Wire Math

Wire is one of those things we take for granted. Hook it up between
two devices and forget about it. But there is a lot more to know about
wire than just hooking it up. Characteristic impedance is one important
thing to check, not so much for AC power lines, or even loudspeaker
lines, but certainly for video lines and matched impedance lines.

The characteristic impedance of any transmission line is the
impedance that must be connected to the far end of the line in order to
make the input impedance of the line equal to the terminating
impedance. If the line is very long, the input impedance will equal the
characteristic impedance of the line, irrespective of the terminating
impedance.

The characteristic impedance depends on the parameters of the line
and the applied frequency. The resistive component is generally high at
the low frequencies as compared to the reactive component, with the
reactive components taking over with an increase of frequency. The
general equation for all lines is:

If the line is of infinite length, the current in the line is hardly
affected by the value of the terminating impedance at the far end of
the line. If there is 20 dB of attenuation in a line, and the far end
is short-circuited, the characteristic impedance as measured at the
sending end will not be affected by more than 2%.

For shielded and unshielded twisted pairs, the characteristic
impedance is