Topic: Network Theory
The voltage gain $v_{out}/v_{in}$ of a circuit shown below is zero. If $\omega=333.33~rad/s,$ the value of C is
(The circuit is a bridged-T network with mutual inductance $M=0.09\text{ H}$, $L_1=0.12\text{ H}$, $L_2=0.27\text{ H}$, $k=0.5$).
- A. 3.33 mF
- B. 33.33 mF
- C. $3.33~\mu F$
- D. $33.33~\mu F$
Answer: D. $33.33~\mu F$
Explanation: When $v_{out}/v_{in} = 0$, the transmission through the parallel paths cancels out. The condition for the null in this bridged-T network with mutual coupling simplifies to $3M = \frac{1}{\omega² C}$.
$M = k\sqrt{L_1 L_2} = 0.5\sqrt{0.12 \times 0.27} = 0.5 \times 0.18 = 0.09\text{ H}$.
$C = \frac{1}{3M\omega²} = \frac{1}{3(0.09)(1000/3)^2} = \frac{1}{0.27 \times 1000000/9} = \frac{9}{270000} = \frac{1}{30000}\text{ F} = 33.33~\mu F$.