Handbook for Sound Engineers

346 Chapter 12

the human perception of loudness. This makes them
particularly useful for audio dynamics control.
Rms is mathematically defined as the square root of
the mean of the square of a waveform. Electronically,
the mean is equal to the average, which can be approxi-
mated by an R-C network or an op-amp-based inte-
grator. However, calculating the square and square root
of waveforms is more difficult.
Designers have come up with a number of clever
techniques to avoid the complexity of numerical rms
calculation. For example, the heat generated by a resis-
tive element may be used to measure power. Power is
directly proportional to the square of the voltage across,
or current through, a resistor, so the heat given off is
proportional to the square of the applied signal level. To
measure large amounts of power having very complex
waveforms, such as the RF output of a television trans-
mitter, a resistor dummy load is used to heat water. The

temperature rise is proportional to the transmitter power.

Such caloric instruments are naturally slow to respond,

and impractical for the measurement of sound. Nonethe-

less, solid-state versions of this concept have been inte-

grated, as, for example U.S. Patent 4,346,291, invented

by Roy Chapel and Macit Gurol.^18 This patent, assigned

to instrumentation manufacturer Fluke, describes the

use of a differential amplifier to match the power dissi-

pated in a resistive element, thus measuring the true rms

component of current or voltage applied to the element.

While very useful in instrumentation, this technique has

not made it into audio products due to the relatively

slow time constants of the heating element.

To provide faster time constants to measure small

rms voltages or currents with complex waveforms such

as sound, various analog computational methods have

been employed. Computing the square of a signal

generally requires extreme dynamic range, which limits

the usefulness of direct analog methods in computing

rms value. As well, the square and square-root opera-

tions require complex analog multipliers, which have

traditionally been expensive to fabricate.

As with VCAs, the analog computation required for

rms level detection is simplified by taking advantage of

the logarithmic properties of bipolar junction transis-

tors. The seminal work on computing rms values for

audio applications was developed by David E.

Blackmer, who received U.S. Patent 3,681,618 for an

“RMS Circuit with Bipolar Logarithmic Converter.”^17

Blackmer’s circuit, discussed later, took advantage of

two important log-domain properties to compute the

square and square root. In the log domain, a number is

squared by multiplying it by 2; the square root is

obtained by dividing it by 2.

For example, to square the signal Vin use

Figure 12-47. VCA state-variable filter. Courtesy THAT Corporation.

IN

1 8

OUT

U1

2180A

3

2

1

U2A

5532

Input

Low Pass

Vcontrol

High Pass

V+

V–

Cset

Cc

Rset

Rbias

R

R

R

3

2

6

U3

LF351

2

4

7

3

6

5

EC+

EC–

SYM

GND

V+

V–

15

Figure 12-48. State-variable filter response. Courtesy THAT
Corporation.