GATE EC 2014 Electromagnetics Solution Q 5

GATE EC 2014 Electromagnetics Solution Q 5

The input impedance of a λ/8 section of a lossless transmission line of characteristic impedance 50 Ω is found to be real when the other end is terminated by a load ZL = (R + jX)Ω. If X is 30Ω , the value of R (in Ω) is _.

Ans :- 40

Explanation

Z_{in}=Z_0\left(\frac{Z_L+jZ_0\;\tan\beta l}{Z_0+jZ_L\;\tan\beta l}\right) l=\frac\lambda8 \Rightarrow\beta l=\frac{2\pi}\lambda\times\frac\lambda8=\frac\pi4 \therefore\tan\beta l=\tan\frac\pi4=1 \therefore Z_{in}=Z_0\left[\frac{Z_L+j50}{Z_0+jZ_L}\right] Z_{in}=50\left[\frac{R+j30+j50}{50+j(R+j30)}\right] =50\left[\frac{R+j80}{(50-30)+jR}\right] =50\left[\frac{R+j80}{20+jR}\right] =50\left[\frac{(R+j80)(20-jR)}{20^2+R^2}\right]

Imaginary part of Z_{in}=0. since only real part of Z_{in} exists.

\therefore Z_{in}=50\left[\frac{20R+1600j-jR^2+50R}{20^2+R^2}\right] \therefore Z_{imaginary}=0=\frac{1600-R^2}{20^2+R^2} \begin{array}{l}\therefore R^2=1600\\R=40\;\Omega\end{array}

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