REV. D
AD641
–8–
FUNDAMENTALS OF LOGARITHMIC CONVERSION
The conversion of a signal to its equivalent logarithmic value
involves a nonlinear operation, the consequences of which can be
very confusing if not fully understood. It is important to realize
from the outset that many of the familiar concepts of linear
circuits are of little relevance in this context. For example, the
incremental gain of an ideal logarithmic converter approaches
infinity as the input approaches zero. Further, an offset at the
output of a linear amplifier is simply equivalent to an offset at
the input, while in a logarithmic converter it is equivalent to a
change of amplitude at the input—a very different relationship.
We assume a dc signal in the following discussion to simplify the
concepts; ac behavior and the effect of input waveform on cali-
bration are discussed later. A logarithmic converter having a
voltage input V
IN
and output V
OUT
must satisfy a transfer func-
tion of the form
V
OUT
= V
Y
LOG (V
IN
/V
X
) Equation (1)
where V
Y
and V
X
are fixed voltages which determine the scaling
of the converter. The input is divided by a voltage because the
argument of a logarithm has to be a simple ratio. The logarithm
must be multiplied by a voltage to develop a voltage output.
These operations are not, of course, carried out by explicit com-
putational elements, but are inherent in the behavior of the
converter. For stable operation, V
X
and V
Y
must be based on
sound design criteria and rendered stable over wide temperature
and supply voltage extremes. This aspect of RF logarithmic
amplifier design has traditionally received little attention.
When V
IN
= V
X
, the logarithm is zero. V
X
is, therefore, called
the Intercept Voltage, because a graph of V
OUT
versus LOG
(V
IN
)—ideally a straight line—crosses the horizontal axis at this
point (see Figure 20). For the AD641, V
X
is calibrated to ex-
actly 1 mV. The slope of the line is directly proportional to V
Y
.
Base 10 logarithms are used in this context to simplify the rela-
tionship to decibel values. For V
IN
= 10 V
X
, the logarithm has a
value of 1, so the output voltage is V
Y.
At V
IN
= 100 V
X
, the
output is 2 V
Y
, and so on. V
Y
can therefore be viewed either as
the Slope Voltage or as the Volts per Decade Factor.
The AD641 conforms to Equation (1) except that its two out-
puts are in the form of currents, rather than voltages:
I
OUT
= I
Y
LOG (V
IN
/V
X
) Equation (2)
ACTUAL
0
INPUT ON
LOG SCALE
Y
Y
2V
Y
IDEAL
V
Y
LOG (V
IN
/V
X
)
V
IN
= V
X
V
IN
= 100V
X
V
IN
= 10V
X
ACTUAL
SLOPE = V
Y
IDEAL
+
–
Figure 20. Basic DC Transfer Function of the AD641
I
Y
, the Slope Current, is 1 mA. The current output can readily
be converted to a voltage with a slope of 1 V/decade, for ex-
ample, using one of the 1 kΩ resistors provided for this purpose,
in conjunction with an op amp, as shown in Figure 21.
9
12
8
13
7
14
6
15
10
11
LOG
OUT
LOG
COM
SIG
+OUT
+V
S
–V
S
ITC
BL2
SIG
–OUT
AD641
C1
330pF
1mA PER DECADE
AD846
R1
48.7V
R2
OUTPUT VOLTAGE
1V PER DECADE
FOR R2 = 1kV
100mV PER dB
FOR R2 = 2kV
Figure 21. Using an External Op Amp to Convert the
AD641 Output Current to a Buffered Voltage Output
Intercept Stabilization
Internally, the intercept voltage is a fraction of the thermal volt-
age kT/q, that is, V
X
= V
XO
T/T
O
, where V
XO
is the value of V
X
at a reference temperature T
O
. So the uncorrected transfer
function has the form:
I
OUT
= I
Y
LOG (V
IN
T
O
/V
XO
T) Equation (3)
Now, if the amplitude of the signal input V
IN
could somehow be
rendered PTAT, the intercept would be stable with tempera-
ture, since the temperature dependence in both the numerator
and denominator of the logarithmic argument would cancel.
This is what is actually achieved by interposing the on-chip
attenuator, which has the necessary temperature dependence to
cause the input to the first stage to vary in proportion to abso-
lute temperature. The end limits of the dynamic range are now
totally independent of temperature. Consequently, this is the pre-
ferred method of intercept stabilization for applications where
the input signal is sufficiently large.
When the attenuator is not used, the PTAT variation in V
X
will
result in the intercept being temperature dependent. Near 300K
(+27°C) it will vary by 20 LOG (301/300) dB/°C, about 0.03 dB/
°C. Unless corrected, the whole output function would drift up
or down by this amount with changes in temperature. In the
AD641 a temperature compensating current I
Y
LOG(T/T
O
) is
added to the output. This effectively maintains a constant inter-
cept V
XO
. This correction is active in the default state (Pin 8
open circuited). When using the attenuator, Pin 8 should be
grounded, which disables the compensation current. The drift
term needs to be compensated only once; when the outputs of
two AD641s are summed, Pin 8 should be grounded on at least
one of the two devices (both if the attenuator is used).
Conversion Range
Practical logarithmic converters have an upper and lower limit
on the input, beyond which errors increase rapidly. The upper
limit occurs when the first stage in the chain is driven into limit-
ing. Above this, no further increase in the output can occur and
the transfer function flattens off. The lower limit arises because
a finite number of stages provide finite gain, and therefore at
low signal levels the system becomes a simple linear amplifier.
