Chapter 8. Natural and Step Responses of RLC 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 8.0. Introduction 8.1. Introduction to the Natural Response of a Parallel RLC Circuit 8.2. The Forms of the Natural Response of a Parallel RLC Circuit 8.3. The Step Response of a Parallel RLC Circuit 8.4. The Natural and Step Response of a Series RLC Circuit 8.5. A Circuit with Two Integrating Amplifiers 8.6. Summary Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 2
8.0. Introduction o Discussion of natural response and step response of circuits containing both inductors and capacitors is limited to 2 simple structures: parallel RLC circuit and series RLC circuit. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 3
8.0. Introduction o Finding natural response of a parallel RLC circuit consists of finding voltage across parallel branches by release of energy stored in L or C or both. o Initial voltage on C, V 0, represents initial energy stored in C. o Initial current in L, I 0, represents initial energy stored in L. o If individual branch currents are of interest, you can find them after determining terminal voltage. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 4
8.0. Introduction o In step response of a parallel RLC circuit, we are interested in voltage across parallel branches as a result of sudden application of a dc current source. o Energy may or may not be stored in circuit when current source is applied. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 5
8.0. Introduction o Finding natural response of a series RLC circuit consists of finding current generated in series connected elements by release of initially stored energy in L, C, or both. o As before, initial L current, I 0 and initial C voltage, V 0 represent initially stored energy. o If any of individual element voltages are of interest, you can find them after determining current. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 6
8.0. Introduction o In step response of a series RLC circuit, we are interested in current resulting from sudden application of dc voltage source. o Energy may or may not be stored in circuit when switch is closed. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 7
Chapter Contents 8.0. Introduction 8.1. Introduction to the Natural Response of a Parallel RLC Circuit 8.2. The Forms of the Natural Response of a Parallel RLC Circuit 8.3. The Step Response of a Parallel RLC Circuit 8.4. The Natural and Step Response of a Series RLC Circuit 8.5. A Circuit with Two Integrating Amplifiers 8.6. Summary Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 8
8.1. Natural Response of a Parallel RLC o First step in finding natural response of circuit is to derive differential equation that voltage v must satisfy. o We choose to find voltage first, because it is same for each component. o A branch current can be found by using current-voltage relationship for component. o We easily obtain differential equation for voltage by summing currents away from top node: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 9
8.1. Natural Response of a Parallel RLC o Equation is an ordinary, 2 nd order differential equation with constant coefficients. o Circuits here contain both L and C. o Differential equation describing these circuits is of the 2 nd order. o We sometimes call such circuits the 2 nd order circuits. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 10
8.1. Natural Response of a Parallel RLC o Classical approach is to assume that solution is of exponential form: A and s are unknown constants. o Equation is called characteristic equation of differential equation. o Roots of this quadratic equation determine mathematical character of v(t). Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 11
8.1. Natural Response of a Parallel RLC o We can show that sum of v 1 and v 2 is also a solution: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 12
8.1. Natural Response of a Parallel RLC o Roots of characteristic equation (s 1 and s 2 )are determined by circuit parameters R, L, and C. o Initialconditions determine constants A 1 and A 2. o Form of v must be modified if 2 roots s 1 and s 2 are equal. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 13
8.1. Natural Response of a Parallel RLC o is the neper frequency. o is the resonant radian frequency. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 14
8.1. Natural Response of a Parallel RLC o s 1 and s 2 are referred to as complex frequencies. o is neper frequency. o is resonant radian frequency. o Exponent of e must be dimensionless. o s 1 and s 2 ( and ) must have dimension of 1/time or frequency. o All these 4 frequencies have dimension of angular frequency per time (rad/s). o Nature of roots s 1 and s 2 depends on and. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 15
8.1. Natural Response of a Parallel RLC o There are 3 possible outcomes: If < 2,both roots will be real and distinct. Voltage response is said to be overdamped. If > 2,both s 1 and s 2 will be complex and conjugates of each other. Voltage response is said to be underdamped. If = 2, s 1 and s 2 will be real and equal. Voltage response is said to be critically damped. o Damping affects the way voltage response reaches its final (or steady-state) value. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 16
Chapter Contents 8.0. Introduction 8.1. Introduction to the Natural Response of a Parallel RLC Circuit 8.2. The Forms of the Natural Response of a Parallel RLC Circuit 8.3. The Step Response of a Parallel RLC Circuit 8.4. The Natural and Step Response of a Series RLC Circuit 8.5. A Circuit with Two Integrating Amplifiers 8.6. Summary Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 17
8.2. Forms of Parallel RLC Natural Response o Behavior of a 2 nd order RLC circuit depends on s 1 and s 2. o s 1 and s 2 depend on circuit parameters R, L, and C. o The 1 st step in finding natural response is to calculate s 1 and s 2, and determine whether response is over-, under-, or critically damped. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 18
8.2. Forms of Parallel RLC Natural Response o The 2 nd step is to find 2 unknown coefficients, such as A 1 and A 2. o Initialconditions determine constants A 1 and A 2. o Natural response should be matched to initialconditions imposed by the circuit. o Initialconditions are: initialvalue of current (or voltage) and initialvalue of the 1 st derivative of current (or voltage). Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 19
8.2. Forms of Parallel RLC Natural Response o Response equations, as well as equations for evaluating unknown coefficients, are slightly different for each of 3 damping configurations. o This is why first we want to determine whether response is overdamped, underdamped, or critically damped. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 20
8.2. Forms of Parallel RLC Natural Response Overdamped Voltage Response o Roots of characteristic equation, s 1 and s 2,are real and distinct. o Voltage response is said overdamped. o Form of voltage is: o A 1 and A 2 are determined by initial conditions: v(0 + ) and dv(0 + )/dt. o v(0 + ) and dv(0 + )/dt are determined from: initial voltage on C, V 0, and initial current in L, I 0 : Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 21
8.2. Forms of Parallel RLC Natural Response Overdamped Voltage Response o v(0 + ) is initial voltage on C, V 0. o We get dv(0 + )/dt by first finding current in C at t = 0 + : o KCL at top node is: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 22
8.2. Forms of Parallel RLC Natural Response Overdamped Voltage Response o Finding overdamped response, v(t): 1. Find s 1 and s 2 using values of R, L, and C: 2. Find v(0 + ) and dv(0 + )/dt using circuit analysis. 3. Find A 1 and A 2 by solving: 4. To determine v(t) for t > 0, substitute s 1, s 2, A 1, and A 2 into: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 23
8.2. Forms of Parallel RLC Natural Response Overdamped Voltage Response o Finding currents using v(t): Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 24
8.2. Forms of Parallel RLC Natural Response o Roots are real and distinct. o Response is overdamped. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 25
8.2. Forms of Parallel RLC Natural Response o C holds initial voltage across parallel elements to 12 V. i R (0 + ) = 12/200 = 60 ma Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 26
8.2. Forms of Parallel RLC Natural Response Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 27
8.2. Forms of Parallel RLC Natural Response Underdamped Voltage Response o When > 2, both s 1 and s 2 will be complex and conjugates of each other. o Voltage response is underdamped. o d is called damped radian frequency. o Form of voltage is: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 28
8.2. Forms of Parallel RLC Natural Response Underdamped Voltage Response o Euler identity is: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 29
8.2. Forms of Parallel RLC Natural Response Underdamped Voltage Response o Constants B 1 and B 2 are real, not complex. Because voltage is a real function. o Don't be misled by the fact that B 2 = j(a( 1 -A 2 ). In this underdamped case, A 1 and A 2 are complex conjugates. o Using B 1 and B 2 yields a simpler expression for voltage. o Like A 1 and A 2, we determine B 1 and B 2 by initial energy stored in circuit: o As with s 1 and s 2, and d are fixed by circuit parameters R, L, and C. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 30
8.2. Forms of Parallel RLC Natural Response Underdamped Voltage Response o General nature of underdamped response: First, trigonometric functions indicate that response is oscillatory. Voltage alternates between positive and negative values. The rate at which voltage oscillates is fixed by d. Second, amplitude of oscillation decreases exponentially. The rate at which amplitude falls off is determined by. is also referred to as damping factor or damping coefficient. d is called damped radian frequency. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 31
8.2. Forms of Parallel RLC Natural Response Underdamped Voltage Response o General nature of underdamped response: If there is no damping: =0. Frequency of oscillation is d = 0. If there is an R in circuit: is not zero. Frequency of oscillation is d < 0. d is said to be damped. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 32
8.2. Forms of Parallel RLC Natural Response Underdamped Voltage Response o General nature of underdamped response: Oscillatory behavior is possible because of 2 types of energy storage elements in circuit: L and C. A mechanical analogy of this electric circuit is that of a mass suspended on a spring. Oscillation is possible because energy can be stored in both spring and moving mass. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 33
8.2. Forms of Parallel RLC Natural Response Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 34
8.2. Forms of Parallel RLC Natural Response i R (0 + ) = 0 ma Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 35
8.2. Forms of Parallel RLC Natural Response o v(t) approaches its final value, alternating between values that are greater and less than final value. o Swings about final value decrease exponentially with time. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 36
8.2. Forms of Parallel RLC Natural Response Critically Damped Voltage Response o When = 2, s 1 and s 2 will be real and equal: o Form of voltage is: o Two simultaneous equations needed to determine D 1 and D 2 are: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 37
8.2. Forms of Parallel RLC Natural Response Critically Damped Voltage Response o You will rarely encounter critically damped systems in practice. o Largely because 0 must equal exactly. o Both 0 and depend on circuit parameters: o In a real circuit, it is very difficult to choose component values that satisfy an exact equality relationship. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 38
8.2. Forms of Parallel RLC Natural Response Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 39
Chapter Contents 8.0. Introduction 8.1. Introduction to the Natural Response of a Parallel RLC Circuit 8.2. The Forms of the Natural Response of a Parallel RLC Circuit 8.3. The Step Response of a Parallel RLC Circuit 8.4. The Natural and Step Response of a Series RLC Circuit 8.5. A Circuit with Two Integrating Amplifiers 8.6. Summary Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 40
8.3. Step Response of a Parallel RLC Circuit o Finding step response of a parallel RLC circuit involves finding v(t) across parallel branches or i(t) in individual branches as a result of sudden application of a dc current source. o There may or may not be energy stored in circuit when current source is applied. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 41
8.3. Step Response of a Parallel RLC Circuit o We focus on finding i L (t). o i L (t) does not approach 0 as t increases. o After switch has been open for a long time, i L (t) equals dc source current I. o We assume that initial energy stored in circuit is 0. o This assumption simplifies calculations and doesn't alter basic process. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 42
8.3. Step Response of a Parallel RLC Circuit Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 43
8.3. Step Response of a Parallel RLC Circuit o The solution for a 2 nd order differential equation with a constant forcing function equals the forced response plus a response function identical in form to the natural response. o The solution for the step response is in form: o I f and V f represent final value of response function. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 44
8.3. Step Response of a Parallel RLC Circuit o No energy is stored in circuit prior to application of dc current source. o Initial current in L is 0. o L prohibits an instantaneous change in i L. o i L (0) = 0 immediately after switch has been opened. o Initial voltage on C is 0 before switch has been opened. o It will be 0 immediately after. o Since v = Ldi L /dt: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 45
8.3. Step Response of a Parallel RLC Circuit o i L will be overdamped: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 46
8.3. Step Response of a Parallel RLC Circuit o Increasing R to 625 decreases to 3.2 10 4 rad/s. o Current response is underdamped, since: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 47
8.3. Step Response of a Parallel RLC Circuit o Current response is critically damped. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 48
8.3. Step Response of a Parallel RLC Circuit o Underdamped response reaches 90% of the final value in the fastest time, is desired response type when the speed is most important. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 49
8.3. Step Response of a Parallel RLC Circuit o Underdamped response overshoots final value. o Neither critically damped nor overdamped response produces currents in excess of 24 ma. o It is the best to use overdamped response. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 50
8.3. Step Response of a Parallel RLC Circuit o It would be impractical to require a design to achieve exact component values that ensure a critically damped response. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 51
8.3. Step Response of a Parallel RLC Circuit o There cannot be an instantaneous change in i L. o Initial value of i L in first instant after dc current source is applied must be 29 ma. o C holds initial voltage across L to 50 V: o Current response is critically damped: o Effect of nonzero initial stored energy is on calculations for D' 1 and D' 2. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 52
Chapter Contents 8.0. Introduction 8.1. Introduction to the Natural Response of a Parallel RLC Circuit 8.2. The Forms of the Natural Response of a Parallel RLC Circuit 8.3. The Step Response of a Parallel RLC Circuit 8.4. The Natural and Step Response of a Series RLC Circuit 8.5. A Circuit with Two Integrating Amplifiers 8.6. Summary Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 53
8.4. Natural & Step Response of a Series RLC Natural Response o Finding natural or step responses of a series RLC circuit are same as those for a parallel RLC circuit. o Both circuits are described by differential equations that have same form. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 54
8.4. Natural & Step Response of a Series RLC Natural Response o When you have obtained natural current response, you can find natural voltage response across any circuit element. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 55
8.4. Natural & Step Response of a Series RLC Step Response o V f is final value of v c, i.e., V f =V. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 56
8.4. Natural & Step Response of a Series RLC Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 57
8.4. Natural & Step Response of a Series RLC o i L is 0 before switch has been closed. o It is 0 immediately after: o There will be no voltage drop across resistor. o Initial voltage on C appears across L: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 58
8.4. Natural & Step Response of a Series RLC Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 59
8.4. Natural & Step Response of a Series RLC o Roots are complex, so voltage response is underdamped: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 60
Chapter Contents 8.0. Introduction 8.1. Introduction to the Natural Response of a Parallel RLC Circuit 8.2. The Forms of the Natural Response of a Parallel RLC Circuit 8.3. The Step Response of a Parallel RLC Circuit 8.4. The Natural and Step Response of a Series RLC Circuit 8.5. A Circuit with Two Integrating Amplifiers 8.6. Summary Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 61
8.5. A Circuit with 2 Integrating Amplifiers o A circuit with 2 integrating amplifiers connected in cascade is also a 2 nd order circuit. o Output voltage of the 2 nd integrator is related to input voltage of the 1 st by a 2 nd order differential equation. o In a cascade connection, output signal of the 1 st amplifier is input signal for the 2 nd amplifier. o We assume that op amps are ideal. o Task is to derive differential equation that relates v o to v g. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 62
8.5. A Circuit with 2 Integrating Amplifiers Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 63
8.5. A Circuit with 2 Integrating Amplifiers o Energy stored in circuit initially is 0, and op amps are ideal: Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 64
8.5. A Circuit with 2 Integrating Amplifiers o The 2 nd integrating amplifier saturates when v o reaches 9 V or t = 3 s. o But it is possible that the 1 st integrating amplifier saturates before t = 3 s: o At t = 3 s, v o1 = -3 V. o Power supply voltage on the 1 st integrating amplifier is ±5 V. o Circuit reaches saturation when the 2 nd amplifier saturates. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 65
8.5. A Circuit with 2 Integrating Amplifiers 2 Integrating Amplifiers with Feedback Resistors o The Reason the op amp in integrating amplifier saturates is feedback capacitor's accumulation of charge. o A resistor is placed in parallel with each feedback capacitor to overcome this problem. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 66
8.5. A Circuit with 2 Integrating Amplifiers 2 Integrating Amplifiers with Feedback Resistors Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 67
8.5. A Circuit with 2 Integrating Amplifiers 2 Integrating Amplifiers with Feedback Resistors Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 68
8.5. A Circuit with 2 Integrating Amplifiers 2 Integrating Amplifiers with Feedback Resistors Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 69
8.5. A Circuit with 2 Integrating Amplifiers Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 70
8.5. A Circuit with 2 Integrating Amplifiers o Final value of v o is input voltage times gain of each stage. o Capacitors behave as open circuits as : Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 71
8.5. A Circuit with 2 Integrating Amplifiers Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 72
8.5. A Circuit with 2 Integrating Amplifiers o Solution assumes neither op amp saturates. o Final value of v o is 5 V, which is less than 6 V. The 2 nd op amp does not saturate. o Final value of v o1 is (250 10-3 )(-500/100), or -1.25 V. The 1 st op amp does not saturate. Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 73
Chapter Contents 8.0. Introduction 8.1. Introduction to the Natural Response of a Parallel RLC Circuit 8.2. The Forms of the Natural Response of a Parallel RLC Circuit 8.3. The Step Response of a Parallel RLC Circuit 8.4. The Natural and Step Response of a Series RLC Circuit 8.5. A Circuit with Two Integrating Amplifiers 8.6. Summary Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 74
8.6. Summary Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 75
8.6. Summary Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 76
8.6. Summary Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 77