A$\mathbf{45}\mathbf{}\mathbf{mH}$inductor has a reactance ofrole="math" localid="1662986007307" $\mathbf{1}\mathbf{.}\mathbf{30}\mathbf{}\mathbf{k\Omega }$ . (a) What is its operating frequency? (b) What is the capacitance of a capacitor with the same reactance at that frequency? If the frequency is doubled, what is the new reactance of (c) the inductor and (d) the capacitor?

Inductor is $L=45\mathrm{mH}$.

Reactance is$R=1.30\text{kΩ}$ .

Reactance withstands currents without loss of power, unlike resistors. Inductive reactance increases with frequency and inductance. Capacitive reactance decreases with frequency and capacitance. Impedance represents the total opposition provided by reactance and resistance.

Formula:

The inductive reactance is,

$X_L=2\pi fL$

The capacitive reactance is,

$X_c=\frac{1}{2\pi fC}$

Write inductive capacitance as

$\begin{array}{rcl}{X}_L& =& 2\pi fL\\ 1300& =& 2\times 3.14\times f\times 45\times {10}^{-3}\\ f& =& 4.6\mathrm{kHz}\end{array}$

Hence, the operating frequency is$4.6\mathrm{kHz}$ .

As given that $X_L=X_c$

Therefore,

$\begin{array}{rcl}2\pi fL& =& \frac{1}{2\pi fC}\\ f^2& =& \frac{1}{4{\pi }^{2}LC}\end{array}$

$\begin{array}{rcl}C& =& \frac{L}{\frac{1}{4{\pi }^2{L}^2{f}^2}}\\ & =& \frac{L}{X_l^2}\end{array}$

$\begin{array}{rcl}C& =& \frac{45\times {10}^{-3}}{{\left(1300\right)}^2}\\ & =& 26.62\times {10}^{-9}\mathrm{F}\\ & =& 26.62\mathrm{nF}\end{array}$

Hence, the capacitance is$26.62\mathrm{nF}$.

When frequency is double

The inductive capacitance is,

$\begin{array}{rcl}{X}_L& =& 2\times 3.14\times 9\times {10}^3\times 45\times {10}^{-3}\\ & =& 2.6\mathrm{k\Omega }\end{array}$

If frequency is doubled, then reactance of inductor is $2.6\mathrm{k\Omega }$.

If frequency is doubled,the reactance of capacitor:

$\begin{array}{rcl}{X}_c& =& \frac{1}{2\pi fC}\\ & =& \frac{1}{2\times 3.14\times 9\times {10}^3\times 26.62\times {10}^{-9}}\\ & =& 0.65\mathrm{k\Omega }\end{array}$

Hence, if frequency is doubled, then reactance of capacitor is $0.65\mathrm{k\Omega }$.