   Chapter 17.3, Problem 13E

Chapter
Section
Textbook Problem

A series circuit consists of a resistor with R = 20 Ω,  an inductor with L= 1 H, a capacitor with C = 0.002 F, and a 12-V battery. If the initial charge and current are both 0, find the charge and current at time t.

To determine

To find: The charge of series RLC circuit at time t and the current of series RLC circuit at time t .

Explanation

Given data:

Series RLC circuit with following values.

R=20Ω , L=1H , C=0.002F , E(t)=12V , Q(0)=0 and Q(0)=I(0)=0 .

Formula used:

Write the expression for differential equation of electric circuit.

Ld2Qdt2+RdQdt+1CQ=E(t)

LQ+RQ+1CQ=E(t) (1)

Write the expression for general solution with complex roots.

Q(t)=eαt[c1cos(βt)+c2sin(βt)] (2)

Write the expression for r .

r=α+βi (3)

Write the expression for auxiliary equation.

ar2+br+c=0 (4)

Write the expression for differential equation.

ay+by+cy=0 (5)

Find the differential equation for electric circuit using equation (1).

Substitute 1 for L, 20 for R, 0.002 for C and 12 for E(t) in equation (1),

1Q+20Q+10.002Q=12

Q+20Q+500Q=12 (6)

Modify equation (5) as follows.

aQ+bQ+cQ=0 (7)

Compare equation (6) and (7).

a=1b=20c=500

Substitute 1 for a, 20 for b and 500 for c in equation (4),

r2+20r+500=0

Find the value of r .

r=20±(20)24(1)(500)2(1)=20±40020002=20±16002=20±40i2

Simplify r as follows.

r=10±20i (8)

Compare equations (3) and (8).

α=10β=20

Substitute 10 for α and 20 for β in equation (2),

Q(t)=e10t[c1cos(20t)+c2sin(20t)] (9)

Modify equation (9) for complementary equation as follows.

Qc(t)=e10t[c1cos(20t)+c2sin(20t)]

Consider the value of Qp(t) as follows.

Qp(t)=A (10)

Differentiate equation (10) with respect to t .

Qp(t)=0

Differentiate equation with respect to t .

Qp(t)=0

Modify equation (6) as follows.

Qp(t)+20Qp(t)+500Qp(t)=12

Substitute 0 for Qp(t) , 0 for Qp(t) and A for Qp(t) ,

1(0)+20(0)+500(A)=12500(A)=12A=12500A=3125

Substitute 3125 for A in equation (10),

Qp(t)=3125

Write the expression for general solution.

Q(t)=Qc(t)+Qp(t)

Substitute e10t[c1cos(20t)+c2sin(20t)] for Qc(t) and 3125 for Qp(t) ,

Q(t)=e10t[c1cos(20t)+c2sin(20t)]+3125 (11)

Substitute 0 for t ,

Q(0)=e10(0)[c1cos(20×0)+c2sin(20×0)]+3125=c1+3125

Substitute 0 for Q(0) ,

0=c1+3125c1=03125c1=3125

Write the expression for I

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