Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Chapter 8 Basic RL and RC Circuits

Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.2 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 McGraw-Hill Companies, Inc. Permission required for reproduction or display.3 Show that the voltage v(t) will be volts at t=200 ms.

Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.4 The time constant τ=L/R determines the rate of decay.

Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.5 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

The time constant is τ=RC Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.6

Show that the voltage v(t) is 321 mV at t=200 μs. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.7

8 The time constant of a single-inductor circuit will be τ=L/R eq where R eq is the resistance seen by the inductor. Example: R eq =R 3 +R 4 +R 1 R 2 / (R 1 +R 2 )

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

 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 McGraw-Hill Companies, Inc. Permission required for reproduction or display.10

Find i 1 (t) and i L (t) for t>0. Answer: τ=20 μs; i 1 =-0.24e -t/τ,i L =0.36e -t/τ for t>0 Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.11

The unit-step function u(t) is a convenient notation to respresent change: Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.12

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

Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.14 Rectangular pulse Pulsed sinewave:

 The two circuits shown both have i(t)=0 for t 0.  We now have to find both the natural response and and the forced response due to the source V 0 Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.15

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

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Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.18 Show that i(t)=25+25(1-e -t/2 )u(t) A

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v C = e -t/1.2 V and i= e −t/1.2 A Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.20

v C = e -t/1.2 V i= e −t/1.2 A Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.21