Page 236 - Analog and Digital Filter Design

Impedance Matching Networks 233

sidering the conversion from a series RL circuit into its parallel equivalent, and
vice versa. Also the conversion from series RC to parallel RC, and vice versa.
This transformation could provide a useful simplification to the mathematics of
an impedance matching circuit where, perhaps, the equivalent series reactance
of a parallel RC load needs to be found.

First find the circuit Q. For series circuits this is the reactance divided by the
resistance. For parallel circuits this is the resistance divided by the reactance.
The Q is equal to tan(@, where 8 is the phase angle of the impedance. The
following equations summarize these statements.

The relationship between series and parallel resistance is given by the equation:

ZS
Rp = R,(l+Q') = -
COS0'

The relationship between series and parallel reactance is given by the equation:

The equivalent parallel or series model is only valid at one particular frequency.
This is simply because the reactive element changes with frequency and, hence,
so does the circuit Q. However, impedance matching circuits are also only valid
for one particular frequency; therefore this is not an issue.

Matching Using L, T, and PI Networks
Networks that comprise two or three reactive components can be constructed
to provide narrowband matching. The networks are described by the shape of
the components, as drawn on a circuit diagram. Thus L networks use two reac-
tive components; the L represents a shunt branch followed by a series branch,
or a series branch followed by a shunt branch (a little imagination may be needed
here, the L is upside-down!). Both T and PI (IT) networks require three com-
ponents and represent either series, shunt, series branches; or shunt, series, shunt
branches. With T and PI networks the physical component layout corresponds
closely to the symbol. The four configurations are given in Figure 8.8. Each con-
figuration can be highpass or lowpass, depending on whether the series elements
are capacitors or inductors (and hence whether the shunt elements are induc-
tors or capacitors).