# Two Port Networks

 Question 1
For a 2-port network consisting of an ideal lossless transformer, the parameter $S_{21}$ (rounded off to two decimal places) for a reference impedance of 10 $\Omega$, is _____.
 A 1 B 0.5 C 1.5 D 0.8
GATE EC 2020   Network Theory
Question 1 Explanation:
For ideal transformer of n : 1, the scattering matrix is
$\begin{bmatrix} S_{11} &S_{12} \\ S_{21} & S_{22} \end{bmatrix}=\begin{bmatrix} \frac{n^{2}-1}{n^{2}+1} &\frac{2n}{n6{}} \\ & \end{bmatrix}$
$S_{21}=\frac{2n}{n^{2}+1}=\frac{2(2)}{2^2+1}=\frac{4}{5}=0.8$
 Question 2
In the given circuit, the two-port network has the impedance matrix$[Z] =\begin{bmatrix} 40 & 60\\ 60 & 120 \end{bmatrix}$. The value of $Z_L$ for which maximum power is transferred to the load is _________ $\Omega$.
 A 48 B 38 C 40 D 54
GATE EC 2020   Network Theory
Question 2 Explanation:
From maximum power transfer theorem
$Z_{L}=Z_{th}$
$Z_{th}=Z_{22}-\frac{Z_{12}\times Z_{21}}{R_{S}+Z_{11}}$
For given data,
$Z_{th}=120-\frac{60\times 60}{10+40}=48\Omega$
$Z_{L}=48\Omega$
 Question 3
The ABCD matrix for a two-port network is defined by:

$\begin{bmatrix} V_{1}\\ I_{1} \end{bmatrix}=\begin{bmatrix} A & B\\ C & D \end{bmatrix}=\begin{bmatrix} V_{2}\\ - l_{2} \end{bmatrix}$

The parameter B for the given two-port network (in ohms, correct to two decimal places) is _______.
 A 4.8 B 5.5 C 3.4 D 6.2
GATE EC 2018   Network Theory
Question 3 Explanation:
$B=-\left.\frac{V_{1}}{I_{2}}\right|_{V_{2}=0}$
When $V_{2}=0$ (i.e., when port-2 is short circuited)

\begin{aligned} I_{1}&=\frac{V_{1}}{2 \Omega+(5 \Omega \| 2 \Omega)}=\frac{7 V_{1}}{24 \Omega}\\ I_{2}&=-I_{1} \times \frac{5 \Omega}{5 \Omega+2 \Omega}=\frac{-5 V_{1}}{24 \Omega}\\ \text { So, } \quad B&=-\frac{V_{1}}{I_{2}}=\frac{24}{5} \Omega=4.80 \Omega \end{aligned}
 Question 4
The z-parameter matrix $\begin{bmatrix} z_{11} &z_{12} \\ z_{21}& z_{22} \end{bmatrix}$ for the two-port network shown is
 A $\begin{bmatrix} 2 & -2\\ -2&2 \end{bmatrix}$ B $\begin{bmatrix} 2 & 2\\ 2&2 \end{bmatrix}$ C $\begin{bmatrix} 9 & -3\\ 6&9 \end{bmatrix}$ D $\begin{bmatrix} 9 & 3\\ 6&9 \end{bmatrix}$
GATE EC 2016-SET-3   Network Theory
Question 4 Explanation:
Redrawing the circuit,

\begin{aligned} Z_{11} &=\frac{V_{1}}{I_{1}}=3 \| 6=2 \Omega \\ V_{2} &=-3 \times I_{3 \Omega}=-3 \times I_{1} \times \frac{6}{9}=-2 I_{1} \\ \therefore Z_{21} &=\frac{V_{2}}{I_{1}}=-2 \\ \text { Now, } Z_{22} &=\left.\quad \frac{V_{2}}{I_{2}}\right|_{I_{1}=0}=3 \| 6=2 \Omega \\ V_{1} &=-6 \times I_{6 \Omega} \\ &=-6 \times I_{2} \times \frac{3}{9} \\ &=-2 \\ \therefore[Z] &=\left[\begin{array}{cc} 2 & -2 \\ -2 & 2 \end{array}\right] \end{aligned}

 Question 5
The z-parameter matrix for the two-port network shown is
$\begin{bmatrix} 2j\omega & j\omega \\ j\omega & 3+2j\omega \end{bmatrix}$
where the entries are in $\Omega$. Suppose $Z_{b}(j\omega )=R_{b}+j\omega$. Then the value of $R_{b}$ (in $\Omega$) equals ________
 A 1 B 2 C 3 D 4
GATE EC 2016-SET-2   Network Theory
Question 5 Explanation:
For T-network,

\begin{aligned} Z_{11} &=Z_{a}+Z_{c} \\ Z_{22} &=Z_{b}+Z_{c} \\ \text { and, } \quad Z_{12} &=Z_{21}=Z_{c} \\ \text { Given, } &[Z]=\left[\begin{array}{cc} 2 j \omega & j \omega \\ j \omega & 3+2 j \omega \end{array}\right] \\ \text { Therefore } Z_{12} &=j \omega \\ \text { and, } \quad Z_{22} &=3+2 j \omega \\ &=3+j \omega+j \omega \\ &=Z_{b}+Z_{c} \\ &=R_{b}+j \omega+Z_{c} \\ & R_{b}=3 \Omega \end{aligned}
 Question 6
Consider a two-port network with the transmission matrix: $T=\begin{pmatrix} A & B\\ C& D \end{pmatrix}$. If the network is reciprocal, then
 A $T^{-1}=T$ B $T^{2}=T$ C Determinant (T) = 0 D Determinant (T) = 1
GATE EC 2016-SET-1   Network Theory
Question 6 Explanation:
|T|=1
 Question 7
The ABCD parameters of the following 2-port network are
 A $\begin{bmatrix} 3.5+j2 &20.5 \\ 20.5 & 3.5-j2 \end{bmatrix}$ B $\begin{bmatrix} 3.5+j2 &30.5 \\ 0.5 & 3.5-j2 \end{bmatrix}$ C $\begin{bmatrix} 10 &2+j0 \\ 2+j0 & 10 \end{bmatrix}$ D $\begin{bmatrix} 7+j4 &0.5 \\ 30.5 & 7-j4 \end{bmatrix}$
GATE EC 2015-SET-3   Network Theory
Question 7 Explanation:

\begin{aligned} \left[\begin{array}{l} V_{1} \\ I_{1} \end{array}\right] &=\left[\begin{array}{ll} A & B \\ C & D \end{array}\right]\left[\begin{array}{l} V_{2} \\ I_{2} \end{array}\right] \\ \text { When } I_{2} &=0 \\ V_{1} &=(5+j 4+2) I_{1} \\ V_{2} &=2 I_{1} \\ A &=\left.\frac{V_{1}}{V_{2}}\right|_{I_{2}=0}=\frac{7+j 4}{2}=3.5+j 2 \\ C &=\left.\frac{I_{1}}{V_{2}}\right|_{I_{2}=0}=\frac{1}{2}=0.5 \end{aligned}
Option (B) matches. From here only
 Question 8
The 2-port admittance matrix of the circuit shown is given by

 A $\begin{bmatrix} 0.3 & 0.2\\ 0.2& 0.3 \end{bmatrix}$ B $\begin{bmatrix} 15 & 5\\ 5& 15 \end{bmatrix}$ C $\begin{bmatrix} 3.33 & 5\\ 5& 3.33 \end{bmatrix}$ D $\begin{bmatrix} 0.3 & 0.4\\ 0.4& 0.3 \end{bmatrix}$
GATE EC 2015-SET-2   Network Theory
Question 8 Explanation:

\begin{aligned} V_{1} &=6 I_{1}+4 I_{2} \\ V_{2} &=4 I_{1}+6 I_{2} \\ [Z] &=\left[\begin{array}{cc} 6 & 4 \\ 4 & 6 \end{array}\right] \\ Y&=\frac{1}{20}\left[\begin{array}{rr} 6 & -4 \\ -4 & 6 \end{array}\right]=\left[\begin{array}{rr} 0.3 & -0.2 \\ -0.2 & 0.3 \end{array}\right] \\ &\text { Ignoring negative sign: } \\ [Y]&=\left[\begin{array}{cc} 0.3 & 0.2 \\ 0.2 & 0.3 \end{array}\right] \end{aligned}
 Question 9
For the two-port network shown in the figure, the impedance (Z) matrix (in $\Omega$) is
 A $\begin{bmatrix} 6 & 24\\ 42&9 \end{bmatrix}$ B $\begin{bmatrix} 9 & 8\\ 8&24 \end{bmatrix}$ C $\begin{bmatrix} 9 & 6\\ 6&24 \end{bmatrix}$ D $\begin{bmatrix} 42 & 6\\ 6&60 \end{bmatrix}$
GATE EC 2014-SET-4   Network Theory
Question 9 Explanation:
converting II-network to Y-network, we get

$\therefore \quad$ Z -parameter
\begin{aligned} Z_{11}&=3 \Omega+6 \Omega=9 \Omega \\ Z_{12}&=Z_{21}=6 \Omega \\ Z_{22}&=18 \Omega+6 \Omega=24 \Omega \\ \left[Z\right]&=\left[\begin{array}{cc}9 & 6 \\ 6 & 24\end{array}\right] \end{aligned}
 Question 10
Consider the building block called 'Network N ' shown in the figure. Let C = 100 $\mu$F and R = 10 k$\Omega$.

Two such blocks are connected in cascade as shown in the figure.

The transfer function $\frac{V_{3}(s)}{V_{1}(s)}$ of the cascaded network is
 A $\frac{s}{1+s}$ B $\frac{s^{2}}{1+3s+s^{2}}$ C $(\frac{s}{1+s})^{2}$ D $\frac{s}{2+s}$
GATE EC 2014-SET-3   Network Theory
Question 10 Explanation:
$I_{2}(\mathrm{s})$
$\begin{array}{l} \left(R+\frac{1}{C s}\right) I_{1}(s)=R I_{2}(s)=V_{1}(s) \qquad \ldots(i)\\ \left(R+\frac{1}{C s}\right) I_{2}(s)-R I_{2}(s)=0 \qquad \ldots(ii) \end{array}$
$V_{3}(s)=I_{2}(s) \times R \qquad \ldots(iii)$
\begin{aligned} &=\left(R+\frac{1}{C s}\right) \times\left(1+\frac{1}{R C s}\right) I_{2}(s)-R I_{2}(s)=V_{1}(s) \\ &=\left(\frac{R C s+1}{C S}\right)\left(\frac{R C s+1}{R C s}\right) I_{2}(s)-R I_{2}(s)=V_{1}(s) \\ \text{or } I_{2}(s) &=\frac{R C^{2} s^{2} V_{1}(s)}{(1+R C s)^{2}-R^{2} C^{2} s^{2}} \;\ldots(iv) \end{aligned}
\begin{aligned} \frac{V_{3}(s)}{V_{1}(s)}&=\frac{s^{2} R^{2} C^{2}}{1+3 R C s+s^{2} R^{2} C^{2}} \\ \because \qquad R&=10 \mathrm{k} \Omega, C=100 \mu \mathrm{F} \text{ and } R C=1 \\ \frac{V_{3}(s)}{V_{1}(s)}&=\frac{s^{2}}{1+3 s+s^{2}} \end{aligned}