AD8307
LOG AMP THEORY
Logarithmic amplifiers perform a more complex operation than
that of classical linear amplifiers, and their circuitry is significantly
different. A good grasp of what log amps do and how they work
can prevent many pitfalls in their application. The essential
purpose of a log amp is not to amplify, though amplification is
utilized to achieve the function. Rather, it is to compress a
signal of wide dynamic range to its decibel equivalent. It is thus
a measurement device. A better term might be logarithmic
converter, since its basic function is the conversion of a signal
from one domain of representation to another, via a precise
nonlinear transformation.
Logarithmic compression leads to situations that can be
confusing or paradoxical. For example, a voltage offset added to
the output of a log amp is equivalent to a gain increase ahead of
its input. In the usual case where all the variables are voltages,
and regardless of the particular structure, the relationship
between the variables can be expressed as:
V
OUT
=
V
Y
log (
V
IN
/
V
X
)
(the log intercept) at the unique value V
IN
= V
X
and ideally
becomes negative for inputs below the intercept. In the ideal
case, the straight line describing V
OUT
for all values of V
IN
continues indefinitely in both directions. The dotted line shows
that the effect of adding an offset voltage V
SHIFT
to the output is
to lower the effective intercept voltage V
X
. Exactly the same
alteration could be achieved by raising the gain (or signal level)
ahead of the log amp by the factor V
SHIFT
/V
Y
. For example, if V
Y
is 500 mV per decade (25 mV/dB), an offset of +150 mV added
to the output appears to lower the intercept by two tenths of a
decade, or 6 dB. Adding an offset to the output is thus
indistinguishable from applying an input level that is 6 dB higher.
The log amp function described by Equation 1 differs from that
of a linear amplifier in that the incremental gain δV
OUT
/δV
IN
is a
very strong function of the instantaneous value of V
IN
, as is
apparent by calculating the derivative. For the case where the
logarithmic base is δ,
δ
V
OUT
V
=
Y
V
IN
δ
V
IN
(2)
(1)
where:
V
OUT
is the output voltage.
V
Y
is the slope voltage; the logarithm is usually taken to base 10
(in which case
V
Y
is also the volts per decade).
V
IN
is the input voltage.
V
X
is the intercept voltage.
All log amps implicitly require two references, here, V
X
and V
Y
,
which determine the scaling of the circuit. The absolute
accuracy of a log amp cannot be any better than the accuracy of
its scaling references. Equation 1 is mathematically incomplete
in representing the behavior of a demodulating log amp such as
the AD8307, where V
IN
has an alternating sign. However, the
basic principles are unaffected, and this can be safely used as the
starting point in the analyses of log amp scaling.
V
OUT
5V
Y
4V
Y
3V
Y
2V
Y
V
Y
V
OUT
= 0
V
IN
= 10
–2
V
X
–40dBc
–2V
Y
V
IN
= V
X
0dBc
V
IN
= 10
2
V
X
+40dBc
LOG V
IN
V
IN
= 10
4
V
X
+80dBc
01082-021
That is, the incremental gain is inversely proportional to the
instantaneous value of the input voltage. This remains true for
any logarithmic base, which is chosen as 10 for all decibel
related purposes. It follows that a perfect log amp is required to
have infinite gain under classical small signal (zero amplitude)
conditions. Less ideally, this result indicates that, whatever
means are used to implement a log amp, accurate response
under small signal conditions (that is, at the lower end of the
dynamic range) demands the provision of a very high gain
bandwidth product. A further consequence of this high gain is
that, in the absence of an input signal, even very small amounts
of thermal noise at the input of a log amp cause a finite output
for zero input. This results in the response line curving away
from the ideal shown in Figure 21 toward a finite baseline,
which can be either above or below the intercept. Note that the
value given for this intercept can be an extrapolated value, in
which case the output can not cross zero, or even reach it, as is
the case for the AD8307.
While Equation 1 is fundamentally correct, a simpler formula is
appropriate for specifying the calibration attributes of a log amp
like the AD8307, which demodulates a sine wave input:
V
OUT
= V
SLOPE
(P
IN
– P
0
)
where:
V
OUT
is the demodulated and filtered baseband (video or
RSSI) output.
V
SLOPE
is the logarithmic slope, now expressed in V/dB (typically
between 15 mV/dB and 30 mV/dB).
(3)
V
SHIFT
LOWER INTERCEPT
Figure 21. Ideal Log Amp Function
Figure 21 shows the input/output relationship of an ideal log
amp, conforming to Equation 1. The horizontal scale is
logarithmic and spans a wide dynamic range, shown here as
over 120 dB, or six decades. The output passes through zero
P
IN
is the input power, expressed in decibels relative to some
reference power level.
P
0
is the logarithmic intercept, expressed in decibels relative to
the same reference level.
Rev. C | Page 9 of 24
AD [ ANALOG DEVICES ]
