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GATE EC 2014 Official Paper: Shift 1

Option 2 : \(\frac{{{s_{11}} + {s_{11}}{s_{22}} - {s_{12}}{s_{21}}}}{{1 + {s_{22}}}}\)

CT 1: Ratio and Proportion

2536

10 Questions
16 Marks
30 Mins

__Concept__**: **

Consider a two-port network as shown:

Where,

a_{1} = Incident wave at port 1

b_{1} = reflected wave at port 1

a_{2} = Incident wave at port 2

b_{2} = reflected wave at port 2

Scattering parameters are defined as:

\({s_{11}} = \frac{{{b_1}}}{{{a_1}}},\;\;{s_{12}} = \frac{{{b_1}}}{{{a_2}}}\)

\({s_{21}} = \frac{{{b_2}}}{{{a_1}}}\;and\;{s_{22}} = \frac{{{b_2}}}{{{a_2}}}\)

\({s_{11}} = \frac{{{b_1}}}{{{a_1}}} = \frac{{V_1^ - }}{{V_1^ + }}\)

\({s_{12}} = \frac{{{b_1}}}{{{a_2}}} = \frac{{V_1^ - }}{{V_2^ + }}\)

\({s_{21}} = \frac{{{b_2}}}{{{a_1}}} = \frac{{V_2^ - }}{{V_1^ + }}\)

\({s_{22}} = \frac{{{b_2}}}{{{a_2}}} = \frac{{V_2^ - }}{{V_2^ + }}\)

V^{+} = Incident voltage wave

V^{-} = Reflected voltage wave

Boundary conditions in a Transmission line:

**At port 2:**

\(V_2^ - = - V_2^ + \)

or b_{2} = -a_{2} ---(1)

Now, b_{1} = s_{11}a_{1} + s_{12} a_{2} ---(2)

b_{2} = s_{21} a_{1} + s_{22} a_{2} ---(3)

From (1) and (2), we get:

-a_{2} = s_{21} a_{1} + s_{22} a_{2}

- (1 + s_{22}) a_{2} = s_{21} a_{1}

\({a_2} = \frac{{ - {s_{21}}\;{a_1}}}{{1 + {s_{22}}}}\) ---(4)

From (1) and (4), we get:

\({b_1} = {s_{11}}\;{a_1} + {s_{12}}\left( {\frac{{ - {s_{21}}{a_1}}}{{1 + {s_{22}}}}} \right)\)

\({b_1} = \frac{{\left( {{s_{11}}\left( {1 + {s_{22}}} \right) - {s_{12}}{s_{21}}} \right){a_1}}}{{1 + {s_{22}}}}\)

\({s_{11}} = \;\frac{{{b_1}}}{{{a_1}}} = \frac{{{s_{11}} + {s_{11}}{s_{22}} - {s_{12}}{s_{21}}}}{{1 + {s_{22}}}}\)

Option B is correct.

__Important__**:**

1) If the port is short-circuited:

Reflected voltage wave = - Incident voltage wave

2) If the port is terminated with load, such that Z_{L} = Z_{0}, with Z_{0} the characteristic impedance of that port, then the reflected wave = 0