Chapter 8 Basic RL and RC Circuits 1

The Source-Free RL Circuit Applying KVL: We can solve for the natural response if we know the initial condition i(0)=I 0: i(t)=I 0 e-Rt/L for t>0 Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 2

Example: RL with a Switch Show that the voltage v(t) will be -12. 99

The Exponential Response The time constant τ=L/R determines the rate of decay. Copyright ©

The Source-Free RC Circuit Applying KCL: We can solve for the natural response if we know the initial condition v(0)=V 0 v(t)=V 0 e-t/RC for t>0 Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 5

RC Natural Response The time constant is τ=RC Copyright © 2013 The Mc. Graw-Hill

The Source Free RC Circuit Show that the voltage v(t) is 321 m. V

General RL Circuits The time constant of a single-inductor circuit will be τ=L/Req where Req is the resistance seen by the inductor. Example: Req=R 3+R 4+R 1 R 2 / (R 1+R 2) Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 8

General RC Circuits The time constant of a single-capacitor circuit will be τ=Req. C where Req is the resistance seen by the capacitor. Example: Req=R 2+R 1 R 3 / (R 1+R 3) Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 9

1 st Order Response Observations The voltage on a capacitor or the current through a inductor is the same prior to and after a switch at t=0. Resistor voltage (or current) prior to the switch v(0 -) can be different from the voltage after the switch v(0+). All voltages and currents in an RC or RL circuit follow the same natural response e-t/τ. Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 10

Example: L and R Current Find i 1(t) and i. L(t) for t>0. Answer: τ=20 μs; i 1=-0. 24 e-t/τ, i. L=0. 36 e-t/τ for t>0 Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 11

The Unit Step Function The unit-step function u(t) is a convenient notation to respresent

Switches and Steps The unit step models a double-throw switch. A single-throw switch is open circuit for t<0, not short circuit. Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 13

Modeling Pulses using u(t) Rectangular pulse Pulsed sinewave: Copyright © 2013 The Mc. Graw-Hill

Driven RL Circuits The two circuits shown both have i(t)=0 for t<0 and are also the same for t>0. We now have to find both the natural response and the forced response due to the source V 0 Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 15

Driven RL Circuits The total response is the combination of the transient/natural response and the forced response: Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 16