An application of Kirchhoff's law shows the relationship between the waveforms across the resistor
and capacitor in a series network. Since the sum of the voltage drops in a closed loop must equal the total
applied voltage, the graphical sum of the voltage waveforms in a closed loop must equal the applied
waveform. Figure 4-39 shows a differentiator circuit with the output taken across a variable resistor.
Figure 4-39.RC circuit as a differentiator.
Short Time-Constant Differentiator
With the variable resistor set at 1,000 ohms and the capacitor value of 0.01 microfarad, the time
constant of the circuit is 10 microseconds. Since the input waveform has a duration of 100 microseconds,
the circuit is a short time-constant circuit.
At the first instant of time in the short time-constant circuit, the voltage across the capacitor is 0.
Current flows through the resistor and causes a maximum voltage to be developed across it. This is shown
at the first instant of time in the graph of figure 4-40.
Figure 4-40.Square wave applied to a short time-constant differentiator.
As the capacitor begins accumulating a charge, the voltage developed across the resistor will begin
to decrease. At the end of the first time constant, the voltage developed across the resistor will have
decreased by a value equal to 63.2 percent of the applied voltage. Since 100 volts is applied, the voltage
across the resistor after 1TC will be equal to 36.8 volts. After the second time constant, the voltage across
the resistor will be down to 13.5 volts. At the end of the third time constant, eR will be 5 volts and at the