Page 182 - Analog and Digital Filter Design

Bandpass Filters 1 79

taken from the tables in Chapter 2. The same value of X must be used for both
components in a single branch. This is because each branch in the lowpass filter
has one component, while branches in the bandpass have two components that
are either series or parallel resonant. Both components in a single branch are
related to a single component value in the lowpass prototype.

It may be helpful to redesign the fifth-order Butterworth filter to illustrate the
use of these formulae. Since it is a symmetrical design, only the first three
branches need to be calculated. As before, R = 50, Fu = (198 + 3.4)kHz =
201.4kHz, FZ= (198 - 3.4)kH~= 194.6kHz.
The first branch has a value X= 0.618 and could be a series arm or a shunt arm.
Taking the shunt arm case first (parallel resonant) gives:

X
CPrrrrrllrl = = 0.61812.136283 x lo6 = 289.3pF
2~.(Fu - FL).R

LPurullcl = (" - FL)*R = 340 x 103/15.218466 x 10" = 2.23413pH
2~ Fu FZX
The second branch has a value X= 1.618. Since the first arm was chosen to be
a shunt arm, this arm must be connected in series. Calculating the values gives:

F" -FL
Cswirs = =6.8 x103/1.992189 x 10" = 341.3pF
2~ F" FL RX
RX
&rrie.s = = 80.9/42,725.66 = 1.8935mH
2a.(F" - FL)

The third branch is a parallel shunt arm, the same as the first branch. This time
the value of X is 2.0. Let's cheat by using the results of the first branch and
multiplying them by a ratio of X, to XI.

C, = 289.3 x 2.010.618 = 936.2pF
L3 = 2.23413 x 0.618/2.0= 0.69035pH, or 690.35nH

The differences between these results and those obtained in Figure 6.5 are due
to round-off errors in the tables of normalized values and during the calcula-
tions. The calculations were done by hand using a calculator. Floating-point
arithmetic in a computer program such as FILTECH achieves more accurate
results.

To obtain the circuit given in Figure 6.6, it is necessary to calculate the series
arm first. This will use a value of X= 0.618.