AD8307
of A. At the next critical point (labeled 3 in Figure 24), the input
is again A times larger and VOUT has increased to (3A–2)EK, that
is, by another linear increment of (A–1)EK.
SLOPE = 0
AE
K
tanh
Further analysis shows that right up to the point where the
input to the first cell is above the knee voltage, VOUT changes by
(A–1)EK for a ratio change of A in VIN. This can be expressed as
a certain fraction of a decade, which is simply log10(A). For
example when A = 5, a transition in the piecewise linear output
function occurs at regular intervals of 0.7 decade (log10(A), or
14 dB divided by 20 dB). This insight allows us to immediately
write the volts per decade scaling parameter, which is also the
scaling voltage, VY, when using base 10 logarithms, as
A/0
SLOPE = A
0
E
K
INPUT
Figure 25. A/0 Amplifier Functions (Ideal and Tanh)
The ADꢀ40, ADꢀ0ꢀ, ADꢀ08, AD8307, and various other
Analog Devices, Inc. communications products incorporating a
logarithmic IF amplifier all use this technique. It becomes
apparent that the output of the last stage can no longer provide
the logarithmic output, since this remains unchanged for all inputs
Linear Change in VOUT
Decades Change in VIN
(
A −1
)
EK
VY =
=
(4)
log10(A)
Note that only two design parameters are involved in
above the limiting threshold, which occurs at VIN = EK/AN−1
.
determining VY, namely, the cell gain A and the knee voltage EK,
while N, the number of stages, is unimportant in setting the
slope of the overall function. For A = 5 and EK = 100 mV, the
slope would be a rather awkward 572.3 mV per decade
(28.ꢀ mV/dB). A well designed log amp has rational scaling
parameters.
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 VX is identical to that given in Equation 5, while the
slope voltage is
The intercept voltage can be determined by using two pairs of
transition points on the output function (consider Figure 24).
The result is
AEK
VY =
(ꢀ)
log10 A
( )
EK
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, RL, and an
emitter current source, IE. This exhibits an equivalent knee
voltage of EK = 2 kT/q and a small signal gain of A = IERL/EK.
The large signal transfer function is the hyperbolic tangent
(see dotted 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.
VX =
(5)
(
A−1))
A(N +1/
For the case under consideration, using N = ꢀ, calculate
VZ = 4.28 μV. However, be careful about the interpretation of
this parameter, since 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,
VIN = 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 (point 1 in Figure 24).
Closer analysis shows that the voltage given by Equation 5
represents the extrapolated, rather than actual, intercept.
An amplifier built 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 (gm) cell, which 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 gm (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 2ꢀ.
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 EK 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.
Rev. C | Page ±± of 24
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