# Network Theory

 Question 1
A linear 2-port network is shown in Fig. (a). An ideal DC voltage source of 10 V is connected across Port 1. A variable resistance R is connected across Port 2. As R is varied, the measured voltage and current at Port 2 is shown in Fig. (b) as a $V_2$ versus $-I_2$ plot. Note that for $V_2=5V,I_2=0mA$, and for $V_2=4V,I_2=-4mA$.
When the variable resistance R at Port 2 is replaced by the load shown in Fig. (c), the current $I_2$ is _______ mA (rounded off to one decimal place).

 A 3.2 B 6.8 C 4 D 8
GATE EC 2022      Two Port Networks
Question 1 Explanation:

CASE-1:
$I_2=0$
$V_{oc}=5V$

CASE-2: With help of linearity
$\frac{4-5}{4-0}=\frac{I_{sc}-5}{I_{sc}-0}\Rightarrow I_{sc}=20mA$
$R_{th}=\frac{V_{oc}}{I_{sc}}=\frac{5}{20}k=250\Omega =0.25k\Omega$

CASE-3

$I_2=\frac{10-5}{1.25}=4mA$
 Question 2
Consider the circuit shown in the figure with input $V(t)$ in volts. The sinusoidal steady state current $I(t)$ flowing through the circuit is shown graphically (where $t$ is in seconds). The circuit element $Z$ can be ________.

 A a capacitor of 1 F B an inductor of 1 H C a capacitor of $\sqrt{3}$ F D an inductor of $\sqrt{3}$ H
GATE EC 2022      Sinusoidal Steady State
Question 2 Explanation:
$i(t)=\frac{V(t)}{1+Z}$ where, $V(t)= \sin t$
$\because$ Given i(t) is lagging (from plot)
\begin{aligned} I_{max}&=\frac{V_{max}}{Z}=\frac{1}{Z}\\ \Rightarrow Z&=1/\sqrt{2}\\ Z&=(1+j\omega L)\Rightarrow L=1\\ as\; \omega &=1(\sin \omega t)=\sin t \end{aligned}
 Question 3
For the circuit shown, the locus of the impedance $Z(j\omega )$ is plotted as $\omega$ increases from zero to infinity. The values of $R_1$ and $R_2$ are:

 A $R_1=2k\Omega ,R_2=3k\Omega$ B $R_1=5k\Omega ,R_2=2k\Omega$ C $R_1=5k\Omega ,R_2=2.5k\Omega$ D $R_1=2k\Omega ,R_2=5k\Omega$
GATE EC 2022      Sinusoidal Steady State
Question 3 Explanation:
\begin{aligned} Z(j\omega )&=R_1+\frac{R_2\cdot 1/j\omega c}{R_2+ 1/j\omega c}\\ &=R_1+\frac{R_2}{\frac{1}{2}+jR_2\omega c}\\ Z(j\omega )_{\omega =0}&=R_1+R_2\\ \Rightarrow R_1+R_2&=5k\Omega \\ Z|j\omega |_{\omega \rightarrow \infty }&=R_1\\ \Rightarrow R_1&=2k\Omega \\ R_2&=3k\Omega \end{aligned}
 Question 4
Consider the circuit shown in the figure. The current $I$ flowing through the $10\Omega$ resistor is _________.

 A 1A B 0A C 0.1A D -0.1A
GATE EC 2022      Basics of Network Analysis
Question 4 Explanation:
Here, there is no any return closed path for Current (I) . Hence I=0
Current always flow in loop.
 Question 5
The current $I$ in the circuit shown is ________

 A $1.25 \times 10^{-3}A$ B $0.75 \times 10^{-3}A$ C $-0.5 \times 10^{-3}A$ D $1.16 \times 10^{-3}A$
GATE EC 2022      Basics of Network Analysis
Question 5 Explanation:

Applying Nodal equation at Node-A
\begin{aligned} \frac{V_A}{2k}+\frac{V_A-5}{2k}&=10^{-3}\\ \Rightarrow 2V_A-5&=2k \times 10^{-3}\\ V_A&=3.5V\\ Again,&\\ I&=\frac{5-V_A}{2k}\\ &=\frac{5-3.5}{2k}\\ &=0.75 \times 10^{-3}A \end{aligned}
 Question 6
The circuit in the figure contains a current source driving a load having an inductor and a resistor in series, with a shunt capacitor across the load. The ammeter is assumed to have zero resistance. The switch is closed at time t = 0.

Initially, when the switch is open, the capacitor is discharged and the ammeter reads zero ampere. After the switch is closed, the ammeter reading keeps fluctuating for some time till it settles to a final steady value. The maximum ammeter reading that one will observe after the switch is closed (rounded off to two decimal places) is _______________ A.
 A 1.44 B 2.56 C 8.65 D 7.26
GATE EC 2021      Transient Analysis
Question 6 Explanation:
Apply Laplace transform,

\begin{aligned} \frac{1}{s} &=\frac{V(s)}{1 / C s}+\frac{V(s)}{R+L s} \\ \frac{1}{s} &=V(s)\left[C s+\frac{1}{R+L s}\right] \\ V(s) &=\frac{(1 / s)}{C s+\frac{1}{R+L s}} \\ V(s) &=\frac{(1 / s)(R+L S)}{L C s^{2}+R C s+1} \\ I(s) &=\frac{(1 / s)(R+L s)}{L C s^{2}+R C s+1} \cdot \frac{1}{(R+L s)} \\ I(s) &=\frac{1 / L C}{s\left(s^{2}+\frac{R}{L} s+\frac{1}{L C}\right)} \\ L &=10 \mathrm{mH} ; C=100 \mathrm{pF} ; R=5 \times 10^{3} \Omega\\ \xi &=\frac{R}{2} \sqrt{\frac{C}{L}}=\frac{5 \times 10^{3}}{2} \sqrt{\frac{100 \times 10^{-12}}{10 \times 10^{-3}}}=0.25 \\ \text{Max}. \text { over shoot } &=e^{-\pi \xi / \sqrt{1-\xi^{2}}}=0.44 \end{aligned}
Maximum value = Steady state + Max. overshoot
$=1+0.44$
$=1.44 \mathrm{~A}$
 Question 7
In the circuit shown in the figure, the switch is closed at time $t=0$ , while the capacitor is initially charged to $-5\:V (i.e., v_{c}(0)=-5V)$ .

The time after which the voltage across the capacitor becomes zero (rounded off to three decimal places) is ________________ $\text{ms}$.
 A 0.258 B 0.842 C 0.139 D 0.241
GATE EC 2021      Transient Analysis
Question 7 Explanation:
\begin{aligned} V_{c}\left(0^{-}\right)&=-5 \mathrm{~V} \\ V_{c}\left(0^{+}\right)&=-5 \mathrm{~V} \end{aligned}
$t \rightarrow \infty,$ capacitor acts as a O.C.

Write KCL at node
\begin{aligned} \frac{V_{c}(\infty)-5}{250}+\frac{V_{R}}{500}+\frac{V_{c}(\infty)}{250} &=0 \\ V_{c}(\infty) &=\frac{5}{3} \text { Volts } \\ \text { Time constant }(\tau) &=R_{e q} C \end{aligned}

\begin{aligned} I &=\frac{V}{250}+\frac{V_{R}}{500}+\frac{V}{250} \\ V_{R} &=-V \\ I &=\frac{V}{250}-\frac{V}{500}+\frac{V}{250} \\ \frac{V}{I} &=\frac{500}{3} \Omega ; R_{\mathrm{eq}}=\frac{500}{3} \Omega \\ \tau &=\frac{500}{3} \times 0.6 \mu=0.1 \times 10^{-3} \\ v_{c}(t) &=v_{c}(\infty)+\left(v_{c}(0)-v_{c}(\infty)\right) e^{-\sqrt{2}} \\ v_{c}(t) &=\frac{5}{3}+\left(-5-\frac{5}{3}\right) e^{-t / 0.1 \times 10^{-3}} \\ 0 &=\frac{5}{3}-\frac{20}{3} e^{-10000 t} \\ t &=0.1386 \mathrm{msec} \end{aligned}
 Question 8
Consider the two-port network shown in the figure.

The admittance parameters, in siemens, are
 A $y_{11}=2,\:y_{12}=-4,\:y_{21}=-4,\:y_{22}=2$ B $y_{11}=1,\:y_{12}=-2,\:y_{21}=-1,\:y_{22}=3$ C $y_{11}=2,\:y_{12}=-4,\:y_{21}=-1,\:y_{22}=2$ D $y_{11}=2,\:y_{12}=-4,\:y_{21}=-4,\:y_{22}=3$
GATE EC 2021      Two Port Networks
Question 8 Explanation:

Write KCL at $V_{1}$
\begin{aligned} I_{1}&=\frac{V_{1}}{1}+\frac{V_{1}-V_{2}}{1}-3 V_{2} \\ I_{1}&=2 V_{1}-4 V_{2} \\ \text{Write KCL at }V_{2}\qquad I_{2}&=\frac{V_{2}}{1}+\frac{V_{2}-V_{1}}{1}\\ I_{2}&=-V_{1}+2 V_{2} \\ [y]&=\left[\begin{array}{cc} 2 & -4 \\ -1 & 2 \end{array}\right] \mho \end{aligned}
 Question 9
The switch in the circuit in the figure is in position P for a long time and then moved to position Q at time t=0.

The value of $\dfrac{dv\left ( t \right )}{dt}$ at $t=0^{+}$ is
 A $0\:V/s$ B $3\:V/s$ C $-3\:V/s$ D $-5\:V/s$
GATE EC 2021      Transient Analysis
Question 9 Explanation:
Inductor and capacitors are connected to the inductance source for a long time, so these elements have reached steady state.

\begin{aligned} i_{L}\left(0^{-}\right) &=\frac{20}{5 \mathrm{k} \Omega+5 \mathrm{k} \Omega+10 \mathrm{k} \Omega}=1 \mathrm{~mA} \\ \mathrm{~V}\left(0^{-}\right) &=10 \mathrm{~V} \\ t &=0^{+} \end{aligned}

\begin{aligned} i\left(0^{+}\right)+\frac{10}{5 k}+1 \mathrm{~m} \mathrm{~A} &=0 \\ i\left(\mathrm{O}^{+}\right) &=-3 \mathrm{~mA} \\ C \frac{d V\left(\mathrm{O}^{+}\right)}{d t} &=-3 \mathrm{~mA} \\ 1 \times 10^{-3} \frac{d V\left(\mathrm{O}^{+}\right)}{d t} &=-3 \mathrm{~mA} \\ \frac{d V\left(\mathrm{O}^{+}\right)}{d t} &=-3 \mathrm{~V} / \mathrm{s} \end{aligned}
 Question 10
Consider the circuit shown in the figure.

The value of $v_{0}$ (rounded off to one decimal place) is _________ V.
 A 0.5 B 0.8 C 2 D 1
GATE EC 2021      Basics of Network Analysis
Question 10 Explanation:

Write KVL equation first loop,
\begin{aligned} V_{o}-4+1\left(V_{o}+2\right) &=0 \\ V_{o} &=1 \mathrm{~V} \end{aligned}

There are 10 questions to complete.