Ch 9 Lecture Slides

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Chapter 9

The RLC Circuit


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an RLC circuit hasboth an inductor and a

capacitor

these circuits have a wide range ofapplications, including oscillators and

frequency filters

they also can model automobile suspensionsystems, temperature controllers, airplane

responses, and so forth


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Apply KCL and

differentiate to show:

Cd2

v

dt2 1

R

dv

dt 1

Lv 0


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To solve, assume v=Aest .

The solution must then satisfy

which is called the characteristic equation.

Ifs1ands2are the solutions, then the natural response is



Cs2 1Rs1

L 0

v(t)A1es1t A2e

s2t


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The solutions to the characteristic equation are

Define 0

the resonant frequency:

and the damping coefficient:

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12RC

12RC

2

1LC

0

1LC

1

2RC


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With these definitions, the solutions can be

expressed as:

The constantsA1andA2are determined by the

initial conditions.


s1 2 0

2

s2 2 0

2


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If >0the solutions are

real, unequal and the

response is termedoverdamped.

If


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Show that v(t) = 84(et e6t) when i(0+)=10 A

and v(0+)=0 V.



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Show that vC(t) = 80e50,000t 20e200,000t V for t>0.



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FindR1such that the circuit is critically damped

for t>0 andR2so that v(0)=2 V.

Answer: R1= 31.63 k, R2=0.4



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If


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Show for t>0

iL= e1.2t (2.027 cos 4.75t + 2.561 sin 4.75t)



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For the series RLC circuit,

This circuit is the dual of the parallel RLC

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Ld2i

dt2R

di

dt1

Ci 0


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The characteristic equation is

and the solution is

where


Ls

2

Rs

1

C 0

v(t)A1es1t A2e

s2t

s1,s2 R

2L

R

2L

2

1

LC


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Define and

Then if>0 (overdamped):

=0 (critically damped):


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The response of RLC circuits with dc sources

and switches will consist of the natural

response and the forced response:

v(t) = vf(t)+vn(t)

The complete response must satisfy both the

initial conditions and the final conditions or

the forced response.Copyright 2013 The McGraw-Hill Companies, Inc. Permission required forreproduction or display. 19


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Find the labeled voltages and currents at t=0-and

t=0+.

Answer:iR(0

) = 5 A vR(0) = 150 V iR(0

+) = 1 A vR(0+) = 30 V

iL(0) = 5 A vL(0

) = 0 V iL(0+) = 5 A vL(0

+) = 120 V

iC(0) = 0 A vC(0

) = 150 V iC(0+) = 4 A vC(0

+) = 150 V



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Find the first derivatives of the labeled voltages

and currents at t=0+.

Answer:diR/dt(0

+) = 40 A/s dvR/dt(0+) = -1200 V/s

diL/dt(0+) = 40 A/s dvL/dt(0

+) = -1092 V/s

diC/dt(0+) = -40 A/s dvC/dt(0

+) = 108 V/s



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Show that for t>0

vC(t) = 150 + 13.5(et e9t)volts



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The resistor in the RLC circuit serves to

dissipate initial stored energy.

When this resistor becomes 0 in the seriesRLC or infinite in the parallel RLC, the circuit

will oscillate.

Example: for t>0,

v(t) =2 sin 3t

ifi(0)=-1/6 Aandv(0)=0 V

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Ch 9 Lecture Slides