Page 314 - Analog and Digital Filter Design

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Transmission Lines and Printed Circuit Boards as Filters

lines are anti-resonant at multiples of a half wavelength and resonant at odd
multiples of a quarter wavelength. More details can be found in Helszajn' or
Woiff and Kaul.'

The basic design process is to decide the frequency where maximum attenuation
is required, that is, a zero in the frequency response. The open- and short-circuit
lines (stubs) should be a quarter wavelength long at this frequency. These stubs
should be connected to a transmission line having impedance equal to the in-
put and output impedance of the filter. It is not necessary to space the stubs a
quarter wavelength apart, though.

For example, suppose the requirement is for a passband to 100MHz but 200 MHz
must be stopped: the lines must all be a quarter wavelength at 200MHz. The
equations for inductance and capacitance are simplified, as follows:

[; 3 [:I
wL=Z,tan -.- =&tan -

The ratio of passband to stopband frequency (dwQ) was deliberately chosen to
be I/? to simplify the math because, conveniently, tan(d4) = 1.

Find the characteristic impedance of these short- and open-circuit lines by
taking the input and output impedance to be 50R and designing for a 0.25dB
Chebyshev response in the passband. The normalized element values for this
filter are 1.6325, 1.436, and 1.6325 (to four decimal places).

The first and third elements have the same normalized value, so the result will
be the same for both. Let's design for series inductors at either end with a shunt
capacitor in the center. The inductor equivalent line will be designed first.

= Zo, where Z, is the characteristic impedance of the

short circuited line.

OL z
gl = 1.6325 = - where w = 2n x IOb, the passband edge.
=
50 50
Z,, = g, x 50 = 81.6250.
The capacitor equivalent line will be designed now.