Introduction to Electric Circuits

8.3 Circuits containing resistance and capacitance 189

The RC differentiator circuit

v(

B

i c

.~ r I I I^0

V o = M R

o

Figure 8.26

If the output voltage is taken across the resistor as shown in Fig. 8.26, then the

circuit is called a differentiator. At instant t = 0, when the leading edge of the

pulse arrives at the input terminals AB, the voltage across the capacitor cannot

suddenly change so the potential difference across it is zero and a voltage of + V

appears at the output terminals. Thus Vo (=VR) = V. The capacitor then begins

to charge in accordance with Equation (8.14) so Vc = V[1 - exp (-t/CR)]. At

the same time the potential difference across the resistor falls exponentially in

accordance with Equation (8.16).

If the pulse width is greater than five time constants (5CR) then the voltage

Vo will have reached zero before the trailing edge of the pulse arrives (i.e.

before the pulse is removed). At this point Vin--0 SO that Vc + 12R = 0 and

because vc = V and cannot change suddenly then VR must immediately become

equal to - V.

As the capacitor then begins to discharge in accordance with Equation (8.19),

VR will rise towards zero in accordance with Equation (8.21) so that at every

instant Vc + VR = 0. The waveforms of vc and VR (=Vo) are therefore as shown

in Fig. 8.27.

v

V ......

Vc

0

-V

t

Figure 8.27

If the pulse width is less than five time constants then the capacitor will only
be partially charged by the time the pulse is removed, so that vc will be less than
V and VR will not have fallen to zero. The input voltage Vin is now equal to zero