Question

A lossless transmission line of characteristic impedance $150\Omega$, phase velocity $v_p=2\times {10}^8\text{m/sec}$ and length l = 0.5 m is terminated by a load impedance $25\Omega$. If the operating frequency of the line is 50MHz, then the magnitude of input impedance is

• A. $25\Omega$
• B. $150\Omega$
• C. $900\Omega$
• D. $25\Omega$

Answer 1

The correct option is B $150\Omega$

Given:
$Z_0=150\Omega$
$Z_R=25\Omega$
$v_p=2\times {10}^8\text{msec}$
l = 0.5 m
f = 50 MHz

wavelength $\lambda =\frac{v_p}{f}=\frac{2\times {10}^8}{50\times {10}^6}=4m$
hence $l=\frac{\lambda }{8}$
Input impedance of $l=\frac{\lambda }{8}$ transmission line is given by
$Z_{in}=Z_0\left(\frac{Z_R+j{Z}_0}{Z_0+j{Z}_R}\right)=150\left(\frac{25+j150}{150+j25}\right)$
$|Z_{in}|=150\left(\frac{\sqrt{{25}^2+{150}^2}}{\sqrt{{150}^2+{25}^2}}\right)=150\Omega$ (Characteristic impedance)
Note: When a loseless transmission line of length $l=\frac{\lambda }{8}$ is terminated by a purely resistive load ($R_R$), then the magnitude of input impedance of characteristic impedance ($R_0$)