you work with oscilloscopes on a regular basis, you know know one of
the first things you do (or should do) is to calibrate the frequency
response of the probe by hooking it onto the squarewave port and tweaking
the probe capacitor for no overshooting or undershooting at the waveform
edges, and then verify that the displayed amplitude is correct. I remember
being amazed during engineering courses at learning that any periodic
waveform can be described mathematically as the sum of sinewaves at
various frequencies, amplitudes, and phases. Knowing the theory behind
those waveforms - particularly standard ones like squarewaves, trianglewaves,
sawtooths, etc. - really helps in understanding what you see on the
o-scope and in troubleshooting problems. The same goes for interpreting
the impulse and step function responses as influenced by resistance,
capacitance, and inductance effects. Perhaps the most amazing thing
I learned about squarewaves is that, based on the
, anything short of an infinite series of additive
sinewaves when representing a squarewave results in an overshoot - albeit
vanishingly minute - at the edge. In the real world, complex reactive/resistive
effects render the effect undetectable.
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vintage Radio News
Practical Techniques of Square-Wave Testing
By E. G. Louis
A square-wave generator and an oscilloscope are
useful tools in designing and servicing wide-band amplifiers. What troubles
to look for with certain scope patterns.
Fig. 1. A square wave (1) is made up of fundamental sine
wave (2) and odd harmonics (3).
Fig. 2. Setup for square-wave tests.
Fig. 3. Basic triode amplifier circuit.
By means of a Fourier analysis it is possible to show that an "ideal"
square wave, Fig. 1 (Curve 1) consists of a fundamental (Curve 2) sine
wave whose frequency is equal to that of the square wave, together with
the 3rd (Curve 3), 5th, 7th, 9th, and higher odd harmonics, the amplitude
of each decreasing in direct proportion to its order. Theoretically,
a perfect square wave consists of a fundamental together with an infinite
number of higher odd-harmonic signals.
If such a square-wave
signal is applied to the input of an electrical circuit, whether a filter
network, amplifier, or other system, and the system does not respond
equally well to the fundamental and all higher harmonics, then the output
signal obtained will be distorted in a fashion indicative of the response
characteristics of the system under test. This is the basis of the square-wave
Since not only the fundamental, but all higher
harmonics are applied simultaneously, and an indication of the system's
response to this wide range of signals is obtained at once, square-wave
testing provides an extremely rapid method for checking such network
characteristics as frequency response, phase shift, transient response,
etc. Because of the speed with which the square-wave test technique
can be applied and information obtained, this method becomes quite valuable
not only as an aid in the production testing and servicing of electronic
equipment, but can be applied with equal, if not greater, value to the
requirements of the practical design engineer.
In the past,
the technique of square-wave testing has been confined largely to testing
high-fidelity audio amplifiers and wide-band amplifiers with a bandwidth
of perhaps several hundred kilocycles or 1 megacycle. With wide-band
oscilloscopes now commercially available at prices within the reach
of even the moderate sized experimental laboratory, and with square-wave
generators on the market delivering square waves with fast rise times
to frequencies as high as 1 megacycle, this valuable and easily applied
technique can be applied to a very much greater extent. It may be used
for checking not only audio amplifiers, transformers, and similar systems,
but also for the check and design of wide-band scope and radar amplifiers,
video amplifiers, and similar wide-frequency-range networks.
Since the techniques of square-wave testing and analysis can be
applied in the same fashion irrespective of the end result in view,
whether servicing, design, or production test, we will try to simply
outline the basic technique, with the major emphasis on the application
of the technique in the design and service of wide-band amplifier circuits.
only two pieces of equipment are required to apply the square-wave test
technique, a square-wave generator and an oscilloscope. The square-wave
generator may consist of a sine-wave generator and a suitable clipper
amplifier where audio circuits and comparatively narrow-band circuits
are to be checked. Where the response of video amplifiers and similar
wide-frequency-range systems are to be checked, however, it is best
to obtain a specially designed square-wave generator.
In general, the square-wave generator should deliver perfect square
waves with a short rise time at frequencies from the lowest frequency
response of the system to be studied to a frequency one-tenth the highest
frequency response of the system. For practical laboratory work, a
squarewave generator delivering signals from 50 cps to about 500 kc.
or 1 mc. with a rise time of at most .1 microsecond (and preferably
less) will be found suitable. These signals may be available over a
continuously variable range, or only at four or five "spot" values within
this range. The output voltage should be easily varied from under 1
volt to at least 8 to 10 volts. Output impedance should be low, 600
ohms is about the highest that can generally be tolerated, particularly
at high frequencies.
The oscilloscope used must have characteristics
that are superior to the system under test. From a general viewpoint,
its vertical amplifier should be fiat below the lowest frequency square
wave to be used in testing to a frequency ten times higher than the
highest frequency signal to be used (within 1 or 2 db). It should not,
in itself, cause any appreciable tilt or overshoot to any square-wave
signal applied to its input within the range to be used for test purposes.
The vertical amplifier should have a sensitivity of at least .5 volt/inch
(peak-to-peak) and preferably more.
A linear time base should
be available within the scope which permits observation (with expanded
sweep if necessary) of one cycle of both the highest and lowest frequency
square waves to be used in testing. Applying the Technique
The basic set-up used for square-wave testing is illustrated
in block diagram form in Fig. 2. A square-wave signal is applied to
the input of the system to be tested, and the input and output signals
observed on a cathode-ray oscilloscope. Deviations from the original
square-wave shape indicate certain characteristics of the system under
Test leads, both to and from the equipment, should be
as short as possible, otherwise, with high-frequency signals, or signals
with a short rise time, unnatural peaking and overshoot may be introduced
due to resonance in the connecting leads themselves.
signal should be observed a a point where the loading of the CRO will
not appreciably affect the circuit parameters. If a high-impedance,
low-capacity probe is used with the scope, then individual stage characteristics
can be observed.
Limitations of Test Technique
a square wave contains only a fundamental and higher harmonics, it is
not ordinarily employed for checking the response of a system at frequencies
lower than its fundamental value. The exception to this is the case
of a network whose response is such that the fundamental of the square
wave is changed in some manner with respect to its higher frequency
components. Such a condition may cause a change in the square-wave shape
indicative of the system's response at lower frequencies.
odd harmonics of the square wave are present as part of the entire signal,
hence any sharp dips or holes in the response characteristics of the
system at specific frequencies falling between the odd harmonics may
not show up in a square-wave test. However, the response of most amplifiers
varies in a smooth manner and this limitation is minor.
a square-wave test will not indicate distortion due to overload or overdrive
on an amplifier, unless the overload distortion varies with frequency.
The square wave is simply made more "square," and a sine-wave signal
still must be used for such tests.
Finally, since it is almost
physically impossible to produce a "perfect" square wave, and very difficult
to detect changes in the square wave because of deterioration of signals
higher than the tenth harmonic, square-wave signals should be available,
and used, at approximately decade values. The exact number of signals
required for a complete test will depend on the bandwidth of the system
From a practical viewpoint, signals of 50 cps and
1 kc. are suitable for testing usual amplifiers and transformers. For
checking wide-band amplifiers generally used in the lab, frequencies
of 50 cps, 1 kc., 10 kc. and 100 kc. may be used. For checking video
amplifiers, signals of 50 cps, 1 kc., 10 kc., 100 kc., and 500 kc. may
be used. Signals as high as 1 mc. may be used for checking special pulsing
circuits. Response to L. F. Square Waves
Fig. 4. Typical patterns obtained with low-frequency square
Fig. 5. Square-wave patterns that may result from circuit deficiencies
at the high frequencies with a good square-wave input.
Fig. 6. Illustrating rise time measurement on a square waveform.
Fig. 7. Distributed capacities and inductances in basic amplifier
Fig. 8. Series and shunt peaking coils compensate for high-frequency
Fig. 9. Basic amplifier response curves showing effects of compensating
Typical patterns that may be obtained when a low-frequency square wave
is applied to an amplifier or network are shown in Fig. 4. A basic triode,
resistance-coupled amplifier age is shown in Fig. 3.
amplifier responds perfectly to the input square-wave signal, neither
attenuating nor accentuating the higher harmonics and causing no phase
shift, a perfect output square wave will be obtained which, except for
amplitude, is identical with the input signal, as shown in Fig. 4A.
A boost at the fundamental frequency of the square wave with
respect to it higher harmonics, but with no phase shift, will result
in the rounded signal shown in Fig. 4B. Conversely, a loss at the fundamental
frequency will result in a general dip in the square wave as shown in
Fig. 4C, while a dip in the response curve, causing a loss of a particular
harmonic, will result in a dip at one or more points in the square wave,
as shown in Fig. 4D.
Leading phase shift at low frequencies,
but without appreciable signal loss, displaces the fundamental with
respect to the harmonics, resulting in a tilted square top as shown
in Fig. 4E. This is generally due to too low time constant in the RC
coupling network (Cg1
in Fig. 3). If a loss
of signal accompanies the phase shift, then the flat top will curve
downward as well as be slanted, as shown in Fig. 4F. An extreme case
of too low a time constant in the coupling network may cause differentiation
of the signal, allowing only the higher harmonics to pass and resulting
in a peaked signal as shown in Fig. 4G. Such a signal may also be obtained
due to high-frequency leakage around an attenuator circuit.
Where low-frequency compensation is added to the amplifier stage (Rc
in Fig. 3), overcompensation may result in the phase lagging at low
frequencies, causing the square wave to tilt in the opposite direction,
as shown in Fig. 4H.
Irrespective of whether leading or lagging
phase shift causes the square wave flat top to tilt (Figs. 4E or 4H),
the amount of tilt depends on the degree of phase shift. A 10% slope
will be obtained when the phase shift is 2° at the fundamental frequency
of the signal.
From a design viewpoint, conditions shown in
Figs. 4E, 4F, or 4G generally indicate (in Fig. 3) either that Cg1
, both, should be increased in value, that Ck
should be made larger, or that insufficient low-frequency compensation
has been added. This, in turn, means that either Cc
be made smaller or Rc
should be made larger (with respect
). If the condition shown in Fig. 4H is obtained, then
the amount of low-frequency compensation should be lowered, by either
reducing the value of Rc
or increasing the value of Cc
From a servicing viewpoint, conditions indicated in Figs. 4E,
4F, or 4G generally indicate (again in Fig. 3) either that Cg1
has become lower in value (usually Cg1
have partially opened ... fully open would result in Fig. 4G); that
has lost capacity, that Cc
has increased in
capacity (unlikely) or that Rc
has dropped in value (which
may happen due to overload). The condition of Fig. 4H indicates, generally,
that C. has lost capacity or developed high power factor. It may also
indicate that Rc
has increased in value, but this is not
The response of the amplifier to both low- and
high-frequency signals must be considered before a full analysis of
circuit operation can be made. Both tests, when taken together, give
a much better picture of conditions in the system under test.
Response to H. F. Square Waves
deficiencies at high frequencies may result in any of the patterns shown
in Fig. 5 (and even in some of those shown previously, particularly
as far as phase shift is concerned). A typical single-stage, RC-coupled
triode amplifier is shown in Fig. 7, together with some of the factors
affecting its response to high frequencies.
As in Fig. 4A, the
"perfect" signal is shown in Fig. 5A. A loss of higher frequency harmonic
signals will result in a rounding of the leading edge of the square
wave as in Fig. 5B. The degree of rounding is dependent on loss of high
frequencies, and an extreme loss will result in the output square wave
approaching a sine wave in form. If almost all higher harmonics are
lost, the square wave may appear as in Fig. 5C.
in the amplifier (or in the connecting leads) may cause "ringing" and
result in damped oscillations on the leading edge of the square wave,
as shown in Fig. 5D. The frequency at which the oscillations occur can
be determined approximately by multiplying the number of "cycles" that
would be present along the flat square top of the signal (if not damped)
by twice the fundamental frequency of the square wave. Where rapid damping
occurs, or where the resonant frequency is extremely high, only a small
"overshoot," as shown in Fig. 5E, may be obtained.
rise time (time for square wave to go from 10% to 90% of its peak value)
is dependent on the number of higher harmonic signals present without
attenuation, this serves as a good indication of uniform frequency response
of an amplifier or network irrespective of whether rounding of the square
wave occurs or not.
An increase in rise time in a square wave
is shown in Fig. 6.
A simple relationship, accurate enough for
most practical design work, between rise time and uniform frequency
response of an amplifier, is as follows:
Maximum f (uniform
response in mc.) = 1/2TR
is the rise
time in microseconds. (This relation holds true only where artificial
means, such as peaking coils, are not used to shorten the rise time.)
From a design viewpoint, conditions shown in Figs. 5B or 5C
indicate that the high-frequency response of an amplifier is not sufficient.
This can be improved by reducing the effect of distributed capacities
shown in Fig. 7 by making RL
as small as is practicable for
the gain desired, and then by using series and shunt peaking coils,
as shown in Fig. 8, to offset these capacities. To obtain reasonable
gain with a low RL
, it may be necessary to go to tubes having
high mutual conductance. In such a case, the stage gain is equal to
the product of the load resistance and tube mutual conductance.
Distributed capacities are reduced by keeping leads short, parts
and leads above the chassis, and using miniature parts where economically
The inductance of the peaking coils is generally chosen
so that resonance will occur with the distributed capacities in the
circuit at frequencies higher than the highest frequency at which uniform
response is desired in the amplifier. In better amplifiers, these coils
are usually made adjustable so that each unit may be adjusted for best
response for the individual distributed capacities in that unit.
If these coils resonate at too low a frequency, or if insufficient
in Fig. 8) is used, a severe
overshoot may occur, or an oscillatory wave train may be set up, as
shown in Fig. 5D.
Often a slight amount of overshoot (about
5% maximum) is desirable, as it tends to shorten the rise time of the
amplifier. Thus, the condition shown in Fig. 5E would not always be
A loss of higher frequency signals
may occur if there is inductance in the electrolytic capacitor used
for cathode bypass (Lck
in Fig. 7), due to degeneration across
and the loss of gain at these frequencies. In practical
design, this can be offset by bypassing the electrolytic with a small
capacitor (around .005 μfd.), as at Ck1
in Fig. 8.
From a servicing viewpoint, conditions illustrated in Figs. 5B and
5C may indicate that a peaking coil has become shorted or open (if shunted
with a damping resistor) or that .-RL
has increased in value.
In some cases it may indicate an open in Ck1
The condition shown in Fig. 5D may seldom be encountered, but
generally indicates that a damping resistor across a peaking coil has
The condition of Fig. 5B, at higher frequencies, together
with the condition shown in Fig. 4H at lower frequencies, would indicate
(Fig. 3) has become open or dropped in capacity.
Thus, it is practical to use a combination of conditions to indicate
a specific defect.
Square waves, observed with a cathode-ray
oscilloscope, provide an efficient and extremely rapid technique for
testing and servicing systems designed to pass a band of frequencies.
The technique can also be used to good advantage in design and production
engineering, for adjusting circuits for proper operation, and for determining
optimum values of components.
The basic relationship between
the frequency response of an amplifier or network and its effect on
square waves can be obtained by referring to Fig. 9.
resistance-coupled amplifier may have the response shown in Curve 1.
With this type of response curve, a low-frequency square wave will appear
as in Fig. 4E; at middle-range frequencies, as at Figs. 4A or 5A; while
a high-frequency square wave will appear as at Fig. 5B.
compensation is added, resulting in a boost at low frequencies as shown
in Curve 2, middle-range and high-frequency square waves will appear
as previously, but low-frequency square waves will generally be tilted
in the opposite direction as shown in Fig. 4H. With the proper amount
of compensation, a perfectly flat-top low-frequency square wave may
be obtained, but this is frequently difficult to maintain in production
unless each unit is adjusted individually.
When a peaking coil
is used to provide a boost at high frequencies, the over-all response
may appear as shown in Curves 3 or 4, depending on the frequency of
peaking and the amount of damping. With a response as shown in Curve
3, a high-frequency square wave may appear as in Fig. 5E, while middle
and lower frequency square waves will remain as previously. If the response
is as in Curve 4, a high-frequency square wave will appear as in Fig.
5D, and a middle-range frequency square wave as in Fig. 5E.
If high-frequency compensation is obtained by reducing distributed capacities,
a high-frequency square wave can be made to approach the input signal
in form, as in Fig. 5A, and the response curve becomes as shown in Curve
5. Note that the response falls off smoothly at higher frequencies.
The same result can be obtained by reducing RL
this results in reduced overall gain.
Posted February 5, 2014