Circuit Theory and Basic Electronics

Transcription

1 Circuit Theory and Basic Electronics

2 Board of Studies Prof. H. N. Verma Vice- Chancellor Jaipur National University, Jaipur Dr. Rajendra Takale Prof. and Head Academics SBPIM, Pune Prof. M. K. Ghadoliya Director, School of Distance Education and Learning Jaipur National University, Jaipur Subject Expert Panel Dr. Ramchandra G. Pawar Director, SIBACA, Lonavala Pune Vaibhav Bedarkar Subject Matter Expert Content Review Panel Gaurav Modi Subject Matter Expert Shubhada Pawar Subject Matter Expert Copyright This book contains the course content for Circuit Theory and Basic Electronics. First Edition 2013 Printed by Universal Training Solutions Private Limited Address 05 th Floor, I-Space, Bavdhan, Pune All rights reserved. This book or any portion thereof may not, in any form or by any means including electronic or mechanical or photocopying or recording, be reproduced or distributed or transmitted or stored in a retrieval system or be broadcasted or transmitted.

3 Index I. Content...II II. List of Figures... V III. List of Tables...VIII IV. Abbreviations...IX V. Application VI. Bibliography VII. Self Assessment Answers Book at a Glance I

4 Contents Chapter I... 1 Introduction to Basic Electronics... 1 Aim... 1 Objectives... 1 Learning outcome Introduction to Basic Electronics Material Structure of Matter - Charged Particles Structure of an Atom Structure of Semiconductor Materials Ionisation Energy-Band Theory Semiconductor Intrinsic Semiconductor Extrinsic Semiconductor Intrinsic Semiconductors Vs Extrinsic Semiconductors Diode Characteristics of Diode Diode as a Rectifier Half Wave Rectifier Full Wave Rectifier Zener Diode Zener Diode as Voltage Regulator Characteristics Summary References Recommended Reading Self Assessment Chapter II Bipolar Junction Transistor (BJT) Aim Objectives Learning outcome Introduction to Transistor BJT Structure BJT Circuit Configurations Common Base Configuration Construction of Transistor Forward-Reverse Bias of a BJT Transistor Characteristics and Parameters Current and Voltage Analysis Collector Characteristics Curves Collector Characteristic Curves DC Biasing Circuits Summary References Recommended Reading Self Assessment II

5 Chapter III Field Effect Transistor (FET) Aim Objectives Learning outcome Introduction Significance of FET Types of FET JFET: Junction Field Effect Transistor Construction of JFET Theory of JFET Static Characteristics of a JFET JFET Drain Characteristic with V GS = MOSFET - Metal-Oxide-Semiconductor FETs Operation of FET Body Effect P-Channel Enhancement-type MOSFET (PMOS) Complementary MOS (CMOS) Depletion-Type MOSFET Summary References Recommended Reading Self Assessment Chapter IV Op-amplifier and Filters Aim Objectives Learning outcome Introduction to Operational Amplifier Basics of Op-amp Rules of op-amp Ideal Op-amp Basic Ideal Op-amp Assumptions Simple OP-amp Circuits Non-inverting Amplifier Inverting Amplifier Summing Amplifier Difference Amplifier Standard Operational Amplifier Parameters Minimum and Maximum Parameter Ratings Op-amp IC Basic Introduction to Filters Significance of Filters Basic Filter Types Summary References Recommended Reading Self Assessment Chapter V LCR Circuits Aim Objectives Learning outcome III

6 5.1 Introduction to LCR LCR Series Circuit LCR Parallel Circuit Features of LCR Circuits Ohm s Law Basic Circuit Element Important Part of a Circuit Kirchhoff s Current Law (KCL) Kirchhoff s Second Law - The Voltage Law, (KVL) Summary References Recommended Reading Self Assessment Chapter VI Theorems Aim Objectives Learning outcome Theorem - Superposition Theorem Statements of Superposition Theorem Explanation of Superposition Theorem Importance of Superposition Theorem Properties of Superposition Theorem Limitations of Superposition Theorem Relationship with Mesh and Nodal Analysis Proportionality in Elements Linear Dependent Sources Additivity Property Application of Superposition Theorem Steps to Apply Superposition Theorem Examples of Supersposition Theorem Thevenin s Theorem Explanation of Thevenin s Theorem Thevenin s Equivalent Circuit Example of Thevenin's theorem Formal Presentation of the Thevenin s Theorem To Get Open-circuit Voltage VTH Norton Theorem Maximum Power Transfer Theorem Maximum Power Curve Reciprocity Theorem Laplace Transform Define Laplace Trasnform Table Table of Laplace Transform of Different Waveform Table on Trigonometry Term Property of Laplace Transform Summary References Recommended Reading Self Assessment IV

7 List of Figures Fig. 1.1 Structure of an atom... 4 Fig. 1.2 Atomic structure of Germanium... 4 Fig. 1.3 Atomic structure of Silicon... 5 Fig. 1.4 Energy band diagram of Silicon... 6 Fig. 1.5 PN junction... 8 Fig. 1.6 Characteristics of diode... 9 Fig. 1.7 Half wave rectifier...11 Fig. 1.8 Bridge rectifier...11 Fig. 1.9 Full wave rectifier Fig Symbol of Zener diode Fig Characteristic of Zener diode Fig Zener diode shunt regulator Fig. 2.1 Equivalent circuit of transistor Fig. 2.2 npn and pnp BJTs structures Fig. 2.3 npn transistor symbol Fig. 2.4 pnp transistor symbol Fig. 2.5 Common emitter configuration Fig. 2.6 Common collector configuration Fig. 2.7 Common base configuration Fig. 2.8 Basic structure of transistor Fig. 2.9 Construction of BJT Fig Circuit diagram of BJT Fig Flow of current Fig Transistor DC bias circuits Fig Transistor currents and voltages Fig (a) Fig (b) Fig (a) & (b) Collector characteristic curves Fig Base lead open Fig Base-emitter junction forward-biased Fig DC biasing circuits Fig Voltage divider circuit Fig DC bias circuit Fig DC load line Fig Q-point Fig DC load line and AC signal Fig AC signal with different Q-point Fig. 3.1 Symbol of FET Fig. 3.2 Classification of FETs Fig. 3.3 P-Channel JFET symbol Fig. 3.4 N-Channel JFET symbol Fig. 3.5 Construction of JFET Fig. 3.6 JFET with equal thickness depletion region Fig. 3.7 Characteristic curve JFET Fig. 3.8 Basing of JFET Fig. 3.9 Drain characteristics Fig 3.10 Transfer characteristics Fig N-Channel enhancement-type MOSFET Fig Characteristics OF NMOSFET Fig 3.13 Constructor of MOSFET Fig 3.14 Characteristic curve of NMOS Fig PMOS symbol and current flow Fig CMOS construction V

8 Fig Characteristic of depletion and enhancement type MOSFET Fig Depletion MOSFETs Fig. 4.1 Symbol of Op-amp Fig. 4.2 Ideal Op-amp Fig. 4.3 Voltage follower Fig. 4.4 Non-inverting amplifier Fig. 4.5 Inverting amplifier Fig. 4.6 Summing amplifier Fig. 4.7 Differentiator amplifier Fig pin IC Fig. 4.9 Filters Fig Filter network example Fig BPF Fig Notch filter Fig Notch filter frequency response Fig Amplitude and Phasor response curve Fig Example of low-pass filter amplitude curve Fig Simple high pass filter Fig Example of High pass filter amplitude response curves Fig. 5.1 Phasor diagram of LRC Fig. 5.2 Series LCR circuit Fig. 5.3 Variation of VL and V C Fig. 5.4 Variation of VLC and V R Fig. 5.5 Variation of current with frequency for different R values LCR series circuit Fig. 5.6 Parallel LCR circuit Fig. 5.7 Variation current with frequency for different R values LCR parallel circuit Fig. 5.8 DC voltage source Fig. 5.9 AC Voltage source Fig Current source Fig Current entering and leaving the node Fig Sum of all the voltages Fig Circuit diagram Fig. 6.1 Linearity using resistor Fig. 6.2 Linearity using inductor Fig. 6.3 Linearity using capacitor Fig. 6.4 Linear independent sources Fig. 6.5 Principle of additivity Fig. 6.6 Circuit using superposition theorem Fig. 6.7 i 2 = Fig. 6.8 v 1 =0 v 1 shortcut Fig. 6.9 Example for superposition theorem Fig Illustration of superposition theorem Fig Thevenin s theorem-introduction Fig Circuit model Fig Thevenin s theorem Fig Example of Thevenin s theorem Fig A network and its Thevenin s equivalent Fig Thevenin s voltage Fig Thevenin s resistance Fig Norton theorem Fig Norton current Fig Short circuit replacing the load Fig Thevenin s and Norton equivalent Fig Maximum power transfer theorem Fig Maximum power curve VI

9 Fig Circuit diagram Fig Reciprocity theorem Fig Reciprocity theorem VII

10 List of Tables Table 1.1 Properties of an electron... 2 Table 1.2 Properties of fundamental particle... 2 Table 1.3 Differences between intrinsic semiconductor and extrinsic semiconductor... 8 Table 4.1 Ideal Op-amp parameter Table 6.1 Laplace transform Table 6.2 Laplace transform of waveforms Table 6.3 Laplace transform of trigonometry term Table 6.4 Property of Laplace transforms VIII

11 Abbreviations β DC - DC current gain µs - Micro Second A - Gain AC - Alternate Current B - Base BC - Base-Collector BE - Base-Emitter BiMOS - Bipolar Metal Oxide Semiconductor BJT - Bipolar Junction Transistor BW - Band Width C - Collector C - Coulomb C 3 - Bypass Capacitor CMOS - Complementary MOS CMRR - Common Mode Rejection Ratio D - Diode D - Drain db - Decibel DC - Direct Current E - Electron E - Emitter ev - Electron Volt f - Frequency FET - Field Effect Transistor G - Gate Ge - Germanium H - Henry I - Current I B - Base Current I B - DC Base Current I C - Collector Current I C - DC Collector Current IC - Integrated Circuit I CEO - Collector Leakage Current i D - Drain Current/ Diode Current I DSS - Current drain to source with a shorted gate I E - DC Emitter Current I E - Emitter Current I G - Gate Current I IN - Input Current I L - Load Current I S - Source Current I Zmax - Maximum Zener Current I Zmin - Minimum Zener Current JFET - Junction Field Effect Transistor K - Kelvin KCL - Kirchhoff s current law K HZ - Kilo Hertz KVL - Kirchhoff s Voltage Law LCR - Inductor Capacitor Resistor LM741 - Linear Monolithic 741 IX

12 ma - milli Ampere MESFETs - Metal Semiconductor Field Effect Transistors M HZ - Mega Hertz MOS - Metal Oxide Semiconductor MOSFET - Metal Oxide Semiconductor Field Effect Transistor mv - Milli Volt mw - Milli Watt MΩ - Mega Ohm N/C - Not Connect NMOS - N-Channel Enhancement-Type MOSFET NPN - Negative-Positive-Negative Op-Amp - Operational Amplifier Pd - Power Dissipation PMOS - P-Channel Enhancement-type MOSFET PN - Positive-Negative PNP - Positive-Negative-Positive PSRR - Power-supply rejection ratio PZ - Zener Power q - Charge Q-point - Quiescent Point R - Resistor R B - Base Resistor R C - Collector Resistor R E - Emitter Resistor S - Source Sat - Saturation Si - Silicon T - Boltzmann Constant T - Thermal V - Voltage V BE - DC voltage across base-emitter junction V C - Capacitor Voltage V CB - DC voltage across collector-base junction V CE - DC voltage from collector to emitter V D - Diode Voltage/ Drain Voltage V DS - Voltage Drain Source V DSO - Voltage Drain Source Break Down V GS - Gate and Source Voltage V INmin - Input voltage V L - Inductor Voltage V OS - Input Offset Voltage V P - Peak Voltage V PO - Pinch-Off Voltage V R - Resistor Voltage V T - Thermal Voltage V t - Threshold Voltage VZ - Zener Voltage Z IN - Input Impedance Z OUT - Output Impedance α DC - DC Collector Current Ω - Ohm X

13 Chapter I Introduction to Basic Electronics Aim The aim of this chapter is to: introduce the concept of basic electronic elaborate on materials used in making of semiconductor explain the types of semiconductor and their theory Objectives The objectives of this chapter are to: enlist various application of diode explain the concept of Zener diode explain the working of Zener diode as voltage regulator Learning outcome At the end of this chapter, you will be able to: identify the material used in semiconductor describe pentavalent and trivalent impurities understand the importance of energy band theory in semiconductor 1

14 Circuit Theory and Basic Electronics 1.1 Introduction to Basic Electronics To understand the operation, properties and applications of various semiconductor devices, it is necessary to review the basics of semiconductor physics. The basic knowledge of atomic properties of matter helps to study the discrete electronic energy levels in atoms. The transport of charge through a semiconductor not only depends on the atomic properties but also on the arrangement of atoms in the solid. In a crystalline solid, the various energy levels merge to from various energy bands. This band structure separates various materials such as, metals, insulators and semiconductors. This chapter will explain about the basic nature of atom, concept of field intensity, potential energy and energy bands in solids. 1.2 Material The materials such as, copper, aluminium etc. are good conductors of electricity. While material such as, wood, glass, mica etc. are bad conductors of electricity and are called insulators. There is another class of materials, whose conductivity i.e., their ability to carry electricity, lies between those of conductors and insulators. Such materials are called semiconductors. Germanium (Ge) and Silicon (Si) are two well know semiconductor materials. To understand the working of the diode, transistor, integrated circuits, it is necessary to know the basic physics behind the behaviour of semiconductor materials. 1.3 Structure of Matter - Charged Particles According to the modern electron theory, an atom is composed of the three fundamental particles, which are invisible to bare eyes. These are neutron, proton and electron. The electrons are negatively charged particles and are considered to be the most important element from the conduction point of view. The number of electrons passing across a particular point per second represents an electric current at that point. The various parameters related to an electron are given in the table given below: Particle Mass Radius Charge Nature of Charge Electron x kg m x C Negative Table 1.1 Properties of an electron The charge on one electron is x C, where Coulomb (C) is the unit of measurement of charge. Fundamental particle Nature of charge Mass in Kg Neutron No Charge x Proton Positive x Electron Negative x Table 1.2 Properties of fundamental particle 2

15 1.4 Structure of an Atom The atoms have a planetary type of structure, according to classical Bohr Model. All the protons and neutrons are bound together at the centre of an atom, which is called Nucleus. All the electrons move around the nucleus similar to that of planets revolving around the sun. Here nucleus resembles the central sun, about which electrons revolve in a particular fashion like the planets. In a normal atom, the number of protons is equal to the number of electrons. As neutron is electrically neutral, an atom as a whole is electrically neutral. The number of protons in an atom is makes up its atomic number. While atomic weight is approximately equal to the total number of protons and neutrons in the nucleus of an atom. The electrons which are revolving round the nucleus do not move in the same orbit. The electrons are arranged in different orbits or shells at fixed distances from the nucleus. Each shell can contain a fixed number of electrons. n Generally, a shell may contain a maximum of 2 electrons, where n is the number of the shell. The first shell can occupy maximum of two electrons (2x1 2 ) while the second shell can occupy maximum of eight electrons (2x2 2 ) and so on. Each shell has an energy level associated with it. The closer an electron is to the nucleus, the stronger are the forces that bind it to the nucleus. So the first shell which is closest to the nucleus, is always under tremendous force of attraction. While the shell which is the farthest from the nucleus is under very weak force of attraction. Electrons revolving in the last shell are responsible for the electrical and chemical characteristic of an atom. 2 The exception to the 2n rule stated above states that the outermost shell in an atom cannot accommodate more than eight electrons. The valence electrons revolving in the outermost shell are said to be having energy level. The amount of energy required to extract the valence electron from the outer shell is very less. Thus, the energy level of shell one is considered to be the lowermost, while the energy level of valence shell is the highest. More energy level indicates that the electrons of that shell are loosely bound to the nucleus. Hence, valence electrons are loosely bound to the nucleus as having highest energy level. When an atom absorbs energy from a heat source or from a light source or from high atmospheric temperature, the energy levels of the electrons are raised (gets exited). When such an additional energy is imparted to the electrons, the electrons move to the next orbit which is farther form the nucleus. If such energy is imparted to a valence electron, it tries to jump to the next orbit. But as a valence electron is in the outermost orbit, actually it gets completely removed from the force of attraction of the nucleus. 3

16 Circuit Theory and Basic Electronics 1 st Energy Leval 2 nd Energy Leval 3 rd Energy Leval 4 th Energy Leval Nucleus Electrons Neutrons Protons 1.5 Structure of Semiconductor Materials Fig. 1.1 Structure of an atom The semiconductor materials such as, Ge and Si have four electrons in their valence shell (i.e., outermost shell). Germanium has a nucleus with 32 protons. The electrons are distributed in the following manner: 2 electrons in the first orbit 8 electrons in the second orbit 18 electrons in the third orbit The remaining 4 electrons in the valence orbit (shell) 32 Ge Fig. 1.2 Atomic structure of Germanium Silicon has a nucleus with 14 protons and 14 electrons. Their arrangement is given below: 2 electrons in the first orbit 8 electrons in the second orbit The remaining 4 electrons in the valence orbit 4

17 Ionisation Fig. 1.3 Atomic structure of Silicon If electrons are extracted from the outermost shell of an atom, the overall negative charge of that atom decreases. This is because the atom looses negative charge in the form of an electron when exited. The number of protons remains same hence positive charge remains same. So atom as a whole looses its electrical neutral nature and becomes positively charged. Such an atom is called positive ion. Similarly, by any means if electrically neutral atoms gain an additional electron, it becomes negatively charged and is called negative ion. Thus, by loosing or gaining electrons, an atom gets converted into a charged ion. This process of loosing or gaining an electron, which converts electrically neutral atom to a charged ion is called ionisation. 1.7 Energy-Band Theory We have seen that every shell is associated with an energy level. An electron orbiting very close to the nucleus in the first shell is most tightly bound to the nucleus and possesses only a small amount of energy. Hence, the first shell has the lowest energy level. The greater the distance of an electron from the nucleus, the greater is its energy. Hence the energy level of the outermost shell is the highest. Due to such high energy, the valence electrons in the outermost shell can be easily extracted out and hence such electrons take part in chemical reaction and in bounding the atoms together. This whole discussion is related to the electrons and shells of one isolated atom only. In solids, atoms are brought close together the outer shell electrons are shared by more than one atom these electrons come under the influence of force from other atoms too the valence electrons are shared by forming a bound with the valence electrons of an adjacent atom such bounds are called covalent bounds the valence electrons are not free under normal conditions, as they are shared by the adjacent atoms now the valence electrons possess highest energy level when such electrons form the covalent bounds due to the coupling between the valence electrons, the energy levels associated with the valence electrons merge into each other. this merging forms an energy band 5

18 Circuit Theory and Basic Electronics Similarly, the energy levels of various electrons present in the first orbit, second orbit etc. also merge to form the various energy bands. Instead of the presence of widely separated energy levels as that of the isolated atoms, the closely spaced energy levels are present in a solid, which are called energy bands. Out of all the energy bands, three bands, are the most important in order to understand the behaviour of solids. These bands are: Valence Band Conduction Band Forbidden Band or Gap The graphical representation of the energy bands in a solid is called energy band diagram. Such an energy band diagram for a solid Silicon is shown below. Conduction Band Electron Energy in Silicon Band Gap Energy = 1.11 ev Valence Band Fig. 1.4 Energy band diagram of Silicon The electrons in the various orbits revolving around the nucleus occupy the various bands including fully or partly occupied valence band. The conduction band which is normally empty, carriers the electrons which get drifted from the valence band. These electrons present in the conduction band are free electrons and they drift about in the spaces between the atoms. For any given type of material, the forbidden energy gap may be large, small or nonexistent. The classification of materials as insulator, conductor or semiconductor is mainly dependent on the width of the forbidden energy gap. The energy associated with forbidden band is called energy gap (E and measured in the unit electronic-volt G) (ev). 1eV = 1.6 X J Semiconductor A pure semiconductor is something, which has a lower conductivity. A semiconductor has thermally generated current carriers.

19 At 0 K, all the covalent bonds are complete. Therefore, no free electron is available in the crystal for the conduction of current. Hence, silicon crystal behaves as an insulator at 0 K. At room temperature, a covalent bond breaks, an electron becomes free. The electron which leaves the bonds is called free electron and the vacancy created in the covalent bond due to the release of electron is called a hole Intrinsic Semiconductor If potential difference is applied across an intrinsic semiconductor, electrons will move towards the positive terminal, while the holes will drift towards the negative terminal. Ne= Nh = Ni Ne=Number of free electrons per unit volume Nh=Number of holes per unit volume Ni= Number density of intrinsic carriers Total current inside the semiconductor = current due to free electron + current due to holes The process of adding suitable impurities in the intrinsic is called doping. The impurities are added in the intrinsic semiconductor to increase its conductivity and are known as dopant. A semiconductor obtained after adding impurities atoms in the intrinsic semiconductor is called extrinsic or doped semiconductor Extrinsic Semiconductor Pentavalent impurities - The elements whose each atom has five valence electrons. Example - Arsenic, Antimony, Phosphorus etc. Trivalent impurities - The elements whose each atom has three valence electrons. Example, Indium, Gallium, Aluminium etc. When trivalent impurity is added to pure germanium or silicon crystal, we get extrinsic semiconductor known as p- type semiconductor. Majority charge carriers in p-type semiconductor are holes and minority charge carriers are electrons which are thermally generated. Since each trivalent impurities atom accepts one electron from the neighbouring silicon atom, it is known as acceptor impurities. When pentavalent impurities are added to the pure germanium or silicon crystal, we get an extrinsic semiconductor known as n-type semiconductor. Majority charge carriers in n-type semiconductor are electrons and minority charge carriers are holes which are thermally generated. Since each pentavalent impurity atom donates one electron to the crystal, it is known as donor impurities. N e = N h = N i N h > N e. In p-type semiconductor N e > N h. In n-type semiconductor 7

20 Circuit Theory and Basic Electronics 1.9 Intrinsic Semiconductors Vs Extrinsic Semiconductors The difference between intrinsic semiconductor and extrinsic semiconductor is given below: Intrinsic Semiconductor It is pure elements like Ge and Silicon. N = N e h Low conductivity Conductivity mainly depends on their temperature Extrinsic Semiconductor It is impure elements N N e h High conductivity Conductivity depends on the temperature as well as the amount of impurity added in them 1.10 Diode Table 1.3 Differences between intrinsic semiconductor and extrinsic semiconductor The diode is the simplest and most fundamental nonlinear circuit element. Just like resistor, diode has two terminals. Unlike resistor, it has nonlinear current-voltage characteristics. Its use in rectifiers is the most common application. Physical structure - The most important region, called pn junction, is the boundary between n-type and p-type semiconductor. Description of Diode - The diode is fabricated of a semiconductor material, usually silicon, which is doped with two impurities. One side is doped with a donor or n-type impurity, which releases electrons into the semiconductor lattice. These electrons are not bound and are free to move about. Since there is no net charge in the donor impurity, the n-type semiconductor is electrically neutral. The other side is doped with an acceptor or p-type impurity which imparts free holes into the lattice. A hole is created by the absence of an electron which acts as a positive charge. The p-type semiconductor is also electrically neutral because the acceptor material adds no net charge. The figure below illustrates the cross section of the diode. E Depletion Wider P Region P + P n + n n Narrower (a) (b) Fig. 1.5 PN junction (c) The junction is the dividing line between the n-type and p-type sides. Thermal energy causes the electrons and holes to move randomly. Electrons diffuse across the junction into the p-type side and holes diffuse across the junction into the n-type side. This causes development of a net positive charge in the n-type side and a net negative charge in the p-type side. 8

21 These charges set up an electric field across the junction which is directed from the n-type side to the p-type side. The electric field opposes further diffusion of the electrons and holes. The region in which the electric field exists, is called the depletion region. There are no free electrons or holes in this region because the electric field sweeps them out. Fig. 1.5 shows the diode with a battery connected across it. The polarity of the battery is such that, it reinforces the electric field across the junction causing the depletion region to widen. The positive terminal pulls electrons in the n-type side away from the junction. The negative terminal pulls holes in the p-type side away from the junction. No current can flow. The diode is said to be reverse biased. Fig. 1.5 c shows the diode with the battery polarity reversed. The battery now tends to cancel out the electric field in the depletion region, causing its width to decrease. The positive terminal forces holes toward the junction. The negative terminal forces electrons toward the junction. Current flows increases rapidly if the applied voltage is increased. The diode is said to be forward biased Characteristics of Diode Figure given below shows the circuit symbol for the diode. The arrow part of the symbol points in the direction of current flow when the diode is forward biased. The upper terminal is called the anode. The lower terminal is called the cathode. These names come from vacuum tube diodes v D - i D id (ma) 50 (a) v D (volts) (b) The theoretical equation for the diode current is Fig. 1.6 Characteristics of diode i D = I s Where, I S - saturation current n - emission coefficient VT - thermal voltage The emission coefficient accounts for recombination of electrons and holes in the depletion region, which tend to decrease the current. 9

22 Circuit Theory and Basic Electronics For discrete diodes, it has the value n ~2. For integrated circuit diodes, it has the value n ' 1. The reason of the above difference in values is because an integrated circuit diode is fabricated as a bipolar transistor with the collector connected to the base. The impurity doping in transistors is done so as to minimise recombination s. Thus, n ' 1 when recombine can be neglected. A typical plot of i versus v is given in Fig.1.6 b. D D For v 0.6V, the current is very small. D For v > 0.6V, the current increases rapidly with v. D D The voltage at which the diode appears to begin conducting is called the cut-in voltage. This is approximately 0.6V for the plot in Fig. 1.6 b. The thermal voltage is given by, V T = Where, k - Boltzmann s constant T - Kelvin temperature q - electronic charge At T = 290K, the thermal voltage has the value VT = 0.025V. The default value for T in SPICE is, T = 300K. In this case, the thermal voltage has the value V = V. T This value should be used in any hand calculations that are to be compared to SPICE simulations. Since V increases with T, the equation for i seems to imply that increasing T decreases i. T D D However, I increase rapidly with temperature which causes i to increase with T. S D A rule of thumb that is often quoted for silicon diodes is that if i is held constant, v D D for each degree C as T increases Diode as a Rectifier decreases by about 2mV A rectifier is an electrical device, comprising one or more semi conductive devices (such as, diodes) or vacuum tubes arranged for converting alternating current to direct current. When just one diode is used to rectify AC (by blocking the negative or positive portion of the waveform) the difference between the term diode and the term rectifier is merely one of usage, e.g., the term rectifier describes a diode that is being used to convert AC to DC. Rectification is a process whereby alternating current (AC) is converted into direct current (DC). Almost all rectifiers comprise of a number of diodes in a specific arrangement for more efficiently converting AC to DC than is possible with just a single diode. Rectification is commonly performed by semiconductor diodes Half Wave Rectifier A half wave rectifier is a special case of a clipper. In half wave rectification, the positive or negative half of the AC wave is passed easily while the other half is blocked depending on the polarity of the rectifier. Since only one half of the input waveform reaches the output, it is very inefficient if used for power transfer. Half wave rectification can be achieved with a single diode in a one phase supply. 10

23 U U R Fig. 1.7 Half wave rectifier Full Wave Rectifier Full-wave rectification converts both polarities of the input waveform to DC, and is more efficient. However, in a circuit with a non-center tapped transformer, four rectifiers are required instead of the one needed for half-wave rectification. This is due to each output polarity requiring 2 rectifiers each, example, one for when AC terminal 'X' is positive and one for when AC terminal 'Y' is positive. The other DC output requires exactly the same, resulting in four individual junctions. U U D R Fig. 1.8 Bridge rectifier Four rectifiers arranged in this way are called a bridge rectifier. A full wave rectifier converts the whole of the input waveform to one of constant polarity (positive or negative) at its output by reversing the negative (or positive) portions of the alternating current waveform. The positive (negative) portions thus, combine with the reversed negative (positive) portions to produce an entirely positive (negative) voltage/current waveform. For single phase AC, if the AC is center-tapped, then two diodes back-to-back (i.e., anode-to- anode or cathodeto-cathode) form a full wave rectifier. 11

24 Circuit Theory and Basic Electronics U D 1 U R D Zener Diode Fig. 1.9 Full wave rectifier A Zener diode is a type of diode that permits current not only in the forward direction like a normal diode, but also in the reverse direction if the voltage is larger than the breakdown voltage known as "Zener knee voltage" or "Zener voltage". The device was named after Clarence Zener, who discovered this electrical property. A conventional solid- state diode will not allow significant current if it is reverse-biased, below its reverse breakdown voltage. When the reverse bias breakdown voltage is exceeded, a conventional diode is subject to high current due to avalanche breakdown. Unless this current is limited by external circuitry, the diode will be permanently damaged. In case of large forward bias (current in the direction of the arrow), the diode exhibits a voltage drop due to its junction built-in voltage and internal resistance. The amount of the voltage drop depends on the semiconductor material and the doping concentrations. A Zener diode exhibits almost the same properties, except the device is specially designed so as to have a greatly reduced breakdown voltage, the so-called Zener voltage. A Zener diode contains a heavily doped p-n junction allowing electrons to tunnel from the valence band of the p-type material to the conduction band of the n-type material. In the atomic scale, this tunnelling corresponds to the transport of valence band electrons into the empty conduction band states; as a result of the reduced barrier between these bands and high electric fields that are induced due to the relatively high levels of doping on both sides. A reverse- biased Zener diode will exhibit a controlled breakdown and allow the current to keep the voltage across the Zener diode at the Zener voltage. Example, a diode with a Zener breakdown voltage of 3.2 V will exhibit a voltage drop of 3.2 V if reverse bias voltage applied across it is more than its Zener voltage. However, the current is not unlimited, so the Zener diode is typically used to generate a reference voltage for an amplifier stage, or as a voltage stabiliser for low-current applications. Another mechanism that produces a similar effect is the avalanche effect as in the avalanche diode. The two types of diode are in fact constructed the same way and both effects are present in diodes of this type. In silicon diodes up to about 5.6 volts, the Zener effect is the predominant effect and shows a marked negative temperature coefficient. Above 5.6 volts, the avalanche effect becomes predominant and exhibits a positive temperature coefficient. In a 5.6 V diode, the two effects occur together and their temperature coefficients neatly cancel each other, thus, the 5.6 V diode is the component of choice in temperature-critical applications. Modern manufacturing techniques have produced devices with voltages lower than 5.6 V with negligible temperature coefficients, but as higher voltage devices are encountered, the temperature coefficient rises dramatically. 12

25 A 75 V diode has 10 times the coefficient of a 12 V diode. All such diodes, regardless of breakdown voltage, are usually marketed under the umbrella term of "Zener diode" Zener Diode as Voltage Regulator A Zener diode is a PN junction that has been specially made to have a reverse voltage breakdown at a specific voltage. Its characteristics are otherwise very similar to common diodes. In breakdown, the voltage across the Zener diode is close to constant over a wide range of currents, thus making it useful as a shunt voltage regulator. Cathode Characteristics Anode Fig Symbol of Zener diode The figure given below shows the current versus voltage curve for a Zener diode. Observe the nearly constant voltage in the breakdown region. Vz Forward Bias Soft Transition Leakage Region Breakdown Current Iz min Voltage Fig Characteristic of Zener diode The forward bias region of a Zener diode is identical to that of a regular diode. The typical forward voltage at room temperature with a current of around 1 ma is around 0.6 volts. In the reverse bias condition, the Zener diode is an open circuit and only a small leakage current is flowing as shown on the exaggerated plot. As the breakdown voltage is approached, the current will begin to avalanche. The initial transition from leakage to breakdown is soft but then the current rapidly increases as shown on the plot. The voltage across the Zener diode in the breakdown region is very near to constant with only a small increase in voltage with increasing current. At some high current level, the power dissipation of the diode becomes excessive and the part is destroyed. There is a minimum Zener current, I that places the operating point in the desired breakdown region and Zmin there is a maximum Zener current, I Zmax, at which the power dissipation drives the junction temperature to the maximum allowed (typically in the 125 to 150 C range). 13

26 Circuit Theory and Basic Electronics Beyond that current, the diode can be damaged or destroyed. There is no specific value for I although it is typically taken to be ten percent of I. Zmin Zmax It is possible that a lower value could be used particularly at Zener voltages greater than around six. This ensures that the diode operating current is in the breakdown region and not in the soft transition region. The ten percent value is also a historical rule-of-thumb for shunt voltage regulators in general. A shunt regulator has to conduct current in order to be in regulation. To prevent the current from going to zero, shunt regulators are often designed so that at the maximum load current there is at least ten percent of that current in the regulator. Zener diodes are available from about 2.4 to 200 volts typically using the same sequence of values as used for the 5% resistor series 2.4, 2.7, , 3.6, 3.9, 4.3, 4.7, 5.1, 5.6, 6.2, 6.8, 7.5, 8.2, 9.1, 10, 11, 12, 13, 15, 16, 18, 20, 22, 24, etc. All Zener diodes have a power rating, P. From Watt s Law, the maximum current is I = P / V. Z Zmax Z Z Zener diodes are typically available with power ratings of 0.25, 0.4, 0.5, 1, 2, 3, and 5 watts although other values are available. The purpose of a voltage regulator is to maintain a constant voltage across a load regardless of variations in the applied input voltage and variations in the load current. A typical Zener diode shunt regulator is shown in Fig.1.12 given below. The resistor is sized so that, when the input voltage is at V and the load current is at I, the current through INmin Lmax the Zener diode is at least I Zmin. Then for all other combinations of input voltage and load current the Zener diode conducts the excess current thus maintaining a constant voltage across the load. The Zener diode conducts the least current when the load current is the highest and it conducts the most current when the load current is the lowest. Vin Vin Max R IL min I load IL max Vin Min Load V z Fig Zener diode shunt regulator Shunt regulators are normally only used for applications where the load power is not much (no more than a few watts) because under the worst case situation of no load the Zener diode has to dissipate the full load power. Shunt regulators have an inherent current limiting advantage under load fault conditions because the series resistor limits excess current. 14

27 Summary The transport of charge through a semiconductor not only depends on the atomic properties but also on the arrangement of atoms in the solid. In a crystalline solid, the various energy levels merge to from various energy bands. The materials such as, copper, aluminium etc. are good conductors of electricity. While the material such as, wood, glass, mica etc. are bad conductors of electricity and are called insulators. There is another class of materials, whose conductivity lies between that of conductors and insulators. Such materials are called semiconductors. Germanium (Ge) and Silicon (Si) are two well know semiconductor materials. According to the modern electron theory, an atom is composed of the three fundamental particles, which are invisible to bare eyes. These are neutron, proton and electron. The electrons are negatively charged particles and are most important element from the conduction point of view. The number of electrons passing across a particular point per second represents an electric current at that point. The atoms have a planetary type of structure, according to classical Bohr Model. All the protons and neutrons are bound together at the centre of an atom, called nucleus. The first shell can occupy maximum of two electrons (2x1 2 ) while the second shell can occupy maximum of eight electrons (2x2 2 ) and so on. The semiconductor materials such as, Ge and Si have four electrons in their valence shell. If potential difference is applied across an intrinsic semiconductor, electrons will move towards the positive terminal, while the holes will drift toward the negative terminal. The diode is the simplest and most fundamental nonlinear circuit element. A Zener diode is a type of diode that permits current not only in the forward direction like a normal diode, but also in the reverse direction if the voltage is larger than the breakdown voltage known as "Zener Knee Voltage" or "Zener Voltage". References Gates, D. E., Introduction to Electronics 4E. 4th ed. Cengage Learning. An introduction to electronics. 2nd ed. Cambridge University Press. Prof. Natarajan, S. T., Basic Electronics. [Pdf] Available at: < Basic_Electronics_Lab/LECTURE1.pdf> [Accessed 3rd June 2013]. Zulinski, B., Introduction to Electronics. [Pdf] Available at: < onlinetext/elint200.pdf> [Accessed 3rd June 2013]. Prof. Natarajan, T. S., Lecture - 1 Introduction to Basic Electronics. [Video online] Available at: < youtube.com/watch?v=w8dq8bltmsa> [Accessed 3rd June 2013]. Prof. Radhakrishna, R. K., Lecture - 1 Introduction. [Video online] Available at: < watch?v=8ybxghhkr20> [Accessed 3rd June 2013]. Recommended Reading Yacobi, B. G., Semiconductor materials: an introduction to basic principles. Springer. 0Mandal, Basic Electronics. Tata McGraw-Hill Education. 0Adamy, D., Introduction to Electronic Warfare Modeling and Simulation. SciTech Publishing. 15

28 Circuit Theory and Basic Electronics Self Assessment 1. The electrons are charged particles and are most important element from the conduction point of view. a. harmfully b. c. d. negatively positively naturally The charge on one electron is. a x C b x C c x C d. 1.2 x C All the protons and neutrons are bound together at the of an atom, which is called Nucleus. a. midpoint b. medium c. centre d. inside Which of the following statements is true? a. The number of protons in an atom is called as its atomic weight. b. The number of protons in an atom is called as its atomic number. c. The number of protons in an atom is called as its atomic sum. d. The number of protons in an atom is called as its atomic energy. Which of the following statements is true? a. Electrons revolving in the last shell are responsible for the electrical and chemical characteristic of an atom. b. Electrons revolving in the first shell are responsible for the electrical and chemical characteristic of an atom. c. Electrons revolving in the second shell are responsible for the electrical and chemical characteristic of an atom. d. Electrons revolving in the third shell are responsible for the electrical and chemical characteristic of an atom. The amount of required to extract the valence electron from the outer shell is very less. a. power b. force c. energy d. electrons 16

29 Which of the following is responsible for negative charge of an atom? a. if electrons are extracted from the outermost shell of an atom b. c. d. if electrons are extracted from the innermost shell of an atom if electrons are extracted from the inner shell of an atom if electrons are extracted from the first shell of an atom Which of the following makes pentavalent impurities? a. The elements whose each atom has four valence electrons. b. c. d. The elements whose each atom has three valence electrons. The elements whose each atom has five valence electrons. The elements whose each atom has six valence electrons. Majority charge carriers in p-type semiconductor are and minority charge carriers are, which are thermally generated. a. hole, electrons b. c. d. charge carrier, energy force, valence electron ions, charge 10. The diode is the simplest and most fundamental circuit element. a. linear b. nonlinear c. discrete d. analog 17

30 Circuit Theory and Basic Electronics Chapter II Bipolar Junction Transistor (BJT) Aim The aim of this chapter is to: explain the concept of BJT evaluate various configurations of BJT explain construction of BJT Objectives The objectives of this chapter are to: enlist various parameter of BJT explain the concept of biasing explain the DC load line Learning outcome At the end of this chapter, you will be able to: understand the theory of transistor identify various configurations of transistor understand voltage and current distribution in BJT 18

31 2.1 Introduction to Transistor A BJT is a three terminal semiconductor device. It is widely used in discrete circuits as well as in integrated circuits. The main applications of BJTs are analog circuits. Example, BJTs are used for amplifiers in particular for high-speed amplifiers. BJTs can be used for digital circuits as well, but most of the digital circuits are nowadays realised by field effect transistors (FETs). A transistor is a device that can be used either as an amplifier or a switch. There are three operating modes for BJTs. These are: the active mode (amplifying mode) the cut-off mode the saturation mode To apply a BJT as an amplifier, the BJT has to operate in the active mode. To apply a BJT as a digital circuit element, the BJT has to operate in the cut-off mode and the saturation mode. 2.2 BJT Structure Each BJT consists of two anti serial connected diode. The BJT can be either implemented as a npn or a pnp transistor. (a) (c) (b) C (c) C (d) C B B B n p n p n p n B I b base I C collector emitte I e =I b +I c E E E E Fig. 2.1 Equivalent circuit of transistor In both cases, the center region forms the base (B) of the transistor, while the external regions form the collector (C) and the emitter (E) of the transistor. External wire connections to the p and n regions (transistor terminals) are made through metal (e.g., Aluminium) contacts. BJTs consist of one emitter-base junction and one collector-base junction as shown in Fig. 2.2 npn or pnp transistors are called bipolar transistors because both types of carriers (electrons and holes) contribute to the overall current. In the case of a field effect transistor, either the electronics or the holes determine the current flow. Therefore, a field effect transistor is a unipolar device. The current and voltage amplification of a BJT is controlled by the geometry of the device (example width of the base region) and the doping concentrations in the individual regions of the device. 19

32 Circuit Theory and Basic Electronics In order to achieve a high current amplification the doping concentration in the emitter region is typically higher than that of the base region. The base is a lightly doped, very thin region located between the emitter and the collector, which controls the flow of charge carriers (electrons or holes) from the emitter to collector region. Emitter (E) Collector (C) Emitter (E) Collector (C) n p n p n p Emitter-base junction Base (B) Collector-base junction Emitter-base junction Base (B) Collector-base junction Fig. 2.2 npn and pnp BJTs structures 2.3 BJT Circuit Configurations The figure below shows the symbol for the npn transistor, whereas the pnp symbol is shown in fig C B E npn Fig. 2.3 npn transistor symbol E B C pnp Fig. 2.4 pnp transistor symbol The emitter of the BJT is always marked by an arrow, which indicates whether the transistor is an npn or a pnp transistor. 20

33 Configuration of BJT There are three basic ways in which a BJT can be configured. In each case, one terminal is common to both the input and output circuit. Common emitter configuration The common emitter configuration is used for voltage and current amplification and is the most common configuration for transistor amplifiers. Input Output Fig. 2.5 Common emitter configuration Common collector configuration The common collector configuration is often called an emitter follower since its output is taken from the emitter resistor. Input Output 2.4 Common Base Configuration Fig. 2.6 Common collector configuration Common base configuration is useful as an impedance matching device, since its input impedance is much higher than its output impedance. This configuration is used for high frequency applications because the base separates the input and output, minimising oscillations at high frequency. It has a high voltage gain, relatively low input impedance and high output impedance compared to the common collector. Input Output Fig. 2.7 Common base configuration 21

34 Circuit Theory and Basic Electronics 2.5 Construction of Transistor In Bipolar Junction Transistors (BJT), there are three doped semiconductor regions separated by two pn junctions. These three regions are given below: emitter base collector Transistors can be either pnp or npn type. The pn junction joining the base and emitter region is called base emitter junction. The pn junction joining the base region and collector region is called base collector junction. The base region is lightly doped and very thin when compared to the highly doped emitter and moderately doped collector region. C (collector) C B (base) n p n Base-Collector junction Base-Emitter junction B p n n E (emitter) E (b) npn (c) pnp Fig. 2.8 Basic structure of transistor C (collector) C Metalized contacts Oxide Emitter Base Collector B (base) n p n Base-Collector junction Base-Emitter junction B p n n Substrate (a) Basic epitaxial planar structure E (emitter) E (b) npn (c) pnp Fig. 2.9 Construction of BJT 22

35 2.6 Forward-Reverse Bias of a BJT To operate the transistor properly, the two pn junctions must be correctly biased with external DC voltage. In both the circuits, BE is forward biased and BC is reverse biased. The heavily doped n-type emitter region has number of conduction band free electrons. These electrons diffuse through the base emitter junction (p-type material). The base region is lightly doped and has limited holes. Thus, only a small percentage of all the electrons flowing through the BE junction can combine with the available holes in the base. These relatively few recombined electrons flow out of the base lead as valence electrons forming the small base electron current. + - BC reversebaised BC forwardbaised BC reversebaised BE forwardbaised - + (a) npn (b) pnp Fig Circuit diagram of BJT Most of the electrons flowing from the emitter into the thin lightly doped base region do not recombine but diffuse into the base collector depletion layer. In this region, they are pulled through the reverse biased BC junction by the force of attraction of the positive and negative ion. The electrons now move, through the collector region, out through the collector lead and into the positive terminal of the collector voltage source. Based on the assumed current directions, the following current relation can be written as, I E = I B +I C The base current is very small when compared to I E or I C + I C + - I C - I C I C + I B n p n I B - I B p n p I B I E I E I E - I E + Fig Flow of current 23

36 Circuit Theory and Basic Electronics 2.7 Transistor Characteristics and Parameters V forward biases the base emitter junction and V reverse the bias the base collector junction. BB CC The ratio of the DC collector current (I ) to the DC base current (I ) is the DC beta ratio which is also known C B as DC current gain of the transistor. Typical values of β range from 20 to 200 or higher. DC β DC = I C /I B The ratio of the DC collector current (I ) to the DC emitter current (I ) is the DC alpha (α ). The alpha is always C E DC less than one. Typical values range from 0.95 to α = I C / I E R C I C R C I C R B + R B - + I B V CC - - I B V CC + V BB V BB - I E + I E 2.8 Current and Voltage Analysis Fig Transistor DC bias circuits There are three key DC voltages and three key DC currents to be considered. Note that, these measurements are important for troubleshooting. I B : I E : I C : V BE : V CB : V CE : dc base current dc emitter current dc collector current dc voltage across base-emitter junction dc voltage across collector-base junction dc voltage from collector to emitter 24

37 R C I C + R B I B R C V CE + - V CC V BB V BE - I E Fig Transistor currents and voltages When the base emitter junction is forward biased, it acts like a pn junction diode and has a nominal voltage drop of 0.7 volt. V BE = 0.7 V Since emitter is at ground (0 Volt), by applying KVL, the voltage across RB is V RB = V BB V BE Also by Ohm s law, V RB = I B R B I B R B = V BB V BE Hence, I B = (V BB V BE ) / R B The voltage at the collector with respect to the grounded emitter is V CE = V CC V RC, where V RC = I C R C The voltage at the collector is V CE = V CC I C R C where I C = β DC I B The voltage across the reverse biased collector base junction is V CB = V CE - V BE 2.9 Collector Characteristics Curves For proper operation, the base-emitter junction is forward-biased by V and conducts just like a diode. BB The collector-base junction is reverse biased by V and blocks current flow through its junction just like a CC diode. Remember that current flow through the base-emitter junction will help to establish the path for current flow from the collector to emitter. Analysis of this transistor circuit to predict the DC voltages and currents requires use of Ohm s Law, Kirchhoff s Voltage Law and the beta for the transistor. Application of these laws begins with the base circuit to determine the amount of base current. Using Kirchhoff s Voltage Law, subtract the 0.7 volt (corresponding to V ) and the remaining voltage is BE dropped across R B. Determining the current for the base with this information is a matter of applying of Ohm s Law. V RB /R B = I B 25

38 Circuit Theory and Basic Electronics The collector current is determined by multiplying the base current by beta. 0.7 V will be used in most analysis BE examples Collector Characteristic Curves As determine by use of Kirchhoff s Voltage Law for series circuits, the base circuit V is distributed across BB the base-emitter junction and R B. In the collector circuit we determine that V is distributed proportionally across R and the transistor (V ). CC C CE R C I C R B + + V CE V CC + I B - - V BB - ic Fig (a) ic Saturation Region Active (Linear) Region i B3 0.7 V V CE (max) UCE saturation active breakdown region region region i B1 Cutoff Region i B2 < < i B3... i B2 i B1 V CE Fig (b) Fig (a) & (b) Collector characteristic curves Collector characteristic curves give a graphical illustration of the relationship of collector current and V with CE specified amounts of base current. With greater increase in V, V continues to increase until it reaches breakdown, but the current remains about CC CE the same in the linear region from 0.7V to the breakdown voltage. 26

39 Cut-off region When I = 0, the transistor is in the cut-off region of its operation. This is shown in fig below with base B lead open, resulting in a base current of zero. Under this condition, there is a very small amount of collector leakage current, I, due to thermally produced CEO carriers mainly. Because I CEO is extremely small, will usually be neglected in circuit analysis so that V CE = V CC. In cut-off, both the base-emitter and the base-collector junctions are reverse biased. R C R B + I CEO V CE = V CC V CC I B = 0 - Saturation Fig Base lead open When the base-emitter junction becomes forward-biased and the base current increases (IC = β DC I B ) and V CE decreases as a result of more drop across the collector resistor (V CE = V CC - I C R C ). This is illustrated in fig shown below. When V reaches its saturation value, V (sat), the base-collector junction becomes forward biased and I can CE CE C increase no further even with a continued increase in I B. at the point of saturation, the relation I C = β DC I B is no longer valid. V (sat) for a transistor occurs somewhere below the knee of the collector curves and it is usually only a few CE tenths of a volt for silicon transistors. - R C + I C R B + + V CE = V CC - I C R C V CC + I B - - V BB - Fig Base-emitter junction forward-biased 27

40 Circuit Theory and Basic Electronics As I increases due to increasing V, I also increases and V decreases due to the increased voltage drop B BB C CE across R C. When the transistor reaches saturation, I can increase no further regardless of further increase in I. Base-emitter C B and base-collector junctions are forward biased DC Biasing Circuits The AC operation of an amplifier depends on the initial DC values of I, I, and V. B C CE By varying I around an initial DC value, I and V are made to vary around their initial DC values. B C CE DC biasing is a static operation since it deals with setting a fixed (steady) level of current (through the device) with a desired fixed voltage drop across the device. Fig DC biasing circuits Purpose of the DC biasing circuit The two prime purposes of the DC biasing circuit are as follows: to turn the device ON to place it in operation in the region of its characteristic where the device operates most linearly, i.e., to set up the initial DC values of I B, I C, and V CE Voltage-divider bias The voltage-divider (or potentiometer) bias circuit is the most commonly used one. R, R : voltage-divider to set the value of V, I B1 B2 B B C : it is a bypass capacitor, often used to short circuit AC signals to ground, not affecting the DC operating 3 (or biasing) of a circuit R : stabilises the AC signals E 28

41 Fig Voltage divider circuit Graphical DC bias analysis + V C C I C R C R L R 2 I E R E Fig DC bias circuit V CC - I C R C - V CC -I E R E =0 for I C I E I C = V CE + Point-slope form of straight line equation y=mx+c 29

42 Circuit Theory and Basic Electronics I C(t) = V CC /(R C +R E ) DC Load Line I C (ma) V CE(0ff) = V CC V CE DC load line The straight line is known as the DC load line. Fig DC load line Significance of DC load line is that regardless of the behaviour of the transistor, the collector current I and the C collector-emitter voltage V CE must always lie on the load line, depends only on the V CC, R C and R E (i.e., The DC load line is a graph that represents all the possible combinations of I C and V CE for a given amplifier. For every possible value of I C, and amplifier will have a corresponding value of V CE.). Q-point (static operation point) When a transistor does not have an AC input, it will have specific DC values of I and V.These values correspond c CE to a specific point on the DC load line. This point is c alled th e Q-point. The letter Q corresponds to the word (Latent) quiescent, meaning at rest. A quiescent amplifier is one that has no AC signal applied and therefore has constant DC values of I and C V CE. The intersection of the DC bias value of I with the DC load line determines the Q-point. B It is desirable to have the Q-point centered on the load line. Q-point is said to be mid-point. Midpoint biasing allows optimum AC operation of the amplifier. 30

43 pa I C (ma) 6 4 Q 30 pa 20 pa 2 10 pa I B = 0 pa V CE DC biasing and AC signal Fig Q-point When an AC signal is applied to the base of the transistor, both will vary around their Q-point values. When the Q-point is centered, I and V both can make the maximum possible transitions above and below C CE their initial DC values. When the Q-point is above the center on the load line, the input signal may cause the transistor to saturate. When this happens, a part of the output signal will be clipped off. When the Q-point is below midpoint on the load line, the input signal may cause the transistor to cut-off. This can also cause a portion of the output signal to be clipped. 31

44 Circuit Theory and Basic Electronics I C (sat) 1 IC (sat) 2 Q VCC 2 V CC Fig DC load line and AC signal I C I C I C V CEQ R 0 Q V CEQ I CQ VCEQ Q V CEQ + I C R C V CEQ V CE V V CE 0 CE 0 I CQ R C Q V CEQ (a) Limited by saturation (a) Limited by cutoff (c) Centered Q-point Fig AC signal with different Q-point V CEQ 32

45 Summary A BJT is a three terminal semiconductor device. The main applications of BJTs are analog circuits. Each BJT consist of two anti serial connected diode. The BJT can be either implemented as a npn or a pnp transistor. npn or pnp transistors are called bipolar transistors because both types of carriers (electrons and holes) contribute to the overall current. In order to achieve a high current amplification, the doping concentration in the emitter region is typically higher than that of the base region. The base is a lightly doped, very thin region between the emitter and the collector, which controls the flow of charge carriers (electrons or holes) from the emitter to collector region. The emitter of the BJT is always marked by an arrow, which indicates whether the transistor is an npn or a pnp transistor. The common emitter configuration is used for voltage and current amplification and is the most common configuration for transistor amplifiers. The common collector configuration is often called an emitter follower due to its output is taken from the emitter resistor. Common base configuration is useful as an impedance matching device due to its input impedance, which is much higher than its output impedance. In Bipolar Junction Transistors (BJT) there are three doped semiconductor regions separated by two pn junctions. These three regions are emitter, base and collector. The pn junction joining the base and emitter region is called base emitter junction. The pn junction joining the base region and collector region is called base collector junction. To operate the transistor properly, the two pn junctions must be correctly biased with external DC voltage. Typical values of β range from 20 to 200 or higher. DC The ratio of the DC collector current (I ) to the DC emitter current (I ) is the DC alpha (α ). The alpha is always C E DC less than one. Typical values range from 0.95 to For proper operation, the base-emitter junction is forward-biased by V and conducts just like a diode. BB Analysis of this transistor circuit to predict the DC voltages and currents requires use of Ohm s Law, Kirchhoff s Voltage Law and the beta for the transistor. The AC operation of an amplifier depends on the initial DC values of I, I, and V. B C CE Significance of DC load line is that regardless of the behaviour of the transistor, the collector current I and the C collector-emitter voltage V CE must always lie on the load line, depends only on the V CC, R C and R E (i.e., the DC load line is a graph that represents all the possible combinations of I C and V CE for a given amplifier. For every possible value of I C, and amplifier will have a corresponding value of V CE.). References Kal, S., Basic Electronics Devices Circuits And It Fundamentals. PHI Learning Pvt. Ltd. Kumar, B. & Jain, B. S., Electronic Devices and Circuits. PHI Learning Pvt. Ltd. Bipolar Junction Transistors (BJTs). [Pdf] Available at: < microelectronics/ch5.pdf> [Accessed 29 May 2013]. Bipolar Junction Transistor (BJT). [Pdf] Available at: < Electronics_Ch4.pdf> [Accessed 29 May 2013]. Retrobrad Presents! Electronics Tutorial 7 - Introduction to Transistors. [Video online] Available at: < [Accessed 29 May 2013]. 33

46 Circuit Theory and Basic Electronics Electronics 2 : How a BIPOLAR JUNCTION TRANSISTOR (BJT) WORKS. [Video online] Available at: < [Accessed 29 May 2013]. Recommended Reading Maheshwari, L. K. & Anand, M. M. S., Laboratory Manual For Introductory Electronics Experiments. New Age International. Chen, K. W., The Circuits and Filters Handbook, Second Edition. 2nd ed. CRC Press. Sharma, A. K., Semiconductor Electronics. New Age International. 34

47 Self Assessment 1. Which of following statements is true? a. BJT is a three terminal semiconductor device. b. FET is a three terminal semiconductor device. c. JFET is a three terminal semiconductor device. d. Diode is a three terminal semiconductor device Which of the following statements is true? a. The main applications of BJTs are digital circuits. b. c. d. The main applications of BJTs are analog circuits. The main applications of BJTs are timing circuits. The main applications of BJTs are phase circuits. Which of the following statements is true? a. Each BJT consist of three anti serial connected diode. b. c. d. Each BJT consist of four anti serial connected diode. Each BJT consist of two anti serial connected diode. Each BJT consist of some anti serial connected diode. npn or pnp transistors are called transistors because both types of carriers contribute to the overall current. a. bipolar b. c. d. unipolar polar tripolar The current and voltage amplification of a BJT is controlled by the of the device. a. history b. c. d. geometry physics chemistry In order to achieve a high current the doping concentration in the emitter region is typically higher than that of the base region. a. strengthening b. c. d. magnification extension amplification is a lightly doped very thin region between the emitter and the collector. a. Base b. c. d. Gate Collector Emmiter 35

48 Circuit Theory and Basic Electronics The emitter of the BJT is always marked by an. a. stick b. pole c. arrow d. cursor The configuration is used for voltage and current amplification a. common emitter b. common base c. common collector d. common ground 10. Common based configuration is useful as a/an. a. magnification b. impedance matching device c. amplification d. rectification 36

49 Chapter III Field Effect Transistor (FET) Aim The aim of this chapter is to: introduce the concept of FET describe the concept of JFET explain MOSFET Objectives The objectives of this chapter are to: evaluate types of FET explain voltage and current characteristics of JFET evaluate and explain different types of MOSFET Learning outcome At the end of this chapter, the student will be able to: understand the theory of FET identify various types of FET describe the concept of JFET 37

50 Circuit Theory and Basic Electronics 3.1 Introduction The field effect transistor, abbreviated as FET, is another semiconductor device like a BJT which can be used as an amplifier or a switch. Like BJT, FET is a three terminal device; however, the principle of operation of FET is completely different from that of BJT. The three terminals of FET are named as: Drain (D) Source (S) Gate (G) This can be referred to in Fig. 3.1 shown below. Out of these three terminals, gate terminal acts as a controlling terminal. 2 SOURCE 3 GATE Unipolar device 1 DRAIN Fig. 3.1 Symbol of FET As earlier discussed, the current is carried by both electrons and holes in BJT and hence the name bipolar junction transistor. However, in FET, the current is carried by only one type of charged particles, either electrons or holes. Hence FET is named as a unipolar device. Voltage controlled device In BJT, the output current, I C is controlled by the base current I B. Hence BJT is a current controlled device. Whereas, in FET, the voltage applied between gate and source (V GS ) controls the drain current I D. Therefore, FET is a voltage controlled device. The name field effect is derived from the fact that the output current flow is controlled by an electric field set up in the device by an externally applied voltage between gate and source terminals. 3.2 Significance of FET Like BJT, the parameters of FET are also temperature dependent. In FET, as temperature increases, drain resistance also increases, simultaneously reducing the drain current. Thus unlike BJT, thermal runaway does not occur with FET. So it can be said that FET is more temperature stable compared to the BJT. FET has very high output impedance. Typically, it is in the range of one to several mega ohms. Since FETs have higher input impedance than BJT, they are more preferred in amplifiers where high input impedance is required. 38

51 3.3 Types of FET The FETs are categoried as: Junction Field Effect Transistors (JFETs) Metal Semiconductor Field Effect Transistors (MESFETs) Metal Oxide Semiconductor Field Effect Transistors (MOSFETs) The JFETs and MESFETs are further classified into two types, mentioned below: n-channel JFET/ MESFETs p-channel JFET/ MESFETs MOSFETs are further classified into the following two types: Depletion MOSFET Enhancement MOSFET Field Effect Transistor (FETs) Junction FET (JET) MESFET MOSFET n-channel MESFET p-channel MESFET n-channel JFET p-channel JFET Depletion MOSFET Enhancement MOSFET 3.4 JFET: Junction Field Effect Transistor Fig. 3.2 Classification of FETs JFET is a semiconductor device that operates by altering the conductivity of a region of the semiconductor (the channel) between two contacts (source and drain) by application of a voltage to a third terminal (gate). The current flow between source and drain is controlled by the gate voltage. In a JFET device, the gate voltage is applied to the channel across a P-N junction, in contrast to its application across an insulator in a conventional MOSFET. JFETs are of both P-channel and N-channel types. Refer symbol of JFET N-channel and P-channel in the figure below: D G S Fig. 3.3 P-Channel JFET symbol 39

52 Circuit Theory and Basic Electronics D G S 3.5 Construction of JFET Fig. 3.4 N-Channel JFET symbol JFET can be fabricated with either an N-channel or P-channel though N-channel is generally preferred. For fabricating an N-channel JFET, a narrow bar of N-type semiconductor material is first taken and then two P-type junctions are diffused on opposite sides of the middle part. These junctions form two P-N diodes or gates and the area between these gates is called channel. The two P-regions are internally connected and a single lead is brought out which is called the gate terminal. Ohmic contacts are made at the two ends of the bar. One lead is called source terminal and the other drain terminal. When potential difference is established between drain and source, current flows along the length of the bar through the channel located between the two P-regions. In JFET, current consists of only majority carriers either hole or electrons. D D G p CHANNEL p G p CHANNEL p S S D D G G n-channel FET (a) S S p-channel FET (b) Fig. 3.5 Construction of JFET Source Source is the terminal through which majority carriers enter the bar. Drain Drain is the terminal through which majority carriers leave the bar. The drain-to-source voltage V DS drives the drain current I D. 40

53 Gate Gates are two internally-connected heavily doped impurity regions which form two PN junctions. The gate-source voltage V GS reverse biases the gates. Channel Channel is the space between the two gates through which majority carriers pass from source-to-drain when V DS is applied. 3.6 Theory of JFET Gates are always reversed-biased. Hence, gate current I is practically zero. G The source terminal is always connected to the end of the drain supply which provides the necessary charge carriers. When V GS =0 and V DS =0 drain current I D = 0, as V DS = 0 The depletion regions around PN junctions are of equal thickness and symmetrical. D i D Depletion region Depletion region G p n p i G = 0 G + VGS - + (Small) VDS - S i S = i D When V GS =0 and V DS is increased from zero The JFET is connected to the V supply. DD Fig. 3.6 JFET with equal thickness depletion region The electrons flow from S to D whereas conventional drain current I flows through the channel from D to S. D The gate-to-channel bias at any point along the channel is equal to sum of V and V DS GS. Since external bias V = 0, gate-channel reverse bias is provided by V alone. GS DS Since the value of V keeps decreasing, as we go from D to S, the gate-channel bias also decreases DS accordingly. It has a maximum value in the drain-gate region and minimum in the source-gate region. Depletion regions penetrate more deeply into the channel in the drain-gate region than in the source-gate region. As V is gradually increased from zero, I increases proportionally as per Ohm s law. DS D The ohmic relationship between V and I continues till V reaches a certain critical value called pinch-off DS D DS voltage V PO when drain current becomes constant at its maximum value called I DSS. When V is increased beyond V, I remains constant at its maximum value I up to a certain point. DS PO D DSS 41

54 Circuit Theory and Basic Electronics Ultimately, a certain value of V (called V ) is reached when JFET breaks down and I increases to an DS DSO D excessive value. When V DS =0 and V GS is decreased from zero In this case, as V is made more negative over time, the gate reverse bias increases, which then increases the GS thickness of the depletion regions. The channel is said to be cut-off. The value of V which cuts-off the channel and hence the drain current is called V GS GS(off). When VGS is negative and VDS is increased Values of V as well as breakdown voltages are decreased. P V GS =0 I DSS -1V I D -2V -3V V V DS -4V V GS = - 4V=V P Static Characteristics of a JFET Fig. 3.7 Characteristic curve JFET Drain characteristics It gives the relation between I D and V DS for different values of V GS. Transfer characteristics It gives relation between I D and V GS for different values of V DS. 42

55 D I D R L V DS R g V GS V DD V GG JFET Drain Characteristic with V GS = 0 Fig. 3.8 Basing of JFET Ohmic region This part of the characteristic is linear indicating that for low values of V DS, current varies directly with voltage. Curve AB In this region, I D increases at reverse square-law up to point B which is called pinch-off point. Pinch-off region It is also known as saturation region. Ohmic Region Pinch off (Saturation) Region Breakdown Region I DSS B V as =0 C A I D O V p V A V ps Fig. 3.9 Drain characteristics JFET characteristics with external bias Pinch-off voltage is reached at a lower value of I than when VGS = 0. D Value of V for breakdown is decreased. DS Transfer Characteristics I D = I DSS (1-V GS /V P ) 2 43

56 Circuit Theory and Basic Electronics I DSS V DS Constant I D -4V -2V O V P V GS Fig 3.10 Transfer characteristics 3.7 MOSFET - Metal-Oxide-Semiconductor FETs The most widely used FETs are Metal-Oxide-Semiconductor FETs (or MOSFET). MOSFET can be manufactured as enhancement-type or depletion-type MOSFETs. N-Channel Enhancement Type MOSFET (NMOS) The physical structure of an N-Channel Enhancement-Type MOSFET (NMOS) is shown below. The device is fabricated on a p-type substrate (or body). Two heavily doped n-type regions (source and drain) are created in the substrate. A thin (fraction of micron) layer of SiO, which is an excellent electrical insulator, is deposited between the 2 source and drain region. Metal is deposited on the insulator to form the gate of the device (thus, metal-oxide semiconductor). Metal contacts are also made to the source, drain, and body region. Source (S) Gate (G) Drain (D) Oxide (SiO 2 ) Metal S i S = + U GS - i D G i G = 0 i D D + U DS - (small) Channel region L p type substrate (Body) Body (B) i D Induced n-channel p-type substrate B Fig N-Channel enhancement-type MOSFET To see the operation of a NMOS, let us ground the source and the body and apply a voltage V between the GS gate and the source, as shown in the above Fig This voltage repels the holes in the p-type substrate near the gate region, lowering the concentration of the holes. 44

57 As V increases, hole concentration decreases and the region near gate behaves progressively more like intrinsic GS semiconductor material (excess hole concentration zero) and then, finally, like an n-type material as electrons from n+ electrodes (source and drain) enters this region. As a result, when V become larger than a threshold voltage, V, a narrow layer between source and drain GS t regions, is created that is populated with n-type charges. The thickness of this channel is controlled by the applied V. GS As can be seen, this device works as a channel is induced in the semiconductor and this channel contains n-type charges (thus, named as n-channel MOSFET). In addition, increasing V increases channel width (enhances it). Therefore, this is an enhancement-type GS MOSFET. Now for given values of V > V (so that the channel is formed), let us apply a small and positive voltage V GS t DS between drain and source. Then, electrons from n + source region enter the channel and reach the drain. If V is increased, current i flowing through the channel increases. DS D Effectively, the device acts like a resistor; its resistance is set by the dimension of the channel and its n-type charge concentration. In this regime, the graph plotted of i versus V is a straight line (for a given values of V > V ) as shown D DS GS t below. i D (ma) 0.4 u GS = V r + 4 V u GS = V r + 3 V u GS = V r + 2 V 0.1 u GS = V r + 1 V u GS = V r ups(mv) Fig Characteristics OF NMOSFET The slope of i versus V line is the conductance of the channel. D DS Changing the value of V, changes dimension of the channel and its n-type charge concentration and, therefore, GS its conductance. As a result, changing V affects the slope of i versus V line as shown above. GS D DS The above description is correct for small values of V as in such case, DS V = V -V =V and the induced channel is fairly uniform (i.e., it has the same width near the drain as it GD GS DS GS has near the source). For a given V > V, if we still increase V GS t DS, V = V -V becomes smaller than V GD GS DS GS. As the size of channel near drain becomes smaller compared to its size near the source, as shown in Fig. 3.13, its resistance increases and the curve of i D versus V DS starts to roll over, as shown below. 45

58 Circuit Theory and Basic Electronics Source (S) Gate (G) Drain (D) Oxide (SiO 2 ) Metal S i S = + U GS - i D G i G = 0 i D D + U DS - (small) + Channel + region n+ i n+ n n D Induced n-channel L p type substrate p-type substrate (Body) Body (B) B Fig 3.13 Constructor of MOSFET For values of V = V (or V V V ), width of the channel approaches zero near the drain (channel is GD t DS = GS- t pinched off). Increasing V beyond this value has little effect (no effect in our simple picture) on the channel shape, and the DS current through the channel remains constant at the value reached when V DS = V GS -V t. So when the channel is pinched o, i D only depends on V GS i D (ma) 4 U DS U GS - V I U DS U GS - V I Triode region Saturation region U GS = V I + 4 i D (ma) 4 3 U DS = U GS - V I 3 2 U GS = V I U DS U GS - V I 1 U GS = V I + 2 U GS = V I U DS = (V) (b) U GS V I (cut off) NMOS Characteristic Curves V GS (V) V I Plot of i D versus U GS in the active regime 3.8 Operation of FET Fig 3.14 Characteristic curve of NMOS In sum, a FET can operate in three regimes. These include: Cut-off The regime in which no channel exists (V GS < V t for NMOS) and i D = 0 for any V DS is called cut-off. Ohmic or Triode The regime in which the channel is formed and not pinched off (V GS > V t and V DS V GS -V t for NMOS) and FET behaves as a voltage-controlled resistor, is called triode. 46

59 Saturation Lastly, the third regime is called saturation in most electronic books because i D is saturated in this regime and does not increase further. This is a rather unfortunate name as saturation regime in a FET means very different thing than it does in a BJT. Some newer books call this regime active region (as it equivalent to active-linear regime of a BJT). Note that the transition between ohmic and active region is clearly defined by V = V -V, the point where DS GS t the channel is pinched off. Several important points should be noted are mentioned below: No current flows into the gate, i = 0 (note the insulator between gate and the body). G FET acts as a voltage-controlled resistor in the ohmic region. In addition, when V V, FET would act as DS GS a linear resistor. If i = 0, this does not mean that FET is in cut-off. D FET is in cut-off when a channel does not exist (V < V ) and i = 0 for any applied V. GS t D DS On the other hand, FET can be in ohmic region, i.e., a channel is formed, but i = 0 because V = 0. D DS 3.9 Body Effect In deriving NMOS (and other MOS) i versus V characteristics, let us assumed that the body and source are D DS connected. This is not possible in an integrated chip which has a common body and a large number of MOS devices (connection of body to source for all devices means that all sources are connected). The common practice is to attach the body of the chip to the smallest voltage available from the power supply (zero or negative). In this case, the pn junction between the body and source of all devices will be reversed biased. The impact of this is to lower threshold voltage slightly for the MOS devices and is called the body effect. Body effect can degrade device performance. For analysis, let us assume that body effect is negligible P-Channel Enhancement-type MOSFET (PMOS) The physical structure of a PMOS is identical to a NMOS except that the semiconductor types are interchanged, i.e., body and gate are made of n-type material and source and drain are made of p-type material and a p-type channel is formed. D G B S D D G i D G i D S S Fig PMOS symbol and current flow As the sign of the charge carriers are reversed, all voltages and currents in a PMOS are reversed. By convention, the drain current flows out of the drain as shown in the figure. The entire discussion above NMOS above applies to PMOS as long as we multiply all voltages by a minus sign. 47

60 Circuit Theory and Basic Electronics Cut-off: V GS > V t, i D = 0 for any V DS Ohmic: V GS < V t, i D = K[2V DS (V GS -V t ) -V 2 DS ] for V DS > V GS - V t Active: V GS < V t, i D = K(V GS -Vt) 2 for V DS < V GS- V t Note that V t is negative for a PMOS Complementary MOS (CMOS) Complementary MOS technology employs MOS transistors of both polarities. This can be referred from the figure shown below. CMOS devices are more difficult to fabricate than NMOS, but many more powerful circuits are possible with CMOS configuration. As such, most of MOS circuits today employ CMOS configuration and CMOS technology is rapidly taking over many applications that were possible only with bipolar devices a few years ago. S 2 G 2 NMOS Polysilicon PMOS S G D D G S Gate SiO Thick SiO 2 2 oxide (isolation) SiO 2 D 2 i D2 V 1 V 0 p-type body n well G 1 D 1 S 1 i D Depletion-Type MOSFET Fig CMOS construction The depletion-type MOSFET has a structure similar to the enhancement-type MOSFET with only one important difference; depletion-type MOSFET has a physically implanted channel. Thus, an n-type depletion-type MOSFET already has a n-type channel between drain and source. When a voltage V is applied to the device, a current i = I flows even for V =0 (Show I = KV ). DS D DSS GS DDS t2 Similar to NMOS, if V is increased, the channel become wider and i increases. GS D However, in an n-type depletion-type MOSFET, a negative V can also be applied to the device, which makes GS the channel smaller and reduces i D. As such, negative V depletes the channels from n-type carriers leading to the name depletion-type GS MOSFET. If V is reduced further, at some threshold value V (which is negative), the channel disappears and i = 0, this GS t D can be seen in the figure. It should be obvious that a depletion-type MOSFET can be operated either in enhancement mode or in depletion mode. t P-type depletion MOSFET operate similarly to p-type enhancement MOSFET expect that V > 0 for depletion type and V t < 0 for the enhancement type. 48

61 The figure below shows i versus V of four types of MOSFET devices in the active region. D GS Circuit symbols for depletion-type MOSFET devices are also shown. Fig Characteristic of depletion and enhancement type MOSFET Fig Depletion MOSFETs 49

62 Circuit Theory and Basic Electronics Summary The Field effect transistor abbreviated as FET is another semiconductor device like a BJT which can be used as an amplifier or switch. Like BJT, FET is also a three terminal device; however, the principle of operation of FET is completely different from that of BJT. FET is called unipolar device as well as a voltage controlled device. The parameters of FET are also temperature dependent. In FET, as temperature increases, drain resistance also increases, reducing the drain current. Thus unlike BJT, thermal runaway does not occur with FET. Thus we can say that FET is more temperature stable as compared to the BJT. The FETs are categorised into Junction Field Effect transistors (JFETs), Metal Semiconductor Field Effect Transistors (MESFETs) and Metal Oxide Semiconductor Field Effect Transistors (MOSFETs). The JFETs and MESFETs are further classified into n-channel JFET/ MESFETs and p-channel JFET/ MESFETs. MOSFETs are further classified into two types. These are Depletion MOSFET and Enhancement MOSFET. JFET is a semiconductor device that operates by altering the conductivity of a region of the semiconductor (the channel) between two contacts (source and drain) by application of a voltage to a third terminal (gate). JFET can be fabricated with either an N-channel or P-channel though N-channel is generally preferred. Drain characteristics: It gives the relation between I and V for different values of V D DS GS. Transfer characteristics: It gives relation between I and V for different values of V D GS DS. The most widely used FETs are Metal-Oxide-Semiconductor FETs (or MOSFET). MOSFET can be manufactured as enhancement-type or depletion-type MOSFETs. The physical structure of a PMOS is identical to a NMOS except that the semiconductor types are interchanged, i.e., body and gate are made of n-type material and source and drain are made of p-type material and a p-type channel is formed. Complementary MOS technology employs MOS transistors of both polarities. CMOS devices are more difficult to fabricate than NMOS, but many more powerful circuits are possible with CMOS configuration. The depletion-type MOSFET has a structure similar to the enhancement-type MOSFET with only one important difference; depletion-type MOSFET has a physically implanted channel. Thus, an n-type depletion-type MOSFET has already a n-type channel between drain and source. References Chattopadhyay, D., Electronics (fundamentals And Applications), 7th ed., New Age International. Garg, K. R., Dixit, A. & Yadav, P., Basic Electronics, Firewall Media. MOSFET & JFET Theory [Pdf] Available at: < MOSFET%20Theory.pdf> [Accessed 3 June 2013]. Field-Effect (FET) transistors [Pdf] Available at: < NOTES/FET.pdf> [Accessed 3 June 2013]. Prof. Natarajan, T. S., Lecture - 40 Field Effect Transistor [Video online] Available at: < com/watch?v=4m49vm0ryt8> [Accessed 3 June 2013]. Lecture-38-MOS Field Effect Transistor [Video online] Available at: < watch?v=irbgagrcvic> [Accessed 3 June 2013]. 50

63 Recommended Reading Singh, Electronic Devices And Integrated Circuits. Pearson Education India. Widanarto, W. R., Gas Detection with Floating Gate Field Effect Transistor. Cuvillier Verlag. Kal, S., Basic Electronics Devices Circuits And It Fundamentals. PHI Learning Pvt. Ltd. 51

64 Circuit Theory and Basic Electronics Self Assessment 1. Which of the following statements is true? a. FET is called unipolar device. b. BJT is called unipolar device. c. MOSFET is called unipolar device. d. JFET is called unipolar device Which of the following statements is true? a. FET can be used as a op-amp. b. c. d. FET can be used as a hub. FET can be used as a amplifier and switch. FET can be used as a router. Which of the following statements is true? a. FET is a volume control network b. c. d. FET is a voltage control network FET is a current control network FET is a load control network In FET, as temperature increases drain resistance also, reducing the drain current. a. increases b. c. d. decreases reduces amplifies has very high output impedance. a. JFET b. c. d. FET MOSFET MESFET The two P-regions are internally connected and a single lead is brought out which is called the. a. base terminal b. c. d. gate terminal source terminal drain terminal are made at the two ends of the bar. a. Ionic contacts b. c. d. Ohmic contacts Gate contacts Drain contacts 52

65 8. 9. Which of the following is having current, consisting of only majority carriers either hole or electrons? a. FET b. MOSFET c. BJT d. JFET Which of the following is responsible for letting majority carrier enter the bar? a. Source b. Drain c. Gate d. Channel 10. Gates are always reversed-biased. Hence, gate current I is practically. G a. positive b. zero c. negative d. neutral 53

66 Circuit Theory and Basic Electronics Chapter IV Op-amplifier and Filters Aim The aim of this chapter is to: explain Op-amp describe various rules of Op-amp explain the significance of filters Objectives The objectives of this chapter are to: explain the ideal op-amp enlist the types of op-amp explain the concept of filters Learning outcome After reading this chapter student will be able to: identify various type of filters understand the standard parameter of op-amp describe maximum and minimum parameter rating 54

67 4.1 Introduction to Operational Amplifier The operational amplifier is one of the most useful and important components of analog electronics. They are widely used in popular electronics. Their primary limitation is that they are slow. The typical performance degrades rapidly for frequencies greater than about 1 MHz, although some models are designed specifically to handle higher frequencies also. The primary use of op-amps in amplifier and related circuits is closely connected to the concept of negative feedback. Feedback represents a vast and interesting topic in itself. 4.2 Basics of Op-amp Operational amplifiers are convenient building blocks that can be used to build amplifiers, filters, and even an analog computer. Op-amps are integrated circuits composed of many transistors & resistors such that the resulting circuit follows a certain set of rules. The most common type of op-amp is the voltage feedback type, which is discussed below. The schematic representation of an op-amp is shown below. There are two input pins (non-inverting and inverting), an output pin and two power pins. The ideal op-amp has infinite gain. It amplifies the voltage difference between the two inputs and that voltage appears at the output. Without feedback, this op-amp would act like a comparator (i.e., when the non-inverting input is at a higher voltage than the inverting input the output will be high, when the inputs are reversed the output will be low). Vss - Input Output + Input Vss Fig. 4.1 Symbol of Op-amp 55

68 Circuit Theory and Basic Electronics 4.3 Rules of op-amp Two rules will let one to figure out what most simple op-amp circuits do: No current flows into the input pins (i.e., infinite input impedance). The output voltage will adjust to try and bring the input pins to the same voltage. 4.4 Ideal Op-amp The name Ideal Op-amp is given based on the assumption that the salient parameters of the op-amp are perfect. There is no such thing like an ideal op amp, but present day op-amps approach much closer to an ideal op-amp analysis. Op-amps depart from the ideal in two ways. These are: Firstly, DC parameters such as, input offset voltage, are large enough to cause departure from the ideal. The ideal op-amp assumes that input offset voltage is zero. Secondly, AC parameters such as gain, are a function of frequency so they go from large values at DC to smaller values at high frequencies. This assumption simplifies the analysis, clearing the path for insight. Although the ideal op-amp analysis makes use of perfect parameters, the analysis is often valid because some op-amps approach near perfection. In addition, when working at low frequencies of several khz, the ideal op-amp analysis produces accurate answers. 4.5 Basic Ideal Op-amp Assumptions Parameter Name Parameter Symbol Value Input Current I IN 0 Input Offset Voltage V OS 0 Input Impedance Z IN Output Impedance Z OUT 0 Gain A Table 4.1 Ideal Op-amp parameter 56

69 Fig. 4.2 Ideal Op-amp 4.6 Simple OP-amp Circuits Let us discuss simple circuit of voltage follower. Voltage follower Fig. 4.3 Voltage follower No current flows into the input, R =. in The output acts as the feedback to the inverting input. Since the output adjusts to make the inputs of the same voltage V out = V in (i.e., a voltage follower, gain = 1). This circuit is used to buffer a high impedance source (note: the op-amp has low output impedance Ω). 57

70 Circuit Theory and Basic Electronics Application hint The input impedance on some CMOS amplifiers is so high that without any input the non-inverting input can float around to different voltages (i.e., the input pin picks up signals like an antenna). Wherever the input could be disconnected (such as, while coming from an external sensor) it is a good idea to tie the input to ground via a high value resistor (1-10M Ω). This keeps the input at ground potential if the wire to the sensor becomes disconnected and still has high input impedance. 4.7 Non-inverting Amplifier Fig. 4.4 Non-inverting amplifier No current flows into the input, R =. in The output adjusts to bring V - to the same voltage as V +. in in Therefore V - = V and since no current flows in V - the same current must flow through R1 & R2. V therefore in in in out VR1 + VR2 = V in - + IR2 = V in - + (V in /R1) R2. V out = V in (1+ ) Application hint When dealing with larger signals keep in mind that the output can't exceed the power supply voltage i.e., if the op-amp is powered from +/- 15V and have a one volt input and a gain of 20 you won't get 20V the output. The output will most likely stop a few volts below the supply rail (around 13V). There are special op-amps designed to handle inputs and outputs that go all the way to the power supply rail but these op-amps usually only operate at lower voltages (i.e., 0-5V). 58

71 4.8 Inverting Amplifier Fig. 4.5 Inverting amplifier As there is no current flow into the input pins, there can t be a voltage drop across R1 R2. V + is therefore at 0V, called a virtual ground. in The output will adjust such that V - is at zero volts. This makes R = R1 (not ). in in The current through R1 & R2 have to be the same since no current goes into the input pins. Therefore I = V /R1. V = V + - IR2 = 0 - (V /R1) R2. in out in in Therefore V = -V (R2/R1). out in The negative sign is because the current flows from the input to the output whereas, in the earlier examples, the current flows from the output to the input. Application hint Why not connect V + directly to ground? Actually, many people do and the circuit works fine. in The reason to have R1 R2 is because real op-amp is not perfect and draws a small bias current into both inputs. By adding R1 R2 the voltage drop at the + input is offset by the same amount as the voltage drop at the - input. This is called input offset voltage. 59

72 Circuit Theory and Basic Electronics 4.9 Summing Amplifier Fig. 4.6 Summing amplifier Since V - is a virtual ground adding V2 and R2 (and V3 & R3) does not change the current flowing through in R1 from V1. Each input contributes to the output using the following equation, V out = -V1(R4/R1) - V2(R4/R2) - V3(R4/R3) The input impedance for the V1 input is still R1, similarly V2's input impedance is R2 and V3's is R3. Most of the time, the parallel combination of R1-R4 is not used and V + is grounded. in Application hint Some op-amps have null pins that allow you to add a potentiometer and null (remove) the input offset error (Ex: LM741). Some newer op-amps (Ex: LTC1151) are chopper stabilised (i.e., they measure the offset and null it automatically many times a second). Chopper stabilised op-amps work best with slow moving inputs (temperature for example), something well below the rate at which the offset is measured and nulled. 60

73 4.10 Difference Amplifier Fig. 4.7 Differentiator amplifier One can work out the gain before using the two rules mentioned above (i.e., no current flows into the inputs, and the output will adjust to bring V in - to V in +). The result is V = 2(V2-V1)*(R2/R1). Also, R (-) = R1, R (+) = R1 + R2. out in in Application hint Use precision resistors. If the two R2 resistors differ even by 1% difference, will be off by 1%. Same goes for the R1 resistors. Assume V1 = V2. If the resistors are not matched, the output will not be zero. Actually, it is better to buy a difference amplifier or an instrumentation amplifier instead of building the difference amp from an op-amp and discrete resistors. By using a difference amp with the resistors on the IC, they can be later trimmed to better than 0.01% (Ex: LT1990, or LT1168) Standard Operational Amplifier Parameters The following parameters are considered in a standard operational amplifier: Open-loop voltage gain Voltage gain is defined as the ratio of output voltage to an input signal voltage. The voltage gain is a dimensionless quantity. Large signal voltage gain This is the ratio of the maximum allowable output voltage swing (usually one to several volts less than V- and V++) to the input signal required to produce a swing of ± 10 volts (or some other standard). 61

74 Circuit Theory and Basic Electronics Slew rate The slew rate is the maximum rate at which the output voltage of an op-amp can be changed and is measured in terms of voltage change per unit of time. It varies from 0.5 V/µs to 35 V/µs. Slew rate is usually measured in the unity gain non- inverting amplifier configuration. Common mode rejection ratio A common mode voltage is one that is presented simultaneously to both inverting and non-inverting inputs. In an ideal op-amp, the output signal due to the common mode input voltage is zero, but it is nonzero in a practical device. The common mode rejection ratio (CMRR) is the measure of the device's ability to reject common mode signals, and is expressed as the ratio of the differential gain to the common mode gain. The CMRR is usually expressed in decibels, with common devices having ratings between 60 to 120 db. The higher the CMRR is, the better the device is deemed to be. Input offset voltage Input offset voltage is the DC voltage that must be applied at the input terminal to force the quiescent DC output voltage to zero or other level, if specified, given that the input signal voltage is zero. The output of an ideal op-amp is zero, when there is no input signal applied to it. Power-supply rejection ratio The power-supply rejection ratio (PSRR) is the ratio of the change in input offset voltage to the corresponding change in one power-supply, with all remaining power voltages held constant. The PSRR is also called "power supply insensitivity". Typical values are expressed in µv / V or mv/v. Input bias current The average of the currents into the two input terminals with the output at zero volts is called input bias current. Input offset current The difference between the currents into the two input terminals with the output held at zero is called input offset current. Differential input impedance The resistance between the inverting and the non-inverting inputs is called differential input impedance. 12 This value is typically very high, 1 MΩ in low-cost bipolar op-amps and over 10 Ω in premium BiMOS devices. Common-mode input impedance The impedance between the ground and the input terminals, with the input terminals tied together, is called common mode input impedance. This is a large value, of the order of several tens of MS or more. Output impedance The output resistance is typically less than 100 Ω. 62

75 Average temperature coefficient of input offset voltage The ratio of the change in input offset voltage to the change in free-air or ambient temperature is called average temperature coefficient of input offset voltage. This is an average value for the specified range. Output offset voltage The output offset voltage is the voltage at the output terminal with respect to ground when both the input terminals are grounded. Output short-circuit current The current that flows in the output terminal when the output load resistance external to the amplifier is zero ohms (a short to the common terminal) is called output short-circuit current. Channel separation This parameter is used on multiple op-amp ICs (device in which two or more op-amps sharing the same package with common supply terminals). The separation specification describes part of the isolation between the op-amps inside the same package. It is measured in decibels. Example dual op-amps offer 120 db of channel separation. From this specification, it may be stated that a 1µV change will occur in the output of one of the amplifiers when the other amplifier output changes by 1volt Minimum and Maximum Parameter Ratings Operational amplifiers, like all electronic components, are subject to maximum ratings. If these ratings are exceeded, the device failure is the normal consequence. The ratings described below are commonly used. Maximum supply voltage This is the maximum voltage that can be applied to the op-amp without damaging it. The op-amp uses a positive and a negative DC power supply, which are typically ± 18 V. Maximum differential supply voltage This is the maximum difference signal that can be applied safely to the op-amp power supply terminals. Often this is not same as the sum of the maximum supply voltage ratings. Example, 741 has ± 18V as the maximum power supply voltage, whereas the maximum differential supply voltage is only 30V. It means that if the positive supply is 18V, the negative supply can be only 12V. Power dissipation, Pd This rating is the maximum power dissipation of the op-amp in the normal ambient temperature range. A typical rating is 500mW. Maximum power consumption The maximum power dissipation, usually under output short circuit conditions, that the device can survive is called maximum power consumption. This rating includes both internal power dissipation as well as device output power requirements. Maximum input voltage This is the maximum voltage that can be applied simultaneously to both inputs. It is also the maximum common-mode voltage. 63

76 Circuit Theory and Basic Electronics In most bipolar op-amps, the maximum input voltage is nearly equal to the power supply voltage. There is also a maximum input voltage that can be applied to either input when the other input is grounded. Differential input voltage This is the maximum differential-mode voltage that can be applied across the inverting and non-inverting inputs. Maximum operating temperature The maximum operating temperature is the highest ambient temperature at which the device will operate according to specifications with a specified level of reliability. Minimum operating temperature The lowest temperature at which the device operates within specification is called minimum operating temperature. Output short-circuit duration This is the length of time the op-amp will safely sustain a short circuit of the output terminal. Many modern op-amps can carry short circuit current indefinitely. Maximum output voltage The maximum output potential of the op- amp is related to the DC power supply voltages. Typical for a bipolar op-amp with ± 15V power supply, the maximum output voltage is typically about 13 V and the minimum - 13 V. Maximum output voltage swing This is the maximum output swing that can be obtained without significant distortion (at a given load resistance). Full-power bandwidth This is the maximum frequency at which a sinusoid whose size is the output voltage range is obtained Op-amp IC Fig pin IC

77 Pin 1 (offset null) Offset nulling, see Fig Since the op-amp is the differential type, input offset voltage must be controlled so as to minimise offset. Offset voltage is nulled by application of a voltage of opposite polarity to the offset. An offset null-adjustment potentiometer may be used to compensate for offset voltage. The null-offset potentiometer also compensates for irregularities in the operational amplifier manufacturing process which may cause an offset. Consequently, the null potentiometer is recommended for critical applications. See Offset Null Adjustment for method. Pin 2 (inverted input) All input signals at this pin will be inverted at output pin 6. Pins 2 and 3 are very important (obviously) to get the correct input signals or the op amp can not do its work. Pin 3 (non-inverted input) All input signals at this pin will be processed normally without inversion. The rest is the same as pin 2. Pin 4 (-V) The V- pin (also referred to as V ) is the negative supply voltage terminal. SS Supply-voltage operating range for the 741 is -4.5 volts (minimum) to -18 volts (max), and it is specified for operation between -5 and -15 V dc. The device will operate essentially the same over this range of voltages without change in timing period. Sensitivity of time interval to supply voltage change is low, typically 0.1% per volt. ( Note: Do not confuse the -V with ground). Pin 5 (offset null) See pin 1 in Fig Pin 6 (output) Output signal's polarity will be the opposite of the inputs when this signal is applied to the op-amp's inverting input. Example, a sine-wave at the inverting input will output a square-wave in the case of an inverting comparator circuit. Pin 7 (posv) The V+ pin (also referred to as V ) is the positive supply voltage terminal of the 741 Op-amp IC. CC Supply-voltage operating range for the 741 is +4.5 volts (minimum) to +18 volts (maximum), and it is specified for operation between +5 and +15 V dc. The device will operate essentially the same over this range of voltages without change in timing period. Actually, the most significant operational difference is the output drive capability, which increases for both current and voltage range as the supply voltage is increased. Sensitivity of time interval to supply voltage change is low, typically 0.1% per volt. Pin 8 (N/C) The N/C stands for 'Not Connected'. There is no other explanation. There is nothing connected to this pin, it is just there to make it a standard 8-pin package. 65

78 Circuit Theory and Basic Electronics 4.14 Basic Introduction to Filters Filters of some sort are essential to the operation of most electronic circuits. It is therefore in the interest of anyone involved in electronic circuit design to have the ability to develop filter circuits capable of meeting a given set of specifications. Unfortunately, many in the electronics field are uncomfortable with the subject, whether due to a lack of familiarity with it, or a reluctance to grapple with the mathematics involved in a complex filter design. This Application Note is intended to serve as a very basic introduction to some of the fundamental concepts and terms associated with filters. It will not turn a novice into a filter designer, but it can serve as a starting point for those wishing to learn more about filter design Significance of Filters In circuit theory, a filter is an electrical network that alters the amplitude and/or phase characteristics of a signal with respect to frequency. Ideally, a filter will not add new frequencies to the input signal, nor will it change the component frequencies of that signal, but it will change the relative amplitudes of the various frequency components and/or their phase relationships. Filters are often used in electronic systems to emphasise signals in certain frequency ranges and reject signals in other frequency ranges. Such a filter has a gain which is dependent on signal frequency. As an example, consider a situation where a useful signal at frequency f1 has been contaminated with an unwanted signal at f 2. If the contaminated signal is passed through a circuit that has very low gain at f2 compared to f1, the undesired signal can be removed, and the useful signal will remain. Note that in the case of this simple example, here it is not concerned with the gain of the filter at any frequency other than f 1 and f 2. As long as f is sufficiently attenuated relative to f, the performance of this filter will be satisfactory. 2 1 In general, however, a filter's gain may be specified at several different frequencies, or over a band of frequencies. Since filters are defined by their frequency-domain effects on signals, it makes sense that the most useful analytical and graphical descriptions of filters also fall into the frequency domain. Thus, curves of gain vs frequency and phase vs frequency are commonly used to illustrate filter characteristics, and the most widely-used mathematical tools are based in the frequency domain. 66

79 Fig. 4.9 Filters The frequency-domain behaviour of a filter is described mathematically in terms of its transfer function or network function. This is the ratio of the Laplace transforms of its output and input signals. The voltage transfer function H(s) of a filter can therefore be written as follows: H(s)= V out (S) / V in (S)...(1) Where V (s) and V (s) are the input and output signal voltages and s is the complex frequency variable. in out The transfer function defines the filter s response to any arbitrary input signal, but here it is most often concerned with its effect on continuous sine waves. The magnitude of the transfer function is especially as important as a function of frequency, which indicates the effect of the filter on the amplitudes of sinusoidal signals at various frequencies. Knowing the transfer function magnitude (or gain) at each frequency allows determining how well the filter can distinguish between signals at different frequencies. The transfer function magnitude vs frequency is called the amplitude response or sometimes, especially in audio applications, the frequency response. Similarly, the phase response of the filter gives the amount of phase shift introduced in sinusoidal signals as a function of frequency. 67

80 Circuit Theory and Basic Electronics Since a change in phase of a signal also represents a change in time, the phase characteristics of a filter become especially important when dealing with complex signals where the time relationships between signal components at different frequencies are critical. By replacing the variable s in (1) with jw, where j is equal to 1, and w is the radian frequency (2 f), it is found that the filter s effect on the magnitude and phase of the input signal. The magnitude is found by taking the absolute value of equation (1). H(jw) =...(2) and the phase is, arg H(jw)= arg...(3) As an example, the network of Fig. 4.8 has the transfer function. H(s) =...(4) Fig Filter network example This is a 2nd order system. The order of a filter is the highest power of the variable s in its transfer function. The order of a filter is usually equal to the total number of capacitors and inductors in the circuit. (A capacitor built by combining two or more individual capacitors is still one capacitor). Higher-order filters will obviously be more expensive to build, since they use more components, and they will also be more complicated to design. However, higher-order filters can more effectively discriminate between signals at different frequencies. Before actually calculating the amplitude response of the network, it can be observed at very low frequencies (small values of s), the numerator becomes very small, as do the first two terms of the denominator. Thus, as s approaches zero, the numerator approaches zero, the denominator approaches one, and H(s) approaches zero. Similarly, as the input frequency approaches infinity, H(s) also becomes progressively smaller, because the denominator increases with the square of frequency while the numerator increases linearly with frequency. Therefore, H(s) will have its maximum value at some frequency between zero and infinity, and will decrease at frequencies above and below the peak. To find the magnitude of the transfer function, replace s with jw to yield, 68

81 A(w)= H(s) =...(5) = the phase is, θ(w)= arg H(s)= 90 -tan -1...(6) The above relations are expressed in terms of the radian frequency w, in units of radians/second. A sinusoid will complete one full cycle in 2Π radians Basic Filter Types There are five basic filter types: Band pass Notch Low-pass High-pass All-pass Band pass The filter used in the example in the previous section was a band pass. The number of possible band pass response characteristics is infinite, but they all share the same basic form. Several examples of bandpass amplitude response curves are shown given below. The curve in Fig (a) is what might be called an ideal bandpass response, with absolutely constant gain within the pass band, zero gain outside the pass band, and an abrupt boundary between the two. This response characteristic is impossible to realise in practice, but it can be approximated to varying degrees of accuracy by real filters. Curves Fig (b) through Fig (f) are examples of a few bandpass amplitude response curves that approximate the ideal curves with varying degrees of accuracy. Note that while some bandpass responses are very smooth, other have ripple (gain variations in their pass bands. Other have ripple in their stop bands as well. The stop band is the range of frequencies over which unwanted signals are attenuated. Bandpass filters have two stop bands, one above and one below the pass band. 69

82 Circuit Theory and Basic Electronics GAIN GAIN GAIN GAIN GAIN GAIN FREQUENCY (a) FREQUENCY (b) FREQUENCY (c) FREQUENCY FREQUENCY FREQUENCY (d) (e) (f) Fig BPF Just as it is difficult to determine by observation exactly where the pass band ends, the boundary of the stop band is also seldom obvious. Consequently, the frequency at which a stop band begins is usually defined by the requirements of a given system for example, a system specification might require that the signal must be attenuated at least 35dB at 1.5kHz. This would define the beginning of a stop band at 1.5kHz. The rate of change of attenuation between the pass band and the stop band also differs from one filter to the next. The slope of the curve in this region depends strongly on the order of the filter, with higher-order filters having steeper cut-off slopes. The attenuation slope is usually expressed in db/octave (an octave is a factor of 2 in frequency) or db/ decade (a decade is a factor of 10 in frequency). Bandpass filters are used in electronic systems to separate a signal at one frequency or within a band of frequencies from signals at other frequencies. In above mentioned example of point 4.8, a filter whose purpose was to pass a desired signal at frequency f1, while attenuating as much as possible an unwanted signal at frequency f 2. This function could be performed by an appropriate bandpass filter with center frequency f 1. Such a filter could also reject unwanted signals at other frequencies outside of the pass band, so it could be useful in situations where the signal of interest has been contaminated by signals at a number of different frequencies. Notch or band-reject A filter with effectively the opposite function of the bandpass is the band-reject or notch filter. To understand the concept of notch filter refer figure given below: H N (s) = = 70

83 1Ω 1H V IN 1F V OUT Fig Notch filter The amplitude and phase curves for this circuit are shown in figure given below. As can be seen from the curves, the quantities f, f, and f used to describe the behaviour of the band pass filter c l h are also appropriate for the notch filter. A number of notch filter amplitude response curves are shown below: As in Fig. 4.10, curve (a) shows an ideal notch response, while the other curves show various approximations to the ideal characteristic. Notch filters are used to remove an unwanted frequency from a signal, while affecting all other frequencies as little as possible. An example of the use of a notch filter is with an audio program that has been contaminated by 60 Hz power line hum. A notch filter with a center frequency of 60Hz can remove the hum while having little effect on the audio signals. Low-Pass A third filter type is the low-pass. Fig Notch filter frequency response A low-pass filter passes low frequency signals, and rejects signals at frequencies above the filter's cut-off frequency. 71

84 Circuit Theory and Basic Electronics If the components of our example circuit are rearranged as in figure given below, the resultant transfer function is, H LP (s) = = It is easy to see by inspection that this transfer function has more gain at low frequencies than at high frequencies. As 0 approaches 0, HLP approaches 1; as 0 approaches infinity, HLP approaches 0. Amplitude and phase response curves are shown in figure given below, with an assortment of possible amplitude response curves in figure given below. Note that the various approximations to the unrealisable ideal low-pass amplitude characteristics take different forms, some being monotonic (always having a negative slope), and others having ripple in the pass band and/ or stop band. Low-pass filters are used whenever high frequency components must be removed from a signal. An example might be in a light-sensing instrument using a photodiode. If light levels are low, the output of the photodiode could be very small, allowing it to be partially obscured by the noise of the sensor and its amplifier, whose spectrum can extend to very high frequencies. If a low-pass filter is placed at the output of the amplifier, and if its cut-off frequency is high enough to allow the desired signal frequencies to pass, the overall noise level can be reduced. Fig Amplitude and Phasor response curve 72

85 High-pass filter Fig Example of low-pass filter amplitude curve The opposite of the low-pass is the high-pass filter, which rejects signals below its cut-off frequency. A high-pass filter can be made by rearranging the components of our example network as in Fig The transfer function for this filter is given below: H HP (s) = = Fig Simple high pass filter The amplitude and phase curves are found in Fig Note that the amplitude response of the high-pass is a mirror image of the low-pass response. 73

86 Circuit Theory and Basic Electronics Further examples of high-pass filter responses are shown in Fig 4.16, with the ideal response in Fig (a) and various approximations to the ideal shown in Fig (b) through Fig (f). Fig Example of High pass filter amplitude response curves High-pass filters are used in applications requiring the rejection of low-frequency signals. One such application is in high-fidelity loudspeaker systems. Music contains significant energy in the frequency range from around 100 Hz to 2 khz, but high-frequency drivers (tweeters) can be damaged if low-frequency audio signals of sufficient energy appear at their input terminals. A high-pass filter between the broadband audio signal and the tweeter input terminals will prevent low-frequency program material from reaching the tweeter. In conjunction with a low-pass filter for the low-frequency driver (and possibly other filters for other drivers), the high-pass filter is part of what is known as a crossover network. All pass filter The fifth and final filter response type has no effect on the amplitude of the signal at different frequencies. Instead, its function is to change the phase of the signal without affecting its amplitude. This type of filter is called an allpass or phase-shift filter. 74

87 Summary The operational amplifier is one of the most useful and important components of analog electronics. They are widely used in popular electronics. Their primary limitation is that they are slow. The typical performance degrades rapidly for frequencies greater than about 1 MHz, although some models are designed specifically to handle higher frequencies. Operational amplifiers are convenient building blocks that can be used to build amplifiers, filters, and even an analog computer. Op-amps are integrated circuits composed of many transistors & resistors such that the resulting circuit follows a certain set of rules. Two rules will let one to figure out what most simple op-amp circuits do: No current flows into the input pins (i.e., infinite input impedance). The output voltage will adjust to try and bring the input pins to the same voltage. Voltage gain is defined as the ratio of output voltage to an input signal voltage. The voltage gain is a dimensionless quantity. The slew rate is the maximum rate at which the output voltage of an op-amp can change and is measured in terms of voltage change per unit of time. A common mode voltage is one that is presented simultaneously to both inverting and non-inverting inputs. The DC voltage that must be applied at the input terminal to force the quiescent DC output voltage to zero or other level, if specified, given that the input signal voltage is zero. The power-supply rejection ratio PSRR is the ratio of the change in input offset voltage to the corresponding change in one power-supply, with all remaining power voltages held constant. The average of the currents into the two input terminals with the output at zero volts is called input bias current. The difference between the currents into the two input terminals with the output held at zero is called input offset current. The resistance between the inverting and the non-inverting inputs is called differential input impedance. The impedance between the ground and the input terminals, with the input terminals tied together it is nothing but common mode input impedance. This is the maximum differential-mode voltage that can be applied across the inverting and non-inverting inputs. The maximum temperature is the highest ambient temperature at which the device will operate according to specifications with a specified level of reliability. The maximum output potential of the op- amp is related to the DC power supply voltages. Filters of some sort are essential to the operation of most electronic circuits. It is therefore in the interest of anyone involved in electronic circuit design to have the ability to develop filter circuits capable of meeting a given set of specifications. References Peyton, A. & Walsh, V., Analog Electronics with Op-amps: A Source Book of Practical Circuits. Cambridge University Press. Clayton, G. B. & Winder, S., Operational Amplifiers, 5th ed., Newnes. Prof. Kovacs, G., Operational Amplifiers: Basic Concepts. [Pdf] Available at: < ee122/handouts/2-op-amp_concepts.pdf> [Pdf] [Accessed 3 June 2013]. 6 Op-Amp Basics. [Pdf] Available at: < [Accessed 3 June 2013]. 75

88 Circuit Theory and Basic Electronics Dr. Mahanta, C., Module - 4 Lecture - 1 Operational Amplifier (Introduction). [Video online] Available at: < [Accessed 3 June 2013]. Op-Amp Basics Part I. [Video online] Available at: < [Accessed 3 June 2013]. Recommended Reading Carter, B. & Mancini, R., Op Amps for Everyone. 3rd ed., Newnes. Kishore, Operational Amplifiers and Liner Integrated Circuits. Pearson Education India. Rutkowski, B. G., Operational Amplifiers: Integrated and Hybrid Circuits. John Wiley & Sons 76

89 Self Assessment 1. Which of the following statements is true? a. Inverting amplifiers are convenient building blocks that can be used to build amplifiers, filters, and even an analog computer. b. Operational amplifiers are convenient building blocks that can be used to build amplifiers, filters, and even an analog computer. c. Operational amplifiers are not convenient building blocks and cannot be used to build amplifiers, filters, and even an analog computer. d. Difference amplifiers are convenient building blocks that can be used to build amplifiers, filters, and even an analog computer Which of the following statements is true? a. The input impedance on some CMOS amplifiers is so high that without any input the non-inverting input can float around to different voltages. b. The input impedance on some CMOS amplifiers is so low that without any input the inverting input can float around to different voltages. c. The input impedance on some CMOS amplifiers is so low that without any input the non-inverting input can float around to different voltages. d. The input impedance on some CMOS amplifiers is so high that without any input the inverting input can float around to different voltages. Following is the circuit diagram. a. non-inverting amplifier b. summing amplifier c. difference amplifier d. inverting amplifier is the ratio of the maximum allowable output voltage swing. a. Open loop voltage gain b. Slew rate voltage gain c. Large signal voltage gain d. Output offset voltage What is the measure of the device's ability to reject common mode signals? a. Common mode rejection ratio b. Power supply rejection ratio c. Channel separation d. Maximum Supply Voltage What is the difference between the currents into the two input terminals with the output held at zero called? a. Input bias current b. Input offset current c. Output bias current d. Output offset current 77

90 Circuit Theory and Basic Electronics 7. What is the resistance between the inverting and the non-inverting inputs called? a. Common mode impedance b. Input impedance c. Output impedance d. Differential input impedance 8. is the maximum difference signal that can be applied safely to the op-amp power supply terminals. a. Maximum supply voltage b. c. d. Max output supply voltage Maximum differential supply voltage Max input supply voltage 9. Which of the following statements is true? a. High-pass filters are used in applications requiring the rejection of low-frequency signals. b. Band pass filters are used in applications requiring the rejection of low-frequency signals. c. Notch filters are used in applications requiring the rejection of low-frequency signals. d. All pass filters are used in applications requiring the rejection of low-frequency signals. 10. What are often used in electronic systems to emphasise signals in certain frequency ranges and reject signals in other frequency ranges? a. High pass filter b. c. d. Filters Amplifiers Low pass filter 78

91 Chapter V LCR Circuits Aim The aim of this chapter is to: explain the theory of LCR circuit evaluate various types of LCR circuit describe various features of LCR circuit discuss Ohm s Law Objectives The objectives of this chapter are to: explain Ohm s law evaluate various Kirchhoff s Laws explain Kirchhoff s Current Law Learning outcome At the end of this chapter, you will be able to: understand the concept of LCR understand types of Kirchhoff s Law describe Kirchhoff s Voltage and Current Law 79

92 Circuit Theory and Basic Electronics 5.1 Introduction to LCR Circuits containing an inductor L, a capacitor C and a resistor R, have special characteristics useful in many applications. Their frequency characteristics (impedance, voltage, or current vs. frequency) have a sharp maximum or minimum at certain frequencies. These circuits can hence be used for selecting or rejecting specific frequencies and are also called tuning circuits. These circuits are therefore very important in the operation of television receivers, radio receivers, and transmitters. There are two types of LCR circuits namely; series LCR circuits parallel LCR circuits A series LCR circuit includes a series combination of an inductor, resistor and capacitor whereas; a parallel LCR circuit contains a parallel combination of inductor and capacitor with the resistance placed in series with the inductor. Both series and parallel resonant circuits may be found in radio receivers and transmitters. The selectivity of a tuned circuit is its ability to select a signal at the resonant frequency and reject other signals that are close to this frequency. A measure of the selectivity is Q, or the quality factor. The study of these circuits is basically an application of alternating current circuit analysis. Here, complex number notation is used with sinusoidal varying quantities like alternating voltage and current. In general, the impedance Z is a sum of the real part called resistance R and the complex part called the reactance X, i.e., Z = R + jx. The magnitude and phase of the impedance are given by and Ø= tan-1, respectively. Since voltage leads the current by Π/2 in an inductor, the reactance of L is jwl, while in case of a capacitor, voltage lags behind the current by Π/2, the reactance of C is. If the current in the circuit is I, the relative voltage drops across the inductor, capacitor and resistor can be represented in the phasor diagram as shown in figure given below. In this section, property of resonance in context of series as well as parallel configurations of LCR circuit is discussed. It is a very useful property of reactive AC circuits and is employed in a variety of applications. One of the common applications of resonance effect is in radio and television transmissions, e.g., tuning a radio to a particular station by selecting a desired frequency (or band of frequencies). The series resonant circuit can be used for voltage magnification. A parallel resonant circuit provides current magnification and can be used in induction heating. Another application of resonant circuit is screening certain frequencies out of a mix of different frequencies with the help of circuits called filters. 80

93 5.2 LCR Series Circuit Fig. 5.1 Phasor diagram of LRC Let us consider the LCR circuit, which consists of an inductor L, a capacitor C, and a resistor R, all connected in series with a source as shown in below.. First let us derive the condition of resonance and then explain the methods of determination of the resonant frequency and hence the quality factor. Let an alternating voltage V sinwt or V 0 0 ejwt be applied to an inductor L, a resistor R and a capacitor C all in series as shown in the figure below. If I is the instantaneous current flowing through the circuit, the applied voltage in phasor in phasor form is given by, 81

94 Circuit Theory and Basic Electronics V=V R +V L +V C =RI + jwli+ = = I The impedance Z= = R+ j if, Z=Ze i = ZcosØ+jZsinØ then, Z= and tanø= therefore, current, I= = 82

95 Three cases thus arise, Fig. 5.2 Series LCR circuit wl>, tan Ø is positive and applied voltage leads current by phase angle Ø. wl>, tan Ø is negative and applied voltage lags behind current by phase angle Ø. wl>, tan Ø is zero and applied voltage and current are in phase angle Ø. This condition is known as resonance and the frequency is called resonant frequency (w 0 ). wl= = w 2= or w=w 0 = or wl- and V L =V C If L, R and υ (frequency of function generator) are fixed and the capacitance is varied, then for lower values of C, >w LI or V C >V L. As the capacitance is increased in the circuit, the situation is called resonance and is achieved when V = V. C L The point of intersection of V and V versus curves will give resonance condition. C L This is depicted in Fig At resonance, V is maximum while V is minimum as shown in below.. R LC Corresponding to maximum value of V, C is obtained. R Similarly, for minimum value of V, C is obtained. LC 83

96 Circuit Theory and Basic Electronics This value of C makes the given circuit resonant at the supply frequency with constant values of L and R. Fig. 5.3 Variation of V L and V C with 84 Fig. 5.4 Variation of VLC and VR with Theoretically at resonance, V should be zero. This is because the inductor is of negligible resistance and there LC are no other losses. The minimum value of V is a measure of the effective resistance of inductor coil which is equal to the DC LC resistance plus AC resistance corresponding to iron and hysteresis losses. At resonant frequency f 0, the impedance of circuit is minimum. Hence frequencies near f 0 are passed more readily than the other frequencies by the circuit. Due to the above mentioned reason, LCR-series circuit is called acceptor circuit. The band of frequencies which is allowed to pass readily is called pass-band.

97 The band is arbitrarily chosen to be the range of frequencies between which the current is equal to or greater than Let f and f be these limiting values of frequency. Then the width of the band is as follows: 1 2 BW=f 2 -f 1 The Quality factor is defined in the same way as for a mechanical oscillator and is given by, Q= = Q-factor is also defined in terms of reactance and resistance of the circuit at resonance, i.e., Q= = Fig. 5.5 Variation of current with frequency for different R values LCR series circuit Also, 85

98 Circuit Theory and Basic Electronics Q= = The resonance condition is also evident from the resonance curves or the graphs between I R = and f for different values of R. The bandwidth as well as Q-factor can be calculated. 5.3 LCR Parallel Circuit The parallel resonant circuit obeys the same formula for resonant frequency as the series resonant one, but at resonance the parallel resonant circuit has very high impedance. The resistance at resonance offered by the parallel resonant circuit is very high if the resistance of the inductance is very small, and is known as the dynamic resistance. Let us discuss a series LCR circuit, which is different than a parallel LCR circuit. The condition of resonance in this case is known as anti-resonance. Here, we will derive the condition of anti-resonance of a parallel LCR circuit. The laboratory method of determination of the anti-resonant frequency and hence the quality factor is explained. Consider a circuit containing an inductor L and a capacitor C connected in parallel to an AC source. The resistance R is connected in series with the inductor Land includes its resistance. The total admittance of the LCR combination is given by, = + Therefore, = jwc+ = +j 86

99 Fig. 5.6 Parallel LCR circuit For the condition of resonance, current and voltage are in phase and the coefficient of j, i.e., the reactive term which brings about a phase change is zero, hence w 0 C- = 0 2Πf 0 C= Which gives, f 0 = At resonance, the impedance of the circuit is maximum and is given by, Z= = R+ Z= The impedance at resonance is called dynamic resistance. The current I= V/Z has minimum value. It is for this reason that the condition of resonance for a parallel LCR circuit is known as anti-resonance and the corresponding frequency as the anti-resonance frequency. 87

100 Circuit Theory and Basic Electronics Fig. 5.7 Variation current with frequency for different R values LCR parallel circuit The shape of the impedance versus frequency curve in a parallel LCR circuit is same as the shape of the current versus frequency curve in a series LCR circuit. In other words, the circuit has very high impedance at the anti-resonant frequency. The parallel tuned circuit is used to select one particular signal frequency from among others. It does this by rejecting the resonant frequency because of its high impedance. This is the reason why this type of circuit is also known as a rejector circuit. The circuit is more selective if it offers high impedance at resonance and much lower impedance at other frequencies. The Q-factor is defined in the same way as for a series LCR circuit. As in series circuit, Q can also be written as, Q= = = 5.4 Features of LCR Circuits Alternating voltage An alternating voltage is a sinusoidally varying voltage, where the peak is value and is the angular frequency of the voltage. Anti-resonance Anti-resonance is the condition in a parallel LCR circuit when the impedance of the circuit is maximum and the current minimum is termed as anti-resonance. Anti-resonant frequency For a parallel LCR circuit, the frequency at which the current has minimum value, is called anti-resonant frequency. 88

101 Bandwidth The range of frequencies lying within the upper and lower cut-off frequencies which correspond to times the voltage value at resonance is called bandwidth. It is also defined as the difference between the two half power frequencies which correspond to the points where the power has been reduced to one half of its value at resonance. Capacitance The property of a conductor that describes its ability to store electric charge is called capacitance C and is given by Q/V where Q is the charge stored on the conductor and V is the potential difference between the conductor and earth. Dynamic resistance The frequency-dependent resistance of a parallel LCR circuit at resonance is known as the dynamic resistance. Impedance Impedance is a measure of the total opposition that a circuit or a part of a circuit offers to electric current. It includes both resistance and reactance. Inductance It is the property of a conductor, often in the shape of a coil, defined as the electromotive force induced in a conductor per unit rate of change of current flowing through it. Pass-band The electric waves lying within a certain range, or band, of frequencies allowed passing, all other frequencies being blocked by the series LCR circuit is called pass-band. rms An alternating potential difference has a value of one volt rms (root mean square) if it produces the same heating effect when applied to the ends of a resistance as is done by a steady potential difference of one volt applied to the same resistance in the same time. Numerically, rms value is times the maximum value. The AC ammeters and voltmeters measure the root mean square (rms) value of the current and potential difference respectively. Quality factor It is a measure of the selectivity or the sharpness of the resonance curve and is denoted by Q. A low value of resistance in the circuit leads to a high Q. Quality factor is given by the ratio of the voltage across the inductor to the input voltage and is hence a dimensionless quantity. Since Q is ordinarily greater than unity, it is termed as the magnification factor of the circuit. Reactance The frequency-dependent opposition to current flow, which results from energy storage rather than energy loss, is called reactance and is denoted by X L and X C for an inductor and capacitor respectively. Resistance It is a measure of the opposition offered by an electric circuit to the flow of electric current. Resonance Condition in a series LCR circuit, when the impedance is purely resistive and hence minimum and current maximum is called resonance. 89

102 Circuit Theory and Basic Electronics Resonance curve A graph showing the variation of the voltage across a circuit (or a part of it) with frequency in the vicinity of resonance is the response curve or the resonance curve. Resonant frequency For a series LCR circuit, the frequency at which the reactance due to the inductor, X L, is exactly equal and opposite to the reactance due to the capacitor, X C, resulting in the impedance of the circuit being purely resistive, is called the resonant frequency. Selectivity The selectivity of a tuned circuit is its ability to select a signal at resonant frequency and reject other signals that are close to that frequency. 5.5 Ohm s Law Georg Simon Ohm: Georg Simon Ohm (16 March July 1854) was a German physicist. As a high school teacher, Ohm began his research with the recently invented electrochemical cell, invented by Italian Count Alessandro Volta. Using simple electrical circuits containing various lengths of wire, Ohm determined that there is a direct proportionality between the potential differences (voltage) applied across a conductor and the resultant electric current now known as Ohm's law. Using the results of his experiments, Ohm was able to define the fundamental relationship among voltage, current, and resistance, which represents the true beginning of electrical circuit analysis. Statement of Ohm s Law: Ohm's law states that, the current through a conductor between two points is directly proportional to the potential difference or voltage across the two points and inversely proportional to the resistance between them. I= I= Current V= Voltage R= Resistance 5.6 Basic Circuit Element There are five basic circuit elements based on which the circuits are designed. How much ever the complexity of the circuit be, the five fundamental elements are the bricks for that circuit. Those elements are: Resistor Capacitor Inductor Voltage source Current source Resistor It opposes the flow of current through it. Resistance is called the physical property of a resistor by the virtue of which it gives an opposition to the current flow. 90

103 Unit: Resistance is measured in ohms (Ω). Capacitor Capacitor is a device that stores energy. It has two simple parallel plates with dielectric in between in its construction; each plate is connected to one end of the supply. Once the supply is turned on, charges of opposite polarity appear on each plate and thus it stores energy. Capacitor has a special property that it doesn t allow any sudden changes in voltage across it. Unit: Capacitance is measured in Farads (F). Inductor An inductor is a simple curl of windings that doesn t allow sudden changes in current. Unit: Inductance of an inductor is measured in Henry (H). The above mentioned three elements resistor, inductor and capacitor are called passive elements. Voltage source This serves as a source to create potential difference between two points. It has two types of sources, discussed below: DC voltage source AC voltage source Fig. 5.8 DC voltage source Fig. 5.9 AC Voltage source Current source A current source is a source that supplies the current directly. 91

104 Circuit Theory and Basic Electronics Fig Current source Current source and voltage source are called active elements. A combination of the elements is called a CIRCUIT. 5.7 Important Part of a Circuit The following parameters form the basic parts of a circuit: Node A point in a circuit from which two or more circuit elements diverge. Two points between which there is no circuit element cannot be a node. Path The distance traversed between two nodes is called a path. Loop A closed path is called a loop. Branch A path between two nodes is called the branch of the network. We have two laws describing the current and voltage across any particular branch. These are: Kirchhoff s Current Law (KCL) Kirchhoff s Voltage Law (KVL) 5.8 Kirchhoff s Current Law (KCL) KCL states that the algebraic sum of currents entering / leaving a node is zero. Kirchhoff s current law 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 is known as the "Conservation of Charge. 92

105 Fig Current entering and leaving the node Consider a four way junction. The sum of vehicles entering the signal and leaving the signal is zero. Whatever is the number of vehicles entering the signal, all those vehicles will definitely leave the junctions. Hence the no. of vehicles entering is equal to the number of vehicles leaving. Similarly in the network I, and I are the currents entering, I and I are leaving currents. a b c d I a + I b = I c + I d 5.9 Kirchhoff s Second Law - The Voltage Law, (KVL) Kirchhoff s voltage or loop law 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 sum of all voltages within the loop must be equal to zero. This concept is known as the "Conservation of Energy". 93

106 Circuit Theory and Basic Electronics Fig Sum of all the voltages V AB +V BC +V CD +V DA =0 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. Example 1 Find the current flowing in the 40Ω resistor, R3. Fig Circuit diagram 94

107 Solution 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 (1) At node B: I 3 = I 1 + I 2 (2) Using Kirchhoff s Voltage Law, KVL the equations are given as; Loop 1 is given as: 10 = R 1 x I 1 + R 3 x I 3 = 10I I 3 (3) Loop 2 is given as: 20 = R 2 x I 2 + R 3 x I 3 = 20I I 3 (4) Loop 3 is given as: = 10I 1 20 (5) I 2 as I 3 is the sum of I 1 + I 2 Let us rewrite the equations as; Eq. no (3) as 10 = 10I (I 1 + I 2 ) 10 = 50I I 2 (6) Eq. No (4): 20 = 20I (I 1 + I 2 ) 20 =40I I 2 (7) We now have two Simultaneous Equations that can be reduced to give us the value of both 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 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 95

108 Circuit Theory and Basic Electronics Summary Circuits containing an inductor L, a capacitor C, and a resistor R, have special characteristics useful in many applications. Their frequency characteristics (impedance, voltage, or current vs. frequency) have a sharp maximum or minimum at certain frequencies. These circuits can hence be used for selecting or rejecting specific frequencies and are also called tuning circuits. A series LCR circuit includes a series combination of an inductor, resistor and capacitor whereas; a parallel LCR circuit contains a parallel combination of inductor and capacitor with the resistance placed in series with the inductor. The selectivity of a tuned circuit is its ability to select a signal at the resonant frequency and reject other signals that are close to this frequency. A measure of the selectivity is Q, or the quality factor. The study of these circuits is basically an application of alternating current circuit analysis. The LCR circuit, which consists of an inductor L, a capacitor C, and a resistor R, all connected in series with a source. The point of intersection of V and V versus curves will give resonance condition. C L The Quality factor is defined in the same way as for a mechanical oscillator. The parallel resonant circuit obeys the same formula for resonant frequency as the series resonant one, but at resonance the parallel resonant circuit has very high impedance. The resistance at resonance offered by the parallel resonant circuit is very high if the resistance of the inductance is very small, and is known as the dynamic resistance. The impedance at resonance is called dynamic resistance. The current I= V/Z has minimum value. It is for this reason that the condition of resonance for a parallel LCR circuit is known as anti-resonance and the corresponding frequency as the anti-resonance frequency. The shape of the impedance versus frequency curve in a parallel LCR circuit is the same as the shape of the current versus frequency curve in a series LCR circuit. An alternating voltage is a sinusoidally varying voltage, where the peak is value and is the angular frequency of the voltage. Anti-resonance is the condition in a parallel LCR circuit when the impedance of the circuit is maximum and the current minimum is termed as anti-resonance. For a parallel LCR circuit the frequency, at which the current has minimum value, is called anti-resonant frequency. Ohm's law states that, the current through a conductor between two points is directly proportional to the potential difference or voltage across the two points and inversely proportional to the resistance between them. It opposes the flow of current through it. Resistance is called the physical property of a resistor by the virtue of which it gives an opposition to the current flow. Unit: resistance is measured in ohms (Ω) Capacitor is a device that stores energy. An inductor is a simple curl of windings that doesn t allow sudden changes in current. This serves as a source to create potential difference between two points. It has two types of sources. Kirchhoff s current law 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 is known as the "Conservation of Charge". Kirchhoff s voltage or loop law 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. 96

109 References Bakshi, U. A. & Bakshi, A. V., Electric Circuits. Technical Publications. Kumar, S. & Suresh, K. S. K., Electric Circuits and Networks. Pearson Education India. LCR Series Circuits. [Pdf] Available at: < pdf> [Accessed 3 June 2013]. Basic Circuit Element. [Pdf] Available at: < [Accessed 3 June 2013]. 13. LCR Circuits AC Voltage. [Video online] Available at: < [Accessed 3 June 2013]. 12. LCR Circuits DC Voltage. [Video online] Available at: < A2P1I> [Accessed 3 June 2013]. Recommended Reading Bakshi, K. A., Bakshi, A. V. & Bakshi, U. A., Electronic Measurement Systems. Technical Publications. Pilbeam, J., Fritz, W. & Arnold, M., FCS Electronic Control & Digital Electronics L3. Pearson South Africa. Mayergoyz, I. D., Basic Electric Circuit Theory: A One-semester Text. Gulf Professional Publishing. 97

110 Circuit Theory and Basic Electronics Self Assessment 1. Which of the following statements is true? a. The resistance at resonance offered by the series resonant circuit is very high if the resistance of the inductance is very small, and is known as the dynamic resistance. b. The resistance at resonance offered by the parallel resonant circuit is very low if the resistance of the inductance is very small, and is known as the dynamic resistance. c. The resistance at resonance offered by the parallel resonant circuit is very low if the resistance of the inductance is very large, and is known as the dynamic resistance. d. The resistance at resonance offered by the parallel resonant circuit is very high if the resistance of the inductance is very small, and is known as the dynamic resistance Which of the following statements is true? a. Series and parallel are the types of AC circuits. b. Series and parallel are the types of LCR circuits. c. Series and parallel are not the types of LCR circuits. d. Series and parallel are the types of circuits. Give the magnitude and phase of the impedance. a. and Ø= tan -1, respectively b. and Ø= tan 1, respectively c. and Ø= tan -1, respectively d. and Ø= tan 1, respectively 98

111 4. If I is the instantaneous current flowing through the circuit, the applied voltage in phasor in phasor form is given by V=V R +V L +V C = RI + jwli+ = = I The impedance Z= = R+ j if, Z=Ze i = Zcos +jzsinø then, Z= and tanø= therefore, current, I= = a. 99

112 Circuit Theory and Basic Electronics V=V R +V L +V C = RI + jwli+ = = I The impedance Z= = R+ if, Z=Ze i = Zcos +ZsinØ then, Z= and tanø= therefore, current, I= = b. 100

113 V=V R +V L +V C = RI + jwli+ = The impedance Z= = R+ j if, Z=Ze i = Zcos +jzsinø then, Z= and tanø= therefore, current, I= c. 101

114 Circuit Theory and Basic Electronics V=V R +V L +V C = RI + jwli+ = = I The impedance Z= = R+ j if, Z=Ze i then, Z= 5. Which of the following is the Series LCR circuit? a. 102

115 b. c. d. 103

116 Circuit Theory and Basic Electronics Which of the following statements is true? a. Kirchhoff s current law states that the algebraic difference of currents entering / leaving a node is zero. b. Kirchhoff s current law states that the algebraic product of currents entering / leaving a node is zero. c. Kirchhoff s current law states that the algebraic sum of currents entering / leaving a node is zero. d. Kirchhoff s current law states that the algebraic sum of currents entering / leaving a node is one. A path between two nodes is called the of the network. a. path b. c. d. loop node branch Which device stores energy? a. Resistor b. c. d. Capacitor Inductor Voltage source is a measure of the opposition offered by an electric circuit to the flow of electric current. a. Resistance b. c. d. Reactance Resonance Resonance curve 10. The property of a conductor that describes its ability to store electric charge is called. a. impedance b. inductance c. capacitance d. dynamic resistance 104

117 Chapter VI Theorems Aim The aim of this chapter is to: introduce network theory explain the concept of superposition theorem explicate the theory of superposition theorem Objectives The objectives of this chapter are to: explain the concept of Norton theorem explain the theory of max power theorem elucidate Laplace transform and its importance Learning outcome At the end of this chapter, you will be able to: understand the concept of superposition theorem describe theory of Thevenin s theorem understand Laplace transform and its property 105

118 Circuit Theory and Basic Electronics 6.1 Theorem - Superposition Theorem Definition and explanation of the Superposition theorem is discussed below Statements of Superposition Theorem Statement 1 The superposition principle states that, the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltage across (or currents through) that element due to each independent source acting alone. Statement 2 In any linear electrical circuit, any voltage or current value can be obtained by taking the individual contributions to that voltage or current as a result of each source taken alone and summing the contributions together. Statement 3 The superposition theorem states, that in a linear circuit with several sources, the current and voltage for any element in the circuit is the sum of the currents and voltages produced by each source acting independently Explanation of Superposition Theorem Superposition theorem eliminates one source of power in the circuit one at a time, using series/parallel analysis to determine voltage drops (and/or currents) within the modified network for each power source separately. Then, once voltage drops and/or currents have been determined for each power source working separately, the values are all superimposed on top of each other (added algebraically) to find the actual voltage drops/ currents with all sources active Importance of Superposition Theorem Network theorems provide insight into the behaviour and properties of electrical circuits. Superposition theorem is of theoretical importance, because it is fundamental to linear circuit analysis. A circuit is linear only when it behaves in accordance with superposition theorem. This theorem states that the linear responses in a circuit can be obtained as the algebraic sum of responses, due to each of the independent sources acting alone. This theorem defines the behaviour of a linear circuit. Within the context of linear circuit analysis, this theorem provides the basis for all other theorems. Given a linear circuit, it is easy to see how mesh analysis and nodal analysis make use of the principle of superposition Properties of Superposition Theorem There are two guiding properties of superposition theorem. These are: the property of homogeneity or proportionality the property of additivity Limitations of Superposition Theorem As stated earlier, the linear responses in a circuit can be obtained using this theorem as the algebraic sum of responses, due to each of the independent sources acting alone. Current and voltage associated with an element are linear responses. On the other hand, power in an element is not a linear response. It is a non-linear function, varying proportionately either with the square of voltage across the element or with the square of current through the element. 106

119 Hence, it is not possible to apply superposition theorem directly to determine power associated with an element. In addition, application of superposition theorem does not normally lead to simplification of analysis. It is not the best technique to determine all currents and voltages in a circuit, driven by multiple of sources. 6.2 Relationship with Mesh and Nodal Analysis Superposition theorem is valid for linear circuits and analysis of linear circuits is relatively easy. On the other hand, the principle of superposition is not valid for non-linear circuits. And the analysis of non-linear circuits is quite complex and difficult. It is possible to apply mesh and nodal analysis to nonlinear circuits. However, within the context of linear circuits, mesh or nodal analysis of a circuit illustrates how the principle of superposition is ever so pervasive in defining the behaviour of linear circuits. 6.3 Proportionality in Elements A linear circuit consists of linear elements. The passive elements, the dependent sources and the independent sources used in a linear circuit are linear. Let us how linearity is defined for each type of element. A resistor is linear. V S = R x I Fig. 6.1 Linearity using resistor In the case of a resistor, the voltage across a resistor varies proportionally with its current. The ratio of voltage to current is resistance. Power in a resistor varies with the square of its voltage or it is current. 107

120 Circuit Theory and Basic Electronics Fig. 6.2 Linearity using inductor An inductor is linear. λ = L x i v S = = L In the case of an inductor, the linearity is between the flux linkage and the current. The ratio of flux linkage to current is inductance. The product of inductance and the rate of change of current is the inductor voltage. In an ideal inductor, the core of inductor does not get saturated. In an inductor with a magnetic core, saturation of flux occurs at some level, depending on the material used for the core. Fig. 6.3 Linearity using capacitor A capacitor is linear q = C x v C i = = C In the case of a capacitor, the linearity is between the charge stored and the capacitor voltage. The ratio of charge to voltage is capacitance. The product of capacitance and the rate of change of voltage is the capacitor current. 108

121 In an ideal capacitor, the capacitor voltage can be very high. In a real capacitor, the dielectric within the capacitor breaks down a magnetic core at some level, depending on the dielectric. 6.4 Linear Dependent Sources Fig. 6.4 Linear independent sources In the case of dependent source, the output variable varies proportionately with the controlling variable. In the case of ideal independent source, linearity implies that a voltage can supply any current at constant voltage, and that a current source can sustain any load voltage while supplying constant current. The following are the types of dependant sources: voltage-dependent voltage source current-dependent voltage source voltage- dependent current source current- dependent current source 6.5 Additivity Property Fig. 6.5 Principle of additivity 109

122 Circuit Theory and Basic Electronics The principle of additivity can be explained with the help of a sketch. When there are two sources in a circuit, the contribution due to each source can be found out separately and the response due to both sources is the algebraic sum of contributions due to each source acting alone. This aspect is illustrated by the circuit shown above in the Fig The current through the resistor is the sum of currents due to source V and source V, acting alone. 1 2 When the contribution due to source V is to be calculated, the contribution due to source V should be zero. 1 2 Hence source V 2 can be replaced by a short circuit. When source V is replaced by a short circuit, its contribution has to be nil. 2 Similarly, when the contribution due to source V is to be calculated, the contribution due to source V should 2 1 be zero. Hence source V 1 can be replaced by a short circuit. The relevant circuits are shown above. 6.6 Application of Superposition Theorem If the direct application of superposition theorem is not easy, the question arises when it is suitable to use superposition theorem. It is best to use superposition theorem to find a particular current or voltage in a circuit, when the circuit has multiple independent sources. This theorem states that the linear responses in a circuit can be obtained as the algebraic sum of responses, due to each of the independent sources acting alone. A voltage source that makes no contribution is replaced by a short circuit whereas, a current source that makes no contribution is replaced by an open-circuit. The internal resistance of the source is left in the circuit, as it is where it is. The worked examples are used to show how to apply superposition theorem to circuits Steps to Apply Superposition Theorem Turn off all independent sources except one source. Find the output (voltage or current) due to that active source. Repeat step 1 for each of the other independent source. Find the total contribution by adding algebraically all the contributions due to the independent source. 110

123 6.6.2 Examples of Supersposition Theorem Example 1: Find value of V from figure given below: Example 2: Fig. 6.6 Circuit using superposition theorem Solution: Let, V = V 1 + V 2 To find V1: i 2 =0 Fig. 6.7 i 2 =0 KVL : 12 i 1-6 = 0 i 1 = 0.5 A V 1 = 4 i 1 = 2V Voltage division: V 1 = To find V 2 : 2V 111

124 Circuit Theory and Basic Electronics Current division: Fig. 6.8 v 1 =0 v 1 shortcut i 3 = =2A V 2 = 4 i 3 = 8V The value of V: V= V 1 + V 2 = = 10V Fig. 6.9 Example for superposition theorem Solution: A simple circuit is used to illustrate how the principle of superposition can be used to obtain the current through the resistor in the circuit shown in above figure. Here, Fig Illustration of superposition theorem I 1 = I 2 = - When superposition theorem is used, the response due to only one independent source is obtained at a time. The other sources are replaced, by either open-circuits or short-circuits, as the case may be. In this circuit, there are two sources, voltage V and voltage V

125 When response due to source V is calculated, source V is replaced by a short-circuit. 1 2 Let the current through the resistor be I1, as shown in Fig When response due to source V source V 1 is replaced by a short-circuit. 2 Let the current through the resistor be I. 2 V R = V 1 - V 2 (1) I= = (2) I= (3) is calculated, Without using superposition theorem, it can be stated that the voltage across the resistor is as shown by equation (1). Then we can express the current through the resistor by equation (2), and it can be split into two fractions, as shown by equation (3). We see next how we can apply superposition theorem. I= I 1 + I 2 = (4) The expressions for currents I and I are marked in Fig Using superposition theorem, the current through the resistor is obtained as the sum of currents I and I. 1 2 Equation (4) is the same as the previous equation obtained for current through the resistor. It is seen that the total current in the resistor is expressed as the sum of two responses, due to each source acting alone. That is, when the response due to voltage source V is to be calculated, the contribution of voltage source V 1 12 should be zero and then source V is replaced by a short circuit. A voltage source with zero volts allows current to pass through it, and does not contribute anything to a circuit. That is, when contribution of an ideal voltage source is to be zero, it can be 2 replaced by a short circuit. Similarly, when the response due to source V is to be calculated, the contribution of source V should be zero 1 1 and it is replaced by a short circuit. Equation (4) is a linear equation. In this equation, V and V are the independent variables Thevenin s Theorem Statement - Thevenin s theorem states that a linear two terminal circuit can be replaced by an equivalent circuit consisting of a voltage source V Th in series with a resistor R Th,where V Th is the open circuit voltage at terminals a & b and R Th is the input or equivalent resistance at the terminals when the independent sources are turned off Explanation of Thevenin s Theorem Thevenin s theorem is a popular theorem, used often for analysis of electronic circuits. Its theoretical value is due to the insight it offers about the circuit. This theorem states that a linear circuit containing one or more sources and other linear elements can be represented by a voltage source and a resistance. Using this theorem, a model of the circuit can be developed based on its output characteristic. Let us try to find out what Thevenin s theorem is by using an investigative approach. 113

126 Circuit Theory and Basic Electronics Thevenin s Equivalent Circuit Fig Thevenin s theorem-introduction In this section, the model of a circuit is derived based on its output characteristics. Let a circuit be represented by a box, as shown in Fig Its output characteristic is also displayed. As the load resistor is varied, the load current varies. The load current is bounded between two limits, zero and I, and the load voltage is bounded between limits, m E Volts and zero volts. When the load resistor is infinite, it is an open circuit. In this case, the load voltage is at its highest, which is E volts and the load current is zero. This is the point at which the output characteristic intersects with the Y axis. When the load resistor is of zero value, there is a short circuit across the output terminals of the circuit and in this instance, the load current is maximum, specified as I m and the load voltage is zero. It is the point at which the output characteristic intersects with the X axis. Fig Circuit model R TH = The circuit in Fig reflects the output characteristic, displayed in Fig

127 It has an output of E volts, when the load current is zero. Hence the model of the circuit can have a voltage source of E volts. When the output terminals are short circuited, it can be stated that the internal resistance of circuit absorbs E volts at a current of I m. This means that the internal resistance of the circuit, called as R, has a value of E over I, as shown by the TH m equation mentioned above. Hence the circuit model consists of a voltage source of value E volts and a resistor R. This resistor is the resistance of the circuit, as viewed from the load terminals. Let us see how we can apply what we have learnt. A simple circuit is presented in Fig Fig Thevenin s theorem The task is to get an expression for the load current I and express it in terms of Thevenin s voltage and Thevenin s L resistance. Thevenin s voltage is the voltage obtained across the load terminals, with the load resistor removed. In this case, the load resistor is named as R Example of Thevenin's theorem Example 3: Find the equivalent resistance seen by the voltage source. Find current delivered by it. Use current division rule to find current through R 3 due to voltage source. Fig Example of Thevenin s theorem 115

128 Circuit Theory and Basic Electronics Solution: At first, an expression for the load current is obtained without the use of Thevenin s theorem. To get the load current, the steps involved are as follows: Get an expression for the equivalent resistance R, seen by the source, as shown in Fig Divide the source voltage by the equivalent resistance to get current Is supplied by the source. These source current flows through resistors, R2 and R3 connected in parallel. Use the current division rule to get an expression for the load current. Req = R 1 + (1) I s = (2) I L = (3) As shown by equation (1), the equivalent resistance is obtained by adding resistor R to the parallel value of 1 resistors, R 2 and R 3. The source current is the ratio of source voltage to the equivalent resistance, as expressed by equation (2). Then the load current through resistor R3 is obtained using the current division rule, as shown by equation (3). I L = V S. =V S. (4) = = (5) Now some mathematical manipulations are required to get Thevenin s voltage and Thevenin s resistance. The expression for the load current is expressed by equation (4). Divide both the numerator and the denominator of equation (4) by the sum of resistors, R and R2, and then 1 we get equation (5). The numerator of equation (5) is Thevenin s voltage. The first part of the denominator, containing resistors R and R, is Thevenin s resistance

129 6.7.4 Formal Presentation of the Thevenin s Theorem Fig A network and its Thevenin s equivalent Thevenin s theorem represents a linear network by an equivalent circuit. Let a network with one or more sources supply power to a load resistor as shown in Fig Thevenin s theorem states that the network can be replaced by a single equivalent voltage source, marked as Thevenin s voltage or open-circuit voltage and a resistor marked as Thevenin s resistance. Proof of this theorem is presented below. Thevenin s theorem can be applied to linear networks only. Thevenin s voltage is the algebraic sum of voltages across the load terminals, due to each of the independent sources in the circuit, acting alone. It can be seen that Thevenin s theorem is an outcome of superposition theorem. Thevenin s Theorem equivalent circuit contains: Open- circuit voltage V TH Thevenin s resistance RTH Thevenin s equivalent circuit consists of Thevenin s voltage and Thevenin s resistance. Thevenin s voltage is also referred to as the open-circuit voltage, meaning that it is obtained across the load terminals without any load connected to them. The load is replaced by an open-circuit and hence Thevenin s voltage is called the open-circuit voltage. Fig Thevenin s voltage Fig shows how Thevenin s voltage is to be obtained. 117

130 Circuit Theory and Basic Electronics Here it is assumed that we have a resistive circuit with one or more sources. As shown in Fig. 6.18, Thevenin s voltage is the open-circuit voltage across the load terminals. The voltage obtained across the load terminals without the load being connected is the open-circuit voltage. This open-circuit voltage can be obtained as the algebraic sum of voltages, due to each of the independent sources acting alone. Given a circuit, Thevenin s voltage can be obtained as outlined below To Get Open-circuit Voltage V TH Fig Thevenin s resistance Fig shows how Thevenin s resistance is to be obtained. Thevenin s resistance is the resistance as seen from the load terminals. To obtain this resistance, replace each independent ideal voltage source in the network by a short circuit, and replace each independent ideal current source by an open circuit. If a source is not ideal, only the ideal part of that source is replaced by either a short circuit or an open circuit, as the case may be. The internal resistance of the source, reflecting the non ideal aspect of the circuit, is left in the circuit, as it is where it is. A voltage source is connected across the load terminals. Then Thevenin s resistance is the ratio of this source voltage to its current. A few examples are presented after this page to illustrate the use of Thevenin s theorem. 6.8 Norton Theorem Norton s Theorem states that, any resistive circuit or network, no matter how complex, can be represented as a current source in parallel with a source resistance. 118