(1) Using the transientresponse method of analysis, the square
wave in the illustration can be considered as a voltage which rises
instantaneously from 50 volts to +50 volts at time t0. It remains
at this value until time t1, then drops instantaneously to minus 50
volts and remains at this value until time t2, and so on.
(2) Using the frequencyresponse method, the square wave can be
analyzed by determining what sine waves are required to reproduce it.
To reproduce a symmetrical square wave, it is necessary to start with
a sine wave having the same frequency as the square wave repetition
frequency, and add to it the odd harmonics of this frequency as shown
graphically in Figures 2b, 2c, and 2d. Waveshape C (Figure 2b) is
formed by adding the fundamental frequency A and its third harmonic
B, which has an amplitude equal to onethird of the amplitude of the
fundamental. The resultant waveform already slightly resembles a
square wave, as can be seen by the square wave superimposed on the
diagram. Figure 2c shows the result of adding the fifth harmonic, at
onefifth of the amplitude of the fundamental, to the resultant
waveform C. In the resultant waveform E, the corners are much
sharper and the top is somewhat flatter. The seventh harmonic, at
oneseventh of the amplitude of the fundamental, is added (Figure 2d)
to form the resultant waveform G. This wave is fairly smooth across
the top and fairly sharp at the corners. Adding the ninth harmonic
at oneninth of the amplitude of the fundamental, the eleventh
harmonic at oneeleventh of the amplitude, etc., would further
sharpen the corners and flatten the top of the wave. An infinite
number of odd harmonics would produce a perfect square wave.
However, in practice, the addition of 10 odd harmonics is usually
sufficient for a satisfactory reproduction of a square wave.
(3) Many waveforms used in electronic circuits consist of short
pulses separated by long time intervals. These pulses, called
rectangular pulses, are constructed in a manner similar to the
construction of symmetrical square waves. However, in addition to
the sine waves of the fundamental pulse frequency and many harmonics
of fractional amplitudes, a small DC voltage usually is added to
create a reference level other than zero. In the construction of
this type of waveform, both odd and even harmonics are used. The
addition of the DC voltages causes the resultant wave to be formed
about the DC voltage axis, thus increasing the amplitude of the
positive portion and decreasing the amplitude of the negative portion
with reference to the zero axis. The duration of the desired pulse
in Figure 3 is onethird of the time required to complete one full
cycle. To reconstruct this pulse, a DC voltage equal to onethird of
the desired pulse amplitude is added to a sine wave of the
fundamental frequency with an amplitude twothirds of the desired
pulse amplitude, the second harmonic with an amplitude of onethird
the desired pulse amplitude, etc.
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