Circuit Surgery
Regular clinic by Ian Bell
Op Amp Logarithmic and Exponential Amplifiers – Part 1
T
his month we will start to
look at op amp-based logarithmic and exponential (also called
antilog) amplifiers. This was originally
inspired by the use of exponential amplifiers in the MIDI Ultimate Synthesiser
project, which concluded in July 2019 –
so it has taken a while to get around to
it! Much more recently, in the September
2021 issue of PE, we looked at multistage
log amplifiers in RF power measurement.
These circuits use a cascade of limiting
amplifiers with summed outputs to approximate a logarithmic relationship
between input and output. That article
was inspired by the Low-cost Wideband
Digital RF Power Meter by Jim Rowe in
the August issue. In both cases the original articles are project-based and are not
able to go into a lot of background detail
on circuit operation.
As noted in the September Circuit
Surgery article, using cascaded limiting
amplifiers is not the only way to
implement a circuit with a logarithmic
response (one where the output voltage
is related to the logarithm of the input
voltage). Logarithmic and exponential
amplifiers can also be built using the fact
that the base-emitter voltage of a bipolar
transistor (or just a diode’s forward
voltage) is proportional to the logarithm
of the current through it. Or conversely,
the current through a transistor or diode
has an exponential relationship to the
applied base-emitter voltage, also called
the ‘forward’ voltage.
Fig.1. Ideal logarithmic (log10) circuit input-output relationship for unscaled (y1, green)
and scaled by 0.5 (y2, yellow) input amplitudes.
Adding the log of two numbers and taking
Logarithmic amplifiers have a range of
the antilog is equivalent to multiplication:
uses, including measurement of signal
levels in decibels, RMS (root mean square)
a × b = antilog(log(a) + log(b))
measurements, compression of signal
dynamic range (eg, at analogue-to-digital
Thus, you can make a multiplier using log
converter (ADC) inputs), and multiplying
and antilog circuits (and similarly divide
signals and other mathematical operations.
Logarithms
Mathematically, the logarithm (or
‘log’ for short) and exponential are
inverse functions – that is, if you take
the logarithm of a value then take the
exponential of that result you get the
original value back, and vice versa.
Simulation files
Most, but not every month, LTSpice
is used to support descriptions and
analysis in Circuit Surgery.
The examples and files are available
for download from the PE website.
54
Fig.2. Ideal logarithmic (log10) circuit input-output relationship for unscaled (y1, green)
and scaled by 2 (y3, red) output amplitudes.
Practical Electronics | December | 2021
Fig.3. The three
logarithmic
amplifier
responses from
Fig.1 and Fig.2
(y1, y2, y3) shown
on a logarithmic
input axis.
and raise to the power, such as squaring).
Multiplication is required in a number
of contexts, such as power measurement
(multiply current and voltage) and
signal processing operations based on
multiplying signals together. However,
op amp-based logarithmic amplifiers are
not necessarily the best way of achieving
signal multiplication, as we discussed in
the November 2021 Circuit Surgery article.
Logarithmic response basics
Before looking at circuit details it is worth
getting a feel for what the input-output
relationship of a logarithmic amplifier
looks like. Fig.1 shows two responses
of ideal logarithmic amplifiers to input
voltage x, where:
y1 = log10(x)
y2 = log10(x/2)
The shape of both curves illustrates
the general behaviour of a logarithmic
response. As input amplitude increases,
further increases in amplitude result in a
diminishing increase in output amplitude
– this is the compressing effect of a log
response. Small input amplitudes produce
large negative output amplitudes, tending
towards minus infinity for zero input for
an ideal logarithmic function. Of course,
real logarithmic amplifiers will have a
Fig.4. LTspice
schematic for
plotting Fig.1 to
Fig.3.
limited output range and may deviate
from an accurate logarithmic response for
both small and large input amplitudes.
The two responses
Fig.1
show that
Op Amp in
Log
Amps
the effect of scaling the input voltage is
to produce an offset (fixed DC difference)
in the output voltage. Specifically, for y
= log(ax), where a is a constant scaling
factor, the curve shifts up by log(a)
(positive output offset in circuit terms)
and for y = log(x/a) the curve shifts down
by log(a) (negative offset). The example
in Fig.1 is for y2 = log10(x/2) so the y2
curve is shifted down by log10(2) = 0.3
with respect to y1.
For y = log(x/a) the curves cross the
axis (y = 0) at x = a. For the examples, y1
= 0 for x = 1; and y2 = 0 at x = 2, as can
be seen on Fig.1.
Fig.2 shows the effect of scaling the
output of a logarithmic amplifier – it
shows two responses of ideal logarithmic
amplifiers to input x, where:
y1 = log10(x)
y3 = 2log10(x)
The effect of this scaling is to change the
slope of the curve by the scaling factor,
but with the curve crossing y = 0 at the
same point. This can be seen in Fig.2 –
the y3 curve is steeper than the y1 curve
at all points, but both cross y = 0 at x = 1.
The change of slope in the input-output
relationship is the same effect as changing
the gain of a linear amplifier.
In general, we can write the input (Vin)
to output (Vout) of a logarithmic voltage
amplifier as:
𝑉𝑉!"# = 𝑉𝑉$ log%& &
𝑉𝑉'(
'
𝑉𝑉)
Here, Va and Vb are constants determined
by the circuit𝑉𝑉 configuration. Both
*
𝐼𝐼* scaling
= 𝐼𝐼+ expvalues
& ' are voltages (for a
these
𝑉𝑉,
logarithmic voltage amplifier). This is to
obtain the correct dimensions (physical
quantities) in the equation. A logarithm is
𝐼𝐼*
taken𝑉𝑉*of=a 𝑉𝑉
pure
which we get by
'
, ln &number,
𝐼𝐼+voltage V by voltage V .
dividing the input
in
a
V a is called the intercept voltage,
because, as discussed above, it determines
𝐼𝐼*
the 𝑉𝑉
point
where the
input-output curve
- = −𝑉𝑉, ln & '
𝐼𝐼+ result of the logarithm
crosses the axis. The
is a pure number which we multiply by
a voltage to get an output voltage. Vb is
𝑉𝑉.
called
the−𝑉𝑉
slope
voltage
𝑉𝑉- =
' because it changes
, ln &
𝐼𝐼+ 𝑅𝑅
the slope of the input-output
relationship.
Fig.3 shows the three different amplifier
responses from Fig.1 and Fig.2 on the
𝐼𝐼1
same
logarithmic
𝑉𝑉/0
= 𝑉𝑉, ln & axis
' for the input (the
𝐼𝐼0+ the effect of intercept
x-axis). This shows
and slope scaling discussed above and
provides more information than Fig.1 and
𝑉𝑉.
Fig.2 on the response
at lower voltages
𝑉𝑉 = −𝑉𝑉, ln &
'
(the-plot is from
to 10V). For a real
𝐼𝐼100nV
𝑅𝑅
0+
circuit, Va and Vb need be set so that the
𝑉𝑉.%
𝑉𝑉.3
𝑉𝑉-2, = 𝑉𝑉-% − 𝑉𝑉-3 = −𝑉𝑉, ln &
' − − 𝑉𝑉, ln &
'
𝐼𝐼0+ 𝑅𝑅
𝐼𝐼0+ 𝑅𝑅
𝑉𝑉.
𝑉𝑉.3
𝑉𝑉.%
𝑉𝑉-2, = −𝑉𝑉, ln &
0
' = −𝑉𝑉, ln & '
𝐼𝐼0+ 𝑅𝑅 𝐼𝐼0+ 𝑅𝑅
𝑉𝑉.3
Practical Electronics | December | 2021
55
ID
IR
V I
R
–
V O
+
Fig.5. Logarithmic amplifier based on a
diode and op amp.
required output range is obtained for the
input range of interest.
The graphs in Figs.1 to 3 represent
mathematical (idealised circuit) functions
rather than real circuit responses. They
were obtained using LTspice behavioural
voltage sources and DC sweep simulations
– see Fig.4.
Amps
Exponential diode response
As mentioned above, logarithmic and
exponential amplifi
𝑉𝑉'( ers can be based on the
𝑉𝑉!"# = 𝑉𝑉$ log%&
& '
exponential
current-voltage
relationship
𝑉𝑉)
of the diode. A simplifi
ed version of the
diode equation (for forward bias) is:
𝑉𝑉*
𝐼𝐼* = 𝐼𝐼+ exp & '
𝑉𝑉,
Here, V D is the voltage across the
diode and I D is the current through
𝐼𝐼*
it. 𝑉𝑉I*
the
= 𝑉𝑉
& ' saturation current –
S is
, ln diode
𝐼𝐼+
a parameter specifi
c to the particular
diode or transistor. VT is the thermal
voltage, which commonly occurs in
𝐼𝐼*
semiconductor
VT depends
𝑉𝑉- = −𝑉𝑉, ln & equations.
'
𝐼𝐼+
on physical constants
(the charge on an
electron and Boltzmann’s constant) and
Amps
temperature. It has a value of about 25 to
𝑉𝑉temperature
.
26mV
at room
(specifically
𝑉𝑉- = −𝑉𝑉
'
, ln &
𝐼𝐼
25.85mV at 27°C
+ 𝑅𝑅 = 300K (kelvin)). The
𝑉𝑉'(
diode
𝑉𝑉!"# = equation
𝑉𝑉$ log%& & is 'commonly written in
the exponential𝑉𝑉)form shown above, but
𝐼𝐼1 it to show the voltage
we can rearrange
𝑉𝑉 = 𝑉𝑉, ln & '
𝐼𝐼of
as a/0function
0+ the current, which is a
𝑉𝑉*
logarithmic
function
(we have to use
𝐼𝐼* = 𝐼𝐼+ exp & '
𝑉𝑉,
the inverse function
of the exponential
to make VD the𝑉𝑉.subject) – specifically:
𝑉𝑉- = −𝑉𝑉, ln &
'
𝐼𝐼0+ 𝑅𝑅
𝐼𝐼*
𝑉𝑉* = 𝑉𝑉, ln & '
𝐼𝐼+
𝑉𝑉.%
𝑉𝑉.3
𝑉𝑉-2, = 𝑉𝑉-% − 𝑉𝑉-3 = −𝑉𝑉, ln &
' − − 𝑉𝑉, ln &
'
𝐼𝐼
𝑅𝑅
𝐼𝐼
0+ 𝑅𝑅
Fig.7. LTspice0+𝐼𝐼
*
𝑉𝑉- = −𝑉𝑉,for
ln & '
schematic
𝐼𝐼+
investigating the
𝑉𝑉.
𝑉𝑉.3
𝑉𝑉.%
𝑉𝑉-2, = −𝑉𝑉,circuit
ln & in Fig.6.
0
' = −𝑉𝑉, ln & '
𝐼𝐼0+ 𝑅𝑅 𝐼𝐼0+ 𝑅𝑅
𝑉𝑉.3
𝑉𝑉.
𝑉𝑉- = −𝑉𝑉, ln & '
𝐼𝐼+ 𝑅𝑅
𝑉𝑉/0 = 𝑉𝑉, ln &
This is similar
in form
to the general
Op Amp
Log Amps
logarithmic amplifier equation above,
except that the ‘input’ and ‘intercept’
parameters are currents. Also, we have
the natural logarithm (ln) rather than the
base ten logarithm (log10).
All logarithms are based on a particular
number base. For base 10 (the number base
we use for counting in everyday life), if y
= log10(x) then we can find x from y using
x = 10y, that is 10 to the power y. Natural
logarithms use
base e,
where
e = 2.71828
Op Amp
Log
Amps
(approximately). The function exp(a)
means ea, so if y = ln(x) then x = ey = exp(y).
e is also known as Euler’s number after
the mathematician Leonhard Euler (1707
– 1783). It is an important mathematical
constant and the ln() and exp() functions
have many interesting properties. The
concepts of ideal logarithmic amplifiers
discussed above are the same for different
log bases. To get a base-10 log from a natural
log we can use:
log10(x) = ln(x)/ln(10)
ln(x)/2.303
IC
R
𝑉𝑉!"# =V I 1 𝑉𝑉$ log%& &
IR
𝑉𝑉'(
'
𝑉𝑉)
V B E
–
V O
+
𝑉𝑉*
𝐼𝐼* = 𝐼𝐼+ exp & '
𝑉𝑉,
Fig.6. Logarithmic amplifier based on an
NPN bipolar transistor
and op amp.
𝐼𝐼*
𝑉𝑉* = 𝑉𝑉, ln & '
𝐼𝐼+
voltage (VO) must be negative and equal
𝑉𝑉'( for a current of ID:
to the diode voltage
𝑉𝑉!"# = 𝑉𝑉$ log%& & '
𝐼𝐼*𝑉𝑉)
𝑉𝑉- = −𝑉𝑉, ln & '
𝐼𝐼+
Assuming an ideal
𝑉𝑉* op amp, no current will
𝐼𝐼* =
𝐼𝐼+ exp
'
flow
into
the&op
𝑉𝑉𝑉𝑉,. amps inputs (assume it
has
nite
input
impedance
and requires
𝑉𝑉- infi
= −𝑉𝑉
ln
&
'
,
+ 𝑅𝑅 current). This means
zero external 𝐼𝐼bias
that all the current
in the diode must
𝐼𝐼*
𝑉𝑉* through
= 𝑉𝑉, ln & the
'
flow
𝐼𝐼𝐼𝐼+1 resistor, so ID = IR. The
resistor
is, ln
connected
between the input
𝑉𝑉/0 = 𝑉𝑉
& '
𝐼𝐼0+ so the voltage across
and virtual earth,
it is equal to the
𝐼𝐼* input voltage (VI) and
𝑉𝑉- =Ohm’s
−𝑉𝑉, lnlaw
& 'the current is ID = IR =
from
𝐼𝐼+.
𝑉𝑉
V𝑉𝑉I/R.
Substituting
' ID with VI/R in the
- = −𝑉𝑉, ln &
𝐼𝐼0+
𝑅𝑅
equation for V
we
get:
O
It terms of the generic circuit discussion
above, a change of base is related to the
slope voltage (we are scaling the result
from the logarithm function). This makes
it straightforward to obtain an output
𝑉𝑉.
related to log10 despite the diode function
𝑉𝑉- = −𝑉𝑉, ln & '
𝐼𝐼
+ 𝑅𝑅
𝑉𝑉.%
𝑉𝑉.3
being a natural logarithm. 𝑉𝑉
' − − 𝑉𝑉, ln &
'
-2, = 𝑉𝑉-% − 𝑉𝑉-3 = −𝑉𝑉, ln &
𝐼𝐼0+is
𝑅𝑅 of the same
𝐼𝐼form
This equation
0+ 𝑅𝑅 as the
generic logarithmic voltage amplifier
Op amp circuit
𝐼𝐼1
discussed
above,
A diode on its own has a logarithmic
𝑉𝑉/0 = 𝑉𝑉, ln
& ' with Va = ISR and Vb
𝐼𝐼
𝑉𝑉.% will have
relationship between a current and
= –V 𝑉𝑉. .The𝑉𝑉choice
of resistor
.3 0+
𝑉𝑉-2, = −𝑉𝑉, ln & T 0
' = −𝑉𝑉 ln & '
applied voltage. Using an op amp allows
some𝐼𝐼0+effect
is.3fixed by VT.
𝑅𝑅 𝐼𝐼0+on
𝑅𝑅 Va, but, Vb 𝑉𝑉
us to make a circuit with a logarithmic
We should also note that with VT being
𝑉𝑉.
relationship between two voltages – a
the
thermal
𝑉𝑉- =
−𝑉𝑉, ln &voltage,
' the circuit’s output
𝐼𝐼0+ 𝑅𝑅
logarithmic amplifier. Such a circuit is
is very much dependent
on temperature.
shown in Fig.5. The op amp has negative
Furthermore, the diode saturation
feedback applied via the diode. The op
voltage also varies significantly with
𝑉𝑉.% both the slope
𝑉𝑉.3
amp’s non-inverting input is grounded
(at
temperature,
𝑉𝑉-2, = 𝑉𝑉-% − 𝑉𝑉-3
= −𝑉𝑉, ln & so
' − − 𝑉𝑉, ln &
' and
𝐼𝐼0+ 𝑅𝑅
𝐼𝐼0+ 𝑅𝑅
0V) which means that the feedback will
intercept voltages
are temperature
adjust the output voltage to try to maintain
dependent, which will certainly be a
0V at the other input (inverting input).
problem in some applications.
𝑉𝑉.3 can be improved
𝑉𝑉.% by using
The non-inverting input behaves as if it
The𝑉𝑉.circuit
𝑉𝑉-2, = −𝑉𝑉, ln &
0
' = −𝑉𝑉, ln & '
is at 0V – that is, as if it is grounded
or
a bipolar
instead𝑉𝑉.3
of a diode, as
𝐼𝐼0+ 𝑅𝑅 transistor
𝐼𝐼0+ 𝑅𝑅
earthed. This is known as a ‘virtual earth’.
shown in Fig.6. This uses the transistor’s
With the diode in Fig.5 forward biased,
base-emitter junction instead of the
the anode is connected to the virtual
diode drop to set the output voltage. The
earth and the cathode is connected to
transistor’s base is connected directly to
the op amp output. The op amp output
ground which forces the output voltage
𝐼𝐼1
'
𝐼𝐼0+
𝑉𝑉.
𝑉𝑉- = −𝑉𝑉, ln &
'
𝐼𝐼0+ 𝑅𝑅
56
𝑉𝑉.%
𝑉𝑉.3
𝑉𝑉-2, = 𝑉𝑉-% − 𝑉𝑉-3 = −𝑉𝑉, ln &
' − − 𝑉𝑉, ln &
'
𝐼𝐼0+ 𝑅𝑅
𝐼𝐼0+ 𝑅𝑅
Practical Electronics | December | 2021
Fig.8. Results
from simulating
the LTspice circuit
in Fig.7.
ideal diode behaviour than an
individual diode. The transistor
circuit also reduces the effects
of offset voltages at the op amp’s
mps
inputs, which (in real circuits)
shift the diode voltage with
respect to VO, but do not affect
𝑉𝑉'(
VBE in the transistor version by
𝑉𝑉!"# = 𝑉𝑉$ log%& & '
𝑉𝑉)
virtue of the directly grounded
base. The transistor version
Amps
has the same problem with
𝑉𝑉*
temperature dependence as the
𝐼𝐼* = 𝐼𝐼+ exp & '
diode version.
𝑉𝑉,
𝑉𝑉'(
F i g . 7 s h o w s a n LTs p i c e
𝑉𝑉!"# = 𝑉𝑉$ log%& & '
𝑉𝑉)
schematic of the circuit in Fig.6.
𝐼𝐼*
There are copies of the circuit
𝑉𝑉* = 𝑉𝑉, ln & '
with different resistors values to
𝐼𝐼+
𝑉𝑉*
observe the effect of changing
𝐼𝐼* = 𝐼𝐼+ exp & '
Fig.9. Modification
𝑉𝑉, of part of the schematic of Fig.7 R in Fig.6. Two values of R are
to show the effect
used: R1 = 100 and R2 = 500 ; the
𝐼𝐼* of transistor temperature.
𝑉𝑉- = −𝑉𝑉, ln & '
circuits are otherwise identical.
𝐼𝐼+
𝐼𝐼*
The op amp is an arbitrarily selected
to be exactly equal
to the base-emitter
𝑉𝑉* = 𝑉𝑉, ln & '
precision op amp, and the transistor is
voltage:
VO = –V
𝐼𝐼+ BE.
a generic (LTspice default) NPN. A DC
The relationship
𝑉𝑉. between the collector
𝑉𝑉- = −𝑉𝑉
& base-emitter
'
sweep simulation is performed, changing
current
(IC,)ln
and
voltage (VBE)
𝐼𝐼+ 𝑅𝑅
the input voltage from source V3 from
of a bipolar transistor
also follows the
𝐼𝐼*
𝑉𝑉- =
−𝑉𝑉, ln & '
100nV to 10V logarithmically, with 20
diode
equation:
𝐼𝐼+
data points per decade of voltage.
𝐼𝐼1
𝑉𝑉/0 = 𝑉𝑉, ln & '
The results are shown in Fig.8, in which
𝐼𝐼0+
–V(out) is plotted so that the graph is
𝑉𝑉.
𝑉𝑉- =I −𝑉𝑉
'
, ln &
more visually similar to Fig.3 than a
Here,
current of the
ES is the 𝐼𝐼saturation
+ 𝑅𝑅
direct plot of the negative output voltages.
transistor’s base-emitter
junction.
The
𝑉𝑉.
Noting that IESR corresponds with Va in
collector
𝑉𝑉- = −𝑉𝑉, current
ln &
'all flows through the
𝐼𝐼0+ 𝑅𝑅
resistor, as just
for the diode
the generic response plotted in Fig.3,
𝐼𝐼discussed
1
𝑉𝑉/0 = So,
𝑉𝑉, lnwith
& 'VO = –VBE and IC = IR
current.
we would expect changing the resistor
𝐼𝐼0+
(with everything else equal) to shift the
= VI/R dropped into the above equation,
𝑉𝑉.%
𝑉𝑉.3
response in a similar way to V(y1) and
we=get:
𝑉𝑉-2, = 𝑉𝑉-% − 𝑉𝑉-3
−𝑉𝑉, ln &
' − − 𝑉𝑉, ln &
'
𝐼𝐼0+ 𝑅𝑅
𝐼𝐼0+ 𝑅𝑅
V(y2) in Fig.3, which is what we see.
𝑉𝑉.
𝑉𝑉- = −𝑉𝑉, ln &
'
However, in comparison with Fig.3 we
𝐼𝐼0+ 𝑅𝑅
observe that the circuit does not provide
𝑉𝑉.
𝑉𝑉.3 is very similar
𝑉𝑉.% the diode
an ideal logarithmic response over the
𝑉𝑉-2, = −𝑉𝑉,This
ln & equation
0
' = −𝑉𝑉, ln & to
'
𝐼𝐼0+ 𝑅𝑅 but
𝐼𝐼0+ 𝑅𝑅the transistor
𝑉𝑉.3 version is
entire input voltage range.
version,
𝑉𝑉.%
𝑉𝑉.3
The V(out1) output levels off at high
better
because
a
transistor’s
𝑉𝑉-2, = 𝑉𝑉-% − a𝑉𝑉-3
= −𝑉𝑉circuit
ln
&
'
−
−
𝑉𝑉
ln
&
'
,
,
𝐼𝐼0+ 𝑅𝑅a more
0+ 𝑅𝑅
input voltages and both versions cease
base-emitter 𝐼𝐼junction
provides
𝑉𝑉.
𝑉𝑉.3
𝑉𝑉.%
Practical
Electronics
| December
𝑉𝑉-2, = −𝑉𝑉
0
' = −𝑉𝑉
' | 2021
, ln &
, ln &
𝐼𝐼0+ 𝑅𝑅 𝐼𝐼0+ 𝑅𝑅
𝑉𝑉.3
to provide a logarithmic response below
about 1µV input. A limitation of range
is to be expected from a real circuit. The
value of R1 was chosen to show the high
voltage levelling off in this example,
but in real designs the input resistor is
likely to be much larger for handling this
level of input voltage, which produces a
100mA current in the circuit in Fig.7. The
circuit produces a logarithmic response
over an approximately 1µV to 1V range
– which is 120dB. The simulation uses
a real op amp model, but overall is still
somewhat idealised; for example, in a
real circuit noise will have a significant
effect for very small input voltages.
Temperature considerations
LTspice can be used to observe the effect
of temperature. LTspice uses a default
temperature of 27°C, but the temperature
of the whole simulation or individual
components can be changed. The circuit
in Fig.7 was modified so that the transistor
in the V(out2) circuit was at a different
temperature. Fig.9 shows the modified
V(out2) circuit in which the resistor is
the same as in the V(out1) circuit, but
the transistor temperature is set to 90°C.
Fig.10 shows the simulation results in
which we see a shift in offset and change
of slope. This is because a change of
temperature alters the value of both VT
and IESR, corresponding to both slope
(Vb) and intercept (Va) voltages in the
generic logarithmic amplifier response.
If it possible to compensate for the
effect of IES varying with temperature by
using two logarithmic amplifiers with
matched transistors – they must have
the same characteristics (and hence IES
value) and temperature for this to work.
With discrete transistors, matching
can be a challenge, but it is relatively
57
Op Amp Log Amps
Op Amp Log Amps
𝑉𝑉'(
'
𝑉𝑉𝑉𝑉'(
𝑉𝑉'(
)
𝑉𝑉!"# = 𝑉𝑉$ log%& & ' 𝑉𝑉!"# = 𝑉𝑉$ log%& & '
𝑉𝑉)
𝑉𝑉)
𝑉𝑉!"# = 𝑉𝑉$ log%& &
𝑉𝑉*
𝐼𝐼* = 𝐼𝐼+ exp & '
𝑉𝑉,
𝐼𝐼*
𝑉𝑉* = 𝑉𝑉, ln & '
𝐼𝐼+
𝐼𝐼*
𝑉𝑉- = −𝑉𝑉, ln & '
𝐼𝐼+
𝑉𝑉- = −𝑉𝑉, ln &
𝑉𝑉/0 = 𝑉𝑉, ln &
𝑉𝑉.
'
𝐼𝐼+ 𝑅𝑅
𝐼𝐼1
'
𝐼𝐼0+
Fig.10. Results of
the temperatureeffect simulation.
𝑉𝑉*
𝐼𝐼* = 𝐼𝐼+ exp & '
𝑉𝑉*,
𝐼𝐼* = 𝐼𝐼+ exp & '
𝑉𝑉,
𝐼𝐼*
𝑉𝑉* = 𝑉𝑉, ln & '
𝐼𝐼*+
𝑉𝑉* = 𝑉𝑉, ln & '
𝐼𝐼+
𝐼𝐼*
𝑉𝑉- = −𝑉𝑉, ln & '
𝐼𝐼*+
𝑉𝑉- = −𝑉𝑉, ln & '
𝐼𝐼+
𝑉𝑉.
'
𝐼𝐼𝑉𝑉
+ 𝑅𝑅
.
𝑉𝑉- = −𝑉𝑉, ln & '
𝐼𝐼+ 𝑅𝑅
𝑉𝑉- = −𝑉𝑉, ln &
𝐼𝐼1
'
𝐼𝐼𝐼𝐼0+
1
𝑉𝑉/0 = 𝑉𝑉, ln & '
𝐼𝐼0+
𝑉𝑉/0 = 𝑉𝑉, ln &
𝑉𝑉.
𝑉𝑉- = −𝑉𝑉, ln &
'
𝐼𝐼0+
𝑉𝑉.𝑅𝑅
𝑉𝑉.
IC designers.
' If we take the difference
𝑉𝑉- =straightforward
−𝑉𝑉, ln &
' 𝑉𝑉for
- = −𝑉𝑉, ln &
𝐼𝐼0+ 𝑅𝑅 in Fig.6 we get:
𝑅𝑅 copies
between𝐼𝐼0+
two
of the circuit
𝑉𝑉.%
𝑉𝑉.3
𝑉𝑉-2, = 𝑉𝑉-% − 𝑉𝑉-3 = −𝑉𝑉, ln &
' − − 𝑉𝑉, ln &
'
𝐼𝐼
𝑅𝑅
𝐼𝐼
𝑉𝑉.%
𝑉𝑉.3𝑅𝑅
𝑉𝑉.%
𝑉𝑉.3 0+
0+
𝑉𝑉-2,
=
𝑉𝑉
−
𝑉𝑉
=
−𝑉𝑉
ln
&
'
−
−
𝑉𝑉
ln
&
'
𝑉𝑉-2, = 𝑉𝑉-% − 𝑉𝑉-3 = −𝑉𝑉
ln
&
'
−
−
𝑉𝑉
ln
&
'
-3 ,
,
,
,
𝐼𝐼0+ 𝑅𝑅 to dividing
𝐼𝐼0+-%
𝑅𝑅 subtracting
𝐼𝐼0+logarithms
𝑅𝑅 𝐼𝐼0+ 𝑅𝑅
Recalling
that
is equivalent
the values inside the logarithm we get:
𝑉𝑉.
𝑉𝑉.3
𝑉𝑉.%
𝑉𝑉-2, = −𝑉𝑉, ln &
0
' = −𝑉𝑉, ln & '
𝐼𝐼0+
𝑅𝑅 𝐼𝐼0+
𝑉𝑉.3𝑅𝑅
𝑉𝑉.%
𝑉𝑉.
𝑉𝑉.3
𝑉𝑉.%
.3
.
' = −𝑉𝑉, ln & '
𝑉𝑉-2, = −𝑉𝑉, ln &
0𝑉𝑉-2, '==−𝑉𝑉
−𝑉𝑉
'0
, ln
, ln& &
𝐼𝐼0+
𝑉𝑉.3
𝐼𝐼0+ 𝑅𝑅 𝐼𝐼0+
𝑅𝑅
𝑉𝑉.3𝑅𝑅 𝐼𝐼0+ 𝑅𝑅
Q 1
R
V I1
1
–
+
V O 1
R
U 1
R
3
–
+
Q 2
R
5
R
V I2
R
2
4
V O U T
U 3
6
–
+
V O 2
U 2
Fig.11. Logarithmic amplifier with compensation for the
temperature dependence of transistor IES.
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58
The IES values cancel if they are the same for the two transistors.
This approach has the added advantage that VI2 can be used
to control the intercept voltage. Alternatively, the second
input can be from a reference signal source (eg, from a
sensor, as is done in some light intensity measurements using
photodiodes). An implementation of this idea is shown in
Fig.11. The difference between the two logarithmic amplifiers
is obtained using a standard op amp differential amplifier
configuration, for which the gain is R4/R3, and R3 = R5 and R4
= R6. The pairs of resistors must have very closely matched
values for good performance.
The dependence on VT is not removed by this approach, but
as this effect is simply proportional to absolute temperature
it is relatively easy to add another amplifier stage with an
equal and opposite temperature coefficient using a suitable
temperature-dependent resistor in the gain setting.
The MAX4206 Precision Transimpedance Logarithmic
Amplifier IC from Maxim Integrated is an example of a
chip based on the differencing circuit. It does not have
the input resistors built in, so it has current inputs (hence
‘transimpedance’ in the name).
Practical Electronics | December | 2021
