Handbook for Sound Engineers

Consoles 879

Referring again to Fig. 25-59, a slightly different
light shines. The two reactances are working in direct
opposition to each other with the inductive reactance
trying to cancel the capacitative reactance and vice
versa. Arithmetically, it is surprisingly simple with the
two opposing reactance values directly subtracting from
each other; the combination network behaves as a single
reactance of the same reactive character as the one
predominant in the network.
For example, if for a given frequency, the inductive
reactance is a +1.2 k: (the + indicating the phase shift
character of inductance) and the capacitive reactance is
1.5 k:, then the effective reactance of the entire
network is that of a capacitor of  300 : reactance.

25.11.5 Resonance

Resonance is the strange state where the reactances of

both the L and C are equal. For any inductor-capacitor

pair at resonance, the two reactances will be equal. If

you subtract two equal numbers, the answer is zero. So,

for the series tuned-circuit arrangement of Fig. 25-59A

at resonance, there is no impedance. The two reactances

have canceled themselves out. It is a short circuit at that

one frequency of resonance, disallowing component

losses and is, in effect, a frequency-selective short

circuit. Either side of that frequency, of course, one or

the other of the reactances becomes predominant again.

25.11.6 Resonant Q

Like the single-order networks, there is an infinite
number of combinations of C and L at any given
frequency that will achieve resonance (i.e., the two
reactances are equal). Similarly, it is the scale of imped-
ance that alters with such value changes; the magnitude
and rate of change of reactance on either side of reso-
nance (off tune) hinges on the chosen combination.
At resonance, although the two reactances negate
each other, they both still individually have their orig-
inal values. Off resonance, their actual reactances
matter. If each of the reactances is 400: at resonance,
then 10% off tune either way they are going to become
440 : and 360:, respectively. A 10% change in this
instance equates to about a 40: change either way, up
or down. Now imagine that a smaller capacitor and a
larger inductor were used to obtain the same resonant
frequency. Their reactances will be correspondingly
larger. If they’re five times larger with reactances of
2k: each, then at 10% off tune their reactances will
become 2.2 k: and 1.8 k: or 200: change each. The

higher the network impedance, the more dramatic the

reactance shift off tune.

On its own, the series-tuned circuit with whatever

impedances are involved doesn’t amount to much;

however, in relation to the outside world, it becomes

rather exciting. In Fig. 25-59C the series-tuned circuit is

fed via a series resistor with the output being sensed

across the tuned circuit. Fig. 25-58D shows input-output

curves for three different tuned-circuit impedances based

on low, medium, and high reactances with the series

resistor kept the same in all cases. The detune slopes are

steeper with higher reactance networks than with lower

ones. In other words, higher reactance networks have a

sharper notch filter effect, less bandwidth, and are said to

have a higher Q (quality factor) than lower reactance

networks. In all cases, the output sensed voltage would

be the same as measured across a single reactance of the

appropriate and predominant sort; there is no magic

about a series-tuned circuit other than the curious

subtractive behavior of the two reactances.

25.11.7 Bandwidth and Q

There are direct relationships between the network reac-

tances, the series resistance, the bandwidth, and the Q.

Q is numerically equal to the ratio of elemental reac-

tance to series resistance in a series-tuned circuit (Q =

X/R); on a more practical level, the Q can also be deter-

mined as the ratio of filter center frequency to band-

width (Q = f/BW). Bandwidth is measured between the

3 dB down points on either side of resonance (and

usually where the phase has been shifted ±45°). If a

tuned circuit has a center frequency of 1 kHz and 3 dB

down points at 900 Hz and 1.1 kHz (pedantically

905 Hz and 1.105 kHz), the bandwidth is 200 Hz and

the network Q is 5 (frequency/bandwidth). The greater

the Q, the smaller the bandwidth.

The filter resonant frequency may be altered by

changing either the inductance or capacitance. Q is

subject to variation of the resistor or simultaneously

juggling the reactances in the inductance-capacitance

network, while maintaining the same center frequency.

25.11.8 Creating Inductance

It is most efficient (electrically and financially) in the

majority of console-type circuitry for inductance to be

simulated or generated artificially by circuits that are the

practical implementation of a mathematical conjuring

trick. These are known generically as gyrators.