Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits By: FARHAD FARADJI, Ph.D. Assistant Professor, Electrical Engineering, K.N. Toosi University of Technology http://wp.kntu.ac.ir/faradji/electriccircuits1.htm Reference: ELECTRIC CIRCUITS, J.W. Nilsson, S.A. Riedel, 10 th edition, 2015.

Chapter Contents 7.0. Introduction 7.1. The Natural Response of an RL Circuit 7.2. The Natural Response of an RC Circuit 7.3. The Step Response of RL and RC Circuits 7.4. A General Solution for Step and Natural Responses 7.5. Sequential Switching 7.6. The Integrating Amplifier Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 2

7.0. Introduction o Inductors and capacitors are able to store energy. o We now determine currents and voltages that arise when energy is either released or acquired by an inductor or capacitor in response to an abrupt change in a dc voltage or current source. o In this chapter, we will focus on circuits that consist only of sources, resistors, and either (but not both) inductors or capacitors. o Such circuits are called RL (resistor-inductor) or RC (resistor-capacitor) circuits. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 3

7.0. Introduction o Our analysis of RL and RC circuits will be divided into 3 phases. o In 1 st phase, we consider currents and voltages that arise when stored energy in an L or C is suddenly released to a resistive network. o This happens when L or C is abruptly disconnected from its dc source. o We can reduce circuit to one of 2 equivalent forms shown. o Currents and voltages that arise in this configuration are referred to as natural response of circuit. o Since nature of circuit itself, not external sources, determines its behavior. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 4

7.0. Introduction o In 2 nd phase, we consider currents and voltages that arise when energy is being acquired by an L or C due to sudden application of a dc voltage or current source. o This response is referred to as step response. o Process for finding both natural and step responses is the same. o In 3 rd phase, we develop a general method that can be used to find response of RL and RC circuits to any abrupt change in a dc voltage or current source. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 5

7.0. Introduction o Figure shows 4 possibilities for general configuration of RL and RC circuits. o When there are no independent sources in circuit: Thevenin voltage or Norton current is 0. Circuit reduces to one of those shown below. We have a natural-response problem. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 6

7.0. Introduction o RL and RC circuits are also known as 1 st order circuits. o Their voltages and currents are described by 1 st order differential equations. o No matter how complex a circuit may appear. o If circuit can be reduced to a Thevenin or Norton equivalent connected to an equivalent L or C, it is a 1 st order circuit. o If multiple L s or C s exist in original circuit, they must be interconnected so that they can be replaced by a single equivalent element. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 7

7.1. The Natural Response of an RL Circuit o We assume that independent current source generates a dc current of I s A. o Switch has been in a closed position for a long time. o We define phrase a long time more accurately later in this section. o For now, it means that all currents and voltages have reached a constant value. o Only constant (dc) currents can exist just prior to switch's being opened. o L appears as a short circuit (Ldi/dt = 0) prior to release of stored energy. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 9

7.1. The Natural Response of an RL Circuit Because L appears as a short circuit, voltage across it is 0; there can be no current in either R 0 or R; all source current I s appears in L. Finding natural response requires finding voltage and current at terminals of R after switch has been opened, i.e., after source has been disconnected and L begins releasing energy. If we let t = 0 denote the instant when switch is opened, the problem becomes one of finding v(t) and i(t) for t 0. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 10

7.1. The Natural Response of an RL Circuit Deriving Expression for Current o Equation is an ordinary 1 st order differential equation with constant coefficients. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 11

7.1. The Natural Response of an RL Circuit Deriving Expression for Current o An instantaneous change of current cannot occur in L. o In the 1 st instant after switch has been opened, current in L remains unchanged. o If we use 0 - to denote time just prior to switching, and 0 + for time immediately following switching. o Initial current in L is oriented in same direction as direction of i. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 12

7.1. The Natural Response of an RL Circuit Deriving Expression for Current o i(t) starts from an initialvalue I 0 and decreases exponentially toward 0 as t increases. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 13

7.1. The Natural Response of an RL Circuit Deriving Expression for Current Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 14

7.1. The Natural Response of an RL Circuit Deriving Expression for Current v is defined only for t > 0, not at t = 0. A step change occurs in v at t = 0. For t < 0, di/dt = 0, so v = Ldi/dt = 0. v at t = 0 is unknown. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 15

7.1. The Natural Response of an RL Circuit Deriving Expression for Current As t becomes infinite, energy dissipated in R approaches initial energy stored in L. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 16

7.1. The Natural Response of an RL Circuit Time Constant Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 17

7.1. The Natural Response of an RL Circuit Time Constant o Time constant is an important parameter for the 1 st order circuits. o Several of its characteristics is worthwhile. o First, it is convenient to think of time elapsed after switching in terms of multiples of. o One time constant after L has begun to release its stored energy to R, current has been reduced to e -1, or approximately 0.37 of its initial value. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 18

7.1. The Natural Response of an RL Circuit Time Constant o When elapsed time exceeds 5, current is less than 1% of its initial value. o We sometimes say that after 5, currents and voltages have, for most practical purposes, reached their final values. o For single circuits (1 st order circuits) with 1% error, phrase a long time implies that 5 or more have elapsed. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 19

7.1. The Natural Response of an RL Circuit Time Constant o Existence of current in RL circuit shown is a momentary event and referred to as transient response of circuit. o Response that exists a long time after switching has taken place is called steady-state response. o Phrase a long time then also means the time it takes circuit to reach its steady-state value. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 20

7.1. The Natural Response of an RL Circuit Time Constant o Any 1 st order circuit is characterized, in part, by value of its. o If we have no method for calculating of such a circuit, perhaps because we don't know values of its components, we can determine its value from a plot of circuit's natural response. o gives time required for i to reach its final value if i continues to change at its initial rate: o Assume that i continues to change at this rate: o i would reach its final value of 0 in seconds. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 21

7.1. The Natural Response of an RL Circuit o Calculating natural response of an RL circuit is summarized as follows: 1. Find initial current, I 0, through inductor. 2. Find time constant of circuit, = L/R. 3. Use equation I 0 e -t/, to generate i(t) from I 0 and. o All other calculations of interest follow from knowing i(t). Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 22

7.1. The Natural Response of an RL Circuit o Switch has been closed for a long time prior to t = 0. o Voltage across L must be 0 at t = 0 -. o i L (0 - ) = 20 A o i L (0 + ) = 20 A o We replace resistive circuit connected to terminals of L with a single resistor of 10 : Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 23

7.1. The Natural Response of an RL Circuit Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 24

7.1. The Natural Response of an RL Circuit Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 25

7.1. The Natural Response of an RL Circuit Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 26

7.1. The Natural Response of an RL Circuit Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 27

7.1. The Natural Response of an RL Circuit o This result is difference between initially stored energy (320 J) and energy trapped in inductors (32 J). o L eq for parallel inductors predicting terminal behavior of parallel combination has an initial energy of 288 J. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 28

7.1. The Natural Response of an RL Circuit o Energy stored in L eq represents amount of energy that will be delivered to resistive network at terminals of original inductors. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 29

7.2. The Natural Response of an RC Circuit o Natural response of an RC circuit is developed from the circuit shown. o Switch has been in position a for a long time, allowing loop made up of dc voltage source V g, R 1,and C to reach a steady-state condition. o A capacitor behaves as an open circuit in presence of a constant voltage. o Voltage source cannot sustain a current. o Source voltage appears across capacitor terminals. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 31

7.2. The Natural Response of an RC Circuit o At t = 0, when switch is moved from position a to position b, voltage on C is V g. o Because there can be no instantaneous change in voltage at terminals C, problem reduces to solving the circuit shown below. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 32

7.2. The Natural Response of an RC Circuit Deriving the Expression for the Voltage o We can easily find voltage v(t) by thinking in terms of node voltages: Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 33

7.2. The Natural Response of an RC Circuit Deriving the Expression for the Voltage o Natural response of an RC circuit is an exponential decay of initial voltage. o Time constant RC governs rate of decay. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 34

7.2. The Natural Response of an RC Circuit Deriving the Expression for the Voltage Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 35

7.2. The Natural Response of an RC Circuit o Calculating natural response of an RC circuit is summarized as follows: 1. Find initial current, V 0, across capacitor. 2. Find time constant of circuit, = RC. 3. Use equation V 0 e -t/, to generate v(t) from V 0 and. o All other calculations of interest follow from knowing v(t). Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 36

7.2. The Natural Response of an RC Circuit o Switch has been in position x for a longtime. o C will charge to 100 V and be positive at upper terminal. o We can replace resistive network connected to C at t = 0 + with an equivalent resistance of 80 k. o = (0.5 10-6 )(80 10 3 ) = 40 ms Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 37

7.2. The Natural Response of an RC Circuit Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 38

7.2. The Natural Response of an RC Circuit o Once we know v(t), we can obtain i(t) from Ohm's law. o After determining i(t) we can calculate v 1 (t) and v 2 (t). o To find v(t), we replace series-connected C s with a C eq.. o C eq has a capacitance of 4 F and is charged to a voltage of 20 V. o Circuit reduces to one shown below. o = (4)(250) 10-3 = 1 s. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 39

7.2. The Natural Response of an RC Circuit Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 40

7.2. The Natural Response of an RC Circuit o Initial Energy stored in C 1 and C 2 : Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 41

7.2. The Natural Response of an RC Circuit o Energy delivered to R: Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 42

7.2. The Natural Response of an RC Circuit o Energy stored in C eq is 800 J. o Because C eq predicts terminal behavior of original series-connected capacitors, energy stored in C eq is energy delivered to R. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 43

7.3. Step Response of RL and RC Circuits o We now find currents and voltages generated in the 1 st order RL or RC circuits when either dc voltage or current sources are suddenly applied. o Response of a circuit to sudden application of a constant voltage or current source is referred to as step response of circuit. o We show how circuit responds when energy is being stored in L or C. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 45

7.3. Step Response of RL and RC Circuits The Step Response of an RL Circuit o Energy stored in L when switch is closed is given in terms of a nonzero initial current i(0). o Task is to find expressions for i(t) and v(t) after switch has been closed. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 46

7.3. Step Response of RL and RC Circuits The Step Response of an RL Circuit Algebraic sign of I 0 is positive if I 0 is in same direction as i. Otherwise, I 0 carries a negative sign. When initial energy in L is 0, I 0 = 0: Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 47

7.3. Step Response of RL and RC Circuits The Step Response of an RL Circuit o 1 after switch has been closed, current will have reached approximately 63% of its final value: Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 48

7.3. Step Response of RL and RC Circuits The Step Response of an RL Circuit o If current continues to increase at its initial rate, it reaches its final value at t = : Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 49

7.3. Step Response of RL and RC Circuits The Step Response of an RL Circuit o Voltage across L is 0 before switch is closed (t < 0). o Initial current is I 0. o L prevents an instantaneous change in current. o Hence, current is I 0 in instant after switch has been closed. o Voltage drop across resistor is I 0 R. o When switch is closed, v jumps to V s -l 0 R and decays exponentially to 0. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 50

7.3. Step Response of RL and RC Circuits The Step Response of an RL Circuit o When initial inductor current is 0: o Initial current is 0. o v jumps to V s. o v approaches 0 as t increases. o Current is approaching constant value of V s /R. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 51

7.3. Step Response of RL and RC Circuits The Step Response of an RL Circuit o When I 0 =0: Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 52

7.3. Step Response of RL and RC Circuits o Switch has been in position a for a long time. o L is a short circuit across 8 A current source. o L carries an initial current of 8 A. o This current is oriented opposite to reference direction for i: I 0 = -8 A. o When switch is in position b: final value of i will be 24/2 = 12 A. o = L/R = 200/2 = 100 ms. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 53

7.3. Step Response of RL and RC Circuits Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 54

7.3. Step Response of RL and RC Circuits o In instant after switch has been moved to position b, L sustains a current of 8 A counterclockwise around newly formed closed path. o This current causes a 16 V drop across 2 resistor. o This voltage drop adds to drop across source, producing a 40 V drop across L. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 55

7.3. Step Response of RL and RC Circuits Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 56

7.3. Step Response of RL and RC Circuits The Step Response of an RC Circuit o For mathematical convenience, we choose Norton equivalent of network connected to equivalent capacitor. o For RL circuit, we had: o By comparing the equations, we have: Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 57

7.3. Step Response of RL and RC Circuits The Step Response of an RC Circuit o We obtained solutions by using a mathematical analogy to solution for step response of RL circuit. o Let's see whether these solutions for RC circuit make sense in terms of known circuit behavior. Initial voltage across C is V 0. Final voltage across C is I S R. = RC. Solution for v C is valid for t >0. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 58

7.3. Step Response of RL and RC Circuits The Step Response of an RC Circuit Current in C at t = 0 + is I s V 0 /R. This prediction makes sense. Capacitor voltage cannot change instantaneously. Initial current in R is V 0 /R. Capacitor branch current changes instantaneously from 0 at t = 0 - to I s V 0 /R at t = 0 +. Capacitor current is 0 at t = +. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 59

7.3. Step Response of RL and RC Circuits o We find Norton equivalent with respect to terminals of C for t > 0: Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 60

7.3. Step Response of RL and RC Circuits o We check consistency of solutions: Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 61

7.4. Solution for Step & Natural Responses o To generalize solution of these 4 possible circuits, we let x(t) represent unknown quantity. o x(t) can have 4 possible values: i(t) or v(t) of an L or C. o Constant K can be 0. o Sources are constant voltages and/or currents. o Finalvalue of x or x f will be constant. o Finalvalue must satisfy above equation. o When x reaches its final value, dx/dt = 0: Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 63

7.4. Solution for Step & Natural Responses o To obtain a general solution, we use time t 0 as lower limit and t as upper limit. o t 0 corresponds to time of switching or other change. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 64

7.4. Solution for Step & Natural Responses o Previously we assumed that t 0 = 0. o u and v are symbols of integration. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 65

7.4. Solution for Step & Natural Responses Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 66

7.4. Solution for Step & Natural Responses Calculating natural or step response of RL or RC circuits: 1. Identify variable of interest for circuit. For RC circuits, it is best to choose v C. For RL circuits, it is best to choose i L. 2. Determine variable initialvalue (its value at t 0 ). If variable is v C or i L, it is not necessary to distinguish between t = t 0- and t = t 0+. They both are continuous variables. If another variable is chosen, its initial value is defined at t = t 0+. 3. Calculate variable final value (its value as t ). 4. Calculate for circuit. 5. Use: Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 67

7.4. Solution for Step & Natural Responses Calculating natural or step response of RL or RC circuits: o You can then find equations for other circuit variables using: circuit analysis techniques or by repeating preceding steps for other variables. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 68

7.4. Solution for Step & Natural Responses o Switch has been in position a for a long time. o C looks like an open circuit. v C = v 60 = o After switch has been in position b for a long time, C will look like an open circuit in terms of 90 V source. o Final value of v C is +90 V. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 69

7.4. Solution for Step & Natural Responses o doesn'tchange for i(t). o We need to find only initial and final values for i. o When obtaining initial value, we must get i(0 + ). o Current in capacitor can change instantaneously. o This current is equal to current in resistor: o Final value of i(t) is 0. o Alternative solution is i(t) = Cdv C (t)/dt. t Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 70

7.4. Solution for Step & Natural Responses Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 71

7.4. Solution for Step & Natural Responses o Magnetically coupled coils can be replaced by a single L having an inductance of: o By hypothesis, initial value of i o is 0. See Problems 6.40 and 6.41. o Final value of i o will be 120/7.5 or 16 A. o of circuit is 1.5/7.5 or 0.2 s. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 72

7.5. Sequential Switching o Whenever switching occurs more than once in a circuit, we have sequential switching. o For example, a single 2-position switch may be switched back and forth, or multiple switches may be opened or closed in sequence. o Time reference for all switchings cannot be t = 0. o We determine v(t) and i(t) for a given position of switch or switches and then use these solutions to determine initial conditions for next position of switch or switches. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 74

7.5. Sequential Switching o With sequential switching problems, obtaining initial value x(t 0 ) is important. o Anything but inductive currents and capacitive voltages can change instantaneously at time of switching. o Solving first for inductive currents and capacitive voltages is even more pertinent. o Drawing circuit pertaining to each time interval is often helpful. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 75

7.5. Sequential Switching o At instant switch is moved to position b, initial voltage on C is 0. o If switch were to remain in position b, C would eventually charge to 400 V. o when switch is in position b is 10 ms. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 76

7.5. Sequential Switching o Switch remains in position b for only 15 ms. o This expression is valid for 0 < t < 15 ms. o When switch is moved to position c, initial voltage on C is 310.75 V. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 77

7.5. Sequential Switching o With switch in position c, final value of C voltage is 0. o is 5 ms. o t 0 = 15 ms Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 78

7.5. Sequential Switching o C voltage will equal 200 V at 2 different times: Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 79

7.6. The Integrating Amplifier o Output voltage is proportional to integral of input voltage. o We assume that op amp is ideal. o t 0 represents instant in time when we begin integration. o v o (t 0 ) is value of output voltage at that time: v o (t 0 ) = v Cf (t 0 ) Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 81

7.6. The Integrating Amplifier o Output voltage equals initialvalue of voltage on C f plus an inverted (minus sign), scaled (1/R s C f )replica of integral of input voltage. o If no energy is stored in C f when integration starts, output voltage is proportional to integral of input voltage only if op amp operates within its linear range (if it doesn't saturate). Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 82

7.6. The Integrating Amplifier Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 83

7.6. The Integrating Amplifier v O (0.009) = -5 + 9 = 4 V o During this interval, v o is decreasing, and op amp eventually saturates at -6 V: Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 84

7.6. The Integrating Amplifier o Integrating amplifier can perform integration function very well, but only within specified limits that avoid saturating op amp. o Op amp saturates due to accumulation of charge on feedback capacitor. o We can prevent from saturating by placing an R in parallel with C f. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 85

7.6. The Integrating Amplifier o We can convert integrating amplifier to a differentiating amplifier by interchanging R s and C f : o We can design both integrating- and differentiating-amplifier circuits by using an L instead of a C. o Fabrication of capacitors for integrated-circuit devices is much easier. o Inductors are rarely used in integrating amplifiers. Electric Circuits 1 Chapter 7. Response of First-Order RL and RC Circuits 86