows and b columns. Here, rows and columns are corresponding to the nodes and branches of a directed graph. Hence, the order of incidence matrix will be n b. The elements of incidence matrix will be having one of these three values, +1, -1 and 0. If the branch current is leaving from a selected node, then the value of the element will be +1. If the branch current is entering towards a selected node, then the value of the element will be -1. If the branch current neither enters at a selected node nor leaves from a selected node, then the value of element will be 0 45

46 Procedure to find Incidence Matrix Follow these steps in order to find the incidence matrix of directed graph. Select a node at a time of the given directed graph and fill the values of the elements of incidence matrix corresponding to that node in a row. Repeat the above step for all the nodes of the given directed graph. Note If the given graph is an un-directed type, then convert it into a directed graph by representing the arrows on each branch of it. We can consider the arbitrary direction of current flow in each branch Incidence matrix is formed between number of nodes and number of branches. This matrix is useful easy understanding of network and any complex network can be easily feed into system for coding. Order of incidence matrix is (n*b) Procedure to incidence matrix: aij = 1, if jth branch is incidence to ith node and direction is away from node. aij = -1, if jth branch is incidence to ith node and direction is towards from node. aij = 0, if jth branch is not incidence to ith node. Properties of incidence matrix: 1. Number of non zero entries of row indicates degree of the node. 2. The non zero entries of the coloumn represents branch connections. 3. If two coloumns has same entries then they are in parallel. Reduced incidence matrix: Reduced incidence matrix is formed by eliminating one of the row from incidence matrix generally ground node row is eliminating, which is representing by A!. Number of possible trees of for any graph are det(a! A!T). Once we form the incidence matrix, we can write KCL equations any complex without appling KCL as follows, A!.Ib = 0 Where, A! - reduced incidence matrix Ib - branch current matrix( an coloumn matrix) 46

47 PROPERTIES OF INCIDENCE MATRIX For checking correctness of incidence matrix which we have drawn, we should check sum of column If sum of column comes to be zero, then the incidence matrix which we have created is correct else incorrect The incidence matrix can be applied only to directed graph only The number of entries in a row apart from zero tells us the number of branches linked to that node. This is also called as degree of that node The rank of complete incidence matrix is (n-1), where n is the number of nodes of the graph. The order of incidence matrix is (n b), where b is the number of branches of graph. From a given reduced incidence matrix we can draw complete incidence matrix by simply adding either +1, 0, or -1 on the condition that sum of each column should be zero. INCIDENCE MATRIX AND KCL Kirchhoff s current law (KCL) of a graph can be expressed in terms of the reduced incidence matrix as A 1 I = 0. A 1, I is the matrix representation of KCL, a where I represents branch current vectors I 1,I 2, I 6. Consider the graph shown. It has four nodes a, b, c and d. Let node d be taken as the reference node. The positive reference direction of the branch currents, corresponds to the orientation of the graph branches. Let the branch currents be I 1,I 2, I 6. Applying KCL at nodes a, b and c. 47

48 These equations can be written in the matrix form as follows Here, I b represents column matrix or a vector of branch currents. A 1 is the reduced incidence matrix of a graph with n nodes and b branches. And it is a (n 1) x b matrix obtained from the complete incidence matrix of A deleting one of its rows. The node corresponding to the deleted row is called the reference node or datum node. It is to be noted that A 1 I b = 0 gives a set of n I linearly independent equations in branch currents I 1,I 2, I 6. Here n = 4. Hence, there are three linearly independent equations. 48

49 UNIT III SINGLE PHASE AC CIRCUITS AND RESONANCE 49

50 Analysis of AC circuits: Direct Current or D.C. as it is more commonly called, is a form of electrical current or voltage that flows around an electrical circuit in one direction only, making it a Uni-directional supply. Generally, both DC currents and voltages are produced by power supplies, batteries, dynamos and solar cells to name a few. A DC voltage or current has a fixed magnitude (amplitude) and a definite direction associated with it. For example, +12V represents 12 volts in the positive direction, or -5V represents 5 volts in the negative direction. We also know that DC power supplies do not change their value with regards to time, they are a constant value flowing in a continuous steady state direction. In other words, DC maintains the same value for all times and a constant uni-directional DC supply never changes or becomes negative unless its connections are physically reversed. An example of a simple DC or direct current circuit is shown below. DC Circuit and Waveform An alternating function or AC Waveform on the other hand is defined as one that varies in both magnitude and direction in more or less an even manner with respect to time making it a Bi-directional waveform. An AC function can represent either a power source or a signal source with the shape of an AC waveform generally following that of a mathematical sinusoid being defined as: A(t) = A max *sin(2πƒt). The term AC or to give it its full description of Alternating Current, generally refers to a time-varying waveform with the most common of all being called a Sinusoid better known as a Sinusoidal Waveform. Sinusoidal waveforms are more generally called by their short description as Sine Waves. Sine waves are by far one of the most important types of AC waveform used in electrical engineering. The shape obtained by plotting the instantaneous ordinate values of either voltage or current against time is called an AC Waveform. An AC waveform is constantly changing its polarity every half cycle alternating between a positive maximum value and a negative maximum value respectively with regards to time with a common example of this being the domestic mains voltage supply we use in our homes. This means then that the AC Waveform is a time-dependent signal with the most common type of timedependant signal being that of the Periodic Waveform. The periodic or AC waveform is the resulting product of a rotating electrical generator. Generally, the shape of any periodic waveform can be generated using a fundamental frequency and superimposing it with harmonic signals of varying frequencies and amplitudes but that s for another tutorial. Alternating voltages and currents cannot be stored in batteries or cells like direct current (DC) can, it is much easier and cheaper to generate these quantities using alternators or waveform generators when they are needed. The type and shape of an AC waveform depends upon the generator or device producing them, but all AC waveforms consist of a zero voltage line that divides the waveform into two symmetrical halves. The main characteristics of an AC Waveform are defined as: AC Waveform Characteristics The Period, (T) is the length of time in seconds that the waveform takes to repeat itself from start to finish. This can also be called the Periodic Time of the waveform for sine waves, or the Pulse Width for square waves. 50

51 The Frequency, (ƒ) is the number of times the waveform repeats itself within a one second time period. Frequency is the reciprocal of the time period, ( ƒ = 1/T ) with the unit of frequency being the Hertz, (Hz). The Amplitude (A) is the magnitude or intensity of the signal waveform measured in volts or amps. In our tutorial about waveforms,we looked at different types of waveforms and said that Waveforms are basically a visual representation of the variation of a voltage or current plotted to a base of time. Generally, for AC waveforms this horizontal base line represents a zero condition of either voltage or current. Any part of an AC type waveform which lies above the horizontal zero axis represents a voltage or current flowing in one direction. Likewise, any part of the waveform which lies below the horizontal zero axis represents a voltage or current flowing in the opposite direction to the first. Generally for sinusoidal AC waveforms the shape of the waveform above the zero axis is the same as the shape below it. However, for most non-power AC signals including audio waveforms this is not always the case. The most common periodic signal waveforms that are used in Electrical and Electronic Engineering are the Sinusoidal Waveforms. However, an alternating AC waveform may not always take the shape of a smooth shape based around the trigonometric sine or cosine function. AC waveforms can also take the shape of either Complex Waves, Square Waves or Triangular Waves and these are shown below. The Average Value of an AC Waveform The average or mean value of a continuous DC voltage will always be equal to its maximum peak value as a DC voltage is constant. This average value will only change if the duty cycle of the DC voltage changes. In a pure sine wave if the average value is calculated over the full cycle, the average value would be equal to zero as the positive and negative halves will cancel each other out. So the average or mean value of an AC waveform is calculated or measured over a half cycle only and this is shown below Average Value of a Non-sinusoidal Waveform To find the average value of the waveform we need to calculate the area underneath the waveform using the mid-ordinate rule, trapezoidal rule or the Simpson s rule found commonly in mathematics. The approximate area under any irregular waveform can easily be found by simply using the mid-ordinate rule. The zero axis base line is divided up into any number of equal parts and in our simple example above this value was nine, ( V 1 to V 9 ). The more ordinate lines that are drawn the more accurate will be 51

52 the final average or mean value. The average value will be the addition of all the instantaneous values added together and then divided by the total number. This is given as. Average Value of an AC Waveform Where: n equals the actual number of mid-ordinates used. For a pure sinusoidal waveform this average or mean value will always be equal to 0.637*V max and this relationship also holds true for average values of current. The RMS Value of an AC Waveform The average value of an AC waveform that we calculated above as being: 0.637*V max is NOT the same value we would use for a DC supply. This is because unlike a DC supply which is constant and and of a fixed value, an AC waveform is constantly changing over time and has no fixed value. Thus the equivalent value for an alternating current system that provides the same amount of electrical power to a load as a DC equivalent circuit is called the effective value. The effective value of a sine wave produces the same I 2 *R heating effect in a load as we would expect to see if the same load was fed by a constant DC supply. The effective value of a sine wave is more commonly known as the Root Mean Squared or simply RMS value as it is calculated as the square root of the mean (average) of the square of the voltage or current. That is V rms or I rms is given as the square root of the average of the sum of all the squared mid-ordinate values of the sine wave. The RMS value for any AC waveform can be found from the following modified average value formula as shown. RMS Value of an AC Waveform Where: n equals the number of mid-ordinates. For a pure sinusoidal waveform this effective or R.M.S. value will always be equal too: 1/ 2*V max which is equal to 0.707*V max and this relationship holds true for RMS values of current. The RMS value for a sinusoidal waveform is always greater than the average value except for a rectangular waveform. In this case the heating effect remains constant so the average and the RMS values will be the same. One final comment about R.M.S. values. Most multimeters, either digital or analogue unless otherwise stated only measure the R.M.S. values of voltage and current and not the average. Therefore when using a multimeter on a direct current system the reading will be equal to I = V/R and for an alternating current system the reading will be equal to Irms = Vrms/R. Also, except for average power calculations, when calculating RMS or peak voltages, only use V RMS to find I RMS values, or peak voltage, Vp to find peak current, Ip values. Do not mix them together as Average, RMS or Peak values of a sine wave are completely different and your results will definitely be incorrect. Form Factor and Crest Factor Although little used these days, both Form Factor and Crest Factor can be used to give information about the actual shape of the AC waveform. Form Factor is the ratio between the average value and the RMS value and is given as. For a pure sinusoidal waveform the Form Factor will always be equal to Crest Factor is the ratio between the R.M.S. value and the Peak value of the waveform and is given as. 52

53 For a pure sinusoidal waveform the Crest Factor will always be equal to AC Waveform Example No2 A sinusoidal alternating current of 6 amps is flowing through a resistance of 40Ω. Calculate the average voltage and the peak voltage of the supply. The R.M.S. Voltage value is calculated as: The Average Voltage value is calculated as: The Peak Voltage value is calculated as: The use and calculation of Average, R.M.S, Form factor and Crest Factor can also be use with any type of periodic waveform including Triangular, Square, Sawtoothed or any other irregular or complex voltage/current waveform shape. Conversion between the various sinusoidal values can sometimes be confusing so the following table gives a convenient way of converting one sine wave value to another. Phasor Diagrams and Phasor Algebra Phasor Diagrams are a graphical way of representing the magnitude and directional relationship between two or more alternating quantities Sinusoidal waveforms of the same frequency can have a Phase Difference between themselves which represents the angular difference of the two sinusoidal waveforms. Also the terms lead and lag as well as in-phase and out-of-phase are commonly used to indicate the relationship of one waveform to the other with the generalized sinusoidal expression given as: A (t) = A m sin(ωt ± Φ) representing the sinusoid in the time-domain form. But when presented mathematically in this way it is sometimes difficult to visualise this angular or phasor difference between two or more sinusoidal waveforms. One way to overcome this problem is to represent the sinusoids graphically within the spacial or phasor-domain form by using Phasor Diagrams, and this is achieved by the rotating vector method. Basically a rotating vector, simply called a Phasor is a scaled line whose length represents an AC quantity that has both magnitude ( peak amplitude ) and direction ( phase ) which is frozen at some point in time. A phasor is a vector that has an arrow head at one end which signifies partly the maximum value of the vector quantity ( V or I ) and partly the end of the vector that rotates. Generally, vectors are assumed to pivot at one end around a fixed zero point known as the point of origin while the arrowed end representing the quantity, freely rotates in an anti-clockwise direction at an angular velocity, ( ω ) of one full revolution for every cycle. This anti-clockwise rotation of the vector is considered to be a positive rotation. Likewise, a clockwise rotation is considered to be a negative rotation. Although the both the terms vectors and phasors are used to describe a rotating line that itself has both magnitude and direction, the main difference between the two is that a vectors magnitude is the 53

54 peak value of the sinusoid while a phasors magnitude is the rms value of the sinusoid. In both cases the phase angle and direction remains the same. The phase of an alternating quantity at any instant in time can be represented by a phasor diagram, so phasor diagrams can be thought of as functions of time. A complete sine wave can be constructed by a single vector rotating at an angular velocity of ω = 2πƒ, where ƒ is the frequency of the waveform. Then a Phasor is a quantity that has both Magnitude and Direction. Generally, when constructing a phasor diagram, angular velocity of a sine wave is always assumed to be: ω in rad/sec. Consider the phasor diagram below. Phasor Diagram of a Sinusoidal Waveform As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate one complete revolution of 360 o or 2π representing one complete cycle. If the length of its moving tip is transferred at different angular intervals in time to a graph as shown above, a sinusoidal waveform would be drawn starting at the left with zero time. Each position along the horizontal axis indicates the time that has elapsed since zero time, t = 0. When the vector is horizontal the tip of the vector represents the angles at 0 o, 180 o and at 360 o. Likewise, when the tip of the vector is vertical it represents the positive peak value, ( +Am ) at 90 o or π/2 and the negative peak value, ( -Am ) at 270 o or 3π/2. Then the time axis of the waveform represents the angle either in degrees or radians through which the phasor has moved. So we can say that a phasor represent a scaled voltage or current value of a rotating vector which is frozen at some point in time, ( t ) and in our example above, this is at an angle of 30 o. Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the Alternating Quantity at some particular instant in time especially when we want to compare two different waveforms on the same axis. For example, voltage and current. We have assumed in the waveform above that the waveform starts at time t = 0 with a corresponding phase angle in either degrees or radians. But if a second waveform starts to the left or to the right of this zero point or we want to represent in phasor notation the relationship between the two waveforms then we will need to take into account this phase difference, Φ of the waveform. 54

55 Phase Difference of a Sinusoidal Waveform The generalised mathematical expression to define these two sinusoidal quantities will be written as: The current, i is lagging the voltage, v by angle Φ and in our example above this is 30 o. So the difference between the two phasors representing the two sinusoidal quantities is angle Φ and the resulting phasor diagram will be. Phasor Diagram of a Sinusoidal Waveform The phasor diagram is drawn corresponding to time zero ( t = 0 ) on the horizontal axis. The lengths of the phasors are proportional to the values of the voltage, ( V ) and the current, ( I ) at the instant in time that the phasor diagram is drawn. The current phasor lags the voltage phasor by the angle, Φ, as the two phasors rotate in an anticlockwisedirection as stated earlier, therefore the angle, Φ is also measured in the same anticlockwise direction. If however, the waveforms are frozen at time, t = 30 o, the corresponding phasor diagram would look like the one shown on the right. Once again the current phasor lags behind the voltage phasor as the two waveforms are of the same frequency. However, as the current waveform is now crossing the horizontal zero axis line at this instant in time we can use the current phasor as our new reference and correctly say that the voltage phasor is leading the current phasor by angle, Φ. Either way, one phasor is designated as the reference phasor and all the other phasors will be either leading or lagging with respect to this reference. Phasor Addition Sometimes it is necessary when studying sinusoids to add together two alternating waveforms, for example in an AC series circuit, that are not in-phase with each other. If they are in-phase that is, there is no phase shift then they can be added together in the same way as DC values to find the algebraic sum of 55

56 the two vectors. For example, if two voltages of say 50 volts and 25 volts respectively are together inphase, they will add or sum together to form one voltage of 75 volts ( ). If however, they are not in-phase that is, they do not have identical directions or starting point then the phase angle between them needs to be taken into account so they are added together using phasor diagrams to determine their Resultant Phasor or Vector Sum by using the parallelogram law. Consider two AC voltages, V 1 having a peak voltage of 20 volts, and V 2 having a peak voltage of 30 volts where V 1 leads V 2 by 60 o. The total voltage, V T of the two voltages can be found by firstly drawing a phasor diagram representing the two vectors and then constructing a parallelogram in which two of the sides are the voltages, V 1 and V 2 as shown below. Phasor Addition of two Phasors By drawing out the two Phasor to scale onto graph paper, their phasor sum V 1 + V 2 can be easily found by measuring the length of the diagonal line, known as the resultant r-vector, from the zero point to the intersection of the construction lines 0-A. The downside of this graphical method is that it is time consuming when drawing the phasors to scale. Also, while this graphical method gives an answer which is accurate enough for most purposes, it may produce an error if not drawn accurately or correctly to scale. Then one way to ensure that the correct answer is always obtained is by an analytical method. Mathematically we can add the two voltages together by firstly finding their vertical and horizontal directions, and from this we can then calculate both the vertical and horizontal components for the resultant r vector, V T. This analytical method which uses the cosine and sine rule to find this resultant value is commonly called the Rectangular Form. In the rectangular form, the phasor is divided up into a real part, x and an imaginary part, yforming the generalised expression Z = x ± jy. ( we will discuss this in more detail in the next tutorial ). This then gives us a mathematical expression that represents both the magnitude and the phase of the sinusoidal voltage as: Definition of a Complex Sinusoid So the addition of two vectors, A and B using the previous generalised expression is as follows: Phasor Addition using Rectangular Form Voltage, V 2 of 30 volts points in the reference direction along the horizontal zero axis, then it has a horizontal component but no vertical component as follows. 56

57 Horizontal Component = 30 cos 0 o = 30 volts Vertical Component = 30 sin 0 o = 0 volts This then gives us the rectangular expression for voltage V 2 of: 30 + j0 Voltage, V 1 of 20 volts leads voltage, V 2 by 60 o, then it has both horizontal and vertical components as follows. Horizontal Component = 20 cos 60 o = 20 x 0.5 = 10 volts Vertical Component = 20 sin 60 o = 20 x = volts This then gives us the rectangular expression for voltage V 1 of: 10 + j17.32 The resultant voltage, V T is found by adding together the horizontal and vertical components as follows. V Horizontal = sum of real parts of V 1 and V 2 = = 40 volts V Vertical = sum of imaginary parts of V 1 and V 2 = = volts Now that both the real and imaginary values have been found the magnitude of voltage, V T is determined by simply using Pythagoras s Theorem for a 90 o triangle as follows. Then the resulting phasor diagram will be: Resultant Value of V T Phasor Subtraction Phasor subtraction is very similar to the above rectangular method of addition, except this time the vector difference is the other diagonal of the parallelogram between the two voltages of V 1 and V 2 as shown. Vector Subtraction of two Phasors This time instead of adding together both the horizontal and vertical components we take them away, subtraction. 57

58 Types of Periodic Waveform The time taken for an AC Waveform to complete one full pattern from its positive half to its negative half and back to its zero baseline again is called a Cycle and one complete cycle contains both a positive half-cycle and a negative half-cycle. The time taken by the waveform to complete one full cycle is called the Periodic Time of the waveform, and is given the symbol T. The number of complete cycles that are produced within one second (cycles/second) is called the Frequency, symbol ƒ of the alternating waveform. Frequency is measured in Hertz, ( Hz ) named after the German physicist Heinrich Hertz. Then we can see that a relationship exists between cycles (oscillations), periodic time and frequency (cycles per second), so if there are ƒ number of cycles in one second, each individual cycle must take 1/ƒ seconds to complete. Relationship Between Frequency and Periodic Time 58

59 AC Waveform Example No1 1. What will be the periodic time of a 50Hz waveform and 2. what is the frequency of an AC waveform that has a periodic time of 10mS. 1). 2). Frequency used to be expressed in cycles per second abbreviated to cps, but today it is more commonly specified in units called Hertz. For a domestic mains supply the frequency will be either 50Hz or 60Hz depending upon the country and is fixed by the speed of rotation of the generator. But one hertz is a very small unit so prefixes are used that denote the order of magnitude of the waveform at higher frequencies such as khz, MHz and even GHz. Amplitude of an AC Waveform As well as knowing either the periodic time or the frequency of the alternating quantity, another important parameter of the AC waveform is Amplitude, better known as its Maximum or Peak value represented by the terms, Vmax for voltage or Imax for current. The peak value is the greatest value of either voltage or current that the waveform reaches during each half cycle measured from the zero baseline. Unlike a DC voltage or current which has a steady state that can be measured or calculated using Ohm s Law, an alternating quantity is constantly changing its value over time. For pure sinusoidal waveforms this peak value will always be the same for both half cycles ( +Vm = -Vm ) but for non-sinusoidal or complex waveforms the maximum peak value can be very different for each half cycle. Sometimes, alternating waveforms are given a peak-to-peak, Vp-p value and this is simply the distance or the sum in voltage between the maximum peak value, +Vmax and the minimum peak value, -Vmax during one complete cycle. Phasor Diagram of a Sinusoidal Waveform 59

60 As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate one complete revolution of 360o or 2π representing one complete cycle. If the length of its moving tip is transferred at different angular intervals in time to a graph as shown above, a sinusoidal waveform would be drawn starting at the left with zero time. Each position along the horizontal axis indicates the time that has elapsed since zero time, t = 0. When the vector is horizontal the tip of the vector represents the angles at 0o, 180o and at 360o. Likewise, when the tip of the vector is vertical it represents the positive peak value, ( +Am ) at 90o or π/2 and the negative peak value, ( -Am ) at 270o or 3π/2. Then the time axis of the waveform represents the angle either in degrees or radians through which the phasor has moved. So we can say that a phasor represent a scaled voltage or current value of a rotating vector which is frozen at some point in time, ( t ) and in our example above, this is at an angle of 30o. Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the Alternating Quantity at some particular instant in time especially when we want to compare two different waveforms on the same axis. For example, voltage and current. We have assumed in the waveform above that the waveform starts at time t = 0 with a corresponding phase angle in either degrees or radians. But if a second waveform starts to the left or to the right of this zero point or we want to represent in phasor notation the relationship between the two waveforms then we will need to take into account this phase difference, Φ of the waveform. Consider the diagram below from the previous Phase Difference tutorial. Phase Difference of a Sinusoidal Waveform The generalised mathematical expression to define these two sinusoidal quantities will be written as: The current, i is lagging the voltage, v by angle Φ and in our example above this is 30o. So the difference between the two phasors representing the two sinusoidal quantities is angle Φ and the resulting phasor diagram will be. Phasor Diagram of a Sinusoidal Waveform The phasor diagram is drawn corresponding to time zero ( t = 0 ) on the horizontal axis. The lengths of the phasors are proportional to the values of the voltage, ( V ) and the current, ( I ) at the instant in time that the phasor diagram is drawn. The current phasor lags the voltage phasor by the angle, Φ, as the two phasors rotate in an anticlockwisedirection as stated earlier, therefore the angle, Φ is also measured in the same anticlockwise direction. 60

61 If however, the waveforms are frozen at time, t = 30o, the corresponding phasor diagram would look like the one shown on the right. Once again the current phasor lags behind the voltage phasor as the two waveforms are of the same frequency. However, as the current waveform is now crossing the horizontal zero axis line at this instant in time we can use the current phasor as our new reference and correctly say that the voltage phasor is leading the current phasor by angle, Φ. Either way, one phasor is designated as the reference phasor and all the other phasors will be either leading or lagging with respect to this reference. Phase Difference and Phase Shift Phase Difference is used to describe the difference in degrees or radians when two or more alternating quantities reach their maximum or zero values Previously we saw that a Sinusoidal Waveform is an alternating quantity that can be presented graphically in the time domain along an horizontal zero axis. We also saw that as an alternating quantity, sine waves have a positive maximum value at time π/2, a negative maximum value at time 3π/2, with zero values occurring along the baseline at 0, π and 2π. However, not all sinusoidal waveforms will pass exactly through the zero axis point at the same time, but may be shifted to the right or to the left of 0 o by some value when compared to another sine wave. For example, comparing a voltage waveform to that of a current waveform. This then produces an angular shift or Phase Difference between the two sinusoidal waveforms. Any sine wave that does not pass through zero at t = 0 has a phase shift. The phase difference or phase shift as it is also called of a Sinusoidal Waveform is the angle Φ (Greek letter Phi), in degrees or radians that the waveform has shifted from a certain reference point along the horizontal zero axis. In other words phase shift is the lateral difference between two or more waveforms along a common axis and sinusoidal waveforms of the same frequency can have a phase difference. The phase difference, Φ of an alternating waveform can vary from between 0 to its maximum time period, T of the waveform during one complete cycle and this can be anywhere along the horizontal axis between, Φ = 0 to 2π (radians) or Φ = 0 to 360 o depending upon the angular units used. Phase difference can also be expressed as a time shift of τ in seconds representing a fraction of the time period, T for example, +10mS or 50uS but generally it is more common to express phase difference as an angular measurement. Then the equation for the instantaneous value of a sinusoidal voltage or current waveform we developed in the previous Sinusoidal Waveform will need to be modified to take account of the phase angle of the waveform and this new general expression becomes. Phase Difference Equation Where: A m - is the amplitude of the waveform. ωt - is the angular frequency of the waveform in radian/sec. Φ (phi) - is the phase angle in degrees or radians that the waveform has shifted either left or right from the reference point. 61

62 If the positive slope of the sinusoidal waveform passes through the horizontal axis before t = 0 then the waveform has shifted to the left so Φ >0, and the phase angle will be positive in nature, +Φ giving a leading phase angle. In other words it appears earlier in time than 0 o producing an anticlockwise rotation of the vector. Likewise, if the positive slope of the sinusoidal waveform passes through the horizontal x-axis some time after t = 0 then the waveform has shifted to the right so Φ <0, and the phase angle will be negative in nature -Φ producing a lagging phase angle as it appears later in time than 0 o producing a clockwise rotation of the vector. Both cases are shown below. Phase Relationship of a Sinusoidal Waveform Firstly, lets consider that two alternating quantities such as a voltage, v and a current, ihave the same frequency ƒ in Hertz. As the frequency of the two quantities is the same the angular velocity, ω must also be the same. So at any instant in time we can say that the phase of voltage, v will be the same as the phase of the current, i. Then the angle of rotation within a particular time period will always be the same and the phase difference between the two quantities of v and i will therefore be zero and Φ = 0. As the frequency of the voltage, v and the current, i are the same they must both reach their maximum positive, negative and zero values during one complete cycle at the same time (although their amplitudes may be different). Then the two alternating quantities, v and iare said to be in-phase. Two Sinusoidal Waveforms in-phase Now lets consider that the voltage, v and the current, i have a phase difference between themselves of 30 o, so (Φ = 30 o or π/6 radians). As both alternating quantities rotate at the same speed, i.e. they have the same frequency, this phase difference will remain constant for all instants in time, then the phase difference of 30 o between the two quantities is represented by phi, Φ as shown below. Phase Difference of a Sinusoidal Waveform 62

63 The voltage waveform above starts at zero along the horizontal reference axis, but at that same instant of time the current waveform is still negative in value and does not cross this reference axis until 30 o later. Then there exists a Phase difference between the two waveforms as the current cross the horizontal reference axis reaching its maximum peak and zero values after the voltage waveform. As the two waveforms are no longer in-phase, they must therefore be out-of-phase by an amount determined by phi, Φ and in our example this is 30 o. So we can say that the two waveforms are now 30 o out-of phase. The current waveform can also be said to be lagging behind the voltage waveform by the phase angle, Φ. Then in our example above the two waveforms have a Lagging Phase Difference so the expression for both the voltage and current above will be given as. where, i lags v by angle Φ Likewise, if the current, i has a positive value and crosses the reference axis reaching its maximum peak and zero values at some time before the voltage, v then the current waveform will be leading the voltage by some phase angle. Then the two waveforms are said to have a Leading Phase Difference and the expression for both the voltage and the current will be. where, i leads v by angle Φ The phase angle of a sine wave can be used to describe the relationship of one sine wave to another by using the terms Leading and Lagging to indicate the relationship between two sinusoidal waveforms of the same frequency, plotted onto the same reference axis. In our example above the two waveforms are out-of-phase by 30 o. So we can correctly say that i lags v or we can say that v leads i by 30 o depending upon which one we choose as our reference. The relationship between the two waveforms and the resulting phase angle can be measured anywhere along the horizontal zero axis through which each waveform passes with the same slope direction either positive or negative. In AC power circuits this ability to describe the relationship between a voltage and a current sine wave within the same circuit is very important and forms the bases of AC circuit analysis. AC Inductance and Inductive Reactance: The opposition to current flow through an AC Inductor is called Inductive Reactance and which depends lineally on the supply frequency 63

64 Inductors and chokes are basically coils or loops of wire that are either wound around a hollow tube former (air cored) or wound around some ferromagnetic material (iron cored) to increase their inductive value called inductance. Inductors store their energy in the form of a magnetic field that is created when a voltage is applied across the terminals of an inductor. The growth of the current flowing through the inductor is not instant but is determined by the inductors own self-induced or back emf value. Then for an inductor coil, this back emf voltage V L is proportional to the rate of change of the current flowing through it. This current will continue to rise until it reaches its maximum steady state condition which is around five time constants when this self-induced back emf has decayed to zero. At this point a steady state current is flowing through the coil, no more back emf is induced to oppose the current flow and therefore, the coil acts more like a short circuit allowing maximum current to flow through it. However, in an alternating current circuit which contains an AC Inductance, the flow of current through an inductor behaves very differently to that of a steady state DC voltage. Now in an AC circuit, the opposition to the current flowing through the coils windings not only depends upon the inductance of the coil but also the frequency of the applied voltage waveform as it varies from its positive to negative values. The actual opposition to the current flowing through a coil in an AC circuit is determined by the AC Resistance of the coil with this AC resistance being represented by a complex number. But to distinguish a DC resistance value from an AC resistance value, which is also known as Impedance, the term Reactance is used. Like resistance, reactance is measured in Ohm s but is given the symbol X to distinguish it from a purely resistive R value and as the component in question is an inductor, the reactance of an inductor is called Inductive Reactance, ( X L ) and is measured in Ohms. Its value can be found from the formula. Inductive Reactance Where: X L is the Inductive Reactance in Ohms, ƒ is the frequency in Hertz and L is the inductance of the coil in Henries. We can also define inductive reactance in radians, where Omega, ω equals 2πƒ. So whenever a sinusoidal voltage is applied to an inductive coil, the back emf opposes the rise and fall of the current flowing through the coil and in a purely inductive coil which has zero resistance or losses, this impedance (which can be a complex number) is equal to its inductive reactance. Also reactance is represented by a vector as it has both a magnitude and a direction (angle). Consider the circuit below. AC Inductance with a Sinusoidal Supply This simple circuit above consists of a pure inductance of L Henries ( H ), connected across a sinusoidal voltage given by the expression: V(t) = V max sin ωt. When the switch is closed this sinusoidal voltage will cause a current to flow and rise from zero to its maximum value. This rise or change in the 64

65 current will induce a magnetic field within the coil which in turn will oppose or restrict this change in the current. But before the current has had time to reach its maximum value as it would in a DC circuit, the voltage changes polarity causing the current to change direction. This change in the other direction once again being delayed by the self-induced back emf in the coil, and in a circuit containing a pure inductance only, the current is delayed by 90 o. The applied voltage reaches its maximum positive value a quarter ( 1/4ƒ ) of a cycle earlier than the current reaches its maximum positive value, in other words, a voltage applied to a purely inductive circuit LEADS the current by a quarter of a cycle or 90 o as shown below. Sinusoidal Waveforms for AC Inductance This effect can also be represented by a phasor diagram were in a purely inductive circuit the voltage LEADS the current by 90 o. But by using the voltage as our reference, we can also say that the current LAGS the voltage by one quarter of a cycle or 90 o as shown in the vector diagram below. Phasor Diagram for AC Inductance So for a pure loss less inductor, V L leads I L by 90 o, or we can say that I L lags V L by 90 o. There are many different ways to remember the phase relationship between the voltage and current flowing through a pure inductor circuit, but one very simple and easy to remember way is to use the mnemonic expression ELI (pronounced Ellie as in the girls name). ELI stands for Electromotive force first in an AC inductance, L before the current I. In other words, voltage before the current in an inductor, E, L, I equals ELI, and whichever phase angle the voltage starts at, this expression always holds true for a pure inductor circuit. The Effect of Frequency on Inductive Reactance When a 50Hz supply is connected across a suitable AC Inductance, the current will be delayed by 90 o as described previously and will obtain a peak value of I amps before the voltage reverses polarity at the end of each half cycle, i.e. the current rises up to its maximum value in T secs. If we now apply a 100Hz supply of the same peak voltage to the coil, the current will still be delayed by 90 o but its maximum value will be lower than the 50Hz value because the time it requires to reach its maximum value has been reduced due to the increase in frequency because now it only has 1/2 T secs 65

66 to reach its peak value. Also, the rate of change of the flux within the coil has also increased due to the increase in frequency. Then from the above equation for inductive reactance, it can be seen that if either the Frequency OR the Inductance is increased the overall inductive reactance value of the coil would also increase. As the frequency increases and approaches infinity, the inductors reactance and therefore its impedance would also increase towards infinity acting like an open circuit. Likewise, as the frequency approaches zero or DC, the inductors reactance would also decrease to zero, acting like a short circuit. This means then that inductive reactance is directly proportional to frequency and has a small value at low frequencies and a high value at higher frequencies as shown. Inductive Reactance against Frequency The inductive reactance of an inductor increases as the frequency across it increases therefore inductive reactance is proportional to frequency ( X L α ƒ ) as the back emf generated in the inductor is equal to its inductance multiplied by the rate of change of current in the inductor. Also as the frequency increases the current flowing through the inductor also reduces in value. We can present the effect of very low and very high frequencies on a the reactance of a pure AC Inductance as follows: In an AC circuit containing pure inductance the following formula applies: So how did we arrive at this equation. Well the self induced emf in the inductor is determined by Faraday s Law that produces the effect of self-induction in the inductor due to the rate of change of the current and the maximum value of the induced emf will correspond to the maximum rate of change. Then the voltage in the inductor coil is given as: 66

67 then the voltage across an AC inductance will be defined as: Where: V L = IωL which is the voltage amplitude and θ = + 90 o which is the phase difference or phase angle between the voltage and current. In the Phasor Domain In the phasor domain the voltage across the coil is given as: and in Polar Form this would be written as: X L 90 o where: AC Capacitance and Capacitive Reactance The opposition to current flow through an AC Capacitor is called Capacitive Reactance and which itself is inversely proportional to the supply frequency Capacitors store energy on their conductive plates in the form of an electrical charge. When a capacitor is connected across a DC supply voltage it charges up to the value of the applied voltage at a rate determined by its time constant. A capacitor will maintain or hold this charge indefinitely as long as the supply voltage is present. During this charging process, a charging current, i flows into the capacitor opposed by any changes to the voltage at a rate which is equal to the rate of change of the electrical charge on the plates. A capacitor therefore has an opposition to current flowing onto its plates. The relationship between this charging current and the rate at which the capacitors supply voltage changes can be defined mathematically as: i = C(dv/dt), where C is the capacitance value of the capacitor in farads and dv/dt is the rate of change of the supply voltage with respect to time. Once it is fullycharged the capacitor blocks the flow of any more electrons onto its plates as they have become saturated and the capacitor now acts like a temporary storage device. 67

68 A pure capacitor will maintain this charge indefinitely on its plates even if the DC supply voltage is removed. However, in a sinusoidal voltage circuit which contains AC Capacitance, the capacitor will alternately charge and discharge at a rate determined by the frequency of the supply. Then capacitors in AC circuits are constantly charging and discharging respectively. When an alternating sinusoidal voltage is applied to the plates of an AC capacitor, the capacitor is charged firstly in one direction and then in the opposite direction changing polarity at the same rate as the AC supply voltage. This instantaneous change in voltage across the capacitor is opposed by the fact that it takes a certain amount of time to deposit (or release) this charge onto the plates and is given by V = Q/C. Consider the circuit below. AC Capacitance with a Sinusoidal Supply When the switch is closed in the circuit above, a high current will start to flow into the capacitor as there is no charge on the plates at t = 0. The sinusoidal supply voltage, V is increasing in a positive direction at its maximum rate as it crosses the zero reference axis at an instant in time given as 0 o. Since the rate of change of the potential difference across the plates is now at its maximum value, the flow of current into the capacitor will also be at its maximum rate as the maximum amount of electrons are moving from one plate to the other. As the sinusoidal supply voltage reaches its 90 o point on the waveform it begins to slow down and for a very brief instant in time the potential difference across the plates is neither increasing nor decreasing therefore the current decreases to zero as there is no rate of voltage change. At this 90 o point the potential difference across the capacitor is at its maximum ( V max ), no current flows into the capacitor as the capacitor is now fully charged and its plates saturated with electrons. At the end of this instant in time the supply voltage begins to decrease in a negative direction down towards the zero reference line at 180 o. Although the supply voltage is still positive in nature the capacitor starts to discharge some of its excess electrons on its plates in an effort to maintain a constant voltage. This results in the capacitor current flowing in the opposite or negative direction. When the supply voltage waveform crosses the zero reference axis point at instant 180 o the rate of change or slope of the sinusoidal supply voltage is at its maximum but in a negative direction, consequently the current flowing into the capacitor is also at its maximum rate at that instant. Also at this 180 o point the potential difference across the plates is zero as the amount of charge is equally distributed between the two plates. Then during this first half cycle 0 o to 180 o the applied voltage reaches its maximum positive value a quarter (1/4ƒ) of a cycle after the current reaches its maximum positive value, in other words, a voltage applied to a purely capacitive circuit LAGS the current by a quarter of a cycle or 90 o as shown below. 68

69 Sinusoidal Waveforms for AC Capacitance During the second half cycle 180 o to 360 o, the supply voltage reverses direction and heads towards its negative peak value at 270 o. At this point the potential difference across the plates is neither decreasing nor increasing and the current decreases to zero. The potential difference across the capacitor is at its maximum negative value, no current flows into the capacitor and it becomes fully charged the same as at its 90 o point but in the opposite direction. As the negative supply voltage begins to increase in a positive direction towards the 360 o point on the zero reference line, the fully charged capacitor must now loose some of its excess electrons to maintain a constant voltage as before and starts to discharge itself until the supply voltage reaches zero at 360 o at which the process of charging and discharging starts over again. From the voltage and current waveforms and description above, we can see that the current is always leading the voltage by 1/4 of a cycle or π/2 = 90 o out-of-phase with the potential difference across the capacitor because of this charging and discharging process. Then the phase relationship between the voltage and current in an AC capacitance circuit is the exact opposite to that of an AC Inductance we saw in the previous tutorial. This effect can also be represented by a phasor diagram where in a purely capacitive circuit the voltage LAGS the current by 90 o. But by using the voltage as our reference, we can also say that the current LEADS the voltage by one quarter of a cycle or 90 o as shown in the vector diagram below. Phasor Diagram for AC Capacitance So for a pure capacitor, V C lags I C by 90 o, or we can say that I C leads V C by 90 o. There are many different ways to remember the phase relationship between the voltage and current flowing in a pure AC capacitance circuit, but one very simple and easy to remember way is to use the mnemonic expression called ICE. ICE stands for current Ifirst in an AC capacitance, C before Electromotive force. In other words, current before the voltage in a 69

70 capacitor, I, C, E equals ICE, and whichever phase angle the voltage starts at, this expression always holds true for a pure AC capacitance circuit. Capacitive Reactance So we now know that capacitors oppose changes in voltage with the flow of electrons onto the plates of the capacitor being directly proportional to the rate of voltage change across its plates as the capacitor charges and discharges. Unlike a resistor where the opposition to current flow is its actual resistance, the opposition to current flow in a capacitor is called Reactance. Like resistance, reactance is measured in Ohm s but is given the symbol X to distinguish it from a purely resistive R value and as the component in question is a capacitor, the reactance of a capacitor is called Capacitive Reactance, ( X C ) which is measured in Ohms. Since capacitors charge and discharge in proportion to the rate of voltage change across them, the faster the voltage changes the more current will flow. Likewise, the slower the voltage changes the less current will flow. This means then that the reactance of an AC capacitor is inversely proportional to the frequency of the supply as shown. Capacitive Reactance Where: X C is the Capacitive Reactance in Ohms, ƒ is the frequency in Hertz and C is the AC capacitance in Farads, symbol F. When dealing with AC capacitance, we can also define capacitive reactance in terms of radians, where Omega, ω equals 2πƒ. From the above formula we can see that the value of capacitive reactance and therefore its overall impedance ( in Ohms ) decreases towards zero as the frequency increases acting like a short circuit. Likewise, as the frequency approaches zero or DC, the capacitors reactance increases to infinity, acting like an open circuit which is why capacitors block DC. The relationship between capacitive reactance and frequency is the exact opposite to that of inductive reactance, ( X L ) we saw in the previous tutorial. This means then that capacitive reactance is inversely proportional to frequency and has a high value at low frequencies and a low value at higher frequencies as shown. Capacitive Reactance against Frequency Capacitive reactance of a capacitor decreases as the frequency across its plates increases. Therefore, capacitive reactance is inversely proportional to frequency. Capacitive reactance opposes current flow but the electrostatic charge on the plates (its AC capacitance value) remains constant. This means it becomes easier for the capacitor to fully absorb the change in charge on its plates during each half cycle. Also as the frequency increases the current flowing into the capacitor increases in value because the rate of voltage change across its plates increases. We can present the effect of very low and very high frequencies on the reactance of a pure AC Capacitance as follows: 70

71 In an AC circuit containing pure capacitance the current (electron flow) flowing into the capacitor is given as: and therefore, the rms current flowing into an AC capacitance will be defined as: Where: I C = V/(1/ωC) (or I C = V/X C ) is the current magnitude and θ = + 90 o which is the phase difference or phase angle between the voltage and current. For a purely capacitive circuit, Ic leads Vc by 90 o, or Vc lags Ic by 90 o. Phasor Domain In the phasor domain the voltage across the plates of an AC capacitance will be: and in Polar Form this would be written as: X C -90 o where: 71

72 AC Resistance and Impedance Impedance, measured in Ohms, is the effective resistance to current flow around an AC circuit containing resistances and reactances AC Resistance with a Sinusoidal Supply When the switch is closed, an AC voltage, V will be applied to resistor, R. This voltage will cause a current to flow which in turn will rise and fall as the applied voltage rises and falls sinusoidally. As the load is a resistance, the current and voltage will both reach their maximum or peak values and fall through zero at exactly the same time, i.e. they rise and fall simultaneously and are therefore said to be inphase. Then the electrical current that flows through an AC resistance varies sinusoidally with time and is represented by the expression, I(t) = Im x sin(ωt + θ), where Im is the maximum amplitude of the current and θ is its phase angle. In addition we can also say that for any given current, i flowing through the resistor the maximum or peak voltage across the terminals of R will be given by Ohm s Law as: and the instantaneous value of the current, i will be: So for a purely resistive circuit the alternating current flowing through the resistor varies in proportion to the applied voltage across it following the same sinusoidal pattern. As the supply frequency is common to both the voltage and current, their phasors will also be common resulting in the current being in-phase with the voltage, ( θ = 0 ). In other words, there is no phase difference between the current and the voltage when using an AC resistance as the current will achieve its maximum, minimum and zero values whenever the voltage reaches its maximum, minimum and zero values as shown below. 72

73 Sinusoidal Waveforms for AC Resistance This in-phase effect can also be represented by a phasor diagram. In the complex domain, resistance is a real number only meaning that there is no j or imaginary component. Therefore, as the voltage and current are both in-phase with each other, there will be no phase difference ( θ = 0 ) between them, so the vectors of each quantity are drawn super-imposed upon one another along the same reference axis. The transformation from the sinusoidal time-domain into the phasor-domain is given as. Phasor Diagram for AC Resistance As a phasor represents the RMS values of the voltage and current quantities unlike a vector which represents the peak or maximum values, dividing the peak value of the time-domain expressions above by 2 the corresponding voltage-current phasor relationship is given as. RMS Relationship Phase Relationship This shows that a pure resistance within an AC circuit produces a relationship between its voltage and current phasors in exactly the same way as it would relate the same resistors voltage and current relationship within a DC circuit. However, in a DC circuit this relationship is commonly called Resistance, as defined by Ohm s Law but in a sinusoidal AC circuit this voltage-current 73

74 relationship is now called Impedance. In other words, in an AC circuit electrical resistance is called Impedance. In both cases this voltage-current ( V-I ) relationship is always linear in a pure resistance. So when using resistors in AC circuits the term Impedance, symbol Z is the generally used to mean its resistance. Therefore, we can correctly say that for a resistor, DC resistance = AC impedance, or R = Z. The impedance vector is represented by the letter, ( Z ) for an AC resistance value with the units of Ohm s ( Ω ) the same as for DC. Then Impedance ( or AC resistance ) can be defined as: AC Impedance Impedance can also be represented by a complex number as it depends upon the frequency of the circuit, ω when reactive components are present. But in the case of a purely resistive circuit this reactive component will always be zero and the general expression for impedance in a purely resistive circuit given as a complex number will be: Z = R + j0 = R Ω s Since the phase angle between the voltage and current in a purely resistive AC circuit is zero, the power factor must also be zero and is given as: cos 0 o = 1.0, Then the instantaneous power consumed in the resistor is given by: However, as the average power in a resistive or reactive circuit depends upon the phase angle and in a purely resistive circuit this is equal to θ = 0, the power factor is equal to one so the average power consumed by an AC resistance can be defined simply by using Ohm s Law as: which are the same Ohm s Law equations as for DC circuits. Then the effective power consumed by an AC resistance is equal to the power consumed by the same resistor in a DC circuit. Many AC circuits such as heating elements and lamps consist of a pure ohmic resistance only and have negligible values of inductance or capacitance containing on impedance. In such circuits we can use both well as simple circuit rules for calculating and finding the voltage, current, impedance and power as in DC circuit analysis. When working with such rules it is usual to use RMS values only. AC Resistance Example No1 An electrical heating element which has an AC resistance of 60 Ohms is connected across a 240V AC single phase supply. Calculate the current drawn from the supply and the power consumed by the heating element. Also draw the corresponding phasor diagram showing the phase relationship between the current and voltage. 1. The supply current: 2. The Active power consumed by the AC resistance is calculated as: 3. As there is no phase difference in a resistive component, ( θ = 0 ), the corresponding phasor diagram is given as: 74

75 \ AC Resistance Example No2 A sinusoidal voltage supply defined as: V(t) = 100 x cos(ωt + 30 o ) is connected to a pure resistance of 50 Ohms. Determine its impedance and the peak value of the current flowing through the circuit. Draw the corresponding phasor diagram. The sinusoidal voltage across the resistance will be the same as for the supply in a purely resistive circuit. Converting this voltage from the time-domain expression into the phasor-domain expression gives us: Applying Ohms Law gives us: The corresponding phasor diagram will therefore be: j-notation: The mathematics used in Electrical Engineering to add together resistances, currents or DC voltages use what are called real numbers used as either integers or as fractions. But real numbers are not the only kind of numbers we need to use especially when dealing with frequency dependent sinusoidal sources and vectors. As well as using normal or real numbers, Complex Numbers were introduced to allow complex equations to be solved with numbers that are the square roots of negative numbers, -1. In electrical engineering this type of number is called an imaginary number and to distinguish an imaginary number from a real number the letter j known commonly in electrical engineering as the j-operator, is used. Thus the letter j is placed in front of a real number to signify its imaginary number operation. Examples of imaginary numbers are: j3, j12, j100 etc. Then a complex number consists of two distinct but very much related parts, a Real Number plus an Imaginary Number. Complex Numbers represent points in a two dimensional complex or s-plane that are referenced to two distinct axes. The horizontal axis is called the real axis while the vertical axis is called the imaginary axis. The real and imaginary parts of a complex number are abbreviated as Re(z) and Im(z), respectively. 75

76 Complex numbers that are made up of real (the active component) and imaginary (the reactive component) numbers can be added, subtracted and used in exactly the same way as elementary algebra is used to analyse DC circuits. The rules and laws used in mathematics for the addition or subtraction of imaginary numbers are the same as for real numbers, j2 + j4 = j6 etc. The only difference is in multiplication because two imaginary numbers multiplied together becomes a negative real number. Real numbers can also be thought of as a complex number but with a zero imaginary part labelled j0. The j-operator has a value exactly equal to -1, so successive multiplication of j, ( j x j ) will result in j having the following values of, -1, -j and +1. As the j-operator is commonly used to indicate the anticlockwise rotation of a vector, each successive multiplication or power of j, j 2, j 3 etc, will force the vector to rotate through an angle of 90 o anticlockwise as shown below. Likewise, if the multiplication of the vector results in a -j operator then the phase shift will be -90 o, i.e. a clockwise rotation. Vector Rotation of the j-operator So by multiplying an imaginary number by j2 will rotate the vector by 180o anticlockwise, multiplying by j3 rotates it 270o and by j4 rotates it 360o or back to its original position. Multiplication by j10 or by j30 will cause the vector to rotate anticlockwise by the appropriate amount. In each successive rotation, the magnitude of the vector always remains the same. In Electrical Engineering there are different ways to represent a complex number either graphically or mathematically. One such way that uses the cosine and sine rule is called the Cartesian or Rectangular Form. Complex Numbers using the Rectangular Form In the last tutorial about phasors, we saw that a complex number is represented by a real part and an imaginary part that takes the generalised form of: Where: Z - is the Complex Number representing the Vector x - is the Real part or the Active component y - is the Imaginary part or the Reactive component j - is defined by -1 In the rectangular form, a complex number can be represented as a point on a two dimensional plane called the complex or s-plane. So for example, Z = 6 + j4 represents a single point whose coordinates represent 6 on the horizontal real axis and 4 on the vertical imaginary axis as shown. 76

77 Complex Numbers using the Complex or s-plane But as both the real and imaginary parts of a complex number in the rectangular form can be either a positive number or a negative number, then both the real and imaginary axis must also extend in both the positive and negative directions. This then produces a complex plane with four quadrants called an Argand Diagram as shown above. Four Quadrant Argand Diagram On the Argand diagram, the horizontal axis represents all positive real numbers to the right of the vertical imaginary axis and all negative real numbers to the left of the vertical imaginary axis. All positive imaginary numbers are represented above the horizontal axis while all the negative imaginary numbers are below the horizontal real axis. This then produces a two dimensional complex plane with four distinct quadrants labelled, QI, QII, QIII, and QIV. The Argand diagram above can also be used to represent a rotating phasor as a point in the complex plane whose radius is given by the magnitude of the phasor will draw a full circle around it for every 2π/ω seconds. Then we can extend this idea further to show the definition of a complex number in both the polar and rectangular form for rotations of 90o. Complex Numbers can also have zero real or imaginary parts such as: Z = 6 + j0 or Z = 0 + j4. In this case the points are plotted directly onto the real or imaginary axis. Also, the angle of a complex number can be calculated using simple trigonometry to calculate the angles of right-angled triangles, or measured anti-clockwise around the Argand diagram starting from the positive real axis. Then angles between 0 and 90o will be in the first quadrant ( I ), angles ( θ ) between 90 and 180o in the second quadrant ( II ). The third quadrant ( III ) includes angles between 180 and 270o while the fourth and final quadrant ( IV ) which completes the full circle, includes the angles between 270 and 360o and so on. In all the four quadrants the relevant angles can be found from: tan-1(imaginary component real component) Addition and Subtraction of Complex Numbers: 77

78 The addition or subtraction of complex numbers can be done either mathematically or graphically in rectangular form. For addition, the real parts are firstly added together to form the real part of the sum, and then the imaginary parts to form the imaginary part of the sum and this process is as follows using two complex numbers A and B as examples. Complex Addition and Subtraction Complex Numbers Example No1 Two vectors are defined as, A = 4 + j1 and B = 2 + j3 respectively. Determine the sum and difference of the two vectors in both rectangular ( a + jb ) form and graphically as an Argand Diagram. Mathematical Addition and Subtraction Addition Subtraction Graphical Addition and Subtraction Multiplication and Division of Complex Numbers The multiplication of complex numbers in the rectangular form follows more or less the same rules as for normal algebra along with some additional rules for the successive multiplication of the j- operator where: j2 = -1. So for example, multiplying together our two vectors from above of A = 4 + j1 and B = 2 + j3 will give us the following result. Mathematically, the division of complex numbers in rectangular form is a little more difficult to perform as it requires the use of the denominators conjugate function to convert the denominator of the equation into a real number. This is called rationalising. Then the division of complex numbers is best carried out using Polar Form, which we will look at later. However, as an example in rectangular form lets find the value of vector Adivided by vector B. 78

79 The Complex Conjugate The Complex Conjugate, or simply Conjugate of a complex number is found by reversing the algebraic sign of the complex numbers imaginary number only while keeping the algebraic sign of the real number the same and to identify the complex conjugate of z the symbol z is used. For example, the conjugate of z = 6 + j4 is z = 6 j4, likewise the conjugate of z = 6 j4 is z = 6 + j4. The points on the Argand diagram for a complex conjugate have the same horizontal position on the real axis as the original complex number, but opposite vertical positions. Thus, complex conjugates can be thought of as a reflection of a complex number. The following example shows a complex number, 6 + j4 and its conjugate in the complex plane. Conjugate Complex Numbers The sum of a complex number and its complex conjugate will always be a real number as we have seen above. Then the addition of a complex number and its conjugate gives the result as a real number or active component only, while their subtraction gives an imaginary number or reactive component only. The conjugate of a complex number is an important element used in Electrical Engineering to determine the apparent power of an AC circuit using rectangular form. Complex Numbers using Polar Form Unlike rectangular form which plots points in the complex plane, the Polar Form of a complex number is written in terms of its magnitude and angle. Thus, a polar form vector is presented as: Z = A ±θ, where: Z is the complex number in polar form, A is the magnitude or modulo of the vector and θ is its angle or argument of A which can be either positive or negative. The magnitude and angle of the point still remains the same as for the rectangular form above, this time in polar form the location of the point is represented in a triangular form as shown below. Polar Form Representation of a Complex Number As the polar representation of a point is based around the triangular form, we can use simple geometry of the triangle and especially trigonometry and Pythagoras s Theorem on triangles to find both 79

80 the magnitude and the angle of the complex number. As we remember from school, trigonometry deals with the relationship between the sides and the angles of triangles so we can describe the relationships between the sides as: Using trigonometry again, the angle θ of A is given as follows. Then in Polar form the length of A and its angle represents the complex number instead of a point. Also in polar form, the conjugate of the complex number has the same magnitude or modulus it is the sign of the angle that changes, so for example the conjugate of 6 30 o would be 6 30 o. Converting between Rectangular Form and Polar Form In the rectangular form we can express a vector in terms of its rectangular coordinates, with the horizontal axis being its real axis and the vertical axis being its imaginary axis or j-component. In polar form these real and imaginary axes are simply represented by A θ. Then using our example above, the relationship between rectangular form and polar form can be defined as. Converting Polar Form into Rectangular Form, ( P R ) We can also convert back from rectangular form to polar form as follows. Converting Rectangular Form into Polar Form, ( R P ) Polar Form Multiplication and Division Rectangular form is best for adding and subtracting complex numbers as we saw above, but polar form is often better for multiplying and dividing. To multiply together two vectors in polar form, we must first multiply together the two modulus or magnitudes and then add together their angles. Multiplication in Polar Form Multiplying together 6 30o and 8 45o in polar form gives us. Division in Polar Form Likewise, to divide together two vectors in polar form, we must divide the two modulus and then subtract their angles as shown. 80

81 Fortunately today s modern scientific calculators have built in mathematical functions (check your book) that allows for the easy conversion of rectangular to polar form, ( R P ) and back from polar to rectangular form, ( R P ). Complex Numbers using Exponential Form So far we have considered complex numbers in the Rectangular Form, ( a + jb ) and the Polar Form, ( A ±θ ). But there is also a third method for representing a complex number which is similar to the polar form that corresponds to the length (magnitude) and phase angle of the sinusoid but uses the base of the natural logarithm, e = to find the value of the complex number. This third method is called the Exponential Form. The Exponential Form uses the trigonometric functions of both the sine ( sin ) and the cosine ( cos ) values of a right angled triangle to define the complex exponential as a rotating point in the complex plane. The exponential form for finding the position of the point is based around Euler s Identity, named after Swiss mathematician, Leonhard Euler and is given as: Then Euler s identity can be represented by the following rotating phasor diagram in the complex plane. We can see that Euler s identity is very similar to the polar form above and that it shows us that a number such as Ae jθ which has a magnitude of 1 is also a complex number. Not only can we convert complex numbers that are in exponential form easily into polar form such as: 2e j30 = 2 30, 10e j120 = or -6e j90 = -6 90, but Euler s identity also gives us a way of converting a complex number from its exponential form into its rectangular form. Then the relationship between, Exponential, Polar and Rectangular form in defining a complex number is given as. Complex Number Forms Phasor Notation So far we have look at different ways to represent either a rotating vector or a stationary vector using complex numbers to define a point on the complex plane. Phasor notation is the process of constructing a single complex number that has the amplitude and the phase angle of the given sinusoidal waveform. Then phasor notation or phasor transform as it is sometimes called, transfers the real part of the sinusoidal function: A(t) = Am cos(ωt ± Φ) from the time domain into the complex number domain which is also called the frequency domain. For example: Please note that the 2 converts the maximum amplitude into an effective or RMS value with the phase angle given in radians, ( ω ) 81

82 AC Resistance and Impedance AC Resistance with a Sinusoidal Supply When the switch is closed, an AC voltage, V will be applied to resistor, R. This voltage will cause a current to flow which in turn will rise and fall as the applied voltage rises and falls sinusoidally. As the load is a resistance, the current and voltage will both reach their maximum or peak values and fall through zero at exactly the same time, i.e. they rise and fall simultaneously and are therefore said to be inphase. Then the electrical current that flows through an AC resistance varies sinusoidally with time and is represented by the expression, I(t) = Im x sin(ωt + θ), where Im is the maximum amplitude of the current and θ is its phase angle. In addition we can also say that for any given current, i flowing through the resistor the maximum or peak voltage across the terminals of R will be given by Ohm s Law as: and the instantaneous value of the current, i will be: So for a purely resistive circuit the alternating current flowing through the resistor varies in proportion to the applied voltage across it following the same sinusoidal pattern. As the supply frequency is common to both the voltage and current, their phasors will also be common resulting in the current being in-phase with the voltage, ( θ = 0 ). In other words, there is no phase difference between the current and the voltage when using an AC resistance as the current will achieve its maximum, minimum and zero values whenever the voltage reaches its maximum, minimum and zero values as shown below. Sinusoidal Waveforms for AC Resistance This in-phase effect can also be represented by a phasor diagram. In the complex domain, resistance is a real number only meaning that there is no j or imaginary component. Therefore, as the voltage and current are both in-phase with each other, there will be no phase difference ( θ = 0 ) between them, so the vectors of each quantity are drawn super-imposed upon one another along the same reference axis. The transformation from the sinusoidal time-domain into the phasor-domain is given as. Phasor Diagram for AC Resistance 82

83 As a phasor represents the RMS values of the voltage and current quantities unlike a vector which represents the peak or maximum values, dividing the peak value of the time-domain expressions above by 2 the corresponding voltage-current phasor relationship is given as. RMS Relationship Phase Relationship This shows that a pure resistance within an AC circuit produces a relationship between its voltage and current phasors in exactly the same way as it would relate the same resistors voltage and current relationship within a DC circuit. However, in a DC circuit this relationship is commonly called Resistance, as defined by Ohm s Law but in a sinusoidal AC circuit this voltage-current relationship is now called Impedance. In other words, in an AC circuit electrical resistance is called Impedance. In both cases this voltage-current ( V-I ) relationship is always linear in a pure resistance. So when using resistors in AC circuits the term Impedance, symbol Z is the generally used to mean its resistance. Therefore, we can correctly say that for a resistor, DC resistance = AC impedance, or R = Z. The impedance vector is represented by the letter, ( Z ) for an AC resistance value with the units of Ohm s ( Ω ) the same as for DC. Then Impedance ( or AC resistance ) can be defined as: AC Impedance: Z = V Ω I Impedance can also be represented by a complex number as it depends upon the frequency of the circuit, ω when reactive components are present. But in the case of a purely resistive circuit this reactive component will always be zero and the general expression for impedance in a purely resistive circuit given as a complex number will be: Z = R + j0 = R Ω s Since the phase angle between the voltage and current in a purely resistive AC circuit is zero, the power factor must also be zero and is given as: cos 0 o = 1.0, Then the instantaneous power consumed in the resistor is given by: However, as the average power in a resistive or reactive circuit depends upon the phase angle and in a purely resistive circuit this is equal to θ = 0, the power factor is equal to one so the average power consumed by an AC resistance can be defined simply by using Ohm s Law as: which are the same Ohm s Law equations as for DC circuits. Then the effective power consumed by an AC resistance is equal to the power consumed by the same resistor in a DC circuit. 83

84 Many AC circuits such as heating elements and lamps consist of a pure ohmic resistance only and have negligible values of inductance or capacitance containing on impedance. In such circuits we can use both Ohm s Law,Kirchoff s Law as well as simple circuit rules for calculating and finding the voltage, current, impedance and power as in DC circuit analysis. When working with such rules it is usual to use RMS values only. AC Resistance Example No2 A sinusoidal voltage supply defined as: V(t) = 100 x cos(ωt + 30o) is connected to a pure resistance of 50 Ohms. Determine its impedance and the peak value of the current flowing through the circuit. Draw the corresponding phasor diagram. The sinusoidal voltage across the resistance will be the same as for the supply in a purely resistive circuit. Converting this voltage from the time-domain expression into the phasor-domain expression gives us: Applying Ohms Law gives us: The corresponding phasor diagram will therefore be: AC Inductance and Inductive Reactance Inductors store their energy in the form of a magnetic field that is created when a voltage is applied across the terminals of an inductor. The growth of the current flowing through the inductor is not instant but is determined by the inductors own self-induced or back emf value. Then for an inductor coil, this back emf voltage VL is proportional to the rate of change of the current flowing through it. This current will continue to rise until it reaches its maximum steady state condition which is around five time constants when this self-induced back emf has decayed to zero. At this point a steady state current is flowing through the coil, no more back emf is induced to oppose the current flow and therefore, the coil acts more like a short circuit allowing maximum current to flow through it. However, in an alternating current circuit which contains an AC Inductance, the flow of current through an inductor behaves very differently to that of a steady state DC voltage. Now in an AC circuit, the opposition to the current flowing through the coils windings not only depends upon the inductance of the coil but also the frequency of the applied voltage waveform as it varies from its positive to negative values. The actual opposition to the current flowing through a coil in an AC circuit is determined by the AC Resistance of the coil with this AC resistance being represented by a complex number. But to distinguish a DC resistance value from an AC resistance value, which is also known as Impedance, the term Reactance is used. Like resistance, reactance is measured in Ohm s but is given the symbol X to distinguish it from a purely resistive R value and as the component in question is an inductor, the reactance of an inductor is called Inductive Reactance, ( XL ) and is measured in Ohms. Its value can be found from the formula. Inductive Reactance X C = 2πfL 84

85 Where: XL is the Inductive Reactance in Ohms, ƒ is the frequency in Hertz and L is the inductance of the coil in Henries. We can also define inductive reactance in radians, where Omega, ω equals 2πƒ. X C = ωl So whenever a sinusoidal voltage is applied to an inductive coil, the back emf opposes the rise and fall of the current flowing through the coil and in a purely inductive coil which has zero resistance or losses, this impedance (which can be a complex number) is equal to its inductive reactance. Also reactance is represented by a vector as it has both a magnitude and a direction (angle). Consider the circuit below. AC Inductance with a Sinusoidal Supply This simple circuit above consists of a pure inductance of L Henries ( H ), connected across a sinusoidal voltage given by the expression: V(t) = Vmax sin ωt. When the switch is closed this sinusoidal voltage will cause a current to flow and rise from zero to its maximum value. This rise or change in the current will induce a magnetic field within the coil which in turn will oppose or restrict this change in the current. But before the current has had time to reach its maximum value as it would in a DC circuit, the voltage changes polarity causing the current to change direction. This change in the other direction once again being delayed by the self-induced back emf in the coil, and in a circuit containing a pure inductance only, the current is delayed by 90o. The applied voltage reaches its maximum positive value a quarter ( 1/4ƒ ) of a cycle earlier than the current reaches its maximum positive value, in other words, a voltage applied to a purely inductive circuit LEADS the current by a quarter of a cycle or 90o as shown below. Sinusoidal Waveforms for AC Inductance This effect can also be represented by a phasor diagram were in a purely inductive circuit the voltage LEADS the current by 90 o. But by using the voltage as our reference, we can also say that the current LAGS the voltage by one quarter of a cycle or 90 o as shown in the vector diagram below. Phasor Diagram for AC Inductance So for a pure loss less inductor, V L leads I L by 90 o, or we can say that I L lags V L by 90 o. There are many different ways to remember the phase relationship between the voltage and current flowing through a pure inductor circuit, but one very simple and easy to remember way is to use the mnemonic expression ELI (pronounced Ellie as in the girls name). ELI stands for Electromotive force first in an AC inductance, L before the current I. In other words, voltage before the current in an 85

86 inductor, E, L, I equals ELI, and whichever phase angle the voltage starts at, this expression always holds true for a pure inductor circuit. AC Capacitance and Capacitive Reactance Capacitors store energy on their conductive plates in the form of an electrical charge. When a capacitor is connected across a DC supply voltage it charges up to the value of the applied voltage at a rate determined by its time constant. A capacitor will maintain or hold this charge indefinitely as long as the supply voltage is present. During this charging process, a charging current, i flows into the capacitor opposed by any changes to the voltage at a rate which is equal to the rate of change of the electrical charge on the plates. A capacitor therefore has an opposition to current flowing onto its plates. The relationship between this charging current and the rate at which the capacitors supply voltage changes can be defined mathematically as: i = C(dv/dt), where C is the capacitance value of the capacitor in farads and dv/dt is the rate of change of the supply voltage with respect to time. Once it is fullycharged the capacitor blocks the flow of any more electrons onto its plates as they have become saturated and the capacitor now acts like a temporary storage device. A pure capacitor will maintain this charge indefinitely on its plates even if the DC supply voltage is removed. However, in a sinusoidal voltage circuit which contains AC Capacitance, the capacitor will alternately charge and discharge at a rate determined by the frequency of the supply. Then capacitors in AC circuits are constantly charging and discharging respectively. When an alternating sinusoidal voltage is applied to the plates of an AC capacitor, the capacitor is charged firstly in one direction and then in the opposite direction changing polarity at the same rate as the AC supply voltage. This instantaneous change in voltage across the capacitor is opposed by the fact that it takes a certain amount of time to deposit (or release) this charge onto the plates and is given by V = Q/C. Consider the circuit below. AC Capacitance with a Sinusoidal Supply When the switch is closed in the circuit above, a high current will start to flow into the capacitor as there is no charge on the plates at t = 0. The sinusoidal supply voltage, V is increasing in a positive direction at its maximum rate as it crosses the zero reference axis at an instant in time given as 0o. Since the rate of change of the potential difference across the plates is now at its maximum value, the flow of current into the capacitor will also be at its maximum rate as the maximum amount of electrons are moving from one plate to the other. As the sinusoidal supply voltage reaches its 90 o point on the waveform it begins to slow down and for a very brief instant in time the potential difference across the plates is neither increasing nor decreasing therefore the current decreases to zero as there is no rate of voltage change. At this 90 o point the potential difference across the capacitor is at its maximum ( Vmax ), no current flows into the capacitor as the capacitor is now fully charged and its plates saturated with electrons. At the end of this instant in time the supply voltage begins to decrease in a negative direction down towards the zero reference line at 180 o. Although the supply voltage is still positive in nature the 86

87 capacitor starts to discharge some of its excess electrons on its plates in an effort to maintain a constant voltage. This results in the capacitor current flowing in the opposite or negative direction. When the supply voltage waveform crosses the zero reference axis point at instant 180 o, the rate of change or slope of the sinusoidal supply voltage is at its maximum but in a negative direction, consequently the current flowing into the capacitor is also at its maximum rate at that instant. Also at this 180 o point the potential difference across the plates is zero as the amount of charge is equally distributed between the two plates. Then during this first half cycle 0 o to 180 o, the applied voltage reaches its maximum positive value a quarter (1/4ƒ) of a cycle after the current reaches its maximum positive value, in other words, a voltage applied to a purely capacitive circuit LAGS the current by a quarter of a cycle or 90 o as shown below. Sinusoidal Waveforms for AC Capacitance During the second half cycle 180 o to 360 o, the supply voltage reverses direction and heads towards its negative peak value at 270 o. At this point the potential difference across the plates is neither decreasing nor increasing and the current decreases to zero. The potential difference across the capacitor is at its maximum negative value, no current flows into the capacitor and it becomes fully charged the same as at its 90 o point but in the opposite direction. As the negative supply voltage begins to increase in a positive direction towards the 360 o point on the zero reference line, the fully charged capacitor must now loose some of its excess electrons to maintain a constant voltage as before and starts to discharge itself until the supply voltage reaches zero at 360 o at which the process of charging and discharging starts over again. From the voltage and current waveforms and description above, we can see that the current is always leading the voltage by 1/4 of a cycle or π/2 = 90 o out-of-phase with the potential difference across the capacitor because of this charging and discharging process. Then the phase relationship between the voltage and current in an AC capacitance circuit is the exact opposite to that of an AC Inductance we saw in the previous tutorial. This effect can also be represented by a phasor diagram where in a purely capacitive circuit the voltage LAGS the current by 90 o. But by using the voltage as our reference, we can also say that the current LEADS the voltage by one quarter of a cycle or 90 o as shown in the vector diagram below. Phasor Diagram for AC Capacitance So for a pure capacitor, V C lags I C by 90o, or we can say that I C leads V C by 90 o. There are many different ways to remember the phase relationship between the voltage and current flowing in a pure AC capacitance circuit, but one very simple and easy to remember way is to use the mnemonic expression called ICE. ICE stands for current Ifirst in an AC capacitance, C before Electromotive force. In other words, current before the voltage in a 87

88 capacitor, I, C, E equals ICE, and whichever phase angle the voltage starts at, this expression always holds true for a pure AC capacitance circuit. Capacitive Reactance So we now know that capacitors oppose changes in voltage with the flow of electrons onto the plates of the capacitor being directly proportional to the rate of voltage change across its plates as the capacitor charges and discharges. Unlike a resistor where the opposition to current flow is its actual resistance, the opposition to current flow in a capacitor is called Reactance. Like resistance, reactance is measured in Ohm s but is given the symbol X to distinguish it from a purely resistive R value and as the component in question is a capacitor, the reactance of a capacitor is called Capacitive Reactance, ( X C ) which is measured in Ohms. Since capacitors charge and discharge in proportion to the rate of voltage change across them, the faster the voltage changes the more current will flow. Likewise, the slower the voltage changes the less current will flow. This means then that the reactance of an AC capacitor is inversely proportional to the frequency of the supply as shown. Capacitive Reactance X C = 1 2πfC Where: X C is the Capacitive Reactance in Ohms, ƒ is the frequency in Hertz and C is the AC capacitance in Farads, symbol F. When dealing with AC capacitance, we can also define capacitive reactance in terms of radians, where Omega, ω equals 2πƒ. X C = 1 ωc From the above formula we can see that the value of capacitive reactance and therefore its overall impedance ( in Ohms) decreases towards zero as the frequency increases acting like a short circuit. Likewise, as the frequency approaches zero or DC, the capacitors reactance increases to infinity, acting like an open circuit which is why capacitors block DC. AC through a Series RL Circuit We have seen above that the current flowing through a purely inductive coil lags the voltage by 90o and when we say a purely inductive coil we mean one that has no ohmic resistance and therefore, no I2R losses. But in the real world, it is impossible to have a purely AC Inductance only. All electrical coils, relays, solenoids and transformers will have a certain amount of resistance no matter how small associated with the coil turns of wire being used. This is because copper wire has resistivity. Then we can consider our inductive coil as being one that has a resistance, R in series with an inductance, L producing what can be loosely called an impure inductance. If the coil has some internal resistance then we need to represent the total impedance of the coil as a resistance in series with an inductance and in an AC circuit that contains both inductance, L and resistance, R the voltage, V across the combination will be the phasor sum of the two component voltages, VR and VL. This means then that the current flowing through the coil will still lag the voltage, but by an amount less than 90o depending upon the values of VR and VL, the phasor sum. The new angle between the voltage and the current waveforms gives us their Phase Difference which as we know is the phase angle of the circuit given the Greek symbol phi, Φ. Consider the circuit below were a pure non-inductive resistance, R is connected in series with a pure inductance, L. Series Resistance-Inductance Circuit 88

89 In the RL series circuit above, we can see that the current is common to both the resistance and the inductance while the voltage is made up of the two component voltages, VR and VL. The resulting voltage of these two components can be found either mathematically or by drawing a vector diagram. To be able to produce the vector diagram a reference or common component must be found and in a series AC circuit the current is the reference source as the same current flows through the resistance and the inductance. The individual vector diagrams for a pure resistance and a pure inductance are given as: Vector Diagrams for the Two Pure Components We can see from above and from our previous tutorial about AC Resistance that the voltage and current in a resistive circuit are both in phase and therefore vector VR is drawn superimposed to scale onto the current vector. Also from above it is known that the current lags the voltage in an AC inductance (pure) circuit therefore vector V L is drawn 90 o in front of the current and to the same scale as V R as shown. Vector Diagram of the Resultant Voltage From the vector diagram above, we can see that line OB is the horizontal current reference and line OA is the voltage across the resistive component which is in-phase with the current. Line OC shows the inductive voltage which is 90 o in front of the current therefore it can still be seen that the current lags the purely inductive voltage by 90 o. Line OD gives us the resulting supply voltage. Then: V equals the r.m.s value of the applied voltage. I equals the r.m.s. value of the series current. V R equals the I.R voltage drop across the resistance which is in-phase with the current. V L equals the I. XL voltage drop across the inductance which leads the current by 90 o. As the current lags the voltage in a pure inductance by exactly 90 o the resultant phasor diagram drawn from the individual voltage drops V R and V L represents a right angled voltage triangle shown above as OAD. Then we can also use Pythagoras theorem to mathematically find the value of this resultant voltage across the resistor/inductor ( RL ) circuit. As V R = I. R and V L = I.XL the applied voltage will be the vector sum of the two as follows: V 2 = V R 2 + V L 2 V = V R 2 + V L 2 V = (IR) 2 + (IX L ) 2 V I = (R) 2 + (X L ) 2 The quantity (R) 2 + (X L ) 2 represents the impedance, Z of the circuit. The Impedance of an AC Inductance 89

90 Impedance, Z is the TOTAL opposition to current flowing in an AC circuit that contains both Resistance, ( the real part ) and Reactance ( the imaginary part ). Impedance also has the units of Ohms, Ω s. Impedance depends upon the frequency, ω of the circuit as this affects the circuits reactive components and in a series circuit all the resistive and reactive impedance s add together. Impedance can also be represented by a complex number, Z = R + jxl but it is not a phasor, it is the result of two or more phasors combined together. If we divide the sides of the voltage triangle above by I, another triangle is obtained whose sides represent the resistance, reactance and impedance of the circuit as shown below. The RL Impedance Triangle Then: ( Impedance ) 2 = ( Resistance ) 2 + ( j Reactance ) 2 where j represents the 90 o phase shift. This means that the positive phase angle, θ between the voltage and current is given as. Phase Angle Z = (R) 2 + (X L ) 2 Ω cos 1 φ = R Z sin 1 φ = X L Z tan 1 φ = X L R While our example above represents a simple non-pure AC inductance, if two or more inductive coils are connected together in series or a single coil is connected in series with many non-inductive resistances, then the total resistance for the resistive elements would be equal to: R 1 + R 2 + R 3 etc, giving a total resistive value for the circuit. Likewise, the total reactance for the inductive elements would be equal to: X 1 + X 2 + X 3 etc, giving a total reactance value for the circuit. This way a circuit containing many chokes, coils and resistors can be easily reduced down to an impedance value, Z comprising of a single resistance in series with a single reactance, Z 2 = R 2 + X 2. Series resonance: series is related to series RLC circuit. In an series RLC circuit resonance occurs when voltage across L and C are same in magnitude and 180 degrees out of phase. Parallel resonance: series is related to Parallel RLC circuit. In an Parallel RLC circuit resonance occurs when current flowing through L and C are same in magnitude and 180 degrees out of phase. Series Resonance: I 90

91 Let us consider series RLC circuit as shown, Here, Z = R+ j( XL XC) Where, XL = 2ΠfL XC = 1/ 2ΠfC To say that circuit is under resonance, Z = R This happens only when, XL = XC i.e imaginary part of total impedance is zero. ( XL XC = 0) XL = XC 2ΠfL = 1/ 2ΠfC wl = 1/ wc ( w= 2Πf, angular frequency rad/sec) w 2 = 1/ LC fr = 1/ 2Π LC--- resonant frequency. IMPEDANCE CURVE: We know that, Z = R+ j( XL XC) Z = (R2 + XL XC 2 For the lower frequencies, XC > XL the total impedance is Z = (R2 + XC XL 2 And current through circuit is, I = V/ (R2 + XC XL 2 Here we can say that at lower frequencies Z is very high as XC is infinitely high and current is very low, but as frequency increases towards fr Xc value decreases and (XC-XL) decreases and Z decreases their by current increases. At resonant frequency, XL=XC Z =R (minimum value) 91

92 And current is, I = V / R = maximum current For the higher frequencies, XL > XC the total impedance is Z = (R2 + XL XC 2 And current through circuit is, I = V/ (R2 + XL XC 2 Here we can say that at higher frequencies Z is decreases as XL increases and current is also decreases, as frequency increases towards very high frequencies XL value increases and (XL- XC) increases and Z increases their by current decreases. Here for frequencies < fr circuit is said to be dominantly capacitive and for frequencies > fr circuit is said to be dominantly inductive. BANDWIDTH : Let f1, f2 --- lower and higher cut-off frequencies At f1, I = V / 2R And also at f2, I = V / 2R This is possible only when, At f1, 1/ w1 C w1. L = R f2, w2 L 1/w2. C = R equate 1 and 2 1/ w1 C w1. L = w2 L 1/w2. C 1/ w1 C w1. L = w2 L 1/w2. C w1.w2 = 1/ LC w1.w2 = wr 2 now add two equations, 92

93 1/ w1 C w1. L + w2 L 1/w2. C =2R (w2-w1)l + (w2-w1)/w1w2c = 2R By sloving above equation, f2 f1 = R / 2ΠL Lower cut off frequency Upper cut off frequency (f 1 ) = fr-r/4 L (f 2 ) = fr+r/4 L Quality factor: For inductor. Q = 2Π * energy stored in the element Energy dissipated in one cycle. Q = 2Π * ½ LI I 2. R.t Q = 2Π * ½ LI I 2. R.1/f Q = 2ΠL /R = XL /R For capacitor. Q = 2Π * energy stored in the element Energy dissipated in one cycle. Q = 2Π * ½ CV I 2. R.t Q = 2Π * ½ CV (V/ 2R) 2. R.1/f Q = 1 /2Π fc R = XC /R 93

94 MAGNIFICATION: Magnification is defined ratio voltage across energy storing elements and input voltage under resonance. VL / Vi = IXL / IR = XL / R =Q VC / Vi = IXC / IR = XC / R =Q To say that life of the circuit is high the magnification must be high Parallel Resonance: Signal Generator Let us consider parallel RLC circuit as shown, Here, Y =1/ R+ j( 1/XL 1/XC) = G+ j(bl BC) Where, BL = 1 / 2ΠfL BC = 2ΠfC To say that circuit is under resonance, Y = G This happens only when, BL = BC i.e imaginary part of total impedance is zero. ( BL BC = 0) BL = BC 1 / 2ΠfL = 2ΠfC 1 / wl = wc ( w= 2Πf, angular frequency rad/sec) w 2 = 1/ LC fr = 1/ 2Π LC--- resonant frequency. ADMITTANCE CURVE: 94

95 We know that, Y = G+ j(bl BC) Y = (G2 + BL BC 2 For the lower frequencies, BL > BC the total admittance is Y = (G2 + BL BC 2 And current through circuit is, V = I / (G2 + BL BC 2 Here we can say that at lower frequencies Y is very high as BL is infinitely high and voltage is very low, but as frequency increases towards fr BL value decreases and (BL-BC) decreases and Y decreases their by voltage increases. At resonant frequency, BL=BC Y =1 / R (minimum value) And voltage is, V = I / G = maximum current For the higher frequencies, BC > BL the total admittance is Y = (G2 + BC BL 2 And current through circuit is, V = I/ (G2 + BC BL 2 Here we can say that at higher frequencies Y is decreases as BC increases and voltage is also decreases, as frequency increases towards very high frequencies BC value increases and (BC- BL) increases and Y increases their by voltage decreases. Here for frequencies < fr circuit is said to be dominantly inductive and for frequencies > fr circuit is said to be dominantly capacitive. BANDWIDTH : Let f1, f2 --- lower and higher cut-off frequencies 95

96 At f1, V = I / 2G And also at f2, V = I / 2G This is possible only when, At f1, w1 C 1/w1. L = G f2, 1 / w2 L w2. C = G equate 1 and 2 1w1 C 1 / w1. L = 1 / w2 L w2. C 1/ w1 C w1. L = w2 L 1/w2. C w1.w2 = 1/ LC w1.w2 = wr 2 now add two equations, w1 C 1 / w1. L + 1 / w2 L w2. C =2G By sloving above equation, f2 f1 = 1 / 2ΠRC Lower cut off frequency (f 1 ) =fr-1/4 RC Upper cut off frequency (f 2 ) = fr+1/4 RC Quality factor: For inductor. Q = 2Π * energy stored in the element Energy dissipated in one cycle. Q = 2Π * ½ L(V/XL) (V / 2R) 2..t Q = 2Π * ½ LI I 2. R.1/f Q = R / XL For capacitor. Q = 2Π * energy stored in the element Energy dissipated in one cycle. 96

97 Q = 2Π * ½ CV I 2. R.t Q = 2Π * ½ CV (V/ 2R) 2. R.1/f Q = R / BC = XC.R MAGNIFICATION: Magnification is defined ratio voltage across energy storing elements and input voltage under resonance. IL / I = V/BL / V/R = R / BL IC / Ii = V / BC / V / R = R / BC =Q To say that life of the circuit is high the magnification must be high. 97

98 UNIT-IV MAGNETIC CIRCUITS AND THREE PHASE CIRCUITS 98

99 MAGNETIC CIRCUITS Let us consider an coil allowing current of IA which develops magnetic lines of force forming north and south poles, the flow of magnetic lines from north pole to south pole. If coil is wounded on some core allowing current IA develops flux Ф and this flux follows the path of core to form magnetic circuit. Definitions: Magnetic flux density: Flux developed per unit area. (Ф / A). Represented with B. Whose units are webers/mt2 or tesla. MMF( Magneto motive force): It is the measure of ability of amount of flux can be developed in the coil.(j), which is given as product of number of turns and current flowing through coil. J = N.I (A-turns) Field intensity: It is defined as mmf per unit length.(h) H = mmf / l ( A-turns / mt) Reluctance: It is the property of core which opposes magnetic flux. Generally cores of two types,they are air and iron core. When compare to iron core air core has more reluctance Property. (or) Reluctance is ratio of mmf to the flux.( J / Ф) R = (l / µ.a). Where, l mean length of magnetic circuit. A area of cross of core. µ - permeability of the core. - µo. µr - absolute permeability(µo) - relative permeability(µr) varies for different types of cores. Hence mmf is also given as,j = R. Ф Fringing and leakage effect: Let us consider ring core with small air gap. When flux developed in core, during flow of flux if there is a sudden change in core whose values are largely differ, flux suddenly bulges out which is called as fringing. 99

100 Generally core laminated and these lamination may consists of some weak points and flux leaked through these weak points is called as leakage flux. The use of inductors somehow is restricted due to its ability of radiation of electromagnetic interference. In addition, it is a side effect which makes inductor deviate a little bit from it is real behavior. Definition of Self Inductance: Whenever, current flows through a circuit or coil, flux is produced surround it and this flux also links with the coil itself. Self induced emf in a coil is produced due to its own changing flux and changing flux is caused by changing current in the coil. So, it can be concluded that self-induced emf is ultimately due to changing current in the coil itself. And self inductance is the property of a coil or solenoid, which causes a self-induced emf to be produced, when the current through it changes. Explanation of Self Inductance of a Coil Whenever changing flux, links with a circuit, an emf is induced in the circuit. This is Faraday s laws of electromagnetic induction. According to this law, Where, e is the induced emf. N is the number of turns. (dφ/dt) is the rate of change of flux leakage with respect to time. The negative sign of the equation indicates that the induced emf opposes the change flux linkage. This is according to Len z law of induction. The flux is changing due to change in current of the circuit itself. The produced flux due to a current, in a circuit, always proportional to that current. That means, Where, i is the current in the circuit and K is the proportional constant. Now, from equation (1) and (2) we get, The above equation can also be rewritten as Where, L (= NK) is the constant of proportionality and this L is defined as the self inductance of the coil or solenoid. This L determines how much emf will be induced in a coil for a specific rate of change of current through it. Now, from equation (1) and (3), we get, Integrating, both sides we get, From the above expression, inductance can be also be defined as, If the current I through an N turn coil produces a flux of Ø Weber, then its self-inductance would be L. A coil can be designed to have a 100

101 specific value of self-inductance (L). In the view of self-inductance, a coil or solenoid is referred as an inductor. Now, if cross-sectional area of the core of the inductor(coil) is A and flux density in the core is B, then total flux inside the core of inductor is AB. Therefore, equation (4) can be written as Now, B = μoμrh Where, H is magnetic field strength, µo and μr are permeability of free space and relative permeability of the core respectively. Now, H = mmf/unit length = Ni/l Where l is the length of the coil. Therefore, Self Inductance Formula Mutual Inductance is the basic operating principal of the transformer, motors, generators and any other electrical component that interacts with another magnetic field. Then we can define mutual induction as the current flowing in one coil that induces a voltage in an adjacent coil. But mutual inductance can also be a bad thing as stray or leakage inductance from a coil can interfere with the operation of another adjacent component by means of electromagnetic induction, so some form of electrical screening to a ground potential may be required. The amount of mutual inductance that links one coil to another depends very much on the relative positioning of the two coils. If one coil is positioned next to the other coil so that their physical distance apart is small, then nearly all of the magnetic flux generated by the first coil will interact with the coil turns of the second coil inducing a relatively large emf and therefore producing a large mutual inductance value. Likewise, if the two coils are farther apart from each other or at different angles, the amount of induced magnetic flux from the first coil into the second will be weaker producing a much smaller induced emf and therefore a much smaller mutual inductance value. So the effect of mutual inductance is very much dependant upon the relative positions or spacing, ( S ) of the two coils and this is demonstrated below. Mutual Inductance between Coils The mutual inductance that exists between the two coils can be greatly increased by positioning them on a common soft iron core or by increasing the number of turns of either coil as would be found in a transformer. If the two coils are tightly wound one on top of the other over a common soft iron core unity coupling is said to exist between them as any losses due to the leakage of flux will be extremely small. Then 101

102 assuming a perfect flux linkage between the two coils the mutual inductance that exists between them can be given as. Where: µ o is the permeability of free space (4.π.10-7 ) µ r is the relative permeability of the soft iron core N is in the number of coil turns A is in the cross-sectional area in m 2 l is the coils length in meters Mutual Induction Here the current flowing in coil one, L 1 sets up a magnetic field around itself with some of these magnetic field lines passing through coil two, L 2 giving us mutual inductance. Coil one has a current of I 1 and N 1 turns while, coil two has N 2 turns. Therefore, the mutual inductance, M 12 of coil two that exists with respect to coil one depends on their position with respect to each other and is given as: 102

103 Likewise, the flux linking coil one, L 1 when a current flows around coil two, L 2 is exactly the same as the flux linking coil two when the same current flows around coil one above, then the mutual inductance of coil one with respect of coil two is defined as M 21. This mutual inductance is true irrespective of the size, number of turns, relative position or orientation of the two coils. Because of this, we can write the mutual inductance between the two coils as: M 12 = M 21 = M. Then we can see that self inductance characterises an inductor as a single circuit element, while mutual inductance signifies some form of magnetic coupling between two inductors or coils, depending on their distance and arrangement, an hopefully we remember from our tutorials on Electromagnets that the self inductance of each individual coil is given as: and Dot Convention: Dot convention is the method used to find whether mutually induced emf is positive or negative. Dot convention method is based on right hand thumb rule.right hand thumb rule states that if thum indictaes direction of current then remaining folded fingers indicates how the coil is wounded. that, When we represent coupled coils, they may indicated with dots. Dot convention says If both the currents enter the dot then mutually indued emf is positive. If both the currents leaving the dot then mutually indued emf is positive. If one of the current enter the dot and other leaving the dot then mutually indued emf is negative. 103

104 co-efficient of coupling: Let us consider coupled coil as shown below Here, v1= L1 di1/dt + Mdi2/dt v2= L2 di2/dt + Mdi1/dt total energy stored is, w = ʃv1.i1dt + ʃv2.i2dt = ʃ (L1 di1/dt + Mdi2/dt).i1dt + ʃ.i2d(l1 di1/dt + Mdi2/dt)dt w = (1/2 L1i1 2 )+ (1/2 L2i2 2 )+Mi1i2 similarly, 104

105 Total energ stored is, w = (1/2 L1i1 2 )+ (1/2 L2i2 2 )-Mi1i2 To say energy is positive, (L1.L2) M >= 0. By removing the proportionality, K = M / (L1.L2). Here K is defined as co-efficient of coupling maximum value of 1. If K=1 then coils are said to be perfectly coupled i.e maximum mutual flux linkage takes place. co-efficient of coupling: Let us consider coupled coil as shown below Here, v1= L1 di1/dt + Mdi2/dt v2= L2 di2/dt + Mdi1/dt total energy stored is, w = ʃv1.i1dt + ʃv2.i2dt = ʃ (L1 di1/dt + Mdi2/dt).i1dt + ʃ.i2d(l1 di1/dt + Mdi2/dt)dt w = (1/2 L1i1 2 )+ (1/2 L2i2 2 )+Mi1i2 similarly, Total energ stored is, w = (1/2 L1i1 2 )+ (1/2 L2i2 2 )-Mi1i2 To say energy is positive, (L1.L2) M >=

106 By removing the proportionality, K = M / (L1.L2). Here K is defined as co-efficient of coupling maximum value of 1. If K=1 then coils are said to be perfectly coupled i.e maximum mutual flux linkage takes place. Series magnetic circuit: Let us consider an coil of N turns wounded on ring core. When some current I A is allowed through coil flux Ф is developed in it. Let, mmf required to develop Ф is J R is reluctance of core. N- number of turns. I- Current through coil. Hence, mmf, J = N.I Drop in core is = Ф.R J = Ф.R = N.I Therefore flux developed in coil is given as, Ф = N.I / R Composite magnetic circuit: Let us consider ring core which comprises of three different materials with different lengths and areas. An coil of N turns is wounded on such core as described above, allowing current I A. Let, Ф1 = flux developed in the first part of core R1 = reluctance of first part of the core l1 = length of first part of the core A1 = area of first part of core J1 = mmf drop in first part of core Ф2 = flux developed in the 2nd part of core R2 = reluctance of 2nd part of the core l2= length of 2nd part of the core A2 = area of 2nd part of core J2= mmf drop in 2nd part of core Ф3 = flux developed in the 3 rd part of core 106

107 R3= reluctance of 3 rd part of the core l3= length of 3 rd part of the core A3 = area of 3 rd part of core J3 mmf drop in 3 rd part of core Hence total mmf required, J = J1+J2+J3 = Ф1.R1+ Ф2.R2+ Ф3.R3=N.I Total flux developed is = NI / (R1+R2+R3) Where, R1 = (l1 / µ1.a1) R2 = (l2 / µ2.a2) R3 = (l3 / µ3.a3) Coupled circuits: When two coils are brought together as close as possible then they form coupled coils. Here when current(i1) is allowed through first coil then magnetic flux Ф1 is developed in it, as other coil brought to close proximity some of Ф1 links with second coil called as Фm1 their by inducing voltage in it and when we close the second coil current flows in it (i2). This current i2 develops Ф2 in it and some of Ф2 links with 1 st coil called as Фm2. If the two coils are of same dimensions Фm1= Фm2 = Фm. Here we define two inducatnces slef inductance of coils L1 and L2, mutal inductance between the coils M12=M21=M. Now we can say that total emf induced in coil is the combination of self and mutually induced emf. Emf in 1 st coil, v1= L1 di1/dt + Mdi2/dt Emf in 2 nd coil, v2= L2 di2/dt + Mdi1/dt 107

108 Types of coupled coils: Coupled coils are of three types, they are 1. Conductively coupled: Here an voltage is fed to the potential divider circuit Called as conductively coupled 2. Inductively coupled: where there is no electrical cconnection, i.e electrically Isolated but magnetically coupled. Eg: Transformer. 3. Conductively and inductively coupled: an best device which can be as conductively and inductively coupled is auto-transformer. 108

109 Introduction : There are two types of system available in electric circuit, single phase and three phase system. In single phase circuit, there will be only one phase, i.e the current will flow through only one wire and there will be one return path called neutral line to complete the circuit. So in single phase minimum amount of power can be transported. Here the generating station and load station will also be single phase. This is an old system using from previous time.in 1882, new invention has been done on poly-phase system, that more than one phase can be used for generating, transmitting and for load system. Three phase circuit is the poly-phase system where three phases are send together from the generator to the load. There are various reasons for this question because there are numbers of advantages over single phase circuit. The three phase system can be used as three single phase line so it can act as three single phase system. The three phases generation and single phase generation is same in the generator except the arrangement of coil in the generator to get 120 phase difference. The conductor needed in three phase circuit is 75% that of conductor needed in single phase circuit. And also the instantaneous power in single phase system falls down to zero as in single phase we can see from the sinusoidal curve but in three phase system the net power from all the phases gives a continuous power to the load. Three phase system: The system which has three phases, i.e., the current will pass through the three wires, and there will be one neutral wire for passing the fault current to the earth is known as the three phase system. In other words, the system which uses three wires for generation, transmission and distribution is kn109own as the three phase system. The three phase system is also used as a single phase system if one of their phase and the neutral wire is taken out from it. The sum of the line currents in the 3-phase system is equal to zero, and their phases are differentiated at an angle of 120º The three-phase system has four wires, i.e., the three current carrying conductors and the one neutral. The cross section area of the neutral conductor is half of the live wire. The current in the neutral wire is equal to the sum of the line current of the three wires and consequently equal to 3 times the zero phase sequence components of current. The three-phase system has several advantages like it requires fewer conductors as compared to the single phase system. It also gives the continuous supply to the load. The three-phase system has higher efficiency and minimum losses. The three phase system induces in the generator which gives the three phase voltage of equal magnitude and frequency. It provides an uninterruptible power, i.e., if one phase of the system is disturbed, then the remaining two phases of the system continue supplies the power. The magnitude of the current in one phase is equal to the sum of the current in the other two phases of the system. 109

110 The 120º phase difference of the three phases is must for the proper working of the system. Otherwise, the system becomes damage Generation of 3 Phase E.M.Fs in a 3 Phase Circuit: In a 3 phase system, there are three equal voltages or EMFs of the same frequency having a phase difference of 120 degrees. These voltages can be produced by a three-phase AC generator having three identical windings displaced apart from each other by 120 degrees electrical. When these windings are kept stationary, and the magnetic field is rotated as shown in the figure A below or when the windings are kept stationary, and the magnetic field is rotated as shown below in figure B, an emf is induced in each winding. The magnitude and frequency of these EMFs are same but are displaced apart from one another by an angle of 120 degrees. Consider three identical coils a 1 a 2, b 1 b 2 and c 1 c 2 as shown in the above figure. In this figure a 1, b 1 and c 1 are the starting terminals, whereas a 2, b 2 and c 2 are the finish terminals of the three coils. The phase difference of 120 degrees has to be maintained between the starts terminals a 1, b 1 and c 1.Now, let the three coils mounted on the same axis, and they are rotated by either keeping coil stationary and moving the magnetic field or vice versa in an anticlockwise direction at (ω) radians per seconds. Three EMFs are induced in the three coils respectively. 110

111 Considering the figure C, the analysis about their magnitudes and directions are given as follows. The emf induced in the coil a 1 a 2 is zero and is increasing in the positive direction as shown by the waveform in the above figure C represented as e a1a2.the coil b 1 b 2 is 120 degrees electrically behind the coil a 1 a 2. The emf induced in this coil is negative and is becoming maximum negative as shown by the wave e b1b2. Similarly, the coil c 1 c 2 is 120 degrees electrically behind the coil b 1 b 2, or we can also say that the coil c 1 c 2 is 240 degrees behind the coil a 1 a 2. The emf induced in the coil is positive and is decreasing as shown in the figure C represented by the waveform e c1c2. Phasor Diagram: The EMFs induced in the three coils in a 3 phase circuits are of the same magnitude and frequency and are displaced by an angle of 120 degrees from each other as shown below in the Phasor diagram. These EMFs of a 3 phase circuits can be expressed in the form of the various equations given below. Phase Sequence: In three phase system the order in which the voltages attain their maximum positive value is called Phase Sequence. There are three voltages or EMFs in three phase system with the same magnitude, but the frequency is displaced by an angle of 120 deg electrically. Taking an example, if the phases of any coil are named as R, Y, B then the Positive phase sequence will be RYB, YBR, BRY also called as clockwise sequence and similarly the Negative phase sequence will be RBY, BYR, YRB respectively and known as an anti-clockwise sequence. 111

112 It is essential because of the following reasons:- The parallel operation of three phase transformer or alternator is only possible when its phase sequence is known. The rotational direction of three phase induction motor depends upon its sequence of phase on three phase supply and thus to reverse its direction the phase sequence of the supply given to the motor has to Types of Connections in Three-Phase System : The three-phase systems are connected in two ways, i.e., the star connection and the delta connection. Their detail explanation is shown below. Star Connection The star connection requires four wires in which there are three phase conductors and one neutral conductor. Such type of connection is mainly used for long distance transmission because it has a neutral point. The neutral point passes the unbalanced current to the earth and hence make the system balance. Delta Connection The delta connection has three wires, and there is a no neutral point. The delta connection is shown in the figure below. The line voltage of the delta connection is equal to the phase voltage. 112

113 Connection of Loads in Three Phase System: The loads in the three-phase system may also connect in the star or delta. The three phase loads connected in the delta and star is shown in the figure below. The three phase load may be balanced or unbalanced. If the three loads (impedances) Z 1, Z 2 and Z 3 has the same magnitude and phase angle then the three phase load is said to be a balanced load. Under balance condition, all the phases and the line voltages are equal in magnitude. Relationship of Line and Phase Voltages and Currents in a Star Connected System: To derive the relations between line and phase currents and voltages of a star connected system, we have first to draw a balanced star connected system. 113

114 Suppose due to load impedance the current lags the applied voltage in each phase of the system by an angle ϕ. As we have considered that the system is perfectly balanced, the magnitude of current and voltage of each phase is the same. Let us say, the magnitude of the voltage across the red phase i.e. magnitude of the voltage between neutral point (N) and red phase terminal (R) is V R. Similarly, the magnitude of the voltage across yellow phase is V Y and the magnitude of the voltage across blue phase is V B. In the balanced star system, magnitude of phase voltage in each phase is V ph. V R = V Y = V B = V ph We know in the star connection, line current is same as phase current. The magnitude of this current is same in all three phases and say it is I L. I R = I Y = I B = I L, Where, I R is line current of R phase, I Y is line current of Y phase and I B is line current of B phase. Again, phase current, I ph of each phase is same as line current IL in star connected system. I R = I Y = I B = I L = I ph. We know in the star connection, line current is same as phase current. The magnitude of this current is same in all three phases and say it is I L. I R = I Y = I B = I L, Where, I R is line current of R phase, I Y is line current of Y phase and I B is line current of B phase. Again, phase current, I ph of each phase is same as line current IL in star connected system. I R = I Y = I B = I L = I ph. thus, for the star-connected system line voltage = 3 phase voltage. Line current = Phase current As, the angle between voltage and current per phase is φ, the electric power per phase is 114

115 So the total power of three phase system is Relationship of Line and Phase Voltages and Currents in a Delta Connected System: In this system of interconnection, the starting ends of the three phases or coils are connected to the finishing ends of the coil. Or the starting end of the first coil is connected to the finishing end of the second coil and so on (for all three coils) and it looks like a closed mesh or circuit as shown in fig (1).In more clear words, all three coils are connected in series to form a close mesh or circuit. Three wires are taken out from three junctions and the all outgoing currents from junction assumed to be positive. In Delta connection, the three windings interconnection looks like a short circuit, but this is not true, if the system is balanced, then the value of the algebraic sum of all voltages around the mesh is zero. When a terminal is open, then there is no chance of flowing currents with basic frequency around the closed mesh. at any instant, the EMF value of one phase is equal to the resultant of the other two phases EMF values but in the opposite direction. Delta or Mesh Connection System is also called Three Phase Three Wire System (3-Phase 3 Wire) and it is the best and suitable system for AC Power Transmission. 1. Line Voltages and Phase Voltages in Delta Connection It is seen from fig 2 that there is only one phase winding between two terminals (i.e. there is one phase winding between two wires). Therefore, in Delta Connection, the voltage between (any pair of) two lines is equal to the phase voltage of the phase winding which is connected between two lines. Since the phase sequence is R Y B, therefore, the direction of voltage from R phase towards Y phase is positive (+), and the voltage of R phase is leading by 120 from Y phase voltage. Likewise, the voltage of Y phase is leading by 120 from the phase voltage of B and its direction is positive from Y towards B. If the line voltage between; 115

116 Line 1 and Line 2 = V RY Line 2 and Line 3 = V YB Line 3 and Line 1 = V BR Then, we see that V RY leads V YB by 120 and V YB leads V BR by 120. Let s suppose, V RY = V YB = V BR = V L (Line Voltage) Then V L = V PH I.e. in Delta connection, the Line Voltage is equal to the Phase Voltage. 2. Line Currents and Phase Currents in Delta Connection It will be noted from the below (fig-2) that the total current of each Line is equal to the vector difference between two phase currents flowing through that line. i.e.; Current in Line 1= I 1 = I R I B Current in Line 2 =I 2 = I Y I R Current in Line 3 =I 3 = I B I Y The current of Line 1 can be found by determining the vector difference between I R and I B and we can do that by increasing the I B Vector in reverse, so that, I R and I B makes a parallelogram. The diagonal of that parallelogram shows the vector difference of I R and I B which is equal to Current in Line 1= I 1. Moreover, by reversing the vector of I B, it may indicate as (-I B ), therefore, the angle between I R and -I B (I B, when reversed = -I B ) is 60. If, I R = I Y = I B = I PH. The phase currents Then; The current flowing in Line 1 would be; I L or I 1 = 2 x I PH x Cos (60 /2) = 2 x I PH x Cos 30 = 2 x I PH x ( 3/2) Since Cos 30 = 3/2 = 3 I PH i.e. In Delta Connection, The Line current is 3 times of Phase Current Similarly, we can find the reaming two Line currents as same as above. i.e., I 2 = I Y I R Vector Difference = 3 I PH I 3 = I B I Y Vector difference = 3 I PH As, all the Line current are equal in magnitude i.e. I 1 = I 2 = I 3 = I L Hence 116

117 IL = 3 I PH It is seen from the fig above that; The Line Currents are 120 apart from each other Line currents are lagging by 30 from their corresponding Phase Currents The angle Ф between line currents and respective line voltages is (30 +Ф), i.e. each line current is lagging by (30 +Ф) from the corresponding line voltage. 3. Power in Delta Connection We know that the power of each phase Power / Phase = V PH x I PH x CosФ And the total power of three phases; Total Power = P = 3 x V PH x I PH x CosФ.. (1) We know that the values of Phase Current and Phase Voltage in Delta Connection; I PH = I L / / 3.. (From IL = 3 I PH ) V PH = V L Putting these values in power eq. (1) P = 3 x V L x ( I L / 3) x CosФ (I PH = I L / / 3) P = 3 x 3 x V L x ( I L / 3) x CosФ { 3 = 3x 3 } P = 3 x V L x I L x CosФ Hence proved; Power in Delta Connection, P = 3 x V PH x I PH x CosФ. or P = 3 x V L x I L x CosФ CIRCUIT ANALYSIS OF 3 PHASE SYSTEM BALANCED CONDITION: The electrical system is of two types i.e., the single phase system and the three phase system. The single phase system has only one phase wire and one return wire thus it is used for low power transmission. The three-phase system has three live wire and one returns path. The three phase system is used for transmitting a large amount of power. The 3 Phase system is divided mainly into two types. One is Balanced three phase system and another one is unbalanced three phase system. Contents: Analysis of Balanced 3 Phase Circuit Analysis of Unbalanced 3 Phase Circuit Interconnection of 3 Phase System Connection of 3 Phase Loads in 3 Phase System The balance system in one in which the load are equally distributed in all the three phases of the system. The magnitude of voltage remains same in all the three phases and it is separated by an angle of 120º. In unbalance system the magnitude of voltage in all the three phases becomes different. 117

118 ANALYSIS OF BALANCED 3 PHASE CIRCUIT: It is always better to solve the balanced three phase circuits on per phase basis. When the three phase supply voltage is given without reference to the line or phase value, then it is the line voltage which is taken into consideration. The following steps are given below to solve the balanced three phase circuits. Step 1 First of all draw the circuit diagram. Step 2 Determine X LP = X L /phase = 2πf L. Step 3 Determine X CP = X C /phase = 1/2πf C. Step 4 Determine X P = X/ phase = X L X C Step 5 Determine Z P = Z/phase = R 2 P + X 2 P Step 6 Determine cosϕ = R P /Z P ; the power factor is lagging when X LP > X CP and it is leading when X CP > X LP. Step 7 Determine V phase. For star connection V P = V L / 3 and for delta connection V P = V L Step 8 Determine I P = V P /Z P. Step 9 Now, determine the line current I L. For star connection I L = I P and for delta connection I L = 3 I P Step 10 Determine the Active, Reactive and Apparent power. ANALYSIS OF UNBALANCED 3 PHASE CIRCUIT: The analysis of the 3 Phase unbalanced system is slightly difficult, and the load is connected either as Star or Delta. The topic is discussed in detail in the article named as Star to Delta and Delta to Star Conversion. Interconnection of 3 Phase System In a three-phase AC generator, there are three windings. Each winding has two terminals (start and finish). If a separate load is connected across each phase winding as shown in the figure below, then each phase supplies as independent load through a pair of wires. Thus, six wires will be required to connect the load to a generator. This will make the whole system complicated and costly. 118

119 Therefore, in order to reduce the number of line conductors, the three phase windings of an AC generator are interconnected. The interconnection of the windings of a three phase system can be done in following two ways Star or Wye (Y) connection Also See: Star Connection in 3 Phase System Mesh or Delta (Δ) connection. Also See: Delta Connection in 3 Phase System Connection of 3 Phase Loads in 3 Phase System: As the three phase supply is connected in star and delta connections. Similarly, the three-phase loads are also connected either as Star connection or as Delta Connection. The three phase load connected in the star is shown in the figure below. The delta connection of three phase loads is shown in the figure below. The three phase loads may be balanced or unbalanced as discussed above. If the three loads Z 1, Z 2 and Z 3 have the same magnitude and phase angle, then the 3 phase load is said to be a balanced load. Under such connections, all the phase or line currents and all the phase or line voltages are equal in magnitude. 119

120 Star to Delta and Delta to Star Conversion: The Conversion or transformation or replacement of the Star connected load network to a Delta connected network and similarly a Delta connected network to a Star Network is done by Star to Delta or Delta to Conversion Contents: Star to Delta Conversion Star to Delta Conversion Delta to Star Conversion In star to delta conversion, the star connected load is to be converted into delta connection. Suppose we have a Star connected load as shown in the figure An above, and it has to be converted into a Delta connection as shown in figure B. The following Delta values are as follows. Hence, if the values of Z A, Z B and Z C are known, therefore by knowing these values and by putting them in the above equations, you can convert a star connection into a delta connection. Delta to Star Conversion Similarly, a Delta connection network is given as shown above, in figure B and it has to be transformed into a Star connection, as shown above, in the figure A. The following formulas given below are used for the conversion. 120

121 If the values of Z 1, Z 2 and Z 3 are given, then by putting these values of the Impedances in the above equations, the conversion of delta connection into star connection As Impedance (Z) is the vector quantity, therefore all the calculations are done in Polar and Rectangular forming can be performed. TUTORIAL PROBLEMS: 1.The input power to a 3-phase a.c. motor is measured as 5kW. If the voltage and current to the motor are 400V and 8.6A respectively, determine the power factor of the system? Power P=5000W, line voltage VL = 400 V, line current, IL = 8.6A and power, P = 3 VLIL cos φ Hence power factor = cos φ = P 3 VLIL = (400) (8.6) = Two wattmeters are connected to measure the input power to a balanced 3-phase load by the two-wattmeter method. If the instrument readings are 8kW and 4kW, determine (a) the total power input and (b) the load power factor. (a)total input power, P=P1 +P2 =8+4=12kW (b) tan φ = 3(P1 P2)/(P1 + P2) = 3 (8 4) / (8 + 4) = 1/ 3 Hence φ= tan =30 Power factor= cos φ= cos 30 = Two wattmeters connected to a 3-phase motor indicate the total power input to be 12kW. The power factor is 0.6. Determine the readings of each wattmeter. 121

122 If the two wattmeters indicate P1 and P2 respectively Then P1 + P2 = 12kW ---(1) tan φ = 3(P1 P2)/(P1 + P2) And power factor=0.6= cos φ. Hence = 3(P1 P2)/12 From which, Angle φ= cos 10.6=53.13 and tan = P1 P2 = 12(1.3333) / 3 i.e. P1 P2 =9.237kW ----(2) Adding Equations (1) and (2) gives: 2P1 = i.e P1 = /2 = 10.62kW Hence wattmeter 1 reads 10.62kW From Equation (1), wattmeter 2 reads ( )=1.38kW. 122

123 UNIT V COMPONENTS OF ELECTRICAL SYSTEMS 123

124 Components of LT switch gear: Fuses, MCBs, RCDs, and RCBOs are all devices used to protect users and equipment from fault conditions in an electrical circuit by isolating the electrical supply. With fuses and MCBs only the live feed is isolated; with RCDs and RCBOs both the live and neutral feeds are isolated. Switch Fuse Unit (SFU) : The Switch Fuse Units are used for distributing power and protecting electrical devices and cables from damage due to fluctuations. This fuse unit is housed in an enclosure made using quality CR steel sheet. Salient Features: Pretreated or powder coated finish For cable connections, knock outs are provided at bottom, top and rear side High conductivity due to nickel or silver plated contact Durable and rewirable Application: Used in industrial, residential and commercial buildings for electrical fittings Miniature circuit breaker (MCB) : A miniature circuit breaker automatically switches off electrical circuit during an abnormal condition of the network means in overload condition as well as faulty condition. At continuous over current flow through MCB, the bimetallic strip is heated and deflects by bending. This deflection of bimetallic strip releases mechanical latch. As this mechanical latch is attached with operating mechanism, it causes to open the miniature circuit breaker contacts, and the MCB turns off thereby stopping the current to flow in the circuit. To restart the flow of current the MCB must be manually turned ON. This mechanism protects from the faults arising due to over current or over load. Miniature Circuit Breaker Construction Miniature circuit breaker construction is very simple, robust and maintenance free. Generally a MCB is not repaired or maintained, it just replaced by new one when required. A miniature circuit breaker has normally three main constructional parts. These are: 124

125 Frame of Miniature Circuit Breaker The frame of miniature circuit breaker is a molded case. This is a rigid, strong, insulated housing in which the other components are mounted. Operating Mechanism of Miniature Circuit Breaker Miniature circuit breaker provides the means of manual opening and closing operation of miniature circuit breaker. It has three-positions "ON," "OFF," and "TRIPPED". The external switching latch can be in the "TRIPPED" position, if the MCB is tripped due to over-current. When manually switch off the MCB, the switching latch will be in "OFF" position. In close condition of MCB, the switch is positioned at "ON". By observing the positions of the switching latch one can determine the condition of MCB whether it is closed, tripped or manually switched off. Trip Unit of Miniature Circuit Breaker The trip unit is the main part, responsible for proper working of miniature circuit breaker. Two main types of trip mechanism are provided in MCB. A bimetal provides protection against over load current and an electromagnet provides protection against short-circuit current. Operation of Miniature Circuit Breaker There are three mechanisms provided in a single miniature circuit breaker to make it switched off.we will find there are mainly one bi - metallic strip, one trip coil and one hand operated on-off lever. Electric current carrying path of a miniature circuit breaker shown in the picture is like follows. First left hand side power terminal - then bimetallic strip - then current coil or trip coil - then moving contact - then fixed contact and - lastly right had side power terminal. All are arranged in series. Earth Leakage Circuit Breaker (ELCB) : An ECLB is device used for installing an electrical device with high earth impedance to avoid shock. These devices identify small stray voltages of the electrical device on the metal enclosures and intrude the circuit if a dangerous voltage is identified. The main purpose of Earth leakage circuit breaker is to stop damage to humans & animals due to electric shock. 125

126 There are two types of Earth Leakage Circuit Breaker (ELCB) -> Voltage Operated ELCB A voltage-operated ELCB detects a growth in potential between the threatened consistent metalwork and a distant isolated Earth reference electrode. They work as a sensed potential of around 50V to open the main breaker & separate the supply from the threatened premises Current Operated ELCB: A current operated ELCB is called RCCB (Residual current circuit breaker) which comprises of a three winding transformer, that has two primary windings and also one secondary winding. Neutral & line wires work as the two main windings. A wire wound coil is the minor winding. The flow of current through the minor winding is 0 in the stable condition. In this condition, the flux owed to the current over the phase wire will be deactivated by the current through the neutral wire, meanwhile the current that flows from the phase will be refunded to the neutral. When an error occurs, a slight current will run into the ground also. This creates confusion between line and neutral current and that makes an unstable magnetic field. This encourages a current flow through the minor winding, which is associated with the sensing circuit. This will detect the outflow and direct signal to tripping system. 126

127 Molded Case Circuit Breaker (MCCB) : A molded case circuit breaker, abbreviated MCCB, is a type of electrical protection device that can be used for a wide range of voltages, and frequencies of both 50 Hz and 60 Hz. The main distinctions between molded-case and miniature circuit breaker are that the MCCB can have current ratings of up to 2,500 amperes, and its trip settings are normally adjustable. An additional difference is that MCCBs tend to be much larger than MCBs. As with most types of circuit breakers, an MCCB has three main functions: Protection against electrical faults During a fault such as a short circuit or line fault, there are extremely high currents that must be interrupted immediately. Switching a circuit on and off This is a less common function of circuit breakers, but they can be used for that purpose if there isn t an adequate manual switch. Applications of MCCB as follows Main electric feeder protection The electric feeder circuits that supply power to large distribution boards normally have very high currents, of hundreds of amperes. In addition, if more circuits are added to the system in the future, it may be necessary to adjust the circuit breaker trip settings. Therefore, a moldedcase circuit breaker is required. Capacitor bank protection Capacitor banks are a very important component of commercial and industrial electrical systems, since they allow power factor correction reducing line currents and preventing fees from the electric utility company. Large capacitor banks may draw high currents and will require MCCB protection. Generator protection Large electrical generators may provide an output of hundreds of amperes. In addition, gen-sets are normally very expensive. The high current ratings of molded case circuit breakers allow them to provide reliable protection in this application. Welding applications Some welding machines may draw very high currents that exceed the capabilities of miniature circuit breakers, requiring the use of an MCCB. Low current applications that require adjustable trip settings MCCBs are not only for high current applications. There are models rated below 100 amperes for when low current equipment requires the adjustable trip settings provided by MCCBs. Motor protection The reliable protection capabilities of MCCBs make them an adequate choice for motor protection. A molded case circuit breaker can be adjusted to provide overload protection without tripping during the inrush current of an electric motor. Types of Wires Triplex Wires : Triplex wires are usually used in single-phase service drop conductors, between the power pole and weather heads. They are composed of two insulated aluminum wires wrapped with a third bare wire which is used as a common neutral. The neutral is usually of a smaller gauge and grounded at both the electric meter and the transformer. 127

128 Main Feeder Wires : Main power feeder wires are the wires that connect the service weather head to the house. They re made with stranded or solid THHN wire and the cable installed is 25% more than the load required. Panel Feed Wires : Panel feed cables are generally black insulated THHN wire. These are used to power the main junction box and the circuit breaker panels. Just like main power feeder wires, the cables should be rated for 25% more than the actual load. Non-Metallic Sheathed Wires : Non-metallic sheath wire, or Romex, is used in most homes and has 2-3 conductors, each with plastic insulation, and a bare ground wire. The individual wires are covered with another layer of non-metallic sheathing. Since it s relatively cheaper and available in ratings for 15, 20 and 20 amps, this type is preferred for in-house wiring. 128

129 Single Strand Wires : Single strand wire also uses THHN wire, though there are other variants. Each wire is separate and multiple wires can be drawn together through a pipe easily. Single strand wires are the most popular choice for layouts that use pipes to contain wires. Types of cables: There are more than 20 different types of cables available today, designed for applications ranging from transmission to heavy industrial use. Some of the most commonly-used ones include: Non-Metallic Sheathed Cable: These cables are also known as non-metallic building wire or NM cables. They feature a flexible plastic jacket with two to four wires (TECK cables are covered with thermoplastic insulation) and a bare wire for grounding. Special varieties of this cable are used for underground or outdoor use, but NM-B and NM-C non-metallic sheathed cables are the most common form of indoor residential cabling. Underground Feeder Cable: These cables are quite similar to NM cables, but instead of each wire being individually wrapped in thermoplastic, wires are grouped together and embedded in the flexible material. Available in a variety of gauge sizes, UF cables are often used for outdoor lighting and inground applications. Their high water-resistance makes them ideal for damp areas like gardens as well as open-to-air lamps, pumps, etc. 129

130 Metallic Sheathed Cable: Also known as armored or BX cables, metal-sheathed cables are often used to supply mains electricity or for large appliances. They feature three plain stranded copper wires (one wire for the current, one grounding wire and one neutral wire) that are insulated with crosslinked polyethylene, PVC bedding and a black PVC sheathing. BX cables with steel wire sheathing are often used for outdoor applications and high-stress installations. Multi-Conductor Cable: This is a cable type that is commonly used in homes, since it is simple to use and well-insulated. Multi-conductor or multi-core (MC) cables feature more than one conductor, each of which is insulated individually. In addition, an outer insulation layer is added for extra security. Different varieties are used in industries, like the audio multicore snake cable used in the music industry. Coaxial Cable : A coaxial (sometimes heliax) cable features a tubular insulating layer that protects an inner conductor which is further surrounded by a tubular conducting shield, and might also feature an outer sheath for extra insulation. Called coaxial since the two inner shields share the same geometric axis, these cables are normally used for carrying television signals and connecting video equipment. Unshielded Twisted Pair Cable: Like the name suggests, this type consists of two wires that are twisted together. The individual wires are not insulated, which makes this cable perfect for signal transmission and video applications. Since they are more affordable than coaxial or optical fiber cables, UTP cables are often used in telephones, security cameras and data networks. For indoor use, UTP cables with copper wires or solid copper cores are a popular choice, since they are flexible and can be easily bent for in-wall installation. 130

131 Ribbon Cable: Ribbon cables are often used in computers and peripherals, with various conducting wires that run parallel to each other on a flat plane, leading to a visual resemblance to flat ribbons. These cables are quite flexible and can only handle low voltage applications. Earthing: The process of transferring the immediate discharge of the electrical energy directly to the earth by the help of the low resistance wire is known as the electrical earthing. The electrical earthing is done by connecting the non-current carrying part of the equipment or neutral of supply system to the ground. galvanised iron is used for the earthing. The earthing provides the simple path to the leakage current. The shortcircuit current of the equipment passes to the earth which has zero potential. Thus, protects the system and equipment from damage. Types of Electrical Earthing The electrical equipment mainly consists of two non-current carrying parts. These parts are neutral of the system or frame of the electrical equipment. From the earthing of these two non-current carrying parts of the electrical system earthing can be classified into two types. Neutral Earthing In neutral earthing, the neutral of the system is directly connected to earth by the help of the GI wire. The neutral earthing is also called the system earthing. Such type of earthing is mostly provided to the system which has star winding. For example, the neutral earthing is provided in the generator, transformer, motor etc. Equipment Earthing Such type of earthing is provided to the electrical equipment. The non-current carrying part of the equipment like their metallic frame is connected to the earth by the help of the conducting wire. If any fault occurs in the apparatus, the short-circuit current to pass the earth by the help of wire. Thus, protect the system from damage. 131

132 Importance of Earthing: The earthing protects the personnel from the short circuit current. The earthing provides the easiest path to the flow of short circuit current even after the failure of the insulation. The earthing protects the apparatus and personnel from the high voltage surges and lightning discharge. Dry cell battery: A dry cell has the electrolyte immobilized as a paste, with only enough moisture in it to allow current to flow. Unlike a wet cell, a dry cell can operate in any orientation without spilling, as it contains no free liquid. This versatility makes it suitable for portable equipment. By comparison, the first wet-cell batteries were typically fragile glass containers with lead rods hanging from an open top. They, therefore, needed careful handling to avoid spillage. The development of the dry-cell battery allowed for a major advance in battery safety and portability. A common dry-cell battery is the zinc-carbon battery, which uses a cell that is sometimes called the Leclanché cell. The cell is made up of an outer zinc container, which acts as the anode. The cathode is a central carbon rod, surrounded by a mixture of carbon and manganese(iv) dioxide (MnO2). The electrolyte is a paste of ammonium chloride (NH4Cl). A fibrous fabric separates the two electrodes, and a brass pin in the center of the cell conducts electricity to the outside circuit. 132

133 Nickel cadmium batteries The active components of a rechargeable NiCd battery in the charged state consist of nickel hydroxide (NiOOH) in the positive electrode and cadmium (Cd) in the negative electrode. For the electrolyte, potassium hydroxide (KOH) is normally used. Due to their low internal resistance and the very good current conducting properties, NiCd batteries can supply extremely high currents and can be recharged rapidly. These cells are capable of sustaining temperatures down to -20 C. The selection of the separator (nylon or polypropylene) and the electrolyte (KOH, LiOH, NaOH) influence the voltage conditions in the case of a high current discharge, the service life and the overcharging capability. In the case of misuse, a very high-pressure may arise quickly. For this reason, cells require a safety valve. NiCd cells generally offer a long service life thereby ensuring a high degree of economy. Lead acid battery The battery which uses sponge lead and lead peroxide for the conversion of the chemical energy into electrical power, such type of battery is called a lead acid battery. The lead acid battery is most commonly used in the power stations and substations because it has higher cell voltage and lower cost. The container and the plates are the main part of the lead acid battery. The container stores chemical energy which is converted into electrical energy by the help of the plates. Lithium ion batteries 133

134 The term lithium ion battery refers to a rechargeable battery where the negative electrode (anode) and positive electrode (cathode) materials serve as a host for the lithium ion (Li+). Lithium ions move from the anode to the cathode during discharge and are intercalated into (inserted into voids in the crystallographic structure of) the cathode. The ions reverse direction during charging. Since lithium ions are intercalated into host materials during charge or discharge, there is no free lithium metal within a lithium-ion cell. In a lithium ion cell, alternating layers of anode and cathode are separated by a porous film (separator). An electrolyte composed of an organic solvent and dissolved lithium salt provides the media for lithium ion transport. For most commercial lithium ion cells, the voltage range is approximately 3.0 V (discharged, or 0 % state-of-charge, SOC) to 4.2 V (fully charged, or 100% SOC). Nickel metal hydride batteries The active components of a rechargeable NiMH battery in the charged state consist of nickel hydroxide (NiOOH) in the positive electrode and a hydrogen storing metal alloy (MH) in the negative electrode as well as a potassium hydroxide (KOH) electrolyte. Compared to rechargeable NiCd batteries, NiMH batteries have a higher energy density per volume and weight. 134