An Ideal Operation Amplifier would have :
infinite gain
infinite input impedance that prevents any currents to flow at the inputs
zero output impedance.
Integrated circuit OP-AMPS are used as the basic gain block in most active filters. The OP-AMP provides load isolation and a way to route energy from the supply in the proper amounts at the proper places in a resistor / capacitor network to simulate the energy storage of one or more inductors.
Below the ideal operational amplifier is shown, the inverting and the non-inverting inputs and a single output.
figure X : ideal OP-AMP
In the figure Y below, there is an operational amplifier with input impedance Z_{i}(S) and its feedback impedance Z_{f}(S). The impedances can include capacitors and resistors.
Since the Op-Amp has a high gain and input impedance, the current that flows into the negative terminal is negligible. This causes the voltage at point V1 to become zero. Note that the positive terminal is connected to the ground so that the voltage at point V2 becomes zero as well so that the voltage at V1 = voltage at V2 = 0
When summing the input current and the output current at point V1, the result is zero =>
figure Y : example active circuit
The transfer function would be: