R-C or RC circuit differentiator is kind of first order circuits. First order circuit means that it’s behavior is described with differential equation of the first order. R-C circuit looks like it is shown in picture bellow.
When switch is closed, according to KVL:
outcome is that:
Differentiating both sides of given equation,
Solving this first order differential equation yields to exponential function for current in R-C differentiator circuit:
In here, I0 is initial current at the time instant when switch is just closed. Due to exponential decay, initial switch-on moment is the moment when current is largest. Naturally, current through series RC circuit is dropping as time goes, since capacitor is like open switch for steady state DC current. Initial current is
It means that in time instant when switch is turned on, capacitor behaves like short connection. Reason for that is fact that voltage across capacitor plates cannot change instantaneously. Instant change of voltage across capacitor plates would require infinite current. Now, when switch is just turned on, potential of plate A goes to VIN. Potential of plate B can’t stay on ground level, because it would required infinite current, as it was mentioned earlier. So capacitor draw potential of B plate to VIN too. At that time instant current is largest and it is I0. Later, it is dropping according to exponential function. Product RC is called time constant, RC=?. When time equal to time constant is passed, capacitor current drops to approximately 1/3 of initial value. For continuous input voltage, as higher the capacitor impedance to resistor R, R-C circuit is closer to perfect differentiator circuit.
If input voltage is continuous, op-amp R-C circuit differentiator will do the job. But, if input signal is step signal, R-C circuit without op-amp is better solution. Reason for that is the fact that R-C circuit produces spikes when input is step signal. This spike on inverting input in op-amp would bring op-amp into negative saturation, since non-inverting input is on ground level. Pure R-C differentiator circuit without op-amp has no such problems.