Hello,

characteristic impedance of a finite length of line at a given frequency can be derived by taking two impedance measurements, one with the end open circuit and the other with the end short circuited. From these, Zo is calculated as follows:

Z_0 = SQRT ( Z_short * Z_open )

Anyone knows how to derive this equation ?

Most standard textbooks on line transmission cover this. It is derived from the T equivalent circuit of the transmission line, having total series impedance of Z1 and shunt impedance z2, correctly terminated in Z0.

Looking from the left end as the input with the right end short circuited, the input impedance is

Zsc = Z1/2 + Z1*Z2/(Z1 + 2*Z2) .............1

Open circuited, the input impedance at left is

Zoc = Z1/2 + Z2 ......................................2

The Characteristic Impedance is given by

Z0 = SQRT( Z1*Z2 + Z1^2/4) ..................3

which needs a separate proof.

If equations 1 and 2 are multiplied together you get

Zoc * Zsc = Z1^2/4 + Z1 * Z2 = Z0^2

Thus Z0 = SQRT (Zsc * Zoc)

My textbook is an ancient Everitt "Communication Engineering" written with a quill pen on parchment.

Here is a quick way to test.

Connect to your network analyzer or impedance meter. Find the frequency at which the cable is 1/4 or 1/2 wavelength. Then adjust your network analyzer or impedance meter to a frequency so that your cable is 1/8 wavelength.

Note the reactance when you short or open the far end will be equal and opposite. The magnitude of reactance you measure equals the characteristic impedance. Example, you measure 0.5 +j42 and 0.5 -j42 you now know that the cable Zo is 42.

On a vector network analyzer you can do this really quickly without need to KNOW your frequency. Set the analyzer to 0 Hz sweep and polar display. Keep alternating between shorted and open at the far end and watch the display. Vary the frequency until you see that the capacitive reactance is the same as the inductive reactance (the 'dots' on the CRT will be exactly vertically aligned) as you switch from short to open. Then read the reactance and you have the answer.

For example I have found rolls of RG58 as low as 42 ohm and as high as 62 ohm. It is a lesson to never pay attention to "RG" numbers ever again.

thank you nubbage and wb9jtk !