Handbook for Sound Engineers

Filters and Equalizers 793

23.2.4.4 Elliptical

The elliptical filter has a ripple in both the pass band
and the stop band, with the shortest possible transition
band for the order of the filter with a given ripple. The
ripple in the pass band and the stop band are indepen-
dently controllable. This is a generalized form of the
Butterworth and Chebyshev filters. If the pass band and
stop band ripple is set to zero, we have a Butterworth
filter. If the pass band has a ripple and the stop band
does not, we have a Chebyshev Type I. If the stop band
has a ripple and the pass band does not, we have a
Chebyshev Type II.

The transfer function is the same form as Eq. 23-39
with a different polynomial

(23-43)

where,

E(n,[) is the elliptical polynomial for the order n and

selectivity factor [

23.2.4.5 Normalizing

Normalizing is the process of adjusting the values of
filter components to a convenient frequency and imped-
ance. For analysis, the frequency is usually normalized
to 1 rad s^1 and the impedance to 1:. For designing
practical audio circuits the filter is normalized to 1 kHz
and 10 k:.

23.2.4.6 Scaling

Scaling is the design process of changing the normal-

ized frequency or impedance values for a filter by

varying resistor and capacitor values. Frequency can be

changed relative to the normalized frequency by either

changing all of the resistor values or all of the capacitor

values by the ratio U of the desired frequency to the

normalized frequency. From Eq. 23-11, frequency

varies inversely with the product of the capacitor and

resistor value.

(23-44)

where,

U is the scaling factor,

f 1 is the new frequency.

By multiplying all of the resistor values by a factor,

and dividing all of the capacitor values by that same

factor, we can change the normalized impedance of the

network without changing the RC product, thus keeping

the frequency unchanged.

(23-45)

where,

U is the scaling factor,

Z 1 is the impedance.

23.2.5 Q and Damping Factor

A damping factor, d, or its reciprocal, Q, appears in the

design equation of some filters. The circuit behaves

differently depending on the value of d.

When d is 2, the damping is equivalent to the

isolated resistance-capacitance filters.

When d is 1.41 (square root of 2), the filter is criti-

cally damped and gives maximum flatness without

overshoot.

As d decreases between 1.414 and 0, the overshoot

peak increases in level with its being 1 dB at d= 1.059,

3dB at d= 0.776.

When d is 0, the peak becomes so large that the filter

becomes unstable, and if gain is applied it can become

an oscillator.

23.2.6 Impedance Matching

Source and load impedance have an effect on a passive

filter’s response. They can change the cutoff frequency,

attenuation rate, or Q of the filter. Fig. 23-13 shows the

effects of improper source and load impedance on three

different passive filters. The peaks in the response

before the cutoff frequency lead to a ringing in the filter,

making it potentially unstable at these frequencies. The

bridged T filter is not affected by the impedance

mismatch because of the resistors in the filter; however,

these resistors create an insertion loss.

23.3 Active Filters

Any passive filter may be turned into an active filter by

using amplification at the input and output to provide

the option of gain, Fig. 23-14. This also provides impor-

tant buffering, giving the circuit a high-input impedance

and low-output impedance, guarding the circuit against

external impedance mismatches. This allows active

Hn Z^1

1 H

2

En,]

2 Z

Z 0

+ ©¹§·------

=--------------------------------------------

U

fnorm

f 1

------------=

U

Z 1

Znorm

= -------------