Chapter 21, Problem 1CQ

An RLC circuit connected across an AC voltage source at frequency f has resistance R, capacitive reactance X_G and inductive reactance X_L. If the frequency is doubled so that f_new = 2f, find the ratios (a) R_new/R, (b) X_C,new/X_C, and (c) X_L,new/X_L.

To determine

If the frequency of the RLC circuit is doubled, find the ration of the new resistance to the old resistance.

Answer to Problem 1CQ

The ratio $\frac{R_{\text{new}}}{R}$ is $1$.

Explanation of Solution

The resistance is independent of frequency. So, when the frequency is doubled, there is no change in the resistance.

$\frac{R_{\text{new}}}{R}=1$

Conclusion:

Thus, the ratio $\frac{R_{\text{new}}}{R}$ is $\underset{\_}{1}$.

To determine

If the frequency of the RLC circuit is doubled, find the ration of the new capacitive reactance to the old capacitive reactance.

Answer to Problem 1CQ

The ratio $\frac{X_{\text{C,new}}}{X_{\text{C}}}$ is $\frac{1}{2}$.

Explanation of Solution

The relation between capacitive reactance and frequency is,

$X_{\text{C}}=\frac{1}{2\pi fC}$

• $f$ is the frequency
• $C$ is the capacitance

When the frequency is doubled, the capacitive reactance becomes

$\begin{array}{c}{X}_{\text{C,new}}=\frac{1}{2\pi f_{\text{new}}C}\\ =\frac{1}{2\pi \left(2f\right)C}\\ =\frac{1}{4\pi fC}\end{array}$

The ratio is,

$\begin{array}{c}\frac{X_{\text{C,new}}}{X_{\text{C}}}=\frac{\left(\frac{1}{4\pi fC}\right)}{\left(\frac{1}{2\pi fC}\right)}\\ =\frac{2\pi fC}{4\pi fC}\\ =\frac{1}{2}\end{array}$

Conclusion:

Thus, the ratio $\frac{X_{\text{C,new}}}{X_{\text{C}}}$ is $\frac{1}{2}$.

To determine

If the frequency of the RLC circuit is doubled, find the ration of the new inductive reactance to the old inductive reactance.

Answer to Problem 1CQ

The ratio $\frac{X_{L\text{,new}}}{X_L}$ is $2$.

Explanation of Solution

The relation between inductive reactance and frequency is,

$X_L=2\pi fL$

• $f$ is the frequency
• $L$ is the inductance

When the frequency is doubled, the inductive reactance becomes

$\begin{array}{c}{X}_{\text{L,new}}=2\pi f_{\text{new}}L\\ =2\pi \left(2f\right)L\\ =4\pi fL\end{array}$

The ratio is,

$\begin{array}{c}\frac{X_{L\text{,new}}}{X_L}=\frac{4\pi fC}{2\pi fC}\\ =2\end{array}$

Conclusion:

Thus, the ratio $\frac{X_{L\text{,new}}}{X_L}$ is $2$.