# Network Functions

 Question 1
A series $R L C$ circuit has a quality factor $Q$ of 1000 at a center frequency of $10^{6} \mathrm{rad} / \mathrm{s}$. The possible values of $R, L$ and $C$ are
 A $R=1 \Omega, L=1 \mu \mathrm{H}$ and $C=1 \mu \mathrm{F}$ B $R=0.1 \Omega, L=1 \mu \mathrm{H}$ and $C=1 \mu \mathrm{F}$ C $R=0.01 \Omega, L=1 \mu \mathrm{H}$ and $C=1 \mu \mathrm{F}$ D $R=0.001 \Omega, L=1 \mu \mathrm{H}$ and $C=1 \mu \mathrm{F}$
GATE EC 2023   Network Theory
Question 1 Explanation:
Given: $Q=1000$ and $\omega_{0}=10^{6} \mathrm{rad} / \mathrm{sec}$
We know, for series $R L C$ circuit,
\begin{aligned} Q & =\frac{\omega_{0} L}{R} \\ \omega_{0} & =\sqrt{\frac{1}{L C}} \\ Q & =\frac{1}{\sqrt{L C}} \times \frac{1}{R}=\frac{1}{R} \sqrt{\frac{L}{C}} \end{aligned}

So, $L=1 \mu \mathrm{H}, C=1 \mu \mathrm{F}$ and $R=0.001$
 Question 2
The transfer function $\frac{V_{2}\left ( s \right )}{V_{1}\left ( s \right )}$ of the circuit shown below is
 A $\frac{0.5s+1}{s+1}$ B $\frac{3s+6}{s+2}$ C $\frac{s+2}{s+1}$ D $\frac{s+1}{s+2}$
GATE EC 2013   Network Theory
Question 2 Explanation:
\begin{aligned} \frac{V_{2}(s)}{V_{1}(s)}&=\frac{10 \times 10^{3}+\frac{1}{100 \times 10^{-6} s}}{10 \times 10^{3}+\frac{1}{100 \times 10^{-6} s}+\frac{1}{100 \times 10^{-6} s}} \\ \frac{V_{2}(s)}{V_{1}(s)}&=\frac{s \times 10^{4}+10^{4}}{s \times 10^{4}+10^{4}+10^{4}}=\frac{10^{4}(1+s)}{10^{4}(s+2)} \\ \frac{v_{2}(s)}{V_{1}(s)}&=\frac{s+1}{s+2} \end{aligned}

 Question 3
If the transfer function of the following network is $\frac{V_o(s)}{V_i(s)}=\frac{1}{2+sCR}$

The value of the load resistance $R_{L}$ is
 A $\frac{R}{4}$ B $\frac{R}{2}$ C R D 2R
GATE EC 2009   Network Theory
Question 3 Explanation:

\begin{aligned} Z &=\frac{R_{L}}{1+S R_{L} C} \\ H(s) &=\frac{Z}{Z+R}=\frac{R_{L}}{\left(R_{L}+R\right)+S R R_{L} C} \\ \text{if}\quad R &=R_{L} \\ H(s) &=\frac{1}{2+s R C} \end{aligned}
 Question 4
The driving point impedance of the following network is given by $Z(s)=\frac{0.2s}{s^{2}+0.1s+2}$

The component values are
 A L = 5 H,R = 0.5 $\Omega$,C = 0.1 F B L = 0.1H,R = 0.5 $\Omega$,C = 5 F C L = 5 H,R = 2 $\Omega$,C = 0.1 F D L = 0.1H,R = 2 $\Omega$,C = 5 F
GATE EC 2008   Network Theory
Question 4 Explanation:
\begin{aligned} Z(s) &=\frac{0.2 s}{s^{2}+0.1 s+2} \\ r(s) &=\frac{s^{2}+0.1 s+2}{0.2 s} \\ &=\frac{s}{0.2}+\frac{1}{2}+\frac{2}{0.2 s} \\ &=5 s+0.5+\frac{10}{s}\\ \text{Comparing with}\\ n(s) &=C s+\frac{1}{R}+\frac{1}{L s} \\ C &=5 F, R=\frac{1}{0.5}=2 \Omega \\ L &=\frac{1}{10}=0.1 \mathrm{H} \end{aligned}
 Question 5
Two series resonant filters are as shown in the figure. Let the 3-dB bandwidth of Filter 1 be $B_1$ and that of Filter 2 be $B_2$. The value $\frac{B_{1}}{B_{2}}$ is
 A 4 B 1 C $1/2$ D $1/4$
GATE EC 2007   Network Theory
Question 5 Explanation:
Bandwidth of series RLC circuit is R/L
Bandwidth of filter 1; $B_{1}=\frac{R}{L_{1}}$
Bandwidth of filter 2; $B_{2}=\frac{R}{L_{2}}$
$=\frac{R}{L_{1} / 4}=\frac{4 R}{L_{1}}$
So,$\frac{B_{1}}{B_{2}}=\frac{1}{4}$

There are 5 questions to complete.