Chapter 3: Q 3.5-6E (page 121)

Derive a power balance equation for the RLand RCcircuits. (See Problem 5.) Discuss the significance of the signs of the three power terms.

Short Answer

Expert verified

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.

Step by step solution

Important formula.

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

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 $\frac{d}{dt} (\frac{1}{2} {LI}^{2}) + {RI}^{2} = E (t) I$ .

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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Derive a power balance equation for the RLand RCcircuits. (See Problem 5.) Discuss the significance of the signs of the three power terms.

Short Answer

Step by step solution

Important formula.

Determine the power balance for RL circuit

evaluate the power balance for RC circuit.

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