Page 311 - Analog and Digital Filter Design

308 Analog and Digital Filter Design

dimensions of much less than a quarter wavelength at the passband cutoff fre-
quency. The width of a track on a printed circuit board defines its impedance.
Sections of track wider or narrower than the 50Q line become capacitive or
inductive, respectively. Concatenation of narrow and wide track sections can
therefore form an LC filter, with the length of track being proportional to the
reactance of the equivalent inductor or capacitor.

Transmission lines as Filters

Transmission lines are often modeled as lumped elements of series inductors
and shunt capacitors. This is a good model for our purposes. Another way of
thinking about transmission lines is as a delay.

Consider for a moment a sinusoidal wave applied at one instant to one end
of an open circuit coaxial cable. The cable has certain impedance, say 50!2,
so a signal with amplitude of 1 V will produce a current flow of 20mA in the
cable. This current flows towards the other end of the cable, which is open
circuit, so when it arrives there it is reflected back towards the source: it has
nowhere else to go. The reflected wave has a voltage amplitude peak approxi-
mately equal to the incident voltage peak. Now suppose a second sinusoidal
wave is applied just as the start of the first wave is reflected back. If the reflected
wave has the opposite polarity to the second wave, the two signals will cancel
each other to give zero volts at the cable input. The input impedance will be
effectively zero.

Thus, if a continuous sine wave signal is applied to an open-circuit coaxial
cable, which has a length such that reflected signals are equal and opposite to
the incident signal, the input impedance will be zero. This critical length is a
quarter wavelength. The signal transmission time to the end of the cable and
back is exactly one half cycle. Therefore, at the cable input, the reflected signal
is inverted compared with the incident signal. Also, any odd multiples of quarter
wavelengths are critical lengths. Multiples of quarter wavelengths are not so
effective at creating low impedance. This is because the cable has loss and
reflected signals have lower amplitude than the incident signal.
Now consider the opposite effect, a short-circuited coaxial cable. A sinusoidal
wave is applied across one end to produce a current that flows towards the short
circuit. When the signal current arrives at the short circuit it returns back along
the other conductor, reversing the polarity of the signal at that point. The inci-
dent positive voltage is cancelled by the reflected negative voltage, giving zero
volts at the short circuit (as you would expect). As with the open-circuit example,
the critical length for a short-circuited coaxial cable is a quarter wavelength. The
applied signal is delayed by a quarter wavelength in each direction along the
cable. The signal is also inverted by the short circuit. Overall, the reflected wave