(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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