..Because of the increasing popularity of solidstate inverters, the use of
square waves to operate motors has become commonplace. A wide variation of
results has attended such practice. At best, the substitution of such waveforms
for the in tended sinusoidal excitation has caused no problems. At worst, it
has been found that the motor would not start or operate properly. Between
these two extremes, it’s not unusual to find that operation is essentially
satisfactory but with greater temperature rise than is experienced with sinewave
operation. Much depends upon the type of motor and its construction. The harmonic
energy in a square wave con tributes considerably to the eddy current and hysteresis
losses m motors. It also tends to adversely affect commutation and torque characteristics.
Just how adverse these effects are, usually has to be determined empirically.
Fortunately, some motors have sufficient selfinductance to oppose the flow
of heavy harmonic currents. In other instances, a primitive lowpass filter
can provide this function. Thus, a provision such as shown in FIG. 19 can present
a near enough sinusoid to the motor to alleviate certain difficulties.
One factor that is often overlooked when attempting squarewave operation
of motors is the equivalency between sinusoidal and square waves. That
is, what squarewave voltage would substitute for the rated sinusoidal
voltage? The answer is not immediately obvious and if you resort to “common
sense” you can easily be lead astray. In solving this dilemma, it’s best
to assume that the fundamental component of the square wave is useful
in operating the motor. As previously mentioned, the harmonics either
“come along for the ride” or they produce detrimental effects. To simplify
the situation, suppose the motor’s inductance keeps harmonic currents
at a negligible level. The question that must then be answered is what
the RMS value of the square wave’s fundamental frequency is relative
to that of the square wave itself.
FIG. 19 Filtering of motor current by selfinductance of motor and/or
additional filtering. Actual motor current tends to be a rounded trapazoid.
Additional filtering might be needed, but small filter components generally
suffice because a highquality sine wave is not necessary. Motor inductance,
Motor current, Small inductor (Not always needed)
FIG. 20 A closer look at relationships between a square wave and its
fundamental. The peak of the fundamental sine wave is actually greater
than the peak of the square wave, yet the RMS and average values are
lower than those of the square wave. Peak value of sine wave fundamental
component; Peak RMS and average value of square wave; Average value of
sine wave fundamental component
FIG. 21 The harmonic composition of a near square wave. The fundamental
(1st harmonic) produces motor torque. The higher harmonics generate eddy
current and hysteresis losses and tend to interfere with smooth torque.
RMS value of sine wave fundamental component; Ideal square wave; Resultant
of five odd harmonics; Fundamental (1st harmonic); 3rd harmonic; 5th
harmonic
FIG. 20 shows the somewhat surprising fact that the peak value of the
first harmonic in a square waveform exceeds that of the square wave itself.
How can this be? The explanation is that the overall contributions of
the fundamental and the many higher harmonics is such that the flat top
of the square wave is developed. For example, if you considered the third
harmonic, you would find that it subtracts from the peak of the fundamental.
This is visually evident in FIG. 21. Again, the net result of many harmonics
is the square wave, even though the peak of the fundamental actually
exceeds the amplitude of the square wave itself.
It can be shown that the peak value of the fundamental sine wave exceeds
the top of the square wave by the fraction 4/Pi, or 1.27. In other words,
the peak value of the fundamental is 27 percent greater than the flattop,
or peak value, of the square wave. The important thing to note, however,
is that despite this relationship between peak values, the RMS value
of the fundamental sine wave is still below that of the square wave.
This too can be ascertained from FIG. 20. A larger amplitude square wave
is needed in order to yield a fundamental sine wave whose RMS value matches
the rated RMS sinewave voltage of the motor. Thus, a 115volt motor
will require a square wave with a greater value than 115 volts. But,
how much greater?
Assuming only that a particular motor has sufficient inductive reactance
to cause motor current to be “reasonably” sinusoidal, you can calculate
the required squarewave voltage that must be applied to the motor to
simulate its rated sinewave performance. We’re not dealing with the
low distortions expected in stereo equipment. Rather, distortions of
20 or 30 percent will enable entirely satisfactory motor operation. Referring
again to FIG. 20, start with the fact that the peak value of the fundamental
sine wave is 1.27 times the squarewave amplitude. The RMS value of this
sine wave will then be 0.707 x 1.27, or 0.898 times the amplitude of
the square wave. Thus, the sinewave RMS value is actually less than
the squarewave amplitude. (Square waves, RMS, average, and peak values
are identical.)
The previous calculation shows that the requisite square wave must be
1/0.897, or 1.11 times the squarewave amplitude. For example, a motor
rated for 115 volts from a sinewave source will require 115 x 1.11,
or 127.7 volts from a squarewave source, such as a solidstate inverter.
If there is not sufficient selfinductance, or external inductance associated
with the motor, the above reasoning might not prove valid. Indeed, the
first several harmonies of the squarewave might then contribute enough
hysteresis and eddycurrent heating, and might interfere sufficiently
with the torque of the motor so that an actual reduction in squarewave
voltage from the motorrated voltage might prove necessary. Under such
conditions, the optimum performance won’t likely be forth coming. A rudimentary
lowpass filter in conjunction with a squarewave voltage of 110 percent
rated motor voltage will yield good results in most instances.
