Why Ideal Op-Amps Have Infinite Gain and Input Impedance

In analog electronics, assuming an operational amplifier (op-amp) has infinite open-loop gain and infinite input impedance is a foundational concept used to simplify circuit analysis and design. This article explores how these theoretical ideals eliminate complex internal variables, prevent circuit loading, and allow engineers to predict amplifier behavior using only external components like resistors and capacitors. By understanding these two core assumptions, you will grasp the mechanics behind negative feedback and the golden rules of op-amp circuit design.

The Purpose of the Ideal Op-Amp Model

Real-world operational amplifiers are complex integrated circuits containing dozens of transistors, resistors, and capacitors. Analyzing a circuit by calculating the exact behavior of every internal component is impractical and unnecessary for most applications.

To solve this, engineers use the "ideal op-amp" model. This mathematical abstraction pushes the device's best characteristics to their theoretical limits. While no physical device can achieve infinity, modern op-amps get close enough that the ideal equations produce highly accurate results for everyday circuit design.

Why Open-Loop Gain is Assumed to be Infinite

The open-loop gain (\(A_{OL}\)) is the amplification factor of the op-amp when no feedback loop is connected from the output to the input. In an ideal op-amp, this gain is assumed to be infinite.

Enabling the Virtual Short

Op-amps are almost always used with negative feedback, where a portion of the output voltage is routed back to the inverting input. The basic equation for an op-amp's output is:

\(V_{out} = A_{OL} \times (V_+ - V_-)\)

Where \(V_+\) is the non-inverting input voltage and \(V_-\) is the inverting input voltage. We can rearrange this equation to find the difference between the two inputs:

\(V_+ - V_- = \frac{V_{out}}{A_{OL}}\)

If we assume \(A_{OL}\) is infinite, dividing any finite output voltage (\(V_{out}\)) by infinity results in zero. Therefore:

\(V_+ - V_- = 0\) or \(V_+ = V_-\)

This gives us the first "Golden Rule" of op-amps: with negative feedback, the voltage at the inverting input is forced to equal the voltage at the non-inverting input. Engineers call this a "virtual short." It makes circuit analysis vastly simpler because knowing the voltage at one input immediately tells you the voltage at the other.

Relying Solely on External Components

Real op-amps have very high open-loop gains (often \(100,000\) to \(1,000,000\)), but this gain is unstable. It changes with temperature, signal frequency, and manufacturing variations.

By assuming the gain is infinite and applying negative feedback, the unstable internal gain drops out of the math entirely. The overall gain of the circuit (closed-loop gain) becomes strictly dependent on the stable, precise external resistors chosen by the engineer.

Why Input Impedance is Assumed to be Infinite

Input impedance (\(Z_{in}\)) is the opposition to current flow into the op-amp's input terminals. Assuming this value is infinite is just as critical for straightforward circuit design.

Zero Input Current

According to Ohm's Law, current equals voltage divided by resistance (\(I = \frac{V}{R}\)). If the input impedance (resistance) of the op-amp is infinite, then the current flowing into the input terminals must be zero.

This establishes the second "Golden Rule" of op-amps: the inputs draw no current.

In circuit analysis, this means any current flowing through a feedback network has nowhere to go except around the op-amp, never into it. This allows engineers to apply Kirchhoff’s Current Law at the input nodes with absolute simplicity, as the op-amp itself acts as a dead end for current.

Preventing the Loading Effect

In analog electronics, when you connect a measuring device or an amplifier to a signal source, the amplifier draws some current from that source. If the amplifier has low input impedance, it draws too much current, causing the source voltage to drop. This is known as "loading" the circuit, and it distorts the original signal.

By assuming infinite input impedance, the ideal op-amp places zero load on the previous stage of the circuit. It can sense the voltage of a delicate sensor or high-impedance source without altering the signal it is trying to amplify. Real-world op-amps, particularly those with FET (Field-Effect Transistor) inputs, have input impedances in the tera-ohms (\(10^{12}\) ohms), making the infinite assumption practically true in most scenarios.