Op-amp properties and differentiator

Four properties of an ideal op-amp

1. Infinite open-loop voltage gain.
2. Infinite input resistance (zero input current).
3. Zero output resistance.
4. Infinite bandwidth, infinite CMRR and zero offset voltage.

Differentiator using op-amp

               +-----[ Rf ]-----+
               |                |
          C    |     |\         |
 Vin o----||---+-----|- \       |
                     |   >------+----o Vo
               +-----|+ /
               |     |/
              GND

The input Vin is connected through a capacitor C to the inverting input; the feedback resistor Rf connects the output to the inverting input; the non-inverting input is grounded.

Proof

• The inverting input is at virtual ground, V− = 0.
• Current through the capacitor: i = C d(Vin − 0)/dt = C dVin/dt.
• No current enters the op-amp, so the same current flows through Rf: i = (0 − Vo)/Rf = −Vo/Rf.

Equating:

C dVin/dt = −Vo/Rf

Vo = −RfC dVin/dt

So the output is proportional to the derivative of the input, with a minus sign (180° phase shift). The scaling factor is the time constant RfC.

Example: for a sine input, Vin = Vm sin ωt, the output is Vo = −RfCωVm cos ωt. A square wave input gives spike (pulse) outputs, and a triangular input gives a square wave.

Practical note: the gain increases with frequency and amplifies noise, so a small resistor in series with C and a small capacitor in parallel with Rf are added to limit the high-frequency gain.