Handbook for Sound Engineers

788 Chapter 23

23.2.1.2 Inductive Networks

The inductor has impedance that approaches an open
circuit at high frequency and a short circuit at low
frequency. The reactance of an inductor is given by

(23-14)

where,
XL is the inductive reactance in ohms,

f is the frequency in hertz,

L is the inductance in henrys.

Inductors are prone to parasitic resistances especially
for large inductances where a long coil is wound. In a
large value inductor, the parasitic resistance is reduced
by using heavier gauge wire, which causes the size to
grow rapidly as the inductance becomes larger. The full
expression for the impedance of an inductor is

(23-15)

where,

ZL is the impedance of the inductor,

RL is the dc resistance of the inductor.

If an inductor is connected in series with the signal
path as in Fig. 23-4B, the inductor and the resistor form
a potential divider. High frequencies will be attenuated
as the impedance of the inductor increases at higher
frequencies.

(23-16)

The cutoff frequency of this filter is at the frequency
where R=ZL, so substituting into Eq. 23-14, we find
that

(23-17)

Using the complex analysis in Eq. 23-10 and
ignoring the parasitic resistance, we can determine the
phase at this frequency.

So according to Eqs. 23-7 and 23-8 magnitude is 0.707

or 3 dB and the phase angle is  45 q. Note that this is

the opposite phase angle to the capacitor-based

low-pass filter.

23.2.2 Second-Order L-Type Networks

An L-type filter consists of an inductor in series with a

capacitor, with the outputs across one or more of the

components. Since there are two reactive elements in

the circuit, it forms a second-order filter with a roll-off

of 12 dB per octave. There are two configurations of

this network.

The insertion loss for the low-pass configuration as

shown in Fig. 23-5A is given by

. (23-18)

The insertion loss for the high-pass configuration as

shown in Fig. 23-5B is given by

(23-19)

where,

fc is the frequency of a 3 dB insertion loss,

f is any frequency,

ILdB is the insertion loss in decibels.

These configurations are commonly used in basic

loudspeaker crossover networks as in Fig. 23-6. Both a

high-pass and a low-pass response may be derived from

the same circuit. The L-type filter in this application

presents constant impedance to the input port. The

Figure 23-4. Filters using only an inductor and a resistor.

XL= 2 SfL

ZL RL+= j 2 SfL

A. High pass. B. Low pass.

R

R

L

L

Vout Vin

Zc

RZ+ L

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

f L

2 SR

= ----------

Figure 23-5. Two configurations of L-type filters.

Vout

VinuR

RjR+

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

j

1 +j

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

1 – j

2

=----------

C

C

A. High-frequency

attenuated, inverted

L-type filter.

B. Low-frequency

attenuated, inverted

L-type filter.

L

L

ILdB 10 1

f

fc

---

©¹

= log +§·^2

ILdB 10 1

fa

f

©¹§·--- -

2

= log +