Question

A transmission line has an unknown characteristic impedance of Zo. The length L of the transmission line is also unknown. When the transmission line is short circuited (Zi - 0) at one end, the measured input impedance at the other end is Zv Zsc. When the transmission line is open circuited (Zi- oo) at one end, the measured input impedance at the other end is ZiNZoc Find the characteristic impedance Zo of the transmission line.

Answer 1

For a lossy line, Z_IN = Z_0 [Z_L + Z_0 tan h(gamma L)/Z_0 + Z_L tan h(gamma L)] where r = propagation constant -alpha + j beta If the line is loss less, (alpha = 0) gamma = j beta (Therefore Tan h(j beta x) = j Tan(beta x)) Therefore Z_IN = Z_0 [Z_L + jZ_0 tan (beta L)/Z_0 + j Z_C tan(beta L)] If Z_L = 0 Ohm (short circuit) Z_SC = Z_0 [0 + jZ_0 tan(beta L)/Z_0 + 0] = jZ_0 tan(beta L) If Z_L = infinity (open circuit) Z_OC = Z_0 [1 + j Z_0 /Z_C tan(beta L)/Z_0/Z_C_j tan(beta L)] = Z_0 [1/j tan(beta L)] = -jZ_0 cot(beta L) Therefore Z_SC. Z_OC = Z^2_0 Z_0 = squareroot Z_SC. Z_OC