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
Gibbs phenomenon, 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.
July 1957 Radio & TV News
Wax nostalgic about and learn from the history of early electronics.
See articles from Radio &
Television News, published 1919 - 1959. All copyrights hereby acknowledged.
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.
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.
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.
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 test technique.
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.
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.
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
speaking, 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.
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.
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
Since 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.
Only 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.
In general, 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 under test.
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
Response to L. F. Square Waves
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.
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
Fig. 6. Illustrating rise time measurement on a square waveform.
Fig. 7. Distributed capacities and inductances in basic
Fig. 8. Series and shunt peaking coils compensate for high-frequency
Fig. 9. Basic amplifier response curves showing effects
of compensating circuits.
If the 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-Rg 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-Cc 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 or Rg, 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 should be made smaller or Rc
should be made larger (with respect to RL). 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 or Rg
has become lower in value (usually Cg1 will have partially
opened ... fully open would result in Fig. 4G); that Ck
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 very likely.
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
Response to H. F. Square Waves
Circuit 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
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.
Since the 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
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 where TR 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
Distributed capacities are reduced by
keeping leads short, parts and leads above the chassis, and using
miniature parts where economically feasible.
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
damping (RS1 and RS2 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 considered objectionable.
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 Rk 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 (Fig. 8).
shown in Fig. 5D may seldom be encountered, but generally indicates
that a damping resistor across a peaking coil has opened.
The condition of Fig. 5B, at higher frequencies, together with
the condition shown in Fig. 4H at lower frequencies, would indicate
that Cc (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.
A simple 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.
If low-frequency 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.
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, but this results in reduced
Posted February 5, 2014