Page 250 - Analog and Digital Filter Design

Phase-Shift Networks (Ail-Pass Filters) 247

full and more descriptive name; phase-equalizing all-pass filters. These are
described, in varying detail, in textbooks on filter^.'.^.^ Tomlinson in particular
describes how balanced passive designs can be transformed into unbalanced
ones, which is the usual configuration of equalizers. Balanced equalizers are nor-
mally reserved for use with transmission lines.

Phase-equalizing all-pass filters can be used to increase the delay of signals at
certain frequencies. First-order equalizers have more delay at low frequencies,
but second-order filters can be tuned so their peak delay frequency is selec-
table. When connected in series with an amplitude attenuating filter, like the
Butterworth, the overall proup delay can be much flatter. The amount of resid-
ual ripple in the group delay really depends on how many equalizer sections are
added. Generally, using more equalizing filter sections flattens the overall group
delay.

The all-pass filter increases the complexity and size of the circuit. An equalizer
is built up from first-order and second-order sections connected in series. Thus,
a third-order equalizer comprises a first-order section followed by a second-
order section, and a fourth-order equalizer has two second-order sections con-
nected in series. Odd-order equalizers will always have a first-order section, but
even-order equalizers are comprised of only second-order sections.

A first-order equalizer may be adequate to flatten the group delay of, perhaps.
LIP EO fifth- or sixth-order Butterworth filters. A second-order equalizer may be
suitable for equalizing seventh- and eighth-order filters. The degree of equal-
izer required depends on whether the filter being equalized is a Butterworth,
Chebyshev, or other design. It is also a balance between the amount of ripple
and the complexity of the final circuit.

Passive First- Order Equalizers

The group delay for a first-order equalizer is greatest at low frequencies and is
inversely proportional to frequency. Fortunately, this is almost an exact com-
plement of the group delay for many Iowpass filters.

The circuit for a practical unbalanced first-order all-pass filter is given in Figure
9.3. Unfortunately, it requires a center-tapped inductor. The inductor could be
designed as a transformer having a 1 : I turns ratio, with the start of one winding
connected to the finish of the other. Each “half” of the inductor will have an
inductance of one-quarter of the total inductance, because the mutual coupling
between windings is near enough to unity. The total inductance is thus two self-
inductances plus two mutual inductances of the same value.