Determine the output voltage waveform for the RC differentiator in Figure 20-26 with the rheostat set so that the total resistance of R_{1} and R_{2} is 2 kΩ.
Determine the output voltage waveform for the RC differentiator in Figure 20-26 with the rheostat set so that the total resistance of R_{1} and R_{2} is 2 kΩ.
First, calculate the time constant.
\tau =R_{tot}C= (2 \ k\Omega )(1 \ \mu F)= 2 \ m sOn the rising edge, the resistor voltage immediately jumps to +25 V. Because the pulse width is 5 ms, the capacitor charges for 2.5 time constants and therefore does not reach full charge. Thus, you must use the formula for a decreasing exponential in order to calculate to what voltage the output decreases by the end of the pulse,
v_{out}=V_{i}e^{-t/RC}= 25e^{-5ms/2ms} = 25 (0.082) = 2.05 \ Vwhere V_{i}= 25 V and t= 5 ms. This calculation gives the resistor voltage ( V_{out} ) at the end of the 5 ms pulse width interval. On the falling edge, the resistor voltage immediately jumps from +2.05 V down to —22.95 V (a 25 V transition). The resulting waveform of the output voltage is shown in Figure 20-27.