Characteristic impedance of a transmission line is 50 Ω Input impedance of open circuited line is Zoc = 100 + j150 Ω. The value of short circuited input impedance is

Concept:

For a loss-free, uniform transmission line of characteristic impedance \(Z_0\) and length \(l\), the input impedances with the far end open-circuited and short-circuited are related by

\( Z_{\text{sc}} \;=\; \dfrac{Z_0^{\,2}}{\,Z_{\text{oc}}\,}\;.\)

The ratio arises because the open- and short-circuit impedances are complementary (their product equals \(Z_0^{\,2}\)) at the same electrical length.

Given:

• \(Z_0 = 50\;\Omega\)
• \(Z_{\text{oc}} = 100 + j150\;\Omega\)

Calculation:

Compute the short-circuit input impedance:

\( Z_{\text{sc}} = \frac{Z_0^{\,2}}{Z_{\text{oc}}} = \frac{50^{2}}{100 + j150} = \frac{2500}{100 + j150}. \)

Multiply numerator and denominator by the complex conjugate of the denominator:

\( Z_{\text{sc}} = \frac{2500(100 - j150)} {(100 + j150)(100 - j150)} = \frac{2500\,(100 - j150)} {100^{2} + 150^{2}} = \frac{2500\,(100 - j150)} {10\,000 + 22\,500} = \frac{2500\,(100 - j150)} {32\,500}. \)

Simplifying,

\( Z_{\text{sc}} = \bigl(7.69 - j\,11.54\bigr)\;\Omega. \)