In the circuit shown, all the transmission line sections are lossless. The Voltage Standing Wave Ratio (VSWR) on the 60 Ω line is

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GATE EC 2010 Official Paper

Option 2 : 1.64

__Concept:__

The below diagram shows the transmission line along with the generator:

The input impedance of a transmission line is defined as:

\(Z_{in} = Z_0\left (\frac{Z_L+jZ_0 tanh\gamma l}{Z_0+jZ_Ltanh\gamma l} \right)\)

\(Z_{in} = Z_0\left (\frac{Z_L+jZ_0 tanβ l}{Z_0+jZ_Ltanβ l} \right)\)

where \(β = \frac{2\pi}{λ}\)

The below table represents the behaviour of the Transmission line due to variation in the length of line 'l'

length (l) |
Input impedance |

l = ∞ | Z_{in} = Z_{0} |

l = λ | Z_{in} = Z_{L} |

l = λ/2 | Z_{in} = Z_{L} |

l = λ/4 | \(Z_{in}=\frac{Z_0^2}{Z_L}\) |

l = λ/8 | \(Z_{in}=Z_0\left (\frac{Z_L+jZ_0}{Z_0+jZ_L} \right)\; |Z_L| = Z_0\) |

__Calculation:__

From the given transmission line system, first, we calculate the input impedance to the quarter-wave line (l = λ/4)

\(Z_{i1}=\frac{Z_0^2}{Z_L}=\frac{(30\sqrt 2)^2}{30}\)

\(Z_{i1}=\frac{(30\times 30\times 2)}{30}= 60 Ω\)

For a **short-circuited **line:

Z_{i2} = jZ_{0} tan(βl)

For the length l = λ/8, the parameter is:

\(tan(\beta l)= tan(\frac{2\pi}{\lambda}\frac{\lambda}{8}) = 1\)

**Zi2 = j30 Ω **

For Z0 = 60 Ω line

Z_{L} = Z_{i1} + Z_{i2}

Z_{L} = (60 + j30) Ω

Reflection coefficient

\(\Gamma = \frac{Z_L - Z_0}{Z_L +Z_0}\)

\(\Gamma = \frac{60+j30-60}{60+j30+60}= \frac{j30}{120+j30}\)

\(|\Gamma | = \frac{30}{\sqrt {{{120}^2} + {{30}^2}} } = 0.2425\)

\(VSWR = \frac{1 + |\Gamma |}{1 - |\Gamma |} = \frac{1.2425}{0.7575}\)

**VSWR = 1.64**