Electric Circuits (EELE 2312) Chapter 5 The Natural & Step Response of RL and RC Circuits Basil Hamed

Introduction We begin this chapter by introducing the last two ideal circuit elements, namely, inductors and capacitors. Be assured that the circuit analysis techniques introduced in Chapters 2 and 3 apply to circuits containing inductors and capacitors. Therefore, once you understand the terminal behavior of these elements in terms of current and voltage, you can use Kirchhoff s laws to describe any interconnections with the other basic elements. Basil Hamed

Inductor An inductor is an electrical component that opposes any change in electrical current. It is composed of a coil of wire wound around a supporting core whose material may be magnetic or nonmagnetic. The behavior of inductors is based on phenomena associated with magnetic fields The source of the magnetic field is charge in motion, or current. The circuit parameter of inductance relates the induced voltage to the current. Basil Hamed

A capacitor A capacitor is an electrical component that consists of two conductors separated by an insulator or dielectric material. The capacitor is the only device other than a battery that can store electrical charge. The behavior of capacitors is based on phenomena associated with electric fields. The source of the electric field is separation of charge, or voltage The circuit parameter of capacitance relates the displacement current to the voltage Basil Hamed

Energy can be stored in both magnetic and electric fields Energy can be stored in both magnetic and electric fields. Hence you should not be too surprised to learn that inductors and capacitors are capable of storing energy. For example, energy can be stored in an inductor and then released to fire a spark plug. Energy can be stored in a capacitor and then released to fire a flashbulb. In ideal inductors and capacitors, only as much energy can be extracted as has been stored. Because inductors and capacitors cannot generate energy, they are classified as passive elements. Basil Hamed

5.1 The Inductor ʋ≡ voltage in Volts (V) L≡ inductance in Henry (H) i ≡ current in Amperes (A)

Example 5-1 Determine the Voltage, Given the Current The independent current source in the circuit shown generates zero current for t˂0 and a pulse 10te-5t A, for t˃0. Sketch the current waveform. At what instant of time the current is maximum? Express the voltage across the terminals of the 100 mH inductor as a function of time. Sketch the voltage waveform. Are the voltage and the current at a maximum at the same time? At what instant of time does the voltage change polarity? Is there ever an instantaneous change in the voltage across the inductor? If so, at what time?

Example 5-1 a

Example 5-1

Example 5-1

Example 5-1 e) No f) At 0.2 sec g) yes, at t=0

Current in an Inductor In terms of the Voltage Across the Inductor

Example 5-2 Determine the Current, Given the Voltage The voltage pulse applied to the 100 mH inductor is 0 for t˂0 and given by the expression ʋ(t)=20te-10t V, for t˃0. Also assume ʋ=0 for t ≤ 0. Sketch the voltage as a function of time. Find the inductor current as a function of time. Sketch the current as a function of time. ʋ=0 t≤0 ʋ=20te-10t t˃0

Power and Energy in The Inductor

Example 5-3 Determine the Current, Voltage, Power, and Energy The independent current source in the circuit shown generates zero current for t˂0 and a pulse 10te-5t A, for t˃0. Plot i, ʋ, p, and ω versus time. In what time interval is energy being stored in the inductor? In what time interval is energy being extracted from the inductor? What is the maximum energy stored in the inductor? Evaluate the integral Repeat (a)-(c) for a voltage pulse of ʋ(t)=20te-10t V, for t˃0 and ʋ=0 for t ≤ 0. In (f), why is there a sustained current in the inductor as the voltage approaches zero? i=0 t≤0 i=10te-5t t˃0 ʋ=0 t≤0 ʋ=20te-10t t˃0

5.2 The Capacitor ʋ≡ voltage in Volts (V) C≡ capacitance in farad (F) i ≡ current in Amperes (A)

Example 5-4 Determine the Current, Power, and Energy The voltage pulse described by the following equation is impressed across the terminals of a 0.5 µF capacitor: Derive expression for the capacitor current, power, and energy. Sketch the voltage, current, power, and energy as functions of time. Line up the plots vertically. Specify the interval of time when energy is being stored in the capacitor. Specify the interval of time when energy is being delivered by the capacitor. Evaluate the integrals

Example 5-5 Determine the Voltage, Power, and Energy An uncharged 0.2 µF capacitor id driven by a triangular current pulse. The current pulse is described by: Derive the expressions for the capacitor voltage, power, and energy for each of the four time intervals needed to describe the current. Plot i, Ʋ, p, and ω versus t. Align the plots vertically. Why does the voltage remain on the capacitor after the current returns to zero?

Summary

5.3 Series-Parallel Combination Inductances in Series

5.3 Series-Parallel Combination Inductance in Parallel

5.3 Series-Parallel Combination Capacitance in Series

5.3 Series-Parallel Combination Capacitance in Parallel

5.4 Natural Response of RL and RC Circuits

5.4 Natural Response of RL and RC Circuits Step Response of RL Circuit Step Response of RC Circuit

5.4 The Natural Response of an RL Circuit Deriving The Expression For The Current The independent current source generates a constant current Is and the switch has been in a closed position for a long time

5.4 The Natural Response of an RL Circuit The Significance of The Time Constant

Example 5-6 Natural Response of RL Circuit The switch in the circuit shown has been closed for a long time before it is opened at t=0. find: iL(t) for t ≥ 0 i0(t) for t ≥ 0+ Ʋ0(t) for t ≥ 0 The percentage of the total energy stored in the 2 H inductor that is dissipated in the 10 Ω resistor Req=2+(40║10)=10 Ω iL(0-)=iL(0+)=20 A Ʋ0(0-)=0 =(L/Req)=(2/10)=0.2 s

Example 5-7 Natural Response of RL Circuit/Parallel Inductors In the circuit shown, the initial currents in inductors L1 and L2 have been established by sources not shown. The switch is opened at t=0. Find i1,i2, and i3 for t ≥ 0 Calculate the initial energy stored in the parallel inductors Determine how much energy is stored in the inductors as t → ∞ Show that the total energy delivered to the resistive network equals the difference between the results obtained in (b) and (c) Req=((4+(15║10))║40)=8 Ω iLeq(0-)=iL(0+)=12 A Ʋ0(0-)=0 =(Leq/Req)=(4/8)=0.5 s

The Natural Response of an RC Circuit

The Natural Response of an RC Circuit Deriving The Expression For The Voltage

Example 5-8 Natural Response of an RC Circuit The switch in the circuit shown has been in position x for a long time. At t=0, the switch moves instantaneously to position y. find: ʋC(t) for t ≥ 0 ʋ0(t) for t ≥ 0+ i0(t) for t ≥ 0+ The total energy dissipated in the 60 kΩ resistor Req=32+(240║60)=80 kΩ ƲC(0-)= ƲC(0)= ƲC (0+)=100 V =(ReqC)=80(0.5×10-3)=40 ms

Example 5-9 Natural Response of RC Circuit/Series Capacitors The initial voltages on capacitors C1 and C2 in the circuit shown have been established by sources not shown. The switch is closed at t=0. Find ʋ1(t), ʋ2(t), and ʋ(t) for t ≥ 0 and i(t) for t ≥ 0+ Calculate the initial energy stored in the capacitors C1 and C2 Determine how much energy is stored in the capacitors as t → ∞ Show that the total energy delivered to the 250 kΩ resistor is the difference between the results obtained in (b) and (c) R=250 kΩ ƲC(0-)= ƲC(0)= ƲC (0+)=20 V =RC=250(4×10-3)=1 s Ʋ(t)=20e-t V t ≥ 0 i(t)= Ʋ(t)/250000=80e-t µA t ≥ 0+

5.5 The Step Response of RL Circuit

Example 5-10 Determining the Step Response of an RL Circuit The switch in the circuit shown has been in position a for a long time. At t=0, the switch moves from position a to position b. the switch is a make-before-break type; that is, the connection at position b is established before the connection at position a is broken, so there is no interruption of current through the inductor. Find the expression for i(t) for t ≥ 0 What is the initial voltage across the inductor just after the switch has been moved to position b? How many milliseconds after the switch has been moved does the inductor voltage equal 24 V? Does the initial voltage make sense in terms of circuit behavior? Plot both i(t) and ʋ(t) versus t i(0-)= i(0)= i(0+)=I0=-8 A For t ≥ 0, R=2 Ω =L/R=200/2=100 ms i(∞)=(Vs/R)=12 A i(t)=12+(-8-12)e-t/0.1 i(t)=12-20e-10t A, t ≥ 0

5.5 The Step Response of RL Circuit Describing Voltage across the Inductor

5.5 The Step Response of RC Circuit

Example 5-11 Determining the Step Response of an RC Circuit The switch in the circuit shown has been in position 1 for a long time. At t=0, the switch moves to position 2. find: Ʋ0(t) for t ≥ 0 i0(t) for t ≥ 0+

Step and Natural Responses A General Solution Unknown = final value+[initial value-final value]e-(t-switching time)/time constant

Example 5-12 General Solution Method of RC Step Response The switch in the circuit shown has been in position a for a long time. At t=0, the switch is moved to position b. What is the initial value of ʋC? What is the final value of ʋC? What is the time constant of the circuit when the switch in position b? What is the expression for of ʋC when t ≥ 0 ? What is the expression for of i(t) when t ≥ 0+ ? How long after the switch in position b does the capacitor voltage equal zero? Plot ʋC and i(t) versus time.

Example 5-13 General Solution Method with Zero Initial Condition The switch in the circuit shown has been open a for a long time. The initial charge on the capacitor is zero. At t=0, the switch is closed. Find the expression for i(t) for t ≥ 0+ Ʋ(t) for t ≥ 0+

Example 5-14 General Solution Method for RL Step Response The switch in the circuit shown has been open for a long time. At t=0, the switch is closed. Find the expression for Ʋ(t) when t ≥ 0+ i(t) when t ≥ 0

End of Chapter Five