Circuit Surgery
Regular clinic by Ian Bell
Multistage log amplifiers for RF power measurement
T
he August edition of PE
featured a low-cost wideband digital RF Power Meter by Jim Rowe.
This is based around the AD8318 1MHz
to 8GHz, 70dB Logarithmic Detector/
Controller IC from Analog Devices. The
project uses a modified version of a prebuilt module, but essentially the key
analogue functionality of the project is
based on the AD8318 IC. These pre-built
modules are a great way of using a large
range of advanced ICs without having
to worry about the difficulties of designing a PCB and soldering the often very
small surface mount devices needed for
an optimum layout. In the project, instrument functionality is provided by
software running on an Arduino Nano,
which receives a digitised version of the
AD8318’s output via an LTC2400 DAC.
The article provides a brief description
of the AD8318, but there is insufficient
space to go into its principles of operation in much depth, so that is the topic
of this month’s Circuit Surgery. We will
explain the basics of how this class of ICs
work (Analog Devices produce several
devices based on similar principles, not
just the AD8318), illustrating the theory
with some idealised LTspice simulations
based on behavioural sources. Before getting into how the AD8318 works, it is
worth discussing RF power measurement
in general, as there are some potential
points of confusion.
Signal power measurement
An electrical signal producing a voltage
V across, and a current I through a load
delivers instantaneous power of P = VI to
the load. If the load is a resistance of value
R we can also express the power as P =
V2/R or P = I2R by substituting for I or V
using Ohm’s law (V = IR, I = V/R). So, if we
know R, then we can obtain power values
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.
62
just by measuring voltage (or current).
In RF systems the signal path is usually
matched to a specific impedance (for
example 50 ) so it is possible to obtain
signal power from voltage measurements.
This is the assumption used in the RF
Power Meter project, which has a nominal
input impedance of 50 .
The power dissipated in a resistor driven
by a signal such as a sinewave varies from
instant to instant in accordance with the
equations given above; however, we often
need to know the average power over a
period of time (at least one cycle). This
applies to signal strength indicators (eg,
on your mobile phone), or instrumentation
such as the RF Power Meter project, but
also in the internal circuitry of radio
systems. For example, received radio
signal strength varies considerably and it is
often necessary to adjust the gain of parts
of the system to accommodate this (AGC,
or automatic gain control) – this requires
accurate signal strength measurement.
As discussed, with a fixed/assumed
load we can obtain power from voltage,
but finding average power for an AC
signal is not simply a matter of taking the
average voltage or current – it is zero for a
sinewave over any whole number of cycles.
The heating effect produced in a resistor
forms the basis of how we define power
for non-DC signals – AC power is equal
to the DC power which would produce
the same heating effect in a resistor. For a
fixed R, this can be obtained by averaging
V2 over time (P = V2/R). If we take the
square root of the average of V2 we get
a value called the ‘Root Mean Square’
(RMS) voltage – this is the DC voltage,
which, if applied across the load, would
result in the same power dissipation as
that produced by our AC signal.
For a cyclic waveform of fixed
shape (eg, sine, triangle and square)
there is a fixed relationship between
the peak voltage and the RMS value,
which is known as the crest factor.
Mathematically, this can be obtained by
integrating the square of the waveform
function over one cycle, but of course
the results are well known for common
waveforms. For a sinewave Vpeak = 2 ×
VRMS = 1.4142VRMS (the crest (C) factor
is 1.4142). The crest factor for a triangle
wave is: C = 3 = 1.7321, and for a
square wave, C = 1. These values apply
to waveforms which are undistorted and
symmetrical about 0V.
If we know or assume the waveform
shape and have a fixed, known load
resistance it follows that we can obtain
power from a measurement of peak
voltage (or average peak voltage over
multiple cycles). The average power is
V 2RMS/R – which can be found from the
average peak voltage by an appropriate
scaling obtained from R and the crest
factor. However, if the waveform is of
arbitrary shape and we need a ‘true RMS’
measurement then the process is more
complex. The AD8318 IC, and hence the
RF Power Meter project on which it is
based, do not measure true RMS.
Decibels
Power in RF systems is typically
expressed in dBm units – this is power
in decibels (dB) referenced to 1mW –
this is the measurement unit used by
the RF Power Meter project. The decibel
is based on the logarithm of the power
ratio of two signals (say P1 and P2). If
we are interested in gain or attenuation
(input-output relationship) then the
two signals are obviously the input and
output. However, to use decibel units for
expressing the power of an individual
signals one of the power values must
be a fixed reference, and the units are
written to express this (eg, dBm for 1mW
reference and dBW for 1W reference).
Decibels are useful because their
logarithmic nature means very large
ranges of power levels are compressed
into a small range of decibels values.
An instrument, such as the RF Power
Meter which responds directly to the
decibel signal level is able to handle a
very wide range of signal power. Using
decibels on graph scales allows details
to be seen at both large and small signals
levels (the small level details would be
lost on a linear graph). When considering
signal paths, expressing gain and
attenuation in decibels makes calculations
Practical Electronics | September | 2021
– which is exactly what is provided
by the AD8318 and various other
similar devices. Responding to the peak
means that the output tracks the log of
the envelope of the input waveform
amplitude. This is similar to the process
of detection or demodulation in AM radio
receivers, so these types of circuit are
referred to as detecting or demodulating
logarithmic amplifiers.
A=4
A = 20
Piecewise log amplifiers
There is more than one way to make
a circuit with a logarithmic response.
A common approach for DC or low
frequencies is to use the exponential
current-voltage relationship of a diode
or transistor in an op amp feedback loop.
However, for high frequencies such as
those targeted by the AD8318 a piecewiselinear approximation is often used instead.
The circuit has a gain which varies over
segments of the input range and so appears
as a series of straight-line segments when
the input-output relationship is plotted.
This is illustrated in Fig.1, where the
piecewise-linear approximation is the
set of solid blue lines, which closely
approximates the true logarithmic function
shown by the dashed red line.
The piecewise response in Fig.1 has
its highest gain (× 84) for small inputs
(the gain is the slope of the graph, so a
steeper slope is a higher gain). At a certain
point (about Vin = 16mV in Fig.1) the gain
reduces from × 84 to × 20. This segment
is longer than the first one, extending
for about 47mV to Vin = 63mV. The third
segment has an even lower gain of × 4
and covers a longer range again, up to
about Vin = 250mV. Above this input
voltage the output limits at a fixed 3V. The
piecewise response in Fig.1 approximates
the logarithmic function quite well over
the range 3 to 300mV. For higher input
voltages the saturated (fixed final output
level) would deviate more and more
from the log function as the input level
and hence log of the input increased.
For very small input voltages the log
function would produce large negative
values. A real circuit will only produce
a log response over a limited range.
The piecewise log response can be
implemented by using a cascade of
output-limited amplifiers with their
outputs summed, as shown in Fig.2
A = 84
Fig.1. Piecewise-linear approximation (solid blue line) to a logarithmic (dashed red line)
input-out relationship.
straightforward – the multiplication of
gain or attenuation stages is translated to
addition or subtraction in the logarithmic
world of the decibel.
A power ratio in decibels is given
by 10log 10(P 2/P 1) dB, where P 1 is the
reference level (for example 1mW) and
P2 is the value we are measuring. The
term ‘decibel’ means one tenth (deci,
hence d) of a bel (symbol B). One bel is
log10(P2/P1), but as we use 10log10(P2/P1)
we are counting in tenths of a bel. The bel
is named after Alexander Graham Bell.
We can also use decibels to express
voltage levels. Given that P = V2/R, if
we consider a power ratio involving
the same R then P2/P1 = V22/V21 as both
instances of R cancel in the ratio. If we
square something inside a logarithm it
is equivalent to multiplying the log by
two (without the square). That is log(x2)
= 2log(x). So, to express a voltage in
decibels we use 20log10(V 2/V 1). Note
that we are multiplying by 20, not by 10
as we did with the power ratio. Again,
for signals levels rather than gains, we
need to indicate the reference level (for
example dBV is referenced to 1V).
Demodulating log amplifiers for
power measurement
We can write an expression for the average
power of a signal in terms of dBm, related
to the RMS voltage of a signal (VRMS).
Using P2 = V 2RMS/R and P1 = 1mW:
PdBm = 10log10((Vpeak/C)2/R /1mW)
For a sinewave into 50
and R = 50 we get:
PdBm = 10log10(V2peak/2 × 50 × 0.001)
= 10log10(V2peak/0.1)
Dividing inside a logarithm is equivalent
to subtraction of logs, so:
PdBm = 10log10(V2peak) − 10log10(0.1)
= 10log10(V2peak) − 10
Using voltage rather than its square:
PdBm = 20log10(Vpeak) − 10
Thus, if we build a circuit with an input
impedance of 50 that produces an output
proportional to the logarithm of the peak
value of the input signal we obtain an
output voltage which is proportional to the
power in dBm. There is an offset (10 in the
above equation) which is easily accounted
for in calibration. The crest factor just
changes this offset, so calibration could
be done for different fixed wave shapes.
Similarly, the input impedance also
changes the offset in the above equation,
but for a real implementation this has to be
correctly matched, which would usually
be to the reference impedance.
The key thing here is the logarithmic
response to the peak input voltage
PdBm = 10log10(V 2RMS/R /1mW)
Note that R is not cancelled in this
equation, so we should state the value used
(for example, 50 ). For a fixed waveform
with a known crest factor we have:
Vpeak = C × VRMS, so:
V 2RMS = (Vpeak/C)2
Practical Electronics | September | 2021
with C = 2
In p u t
S ta g e 1
S ta g e 2
S ta g e 3
A
A
A
S ta g e N
A
O u tp u t
Fig.2. A cascade of N limiting amplifiers with summed outputs, which approximates a
logarithmic response.
63
V in
Z e r o g a in lim ite d a t V L
V L
G a in A
0
V L/ A
V out
Fig.3. Transfer function of a limiting
amplifier with gain A and output limit VL.
– this form of circuit structure is used in
the AD8318. The amplifiers have gain A
until the output reaches a limiting voltage
(VL), above this the gain is zero. Inputs
greater than or equal to VL/A produce an
output limited at VL. The input-output
relationship of the limiting amplifier
is shown in Fig.3 – this is for positive
inputs, for negative inputs the output
limits at −VL.
If a sufficiently small input signal is
applied to the circuit in Fig.2 none of
the amplifiers will limit and the gain to
the final output will be AN, for example
with three stages the gain is A × A × A =
A3. The penultimate stage output has a
gain of A(N−1), the one before that A(N−2),
and so on. These outputs are summed;
so, for example, the total output with
three stages is (A3 + A2 + A)Vin. If A = 4
then the gain is (43 + 42 + 4)Vin = (64 +
16 + 4)Vin = 84Vin, the same as the first
segment in Fig.1.
If the input signal is increased to the
point where the output of the penultimate
amplifier is equal to V L then at this,
and higher inputs, the final amplifier
will limit. This occurs at ANVin = VL,
or Vin = VL/AN, for example, with three
stages, and VL = 1 at 1/64 = 16mV – the
end of the first segment in Fig.1. Under
these conditions the final amplifier will
contribute a constant VL to the summed
output. For three stages, the output will
be (A2 + A)Vin + VL. If A = 4 and VL =
1 this segment will have the function
20Vin + 1, starting at Vin = 16mV, so Vout
at this point is 20 × 0.016 + 1 = 1.3V, as
seen on Fig.1.
The next segment occurs when the
penultimate amplifier limits at A(N−1)Vin =
VL. For three stages, the second amplifier
limits at VL/A2, which for A = 4 and VL =
1 is at Vin = 1/16 = 63 mV – the end of the
second segment in Fig.1. In this segment
the last two amplifiers are contributing a
constant VL to the output. For three stages
this will be AVin + 2VL. For A = 4 and
VL = 1 this (third) segment will have the
function 4Vin + 2, starting at Vin = 63mV,
so Vout at this point is 4 × 0.063 + 2 = 2.3V,
as seen on Fig.1.
64
Fig.4. LTspice behavioural simulation of a summed cascade of limiting amplifiers.
With a sufficiently large input (Vin
> VL/A) the first amplifier, and all the
others in the cascade, will limit and the
summed output will limit at NVL. For
our example, with N = 3, A = 4 and VL
= 1 this occurs at Vin = 1/4 = 250 mV.
Above this input voltage, the output
is constant at 3VL = 3V. Again, this is
seen in Fig.1. The preceding discussion
has shown that the circuit in Fig.2 can
produce the piecewise approximation
to a log function shown in Fig.1. This
example used just three stages, which is
sufficient for illustration, but much better
performance can be obtained with more
stages – the AD8318 has nine.
LTspice Behavioural Simulation
We can simulate an idealised version
of the amplifier cascade in LTspice
using behavioural sources. These are
voltage or current sources for which
we can write equations – the output
of the source can be expressed as a
mathematical function of other voltages
and currents in the circuit, time and
the constant π. For example, we can
model an amplifier with gain 4 by using
Fig.5. Simulation results for the circuit in Fig.4.
Practical Electronics | September | 2021
Fig.7 Diode envelope detector circuit (part
of the LTspice schematic).
showing the sinewave signal clipping
as Bamp3 goes into limit.
Fig.6. Zoom-in on two of the signals in Fig.5.
Envelope Detector
The circuit described so far has a
logarithmic gain response but does
not do all we need to implement the
RF Power Meter – specifically, it just
amplifies the input signal, it does not
output a voltage equal to the peak value.
A simple way to obtain the peak (but
not that used by the AD8318) is to use
a half-wave rectifier and RC filter – a
diode detector, or envelope demodulator,
often associated with basic AM radio
circuits. To do this, the circuit in Fig.7
is added to the schematic in Fig.4 (Fig.7
only shows the detector, the rest of the
schematic is the same as Fig.4). The
diode is idealised – it has zero forwardvoltage drop. The simulation results
with the detector are shown in Fig.8, in
which V(out) is the same as in Fig.5.
The output voltage V(det) follows
the envelop of the V(out) waveform,
which due to the linear input amplitude
sweep, is close to the piecewise linear
curve in Fig.1. There is some ripple
on the detected output as the RC filter
does not provide perfect smoothing of
the envelope.
Fig.8. Simulation results for the circuit in Fig.7 – V(det) follows the envelope of V(o u t).
a behavioural source with the equation
V=4*V(in). The output of the source
will be four times the voltage on node
in. To create a limiting amplifier, we can
use the limit function. The output from
LTspice function limit(x,min,max) is
x if x is between min and max, otherwise
it limits to either min or max, depending
on which value is exceeded.
We can use V=limit(4*V(in),-1,1)
to produce an amplifier of gain 4, limiting
its output to ±1V, as used in the previous
discussion. The LTspice schematic in
Fig.4 has three such amplifiers (Bamp1,
Bamp2 and Bamp3) cascaded (eg, the
output of B a m p 1 is on node o u t 1 ,
which is the input to Bamp2). Another
behavioural source (Bsum) is used to
add the three outputs from the amplifier
together. To see the response of the circuit
we create a 1GHz sinewave input signal
which increases in amplitude linearly
from 0V to 300mV (the same input range
as in Fig.1) in 200ns.
The results of the simulation are shown
in Fig.5. The simulation is over so many
cycles of the 1GHz sine wave that the
waveform detail cannot be shown –
the plots show blocks of colour, with
the shape of the blocks following the
envelope of the signal amplitude. As
the input (V(in)) increases the three
Practical Electronics | September | 2021
amplifiers limit in turn (the final amplifier
output V(out3) limits first). We see their
output amplitudes increase linearly until
the 1V limit is reached in each case.
The summed output envelope (v(out))
follows the shape of the piecewise curve
in Fig.1. Fig.5 is a zoom-in of the first part
of the V(out3) and V(out) waveforms
V IN
g m
S ta g e 1
S ta g e 2
S ta g e 3
A
A
A
g m
g m
S ta g e N
A
g m
g m
IO U T
Fig.9. Using transconductance amplifiers to sum the amplifier cascade outputs.
Fig.10. Absolute value transconductance, current summing and filtering circuit (part of
the LTspice schematic).
65
Fig.11. Simulation result from the circuit
in Fig.10.
Transconductance Amplifiers
Summing voltages directly is relatively
difficult and the half-wave diode
detector does not provide very high
performance – real diodes may have too
large a voltage drop to be used directly
on low-level signals.
There are better approaches to addition
and detection. Addition of signals is
easily achieved using currents – simply
connect all the current outputs together.
This can be done using the version of
the circuit in Fig.2 shown in Fig.9. The
output of each amplifier in the cascade
is fed to a transconductance amplifier
(gain gm). Transconductance amplifiers
have an input of voltage and an output of
current. The transconductance amplifier
outputs are connected together to sum
the currents. To achieve detection
the transconductance amplifi ers can
have an absolute value characteristic
(both positive and negative inputs
give positive outputs), or a squarelaw characteristic (the square of both
positive and negative values is positive).
The current output can be converted to
a voltage by passing it through a resistor
and then a low-pass filter to provide a
signal related to the amplitude envelope.
Fig.12. Plot of V(det) from Fig.11 vs the logarithm of the peak input voltage.
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The circuit in Fig.10 is a version of the circuit in Fig.9
in LTspice using behavioural current sources to implement
transconductance amplifiers with an absolute function. Each
current source has function I=100e-6*abs(v(out3)) –
note the use of abs(). The gain is gm = 100µA/V. The rest
of this schematic is the same as Fig.4, but without the Bsum
source as this function is replaced by the circuit in Fig.9.
The current sources are paralleled to sum the currents and
a constant current is added to provide positive offset when
transconductance amplifier outputs are zero. The current
is fed to an RC circuit to convert it to a voltage and provide
low-pass filtering. The simulation result for the circuit in
Fig.10 are shown in Fig.11 – this is just the V(det) signal,
the V(in) and V(out1), V(out2) and V(out3) are the
same as in Fig.5.
The negative slope of the V(det) response is like the
response of the AD8318. To verify this is a log function,
Fig.12 shows V(det) plotted against the log of the peak of
V(in). This was obtained by exporting the LTspice data
to Excel. Fig.12 shows a more or less linear negativeslope relationship between the V(det) and the log of the
peak input voltage, which corresponds with the AD8318
response (Fig.2 in the RF Power Meter article). As already
noted, the LTspice circuit, although idealised, is a crude
implementation as it only uses three stages rather than the
nine in the AD8318. Various other details do not match
exactly, but this example is sufficient to illustrate the basic
principles of operation. Readers interested in knowing
more of the specifics of how these devices operate should
read the AD8318 datasheet and also the AD8307 datasheet,
which is a logarithmic amplifier based on similar principles
– the datasheet for AD8307 provides more details on the
theory of operation.
Practical Electronics | September | 2021
