Datasheet

Data Sheet AD8307
Rev. F | Page 11 of 24
Further analysis shows that right up to the point where the input to
the first cell is above the knee voltage, V
OUT
changes by (A − 1)E
K
for a ratio change of A in V
IN
. This can be expressed as a certain
fraction of a decade, which is simply log
10
(A). For example,
when A = 5, a transition in the piecewise linear output function
occurs at regular intervals of 0.7 decade (log
10
(A), or 14 dB
divided by 20 dB). This insight immediately allows the user to
write the volts per decade scaling parameter, which is also the
scaling voltage, V
Y
, when using base 10 logarithms, as

)(log
1
10
A
EA
VinChangeDecades
VinChangeLinear
V
K
IN
OUT
Y


(4)
Note that only two design parameters are involved in determining
V
Y
, namely, the cell gain A and the knee voltage, E
K
, while N,
the number of stages, is unimportant in setting the slope of the
overall function. For A = 5 and E
K
= 100 mV, the slope would be
a rather awkward 572.3 mV per decade (28.6 mV/dB). A well
designed log amp has rational scaling parameters.
The intercept voltage can be determined by using two pairs of
transition points on the output function (consider Figure 24).
The result is

)1/1( 

AN
K
X
A
E
V
(5)
For the case under consideration, using N = 6, calculate V
Z
=
4.28 µV. However, be careful about the interpretation of this
parameter, because it was earlier defined as the input voltage at
which the output passes through zero (see Figure 21). Clearly, in
the absence of noise and offsets, the output of the amplifier chain
shown in Figure 23 can be zero when, and only when, V
IN
= 0.
This anomaly is due to the finite gain of the cascaded amplifier,
which results in a failure to maintain the logarithmic
approximation below the lin-log transition (labeled 1 in Figure 24).
Closer analysis shows that the voltage given by Equation 5
represents the extrapolated, rather than actual, intercept.
DEMODULATING LOG AMPS
Log amps based on a cascade of A/1 cells are useful in baseband
applications because they do not demodulate their input signal.
However, baseband and demodulating log amps alike can be
made using a different type of amplifier stage, called an A/0 cell.
Its function differs from that of the A/1 cell in that the gain
above the knee voltage E
K
falls to zero, as shown by the solid
line in Figure 25. This is also known as the limiter function, and
a chain of N such cells are often used to generate hard-limited
output in recovering the signal in FM and PM modes.
01082-025
SLOPE = A
SLOPE = 0
OUTPUT
AE
K
0
E
K
INPUT
A/0
tanh
Figure 25. A/0 Amplifier Functions (Ideal and Tanh)
The AD640, AD606, AD608, AD8307, and various other Analog
Devices, Inc., communications products incorporating a logarith-
mic intermediate frequency (IF) amplifier all use this technique.
It becomes apparent that the output of the last stage can no longer
provide the logarithmic output because this remains unchanged
for all inputs above the limiting threshold, which occurs at V
IN
=
E
K
/A
N − 1
. Instead, the logarithmic output is now generated by
summing the outputs of all the stages. The full analysis for this
type of log amp is only slightly more complicated than that of
the previous case. It is readily shown that, for practical purposes,
the intercept voltage, V
X
, is identical to that given in Equation 5,
while the slope voltage is

A
AE
V
K
Y
10
log

(6)
Preference for the A/0 style of log amp over one using A/1 cells
stems from several considerations. The first is that an A/0 cell
can be very simple. In the AD8307, it is based on a bipolar
transistor differential pair, having resistive loads, R
L
, and an
emitter current source, I
E
. This exhibits an equivalent knee
voltage of E
K
= 2 kT/q and a small signal gain of A = I
E
R
L
/E
K
.
The large signal transfer function is the hyperbolic tangent (see
the dashed line in Figure 25). This function is very precise, and
the deviation from an ideal A/0 form is not detrimental. In fact,
the rounded shoulders of the tanh function result in a lower
ripple in the logarithmic conformance than that obtained using
an ideal A/0 function.
An amplifier composed of these cells is entirely differential in
structure and can thus be rendered very insensitive to disturbances
on the supply lines and, with careful design, to temperature
variations. The output of each gain cell has an associated
transconductance (g
m
) cell that converts the differential output
voltage of the cell to a pair of differential currents, which are
summed simply by connecting the outputs of all the g
m
(detector)
stages in parallel. The total current is then converted back to a
voltage by a transresistance stage to generate the logarithmic
output. This scheme is depicted in single-sided form in Figure 26.