Page 312 - Analog and Digital Filter Design

30
Transmission Lines and Printed Circuit Boards as Filters

is phase-shifted by 360" compared with the incident wave. The result is the two
signals are in phase.

Consider what happens if a second wave of the same polarity is applied just as
the first wave is reflected back to the line's input. No current can flow because
the source has the same potential across it as the load. Thus a short-circuited
line presents high impedance at the quarter-wavelength frequency. It also pre-
sents high impedance at the three-quarter-wavelength frequency and at further
odd multiples of a quarter wavelength. However, as the cable becomes longer,
the short circuit becomes less effective and the input impedance falls. This
reduced effect is due to attenuation of the signal along the cable. The reflected
wave amplitude will be less than the incident wave so some current will flow into
the cable.
Clearly, the quarter-wavelength line can act as a filter by itself. Consider a line
that has a short-circuit load and is a quarter wavelength long at 100MHz. At
this frequency the cable will present high impedance to signals applied across
the other end. If this line is placed across the antenna input of a broadcast radio
receiver it will allow VHF signals to pass through but will present a low Im-
pedance at frequencies above and below the quarter-wave frequency. This could
be useful, for example, in rejecting high-powered High Frequency (3 MHz to
30 MHz) band transmissions from radio hams that may otherwise overload the
receiver's input stages.

At frequencies below where the cable becomes a quarter-wavelength resonator,
an open-circuit line is capacitive and a short-circuit line is inductive. In fact, an
open-circuit line can be considered to be a series tuned circuit that is operating
below its resonant frequency. Conversely, a short-circuit line can be considered
tG be a parallel tuned circuit that is operating below its resonant frequency.
Richards' equation' gives the relationship between a wrongly terminated trans-
mission line and its equivalent capacitance or inductance.

Open-circuit line

The impedance looking into an open-circuit line is given by the expression:

z,, = -jz, cot(yl)

Z,, is the characteristic impedance of the line, typically 50sL.

yis the line propagation coefficient, given by:

y = J(R + j~ L.)(G + jw C)