Question 19.SP.9: A RC series circuit with a square-wave input is shown (Fig. ......
A RC series circuit with a square-wave input is shown (Fig. 19-15). The input is a periodic train of pulses with an amplitude of 10 V and a width of 1 ms, with each pulse generated every 2 ms. Plot the output voltage curve across the resistor.
Find some values of v_R within the 0 to 1 ms that the pulse is on, say at T = 0.1 ms, 2T = 0.2 ms, and 3T = 0.3 ms. First, write the formula for v_R when the circuit is charging.
v_R=V e^{-t/ R C} (19-16)
When t = T = 0.1 ms, v_R=10 e^{-1}=10(0.368)=3.68\ V
When t = 2T = 0.2 ms, v_R=10 e^{-2}=10(0.135)=1.35\ V
When t = 3T = 0.3 ms, v_R=10 e^{-3}=10(0.05)=0.5\ V
At t = 5T or 0.5 ms, v_R will reach its steady-state value of 0 V.
When the pulse is turned off between 1 and 2 ms, v_R will instantly drop to -10 V because the capacitor charged to 10 V requires a finite time to discharge. Then v_R will gradually rise to 0 V.
The formula for v_R when the circuit is discharging is
v_R=-V e^{-t / R C} (19-21)
where t represents time from 1 to 2 ms. So when t = 1.1 ms from origin,
v_R=-3.68 VWhen t = 1.2 ms, v_R=-1.35\ V
When t = 1.3 ms, v_R=-0.5\ V
When t = 1.5 ms, v_R=0\ V
The symmetry of the output curve is shown in the plot (Fig. 19-16). Because the circuit has changed the waveform of input pulses to peaks, it is called a RC peaker. It is called also a diferentiating circuit because v_R can change instantaneously. We obtain a peak output when the circuit time constant is small compared with the half-period of the input waveform. In this case T = 0.1 ms compared with the half-period of 1 ms, so the ratio of the time constant to the half-period is 1 to 10.
In summary, to explain a short RC time circuit, a 10-V input is applied for 1 ms (10 time constants of the RC circuit), allowing C to become completely charged and v_R to be 0 V (Figs. 19-15 and 19-16). After C is charged, v_C remains at 10 V with v_R at 0 V (V = v_C+v_{R}). After 1 ms, the total voltage V drops to zero, C discharges completely in five time constants, and v_C and v_R remain at 0 V while there is no applied voltage. On the next cycle, C charges and discharges completely again. The interval between pulses when the input voltage is 0 V acts as a “short” to the RC circuit.
