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Nonsinusoidal Voltages Applied to an RC Circuit The harmonic content of a square wave must be complete to produce a pure square wave. If the harmonics of the square wave are not of the proper phase and amplitude relationships, the square wave will not be pure. The term PURE, as applied to square waves, means that the waveform must be perfectly square. Figure 428 shows a pure square wave that is applied to a seriesresistive circuit. If the values of the two resistors are equal, the voltage developed across each resistor will be equal; that is, from one pure squarewave input, two pure square waves of a lower amplitude will be produced. The value of the resistors does not affect the phase or amplitude relationships of the harmonics contained within the square waves. This is true because the same opposition is offered by the resistors to all the harmonics presented. However, if the same square wave is applied to a series RC circuit, as shown in figure 429, the circuit action is not the same. Figure 428.  Square wave applied to a resistive circuit.
Figure 429.  Square wave applied to an RC circuit.
The RC INTEGRATOR is used as a waveshaping network in communications, radar, and computers. The harmonic content of the square wave is made up of odd multiples of the fundamental frequency. Therefore SIGNIFICANT HARMONICS (those that have an effect on the circuit) as high as 50 or 60 times the fundamental frequency will be present in the wave. The capacitor will offer a reactance (X_{C}) of a different magnitude to each of the harmonics
This means that the voltage drop across the capacitor for each harmonic frequency present will not be the same. To low frequencies, the capacitor will offer a large opposition, providing a large voltage drop across the capacitor. To high frequencies, the reactance of the capacitor will be extremely small, causing a small voltage drop across the capacitor. This is no different than was the case for low and highpass filters (discriminators) presented in chapter 1. If the voltage component of the harmonic is not developed across the reactance of the capacitor, it will be developed across the resistor, if we observe Kirchhoff's voltage law. The harmonic amplitude and phase relationship across the capacitor is not the same as that of the original frequency input; therefore, a perfect square wave will not be produced across the capacitor. You should remember that the reactance offered to each harmonic frequency will cause a change in both the amplitude and phase of each of the individual harmonic frequencies with respect to the current reference. The amount of phase and amplitude change taking place across the capacitor depends on the X_{C} of the capacitor. The value of the resistance offered by the resistor must also be considered here; it is part of the ratio of the voltage development across the network. The circuit in figure 430 will help show the relationships of R and X_{C} more clearly. The square wave applied to the circuit is 100 volts peak at a frequency of 1 kilohertz. The odd harmonics will be 3 kilohertz, 5 kilohertz, 7 kilohertz, etc. Table 41 shows the values of X_{C} and R offered to several harmonics and indicates the approximate value of the cutoff frequency (X_{C} = R). The table clearly shows that the cutoff frequency lies between the fifth and seventh harmonics. Between these two values, the capacitive reactance will equal the resistance. Therefore, for all harmonic frequencies above the fifth, the majority of the output voltage will not be developed across the output capacitor. Rather, most of the output will be developed across R. The absence of the higher order harmonics will cause the leading edge of the waveform developed across the capacitor to be rounded. An example of this effect is shown in figure 431. If the value of the capacitance is increased, the reactances to each harmonic frequency will be further decreased. This means that even fewer harmonics will be developed across the capacitor. Figure 430.  Partial integration circuit.
Figure 431.  Partial integration.
Table 41.  Resistive and reactive values
The harmonics not effectively developed across the capacitor must be developed across the resistor to satisfy Kirchhoff's voltage law. Note the pattern of the voltage waveforms across the resistor and capacitor. If the waveforms across both the resistor and the capacitor were added graphically, the resultant would be an exact duplication of the input square wave. When the capacitance is increased sufficiently, full integration of the input signal takes place in the output across the capacitor. An example of complete integration is shown in figure 432 (waveform e_{C}). This effect can be caused by significantly decreasing the value of capacitive reactance. The same effect would take place by increasing the value of the resistance. Integration takes place in an RC circuit when the output is taken across the capacitor. Figure 432.  Integration.
The amount of integration is dependent upon the values of R and C. The amount of integration may also be dependent upon the time constant of the circuit. The time constant of the circuit should be at least 10 TIMES GREATER than the time duration of the input pulse for integration to occur. The value of 10 is only an approximation. When the time constant of the circuit is 10 or more times the value of the duration of the input pulse, the circuit is said to possess a long time constant. When the time constant is long, the capacitor does not have the ability to charge instantly to the value of the applied voltage. Therefore, the result is the long, sloping, integrated waveform. Q.18 What are the requirements for an integration circuit? 
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