Handbook for Sound Engineers

Filters and Equalizers 787

The impedance may be represented as a single
complex number where the real part is the resistance
and the imaginary part is the reactance. The imaginary
part of a complex number is given by the magnitude
multiplied by the square root of negative one. The math-
ematical notation for this number is i but in engineering,
j is commonly used to avoid confusion in expressions
involving current.

(23-6)

Analyzing a network in term of complex impedance

allows the calculation of both magnitude and phase at

any frequency according to

(23-7)

(23-8)

where,

T is the phase angle of the complex number,

A is the magnitude of the complex number.

23.2.1.1 Capacitive Networks

The capacitor has impedance that approaches a short

circuit at high frequency and an open circuit at low

frequency. The reactance of a capacitor is given by:

(23-9)

where;

Xc is the capacitive reactance,

f is the frequency in hertz,

C is the capacitance in farads.

If a capacitor is connected in series with the signal

path as in Fig. 23-3A, the capacitor and the resistor

form a potential divider. Low frequencies will be atten-

uated as the impedance of the capacitor increases at

lower frequencies.

(23-10)

The cutoff frequency of this filter is at the frequency

where R=|ZC|, so substituting into Eq. 23-8, we find

(23-11)

Using the complex analysis in Eq. 23-8, we can deter-

mine the phase at this frequency.

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

0.707 or 3 dB and the phase angle is  45 q.

If a capacitor is connected in parallel with the signal

path as in Fig. 23-3B, the capacitor and the resistor form

a potential divider. High frequencies will be attenuated

as the impedance of the capacitor reduces at higher

frequencies.

(23-12)

The cutoff frequency of this filter is at the frequency

where R=|ZC|, so substituting into Eq. 23-10, we find

that again

(23-13)

Using the complex analysis in Eq. 23-10, we can

determine the phase at this frequency.

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

0.707 or 3 dB and the phase angle is +45q.

Figure 23-3. Simple filter networks using only a capacitor

and a resistor.

ZRjX+=

T tan

1– imaginary

real

= ©¹§·------------------------

Ainary= imag^2 +real^2

XC^1

2 SfC

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

A. High pass. B. Low pass.

C

R C

R

Vout Vin R

RZ+ c

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

f^1

2 SRC

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

Vout

VinuR

RjR+

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

1

1 +j

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

1 – j

2

=----------

Vout Vin

Zc

RZ+ C

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

f^1

2 SRC

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

Vout

VinujR

RjR+

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

j

1 +j

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

1 +j

2

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