EE6201 Circuit Theory Regulation 2013 Lecture Notes

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EE6201 - Circuit Theory - Regulation 2013 - Lecture Notes by Vidyarthiplus Team - www.Vidyarthiplus.com

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UNIT I BASIC CIRCUITS ANALYSIS

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INTRODUCTION

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BASIC ELEMENTS & INTRODUCTORY CONCEPTS

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The interconnection of various electric elements in a prescribed manner comprises as an electric circuit in order to perform a desired function. The electric elements include controlled and uncontrolled source of energy, resistors, capacitors, inductors, etc. Analysis of electric circuits refers to computations required to determine the unknown quantities such as voltage, current and power associated with one or more elements in the circuit. To contribute to the solution of engineering problems one must acquire the basic knowledge of electric circuit analysis and laws. Many other systems, like mechanical, hydraulic, thermal, magnetic and power system are easy to analyze and model by a circuit. To learn how to analyze the models of these systems, first one needs to learn the techniques of circuit analysis. We shall discuss briefly some of the basic circuit elements and the laws that will help us to develop the background of subject.

Electrical Network: A combination of various electric elements (Resistor, Inductor, Capacitor, Voltage source, Current source) connected in any manner what so ever is called an electrical network. We may classify circuit elements in two categories, passive and active elements.

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Passive Element: The element which receives energy (or absorbs energy) and then either converts it into heat (R) or stored it in an electric (C) or magnetic (L ) field is called passive element.

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Active Element: The elements that supply energy to the circuit is called active element. Examples of active elements include voltage and current sources, generators, and electronic devices that require power supplies. A transistor is an active circuit element, meaning that it can amplify power of a signal. On the other hand, transformer is not an active element because it does not amplify the power level and power remains same both in primary and secondary sides. Transformer is an example of passive element.

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Bilateral Element: Conduction of current in both directions in an element (example: Resistance; Inductance; Capacitance) with same magnitude is termed as bilateral element.

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Unilateral Element: Conduction of current in one direction is termed as unilateral (example: Diode, Transistor) element.

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Linear and Nonlinear Circuits

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Meaning of Response: An application of input signal to the system will produce an output signal, the behavior of output signal with time is known as the response of the system

Non-Linear Circuit: Roughly speaking, a non-linear system is that whose parameters change with voltage or current. More specifically, non-linear circuit

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does not obey the homogeneity and additive properties. Volt-ampere characteristics of linear and non-linear elements are shown in figs. 3.2 - 3.3. In fact, a circuit is linear if and only if its input and output can be related by a straight line passing through the origin as shown in fig.3.2. Otherwise, it is a nonlinear system.

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KIRCHHOFF’S LAWS

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Potential Energy Difference: The voltage or potential energy difference between two points in an electric circuit is the amount of energy required to move a unit charge between the two points.

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Kirchhoff’s laws are basic analytical tools in order to obtain the solutions of currents and voltages for any electric circuit; whether it is supplied from a direct-current system or an alternating current system. But with complex circuits the equations connecting the currents and voltages may become so numerous that much tedious algebraic work is involve in their solutions.

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Elements that generally encounter in an electric circuit can be interconnected in various possible ways. Before discussing the basic analytical tools that determine the currents and voltages at different parts of the circuit, some basic definition of the following terms are considered.

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Meaning of Circuit Ground and the Voltages referenced to Ground

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In many cases, such as in electronic circuits, the chassis is shorted to the earth itself for safety reasons.

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Voltage Divider

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Current divider

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Potentiometer and its function

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Ideal and Practical Voltage Sources

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Ideal and Practical Current Sources

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Another two-terminal element of common use in circuit modeling is current source` as depicted in fig.3.17. An ideal current source, which is represented by a model in fig. 3.17(a), is a device that delivers a constant current to any load resistance connected across it, no matter what the terminal voltage is developed across the load (i.e., independent of the voltage across its terminals across the terminals).

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Conversion of a Practical Voltage Source to a Practical Current source and viseversa

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Current source to Voltage Source

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Solution of Electric Circuit Based on Mesh (Loop) Current Method

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Let us consider a simple dc network as shown in Figure 4.1 to find the currents through different branches using Mesh (Loop) current method.

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Solution of Electric Circuit Based on Node Voltage Method

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UNIT II

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NETWORK REDUCTION AND NETWORK THEOREMS FOR DC AND AC CIRCUITS

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Delta – Star conversion

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Conversion from Delta to Star

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Conversion from Star to Delta

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SOLUTION:

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SUPERPOSITION THEOREM

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INTRODUCTION

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Statement of superposition theorem

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In any linear bilateral network containing two or more independent sources (voltage or current sources or combination of voltage and current sources), the resultant current / voltage in any branch is the algebraic sum of currents / voltages caused by each independent sources acting along, with all other independent sources being replaced meanwhile by their respective internal resistances.

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Superposition theorem can be explained through a simple resistive network as shown in fig.7.1 and it has two independent practical voltage sources and one practical current source.

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Limitations of superposition Theorem

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• “ B power calculations involve either the product of voltage and current, the square of current or the square of the voltage, they are not linear operations. This statement can be explained with a simple example as given below.

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Thevenin’s and Norton’s theorems

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The procedure for applying Thevenin’s theorem

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Maximum Power Transfer Theorem

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Remarks: The Thevenin equivalent circuit is useful in finding the maximum power that a linear circuit can deliver to a load.

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Proof of Thevenin Theorem

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The basic concept of this theorem and its proof are based on the principle of superposition theorem. Let us consider a linear system in fig.L.8.8(a).

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Norton's Theorem

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Norton's Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single current source and parallel resistance connected to a load. Just as with Thevenin's Theorem, the qualification of “linear” is identical to that found in the Superposition Theorem: all underlying equations must be linear (no exponents or roots). Contrasting our original example circuit against the Norton equivalent: it looks something like this:

. . . after Norton conversion . . .

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Remember that a current source is a component whose job is to provide a constant amount of current, outputting as much or as little voltage necessary to maintain that constant current.

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As with Thevenin's Theorem, everything in the original circuit except the load resistance has been reduced to an equivalent circuit that is simpler to analyze. Also similar to Thevenin's Theorem are the steps used in Norton's Theorem to calculate the Norton source current (INorton) and Norton resistance (RNorton).

As before, the first step is to identify the load resistance and remove it from the original circuit:

Then, to find the Norton current (for the current source in the Norton equivalent circuit), place a direct wire (short) connection between the load points and determine the resultant current. Note that this step is exactly opposite the respective step in Thevenin's Theorem, where we replaced the load resistor with a break (open circuit):

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With zero voltage dropped between the load resistor connection points, the current through R 1 is strictly a function of B1's voltage and R1's resistance: 7 amps (I=E/R). Likewise, the current through R3 is now strictly a function of B2's voltage and R3's resistance: 7 amps (I=E/R). The total current through the short between the load connection points is the sum of these two currents: 7 amps + 7 amps = 14 amps. This figure of 14 amps becomes the Norton source current (INorton) in our equivalent circuit:

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Remember, the arrow notation for a current source points in the direction opposite that of electron flow. Again, apologies for the confusion. For better or for worse, this is standard electronic symbol notation. Blame Mr. Franklin again! To calculate the Norton resistance (RNorton), we do the exact same thing as we did for calculating Thevenin resistance (RThevenin): take the original circuit (with the load resistor still removed), remove the power sources (in the same style as we did with the Superposition Theorem: voltage sources replaced with wires and current sources replaced with breaks), and figure total resistance from one load connection point to the other:

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Now our Norton equivalent circuit looks like this:

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If we re-connect our original load resistance of 2 simple parallel arrangement:

, we can analyze the Norton circuit as a

    

REVIEW: Norton's Theorem is a way to reduce a network to an equivalent circuit composed of a single current source, parallel resistance, and parallel load. Steps to follow for Norton's Theorem: (1) Find the Norton source current by removing the load resistor from the original circuit and calculating current through a short (wire) jumping across the open connection points where the load resistor used to be. (2) Find the Norton resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open) and calculating total resistance between the open connection points. (3) Draw the Norton equivalent circuit, with the Norton current source in parallel with the Norton resistance. The load resistor re-attaches between the two open points of the equivalent circuit. (4) Analyze voltage and current for the load resistor following the rules for parallel circuits

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As with the Thevenin equivalent circuit, the only useful information from this analysis is the voltage and current values for R2; the rest of the information is irrelevant to the original circuit. However, the same advantages seen with Thevenin's Theorem apply to Norton's as well: if we wish to analyze load resistor voltage and current over several different values of load resistance, we can use the Norton equivalent circuit again and again, applying nothing more complex than simple parallel circuit analysis to determine what's happening with each trial load.



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Nortons Theorem In some ways Norton's Theorem can be thought of as the opposite to "Thevenins Theorem", in that Thevenin

reduces his circuit down to a single resistance in series with a single voltage. Norton on the other hand reduces his

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circuit down to a single resistance in parallel with a constant current source. Nortons Theorem states that "Any linear circuit containing several energy sources and resistances can be replaced by a single Constant Current generator in parallel with a Single Resistor". As far as the load resistance, RL is concerned this single resistance, RS is the value of the resistance looking back into the network with all the current sources open circuited and IS is the short circuit

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current at the output terminals as shown below.

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Nortons equivalent circuit.

The value of this "constant current" is one which would flow if the two output terminals where shorted together while

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the source resistance would be measured looking back into the terminals, (the same as Thevenin).

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For example, consider our now familiar circuit from the previous section.

To find the Nortons equivalent of the above circuit we firstly have to remove the centre 40

load resistor and short

out the terminals A and B to give us the following circuit.

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When the terminals A and B are shorted together the two resistors are connected in parallel across their two respective voltage sources and the currents flowing through each resistor as well as the total short circuit current can

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now be calculated as:

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with A-B Shorted Out

If we short-out the two voltage sources and open circuit terminals A and B, the two resistors are now effectively connected together in parallel. The value of the internal resistor Rs is found by calculating the total resistance at the

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terminals A and B giving us the following circuit.

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Find the Equivalent Resistance (Rs)

Having found both the short circuit current, Is and equivalent internal resistance, Rs this then gives us the following

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Nortons equivalent circuit.

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Nortons equivalent circuit.

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as shown below.

load resistor connected across terminals A and B

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Ok, so far so good, but we now have to solve with the original 40

Again, the two resistors are connected in parallel across the terminals A and B which gives us a total resistance of:

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load resistor can be found as:

which again, is the same value of 0.286

Nortons Theorem Summary

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Then the current flowing in the 40

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The voltage across the terminals A and B with the load resistor connected is given as:

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amps, we found using Kirchoff´s circuit law in the previous tutorials.

The basic procedure for solving a circuit using Nortons Theorem is as follows: 1. Remove the load resistor RL or component concerned.



2. Find RS by shorting all voltage sources or by open circuiting all the current sources.



3. Find IS by placing a shorting link on the output terminals A and B.



4. Find the current flowing through the load resistor RL.

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

In a circuit, power supplied to the load is at its maximum when the load resistance is equal to the source resistance. In the next tutorial we will look at Maximum Power Transfer. The application of the maximum power transfer theorem can be applied to either simple and complicated linear circuits having a variable load and is used to find the

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load resistance that leads to transfer of maximum power to the load.

Example-L.8.5 For the circuit shown in fig.8.10(a), find the current through resistor ( branch) using Norton’s theorem & hence calculate the voltage across the current source (). 21LRR== abI−cgV Solution: Step-1: Remove the resistor through which the current is to be found and short the terminals ‘a’ and ‘b’ (see fig.8.10(b)).

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4

1

2

2

R I = 3 + R I = 3 – 2 × 2 = - 1 慰 I = - 0.5A 4 1

4 2

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Step-2: Any method can be adopted to compute the current flowing through the a-b branch. Here, we apply ‘mesh – current’ method. Loop-1 3 – R (I – I ) = 0, where I = - 2A

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Loop-3 133323333- RI-R(I-I)=0- 3I-4(I+2)=0- 7I-8=08 I=-=7 N138-7+1I=(I-I)=-0.5+=714僅巾慰勤錦均斤 9A14=(current is flowing from ‘a’ to ‘b’) Step-3: To compute R , all sources are replaced with their internal resistances. The equivalent

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resistance between ‘a’ and ‘b’ terminals is same as the value of Thevenin’s resistance of the circuit shown in fig.8.3(d).

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Step-4: Replace the original circuit with an equivalent Norton’s circuit as shown in fig.8.10(d).

R1.555I=×I=×0.643=0.39A (a to b)R+R1.555+1 In order to calculate the voltage across the current source the following procedures are adopted. Redraw the original circuit indicating the current direction in the load.

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V=3-1×0.39=2.61volt2.61 I==1.305A2 I=1.305-0.39=0.915A ('c' to 'b') V=2×1.305+4×.915=6.26volt ('c' is higher potential than 'g')慰

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L.8.8 Test Your Understanding [Marks: 60] T.1 When a complicated dc circuit is replaced by a Thevenin equivalent circuit, it consists of one ------ in series with one --------- . [2]

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T.2 When a complicated dc circuit is replaced by a Norton equivalent circuit, it consists of ------ in ---- with one -------. [2] T.3 The dual of a voltage source is a -----------. [1] T.4 When a Thevenin theorem is applied to a network containing a current source; the current source is eliminated by --------- it. [1] T.5 When applying Norton’s theorem, the Norton current is determined with the output terminals -------------, but the Norton resistance is found with the output terminals ---------.and subsequently all the independent sources are replaced -----------. [3] T.6 For a complicated circuit, the Thevenin resistance is found by the ratio of -------- voltage and ----------- current. [2] T.7 A network delivers maximum power to the load when its -------- is equal to the -------- of circuit at the output terminals. [2] T.8 The maximum power transfer condition is meaningful in ------------ and --------- systems. [2] T.9 Under maximum power transfer conditions, the efficiency of the system is only --------- %. [1]

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UNIT III

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RESONANCE AND COUPLED CIRCUITS

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Ideal Transformer

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In this lesson, we shall study two winding ideal transformer, its properties and working principle under no load condition as well as under load condition. Induced voltages in primary and secondary are obtained, clearly identifying the factors on which they depend upon. The ratio between the primary and secondary voltages are shown to depend on ratio of turns of the two windings. At the end, how to draw phasor diagram under no load and load conditions, are explained. Importance of studying such a transformer will be highlighted. At the end, several objective type and numerical problems have been given for solving. Key Words: Magnetising current, HV & LV windings, no load phasor diagram, reflected current, equivalent circuit

23.2 Introduction

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Transformers are one of the most important components of any power system. It basically changes the level of voltages from one value to the other at constant frequency. Being a static machine the efficiency of a transformer could be as high as 99%. Big generating stations are located at hundreds or more km away from the load center (where the power will be actually consumed). Long transmission lines carry the power to the load centre from the generating stations. Generator is a rotating machines and the level of voltage at which it generates power is limited to several kilo volts only –

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UNIT IV

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TRANSIENT RESPONSE FOR DC CIRCUITS

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