Chapter 8 Basic RL and RC Circuits 1
Chapter 8 Basic RL and RC Circuits 1 Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display.
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 volts at t=200 ms. Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 3
The Exponential Response The time constant τ=L/R determines the rate of decay. Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 4
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 Companies, Inc. Permission required for reproduction or display. 6
The Source Free RC Circuit Show that the voltage v(t) is 321 m. V at t=200 μs. Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 7
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 change: Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 12
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 Companies, Inc. Permission required for reproduction or display. 14
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
Driven RL Circuits Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 17
Example: RL Circuit with Step Show that i(t)=25+25(1 -e-t/2)u(t) A Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 18
Example: Voltage Pulse Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 19
Driven RC Circuits (part 1 of 2) v. C=20 + 80 e-t/1. 2 V and i=0. 1 + 0. 4 e−t/1. 2 A Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 20
Driven RC Circuits (part 2 of 2) v. C=20 + 80 e-t/1. 2 V i=0. 1 + 0. 4 e−t/1. 2 A Copyright © 2013 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 21
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