Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Derive a power balance equation for the RL and RC circuits. (See Problem 5.) Discuss the significance of the signs of the three power terms.

The power balance equation for RL and RC circuit are $\frac{d}{dt} (\frac{1}{2} {LI}^{2}) + R I^{2} = E (t) I$ and ${RI}^{2} + \frac{d}{dt} (\frac{1}{2} C {E_{C}}^{2}) = E (t) I$ respectively.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Important formula.

The governing differential equation for RC circuit is $\frac{dq}{dt} + \frac{q}{CR} = \frac{E}{R}$ .

Step 2: Determine the power balance for RL circuit

The power dissipated by a circuit is $P_{R} = I^{2} (t) R$ and the power dissipated by an inductor is $P_{L} = \frac{1}{2} \frac{d}{dt} [{LI}^{2} (t)]$ and the power associated by a capacitor is $P_{C} = \frac{1}{2} \frac{d}{dt} [C {E_{C}}^{2} (t)]$ .

So, the equation is

$L \frac{dI}{dt} + RI = E (t) LI \frac{dI}{dt} + {RI}^{2} = IE (t) \frac{d}{dt} (\frac{1}{2} {LI}^{2}) + R I^{2} = E (t) I$

Therefore, the power balance equation for RL circuit is $\frac{d}{dt} (\frac{1}{2} {LI}^{2}) + {RI}^{2} = E (t) I$ .

Step 3: evaluate the power balance for RC circuit.

$R \frac{dq}{dt} + \frac{q}{c} = E I = \frac{dq}{dt} RI + \frac{q}{C} = E (t) {RI}^{2} + \frac{q}{C} = E (t) I$

Further, solve the above expression

$q = {CE}_{C} {RI}^{2} + \frac{{CE}_{C}}{C} . \frac{dq}{dt} = E (t) I {RI}^{2} + \frac{d}{dt} (\frac{1}{2} C {E_{C}}^{2}) = E (t) I$

Therefore, the power balance equation for RC circuit is ${RI}^{2} + \frac{d}{dt} (\frac{1}{2} C {E_{C}}^{2}) = E (t) I$

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Derive a power balance equation for the RL and RC circuits. (See Problem 5.) Discuss the significance of the signs of the three power terms.

Step by Step Solution

Step 1: Important formula.

Step 2: Determine the power balance for RL circuit

Step 3: evaluate the power balance for RC circuit.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Derive a power balance equation for the RL and RC circuits. (See Problem 5.) Discuss the significance of the signs of the three power terms.

Step by Step Solution

Step 1: Important formula.

Step 2: Determine the power balance for RL circuit

Step 3: evaluate the power balance for RC circuit.

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