Unit-1(A) Circuit Analysis Techniques

Transcription

1 Unit-1(A Circuit Analysis Techniques Basic Terms used in a Circuit 1. Node :- It is a point in a circuit where two or more circuit elements are connected together. 2. Branch :- It is that part of a network which lies between two junctions. 3. Loop :- It is a close path in a circuit in which no element or node is encountered more than once. 4. Mesh :- It is a loop that contains no other loop within it. 5. Junction:- It is a point in a circuit where three or more circuit elements are connected together Consider the circuit of Fig. (a. It has six branches, five nodes, two meshes And the circuit of Fig (b has four branches, two nodes three meshes. To find the no of branches or loops or nodes by using this relation ( b no. of branches l no. of loops no. of nodes Kirchhoff`s Laws Kirchhoff s laws are more comprehensive than Ohm's law and are used for solving electrical networks which may not be readily solved by the latter. Kirchhoff`s laws, two in number, are particularly useful in determining the equivalent resistance of a complicated network of conductors and for calculating the currents flowing in the various conductors. 1

2 I. Kirchhoff`s Point Law or Current Law (KCL In any electrical network, the algebraic sum of the currents meeting at a point (or junction is equal to Zero. (or The total current entering a junction is equal to the total current leaving that junction. Consider the case of a network shown in Fig (a.a is the I1(-I2(I3(I4(-I5 0 I1I4-I2-I3-I5 0 Or I1I4 I2I3I5 Or Incoming currents Outgoing currents II Kirchhoff's Mesh Law or Voltage Law (KVL Kirchhoff's voltage Law states that the algebraic sum of voltages around any closed path is equal to zero - It should be noted that if we start from a particular junction and go round the mesh till we come back to the starting point, then we must be at the same potential with which we started - Hence, it means that all the sources of emf met on the way must necessarily be equal to the voltage drops in the resistances, every voltage being given its proper sign, plus or minus. (adetermination of Voltage Sign A rise in voltage should be given a ve sign and a fall in voltage a -ve sign. That is, if we go from the -ve terminal of a battery to its ve terminal there is a rise in potential, hence this voltage should be given a ve sign. And on the other hand, we go from ve terminal to -ve terminal, then there is a fall in potential, hence this voltage should be preceded by a -ve sign. 2

3 The sign of the battery e.m.f is independent of the direction (b Sign of IR Drop Now, take the case of a resistor (Fig.. If we go through a resistor in the same direction as the current, then there is a fall in potential because current flows from a higher to a lower potential.. Hence, this voltage fall should be taken -ve. However, if we go in a direction opposite to that of the current, then there is a rise in voltage. Hence, this voltage rise should be given a positive sign Consider the closed path ABCDA in Fig As we travel around the mesh in the clockwise direction, different voltage drops will have the following signs : I1R1 is - ve (fall in potential I2R2 is - ve (fall in potential I3R3 is ve (rise in potential I4R4 is - ve (fall in potential E2 is - ve (fall in potential E1 is ve (rise in potential Using Kirchhoff's voltage law, we get -I1R1 I2R2 I3R3 I4 R4 E2 E1 0 Or I1R1 I2R2 I3R3 I4R4 E1 E2 3

4 Assumed Direction of Current: In applying Kirchhoff's laws to electrical networks, the direction of current flow may be assumed either clockwise or anticlockwise. If the assumed direction of current is not the actual direction, then on solving the question, the current will be found to have a minus sign. If the answer is positive, then assumed direction is the same as actual direction. However, the important point is that once a particular direction has been assumed, the same should be used throughout the solution of the question. Kirchhoff's laws are applicable both to d.c. and a.c. voltages and currents. However, in the case of alternating currents and voltages, any e.m.f. of self-inductance or that existing across a capacitor should be also taken into account. Voltage Division Rule: In a series circuit, same current flows through each of the given resistors and the voltage drop varies directly with its resistance. (V α R By using voltage division rule we can find the different voltages across resistors from single voltage source. In a series circuit Voltage across resistor is given by Current Division Rule: In Parallel circuit, the current divides in all branches (I α 1/R To find branch current in parallel circuits is given by ℎ 4

5 BASIC NODAL AND MESH ANALYSIS: If a circuit consists of n nodes and b branches. Then to analyze a circuit using Ohm s law: requires n equations. Nodal analysis: requires n 1 equation. Mesh analysis: requires b (n 1 equations. Depending on the circuit analysis we adapt different technique among these three. The nodal and mesh analysis are based on Kirchhoff s laws Nodal Analysis: It is based on Kirchhoff s Current Law. A node is a point in a network, when two or more elements meet. A voltage is always defined as the potential difference between two points. When we talk about the voltage at a certain point of a circuit we imply that the measurement is performed between that point and some other point in the circuit. In most cases that other point is referred to as ground. The procedure for analyzing a circuit with the node method: 1. Count a number of nodes of the given network. 2. Choose one node as reference node and write ground symbol at this node 3. Determine voltage variables at each node 4. If there is some voltage source in the middle of couple nodes that aren t the reference node, you have to reduce these couple node as one node called supernode. 5. Use KCL at all node and supernode except the reference to write the KCL equations - The total equations no of equations will be equal to n Solve the variable system 7. Determine the required value by using the solution of variable system There are a few general guidelines that we need to remember as we make the selection of the reference node. 1. A useful reference node is one which has the largest number of elements connected to it. 2. A useful reference node is one which is connected to the maximum number of voltage sources. 5

6 Nodal Analysis with Voltage Sources Voltage source between reference node and a non reference node. Set the voltage of the non reference node to that of the voltage source. Explanation: For the given Figure 1 for a step by step demonstration of the node method. 1 The nodes are identified and the names are given. 2 One of the identified nodes as the reference node. The Selection of node A as the reference node is the best choice 3 Label the voltages at the selected nodes. Figure shows the circuit with the labeled nodal voltages. The reference node is assigned voltage 0 Volts indicated by the ground symbol. The remaining node voltages are labeled V1, VB, VC and V2 4 The problem is completely defined. Once we determine the values for t h e node voltages V1,VB, VC and V2..we w i l l b e a b le t o c o m p le t e ly characterize this circuit. So go on to calculate the node voltages by applying KCL at the designated nodes. 5 For node 1 since the voltage of the voltage source is known we may directly label the voltage V1 similarly V2 as a result we have reduced the number of unknowns from 4 to 2. Applying KCL at node B, the node voltage as VB Substituting the current equations in above expression we get Similarly applying KCL at node C we get By solving the above two equations we get unknown node voltages. 6

7 Super Node Analysis: If any of the branch in the network has a voltage source such that its any terminal is not connected to reference node or ground,then it is difficult to write KCL equations at this nodes to which this voltage source is connected. One way to overcome this difficulty is to apply the supernode technique. In the circuit of Figure voltage source V2 is not connected to the reference node and thus it is a floating voltage source. In This method,two adjacent nodes that are connected by a voltage source are reduced to a single node and then equations are formed by applying Kirchoff s current law as usual. Due to presence of voltage source V x in between nodes 2 and 3it is difficult to find out current. This type of networks Super Node Technique is very useful to find the current. The super node is considered to be a single node and the KCL equations are written. The difference between the node voltages across the voltage source becomes one of the equations to solve the network. We can write the combined equation for node 2 and 3 as below The other equation between node 2 and 3 is 0 0 (1 (2 By solving the above two equations we can find the Unknown voltages. 7

8 Mesh current analysis (or Loop Current Method: A mesh is a loop that does not contain any other loops within it. It is based on Kirchhoff s Voltage Law. If a network has a large number of voltage sources, It is useful to use mesh analysis. Mesh analysis is applicable only for Planar networks. A circuit said to be planar,if it can be drawn on a plane surface without crossover. The procedure for analyzing a circuit with the Mesh Analysis: 1. Use for planar circuits that can be drawn on a plane surface with no wires crossing each other. 2. Count a number of mesh in the given circuit and assign mesh current for each mesh. 3. Determine current variables at each mesh with an arbitrary direction. 4.Indicate the polarities (sign of the elements that are not indicated with polarities. 1. Use KVL at all mesh to write the KVL equations no of mesh equations b (n-1 Example: The above circuit consists two batteries E1,E2 connected with five resistors. Let the mesh currents for three meshes be I 1, I2,& I3 For Mesh-1 : E1 I1R1 R4(I1 I2 0 I1[R1 R4] I2R4 E (1 Mesh-2: -I2R2 R5[I2-I3] R4[I2 I1] 0 I1R4 I2(R2R4R5 I3R (2 Mesh-3 : -I3R3 E2 R5[I3 I2] 0 I2R5-I3[R3 R5] E (3 The above three equations can be solved to find the not only three mesh currents but branch currents as well. If any mesh current results in negative magnitude then the assumed current is opposite to that of the original. Mesh Analysis with Current Sources Current source exists only in one mesh. Current source between two meshes, form Supermesh by excluding the current source any element connected in series with it. 8

9 The Super Mesh: A Supermesh results when two meshes have a (dependent or independent current source in common. The loop represented with a dotted line represents the Supermesh. While applying KVL to solve this circuit with mesh equations analysis, are the expressed two loop in a single equation.the required second expression will be the difference in loop currents at the branch with current source. i.e. KVL of loop 1 KVL of loop 2 0 From the circuit Supermesh equation is I1 R1 I2R2 VA - VB eq.1. And the second equation is I1 - I2 IX eq.2 On solving these two equations we can determine the loop currents and any other parameters. Properties of a Supermesh: The current source in the Supermesh provides the constraint equation necessary to solve for the mesh currents. A Supermesh has no current of its own. A Supermesh requires the application of both KVL and KCL. 9

10 NETWORK THEOREMS: Thevenin's Theorem: Thevenin's theorem states that any two terminal linear network having a number of voltage current sources and resistances can be replaced by a simple equivalent circuit consisting of a single voltage source in series with a resistance where the value of the voltage source is equal to the open circuit voltage across the two terminals of the network, and resistance is equal to the equivalent resistance measured between the terminals To find Thevenin s Voltage: The current through resistance R2 Hence voltage appears across the terminals A and B i.e. To find Thevenin s Resistance: Thus the Thevenin equivalent resistance RTh is viewed from the open terminals A and B is given as. As per Thevenin theorem, when resistance RL is connected across terminals A and B, the network behaves as a source of voltage VT and internal resistance RT and this is called Thevenin equivalent circuit. The current through RL is given by Thevenin s equivalent Cicuit 10

11 Norton's theorem : Norton's theorem, states that any two terminal linear network consisting current sources, voltage sources and resistances can be replaced by an equivalent circuit consisting of a current source in parallel with a resistance. The value of the current source is the short circuit current between the two terminals of the network and the resistance is the equivalent resistance measured the terminals of the network In the example. we have to find out the current through RL by applying Norton theorem. Step 1: First, we have to remove the resistor RL from terminals A and B and make the terminals A and B short circuited by zero resistance.as shown in fig.b Step 2: Second, we have to calculate the short circuit current or Norton equivalent current IN through the points A and B. the equivalent resistance of the network. ( ( // Current supplied from the Source Norton equivalent current ℎ ℎ Step 3: To determine Norton equivalent resistance RN of the network the equivalent resistance as viewed from open terminals A and B is RN ( ( As per Norton theorem, when resistance RL is reconnected across terminals A and B, the network behaves as a source of constant current IN with shunt connected internal resistance RN and this is Norton equivalent circuit. Norton Equivalent Circuit The Current through RL is given as 11

12 Superposition theorem: The superposition theorem states that in any linear network containing two or more sources, the response in any element is equal to the algebraic sum of the responses caused by individual sources acting alone, while the other sources are non-operative while considering the effect of individual sources, other ideal voltage sources and ideal current sources in the network are replaced by short circuit and open circuit across their terminals. This theorem is valid only for linear systems. Suppose there are two voltage sources V1 and V2 acting simultaneously on the circuit. because of these two voltage sources, say current I flows through the resistance R. i When source V1 acting alone: Now replace V2 by short circuit, keeping V 1 at its position and measure current through the resistance, R1. Say it is I1. As shown in fig.1 ii and is given by When source V2 acting alone: ( Figure 1 Now replace V1 by short circuit, keeping V2 at its position and measure current through the resistance, R1. Say it is I2. and is given by ( Figure 2 Now if we add these two currents, I1 and I2 we will get the current which is equal to the current was actually flowing through R, when both voltage sources V1 and V2 were acting on the circuit simultaneously. That is Limitations: 1. Super positions theorem is not valid for Power Calculations 2. Super position theorem is not valid for Non-linear circuits. 12

13 Maximum Power Transfer Theorem: The maximum Power Transfer Theorem states that maximum power is delivered from a source to a load when the load resistance is equal to the source resistance. In circuits basically consist of sources, supplying voltage, current, or power to the load; for example, a radio speaker system, or a microphone supplying the input signals to voltage pre-amplifiers. Sometimes it is necessary to transfer maximum voltage, current or power from the source to the load. It is a fact that more voltage is delivered to the load when the load resistance is high as compared to the resistance of the source. On the other hand, maximum current is transferred to the load when the load resistance is small compared to the source resistance. From the above fig power delivered to load resistance. To find the maximum power, differentiate the above expression with respect to resistance R L and equate it to zero. Thus, [( ( ( ( ( ( ( ( ] 13

14 Reciprocity Theorem : Reciprocity Theorem states that in any linear, bilateral, passive network containing single source (voltage or current, the ratio of excitation (V or I to the response remains unchanged even after interchanging the positions of excitation and response. It is also referred as Principle of Reciprocity Consider example, but with only one source. Determine the current in the 160 Ω branch. Now replace the 160 Ω resistor with the source in series with it and after short- circuit the source at the original location, find the current flowing at the original source location. Show that it verifies the Reciprocality theorem. Solution It is seen that the identical current has appeared verifying the reciprocity theorem.the advantage of the theorem is when a circuit has already been analysed for one solution, it may be possible to find a corresponding solution without further work 14

15 Tellegen theorem : This theorem has been introduced in the year of 1952 by Dutch Electrical Engineer Bernard D.H. Tellegen. This is a very useful theorem in network analysis. Tellegen's theorem is valid for any lumped network which may be linear or non linear, passive or active, time-varying or time-invarient. All branch currents and voltages in that network must satisfy Kirchhoff's laws. This theorem states that in an arbitrary lumped network, the algebraic sum of the powers in all branches at any instant is zero. (or the algebraic sum of the powers delivered by all sources is equal to the algebraic sum of the powers absorbed by all elements. Example: Verifying Tellegens theorem for the circuit hence verified. Millman s Theorem: Millman s theorem states that in any network, if the voltage sources V 1, V2. Vn in series with internal resistances R1,R2..Rn,respectively,are in parallel, then these sources may be replaced by a single voltage source V in series with R eq. 15

16 .. Here Gn is the conductance of the nth branch, 1 Miller Theorem: The Miller theorem states that in a linear circuit,if there exists a branch with impedance Z,connecting two nodes with nodal voltages V1 and V2, we can replace this branch by two branches connecting the corresponding nodes to ground by impedances respectively Z 1 and Z2. ( Where ( 1 (. ( 1 16