October 4, 1965 Electronics
[Table of Contents]
Wax nostalgic about and learn from the history of early electronics.
See articles from Electronics,
published 1930 - 1988. All copyrights hereby acknowledged.
Here is a good, brief introduction
to harmonic and intermodulation distortion measurement methods that were commonly used
in the 1960s. Total harmonic distortion (THD) was used often, especially for audio equipment,
which of course most frequency conversion circuits ultimately were in the era since digital
data transmission over the air was not too common. Author Charles Moore worked for Hewlett-Packard
(HP) and references HP
Application Note 15, "Distortion and Intermodulation" which,
thankfully, is made available by Hewlett-Packard / Agilent / Keysight on their website.
In fact, a complete list of all the vintage app notes are available on
this page by downloading the Excel file at the bottom.
I highly recommend that you download and save all you think you might ever want because as with
most things on the Web, they could disappear at any time.
Two Ways to Measure Distortion
Its cause, system nonlinearity, may be calculated
from the output harmonics of a single signal, or from the intermodulation of two.
By Charles R. Moore
Loveland division, Hewlett-Packard Company Loveland, Colorado
The goal of audio and communications equipment is to reproduce input signals faithfully
at the output. But system nonlinearity changes the waveshape of the signals; the resulting
additional frequencies at the output are a measure of the distortion. Poor reproduction
brought about by distortion will appear to the user of audio equipment as a change in
the quality of musical instruments, or as noise; to the user of communications gear,
it may also appear as channel crosstalk
It is necessary to identify nonlinear distortion before it can be eliminated. Two
methods are in common use - total harmonic distortion analysis and intermodulation distortion
analysis. The choice depends primarily on the characteristics of the system being tested
and the information desired.
Linear System Theory
Output signal of nonlinear system, with the fundamental filtered out,
is the lower trace on the oscilloscope screen. The residual output shows that seemingly
pure sine wave does in fact contain harmonics.
Each input to a perfectly linear system produces a proportional output. For example,
if an input f1(t) produces an output g1(t), and a second input
f2(t) produces an output g2(t), the sum, f1(t) + f2(t),
must produce g1(t) + g2(t) at the output. The output of the system
can then be defined as
G (jω) = H (jω) F (jω)
where F (jω) is the frequency spectrum of f(t), G (jω) the frequency spectrum
of g(t), and H(jω) the transfer function of the system, which has finite gain at
all frequencies. For every perfectly linear system, therefore, all frequencies in the
input will appear at the output, changed only by a scale factor; no frequency that is
not in the input can appear at the output.
For a perfectly linear amplifier, the expression is eo = Aein,
where ein is the voltage at the input, eo the voltage at the output,
and A the transfer function - in this case the gain of the amplifier. A nonlinear amplifier,
however, produces harmonics at the output, which can be characterized by the power series
expansion of its transfer function:
eo = A1ein + A2ein2
+ A3ein3 + ... + Aneinn
The purpose of any distortion measurement is to determine the value of the coefficients
of the terms in the series. As an example, if the input signal is
ein = e1 sin ω1t
+ e2 sin ω2t
then the output, eo, expanded into a Taylor power series, becomes:
2nd and 3rd Harmonic cComponents
If proper care has been taken during the design of a system, nonlinearity will not
be too severe. It is practical to assume that the distortion is less than 10%, so the
terms of the expansion higher than the third power have been neglected.
Analysis of the series expansion shows that the relative amplitude of the second and
third harmonic terms generated will vary directly with the input signal level. For second
harmonics, the amplitude is proportional to e12 or e22.
These terms will, therefore, vary 2 decibels per decibel of signal level change. Correspondingly,
the third harmonic terms will vary 3 db per decibel of signal. level change.
For the intermodulation terms, a frequency of the form aω1 + bω2
varies as e1|a| e2|b|. For example, the frequency
2ω1 - ω2 has an amplitude proportional to e12
e2 and will vary 2 db per decibel of signal-level change in e1,
and 1 db per decibel of change in e2.
Thus, the power series defines the nonlinearity in terms of easily recognized frequency
components, whose dependence on signal level can be readily determined.
Total Harmonic Distortion Analysis
Total harmonic distortion analysis requires only one signal source. Because of the
system nonlinearity, simple harmonics of the input signal are generated at the output.
The measurement technique compares the amplitude of the harmonics to that of the original
signal at the output, where the original signal becomes the fundamental frequency of
the harmonics. The defining equation is
total harmonic distortion =
A frequency-selective voltmeter is needed to measure the fundamental; and either a
selective voltmeter with a wide dynamic range or a frequency rejection circuit with a
true rms detector to measure the harmonics. The frequency rejection circuit nulls the
fundamental and passes its harmonics to the detector with no attenuation, so the ratio
between the fundamental and harmonics can be determined.
A less expensive way to measure the total harmonic distortion, however, is to use
a rejection filter and a broadband detector. Since the fundamental is not directly measured,
the equation becomes
If the distortion is less than 10%, the denominator of equation 2 will be within 1/2%
of the denominator in equation 1, which is as accurate as any frequency selective voltmeter.
To cut costs further, most manufacturers use an averaging detector instead of a broadband
detector. Under certain conditions, this can lead to reading errors in the null that
are 20% to 30% low, since the averaging detector responds to the area of the rectified
waveform and not to the instantaneous power of the wave shape. Even so, these types of
errors are not considered significant; they affect only the over-all percentage of harmonics
present in the output signal and not the individual terms, and the percentage is small.
For example, a 20% error in reading the null of a system with 0.1% harmonics results
in reading of 0.08% instead.
A more important error, much larger than the metering error, is caused by the attenuation
of the harmonics by the rejection circuit. This error normally effects the second harmonics
more than the higher ones. However, manufacturers of these circuits generally specify
the second-harmonic attenuation, which makes it easy to compensate mathematically for
the error in the readings.
There are two difficulties in making total harmonic distortion measurements. First,
to get a measurement within the desired accuracy, the harmonic content of the test signal
must be not more than a third of the distortion expected to be caused by the system.
Second, the chore of nulling the fundamental can be time-consuming. Oscillators that
meet the distortion requirements and automatic nulling equipment, which has recently
become available, can overcome the difficulties.
The total harmonic method is very useful when testing low-distortion circuits, which
require a large amount of negative feedback and must be unconditionally stable. It is
important that oscillations that occur in these circuits be detected. In a TH system,
the harmonics can be viewed on an oscilloscope, with the fundamental filtered out. Not
only can the character of the distortion be easily determined, but residual oscillations
that would have been much harder to find with a wave analyzer, and are too small in comparison
to the fundamental to be detected on an oscilloscope, can be viewed.
In the CCIF method of distortion measurement, two signals are applied
to the system under test. The diagram above shows how the low-pass filter separates the
low-frequency intermodulation terms from these two signals. The amplitudes of the intermodulation,
or difference terms, can be compared with that of the input signals. The lower frequency
intermodulation terms are only even.
Two signals, f1 and f2, are used in the SMPTE
method, with one having 50 times the frequency and one-quarter of the magnitude of the
other. The frequencies of interest are restricted, as shown in the diagram, to a pass-band
that is 20 times the higher frequency and centered around it. The envelope of these terms
is then used to determine the modulation index of the higher frequency, f2
in the above diagram.
Only the SMPTE method of intermodulation distortion measurement and
the harmonic distortion technique measure both odd and even order nonlinearities. When
used to analyze amplifiers, each method defined the non-linearities of the system in
the same manner and gave the same information. They only differed by a scale factor.
There are three major methods of making intermodulation distortion measurements. In
the Comité Consultatif International Téléphonique (CCIF) method,
two high-frequency signals with amplitudes e1 and e2 are applied
to the amplifier under test. The difference between their frequencies must lie within
the amplifier's passband. The low-frequency difference products are extracted from the
output signal with a low-pass filter, and their amplitudes are compared to those of the
two original signals. If the input-signal amplitudes are equal to each other and represented
by e/2, then
There is one serious fault with this method: only the even-order terms in the nonlinearity
are detected. As a result, it is not a good method where the system distortion is expected
to contain primarily odd-order terms, such as in a push-pull amplifier, or an amplifier
that is overdriven.
Comparison of Techniques
Another method, used by the Society of Motion Picture and Television Engineers, also
requires two input signals, one of which has 50 times the frequency and only one-fourth
the amplitude of the other. The output is put through a band-pass filter, which filters
out everything except the intermodulation terms. The latter are envelope-detected, and
then low-pass filtered. The distortion is defined as the modulation index of the higher
input frequency. With this method, both even- and odd-order non-linearities are detected.
The response to the even ones are the sidebands corresponding to f2 + (2n-1)f1
and the response to the odd are the f22 ± 2nf1
sidebands. The bandwidth of the bandpass filter should be approximately 20 f1
to ensure passing all the sidebands.
With the conditions that e1/4 = e2, the intermodulation distortion
of a push-pull amplifier and a single-ended amplifier can be derived from the Taylor
series. They are shown in the table at the top of this column.
A serious drawback of this technique is that the envelope detection process is nonlinear.
If the signal amplitudes are low, as is often the case in transistor circuits, envelope
detection can add significantly to the distortion at the output. Such circuits can be
tested, however, if a wave analyzer - basically a selective voltmeter - replaces the
envelope detector. This procedure requires tuning to and measuring all the spurious frequencies
generated, and then computing the modulation index. The results are very reliable, but
the procedure is time-consuming and the equipment is considerably more expensive than
that used in the total harmonic distortion method. And since all spurious frequencies
must be measured, the upper cutoff frequency of the system being tested must be 50 times
greater than the lower cutoff frequency to pass all the significant frequencies.
In fact, all of the methods discussed so far work only with broadband systems. But
there is one technique of intermodulation distortion measurement that is designed specifically
for such limited passband systems as intermediate-frequency amplifiers. Again, two signals
whose significant intermodulation products lie within the amplifier's passband are applied
to the system. In this technique, if e1 and e2 = e/2 then
This method detects only the odd-order terms of nonlinearity, since the sum of the
coefficients of the terms in the output closest to the test signal is odd. This method
is quite satisfactory in the case of i-f amplifiers, because only the odd terms cause
significant spurious responses. The equipment normally used is a wave or spectrum analyzer.
Odd Or Even
Two signals, close together, are used in the narrow band method of
distortion analysis. The amplifier under test has a narrow pass-band, and the intermodulation
terms measured are restricted to this band. As shown in the diagram above, these terms
are all odd.
To obtain complete distortion data, it is necessary, in most cases, to detect both
even and odd nonlinearities. Of the systems discussed, only total harmonic distortion
analysis and the SMPTE intermodulation methods have this capability. A brief summary
of a comparison between these two methods, made by W. J. Warren and W. R. Hewlett2,
is shown in the table on page 82. The ratios of intermodulation to harmonic distortion
(IM/HD) shown will hold true for any frequency-independent system in a predictable manner.
Both methods give the same information about the coefficients of the power series describing
the amplifier; the answers just differ by a scale factor. Even so, intermodulation measurements
are more difficult to make and generally require more sophisticated equipment than total
Intermodulation measuring requires two test signals which have no prior interaction.
The distortion of these two signals does not have to be low, since their harmonics will
not usually cause any significant intermodulation products. Setting up a measurement
at one set of test frequencies is not difficult; but if measurements are required at
several different sets of frequencies, the procedure becomes very complicated - especially
if it is necessary to tune to each intermodulation term separately.
With the total harmonic distortion method however, both high and low frequency response
can be easily measured, since only one signal frequency need be changed. This is useful
when checking the effects of diminishing feedback gains at either end of the frequency
response characteristic or the effect of load capacitance at the high-frequency end.
In addition, the total harmonic distortion method requires only that the system have
a flat frequency response over a frequency deviation of three to one, whereas the SMPTE
method requires a flat response for a deviation greater than 50 to one.
New Test Instrument
Since the nulling of the fundamental is normally the time-consuming portion of total
harmonic distortion measurement, great savings can be realized, especially in production
line testing with an analyzer which automatically rejects the fundamental. The time saved
is as much as 25 seconds of a 30-second measurement. With automatic nulling, the accuracy
of the null achieved is no longer a function of operator training, manual dexterity or
signal source frequency drift.
Automatic nulling circuitry in a new commercial wave analyzer, the H-P 333A and 334A,
operates on the principle that the fundamental at either side of a Wien bridge off null
follows well-known phase relationships. In this instrument, phase-sensitive feedback
loops are employed which drive photo-cells in parallel with the resistances on either
side of the bridge. These loops reject the fundamental and are not critical to adjust,
since any imbalance on one side of the bridge is automatically compensated for on the
other. Imbalances on either side cause phase errors in the fundamental which are in quadrature,
so the phase-sensitive feedback loops are independent of each other.
The analyzer will maintain a null even though there is a slow drift in the input frequency.
This ability to "pull" the null has opened the door to a number of applications where
the total harmonic distortion measurements were not readily applied in the past. Among
• Single-frequency production line testing of such components as integrated-circuit
amplifiers or transformers. As long as the long-term drift of the signal source is less
than ±1 %, a good null will always be achieved. Therefore, time-consuming rebalancing
operations at the test position are eliminated.
• Optimizing the performance of an oscillator. Here, any variation in the parameters
causes the frequency to shift slightly. The automatic nulling of the analyzer allows
the oscillator performance to be improved on a continuous basis, rather than by relying
on a point-to-point check, which may or may not find the optimum point.
• Correcting distortion in signal generators which produce sine waves either
by mixing or by non-linear shaping. The small frequency shifts that occur in the process
would also cause the loss of the null if it were not for the automatic null feature.
1. B.M. Oliver, "Distortion and intermodulation," Hewlett-Packard Co., Application
2. W.J. Warren and W.R. Hewlett, "Analysis of intermodulation method of distortion
measurement," Proceedings of the IRE, p. 457-466, April, 1948.
Charles R. Moore is a group leader at the research and development laboratories of
the Hewlett-Packard Companys Loveland division. He is a graduate of Princeton University
and earned his master's degree in electrical engineering at the University of California.
Posted October 22, 2018