Handbook for Sound Engineers

Filters and Equalizers 789

impedance of the inductor Eq. 23-12 and the capacitor
Eq. 23-7 vary with frequency and are chosen such that,
at the crossover frequency, their impedances equal the
characteristic impedance, Z 0. Each port is in parallel
with a load, which for simplicity of analysis we will
consider to be constant and of value Z0:

(23-20)

(23-21)

where,

Z 0 is the circuit impedance,

Fx is the crossover frequency.

The total impedance at the input is

. (23-22)

When the frequency is very much lower than the
crossover frequency, the value of ZL becomes 2SfxL,
which is very small. At the same time, the value of ZC
becomes Z 0 as 2 SfxC becomes smaller. The total imped-
ance becomes Z 0.

At the crossover frequency, the inductor and capac-
itor impedances equal Z 0 , so the total circuit impedance
also equals Z 0.

When the frequency is very much higher than the
crossover frequency, the value of ZL becomes Z 0 as
2 SfxL becomes larger. At the same time, the value of ZC
becomes 1 / (2S fxC), which is small. The total imped-
ance becomes Z 0.

23.2.3 T and SNetworks

T and S networks are classes of constant-k filters. They

are formed by combing L-type filters with one leg being

in common. The line impedance Z 0 is a critical param-

eter in the design of these filters. The impedance

presented by a T network to the input and output trans-

mission lines is symmetrical and is designated ZT. This

impedance is equal to the line impedance in the pass-

band and progressively decreases in the stop band. The

impedance presented by a S network to the transmission

lines is also symmetrical and is designated ZP. This

impedance is equal to the line impedance in the pass-

band and progressively increases in the stop band.

The full T and S networks have twice the attenuation

of the L-type half sections.

23.2.3.1 Low Pass

A T-type low-pass filter has two inductances: L 1 in

series with the line and a capacitance C 2 in parallel. As

frequency increases, the inductive reactance increases,

presenting an increasing opposition to transmission. As

frequency increases, capacitive reactance reduces, so

the parallel capacitor becomes more effective at

shunting the signal to ground. The design equations for

the component values are

(23-23)

(23-24)

where,

fc is the cutoff frequency,

Z 0 is the line impedance.

These equations are the same as for the L-type

network. In the T network, the actual value of the

capacitor is 2C 2 , where the capacitors from two

low-pass L-type networks are combined in parallel. In

the Snetwork, the actual value of the indictor is 2L 1 ,

where the inductors from the L-type network are

combined in series.

23.2.3.2 High Pass

The basic designs of constant-k high-pass filters are

shown in Fig. 23-8. The positions of the inductors and

capacitors are opposite to those in the low-pass case.

The design equations are

Figure 23-6. Passive crossover using an L-type filter.

L

C

Loudspeaker

Loudspeaker

ZL^1

1

2 SfxL

--------------^1

Z 0

©¹§·+-----

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

ZC^1

2 SfxC^1

Z 0

©¹§·+-----

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

Zin ZC+= ZL

C 2 1

2 SfcZ 0

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

L 1

Z 0

2 Sfc

=----------