CR · Operational amplifier

The Ideal Op-Amp and the Virtual Short

An operational amplifier has such enormous gain that, with negative feedback applied, it adjusts itself until its two inputs are equal. This unit shows how that single rule lets the gain of an amplifier be read straight from a ratio of resistors.

Design the gain with one resistor ratio

The figure is a non-inverting amplifier: the input enters at the + terminal, and the output returns through two resistors to the − terminal. Change the feedback ratio Rf/Rg and the length of the gold arrow at the output, the gain, grows to (1 + Rf/Rg) times. Find the ratio that makes the output exactly 3 times the input, so the arrow's tip reaches the dashed mark.

Feedback ratio Rf/RgRf/Rg = 0.50

Voltage gain G

G = 1 + 0.50 = 1.50

Far from ×3

Two rules: virtual short and 0 input current

An ideal op-amp amplifies the tiny voltage difference between its two inputs by a nearly infinite amount. When the output returns to the inverting input through feedback, the op-amp adjusts its output in the direction that shrinks that difference toward 0, and as a result the two input voltages become equal. This is the virtual short. In addition, the input resistance is infinite, so no current at all flows into the input terminals.

The two rules turn gain into a resistor ratio

In a non-inverting amplifier the input goes to the + terminal, and the output sends the voltage divided by Rf and Rg back to the inverting input. Because of the virtual short, the inverting input sits at the input voltage, and because the input current is 0, the current through Rg flows on through Rf unchanged. Solving these two conditions gives an output 1 + Rf/Rg times the input. The op-amp's huge gain drops out of the equation, leaving only the ratio of the external resistors.

That is why the circuit is stable and predictable

The gain of each individual op-amp varies from part to part and drifts with temperature, yet the gain of an amplifier built with feedback is set by the ratio of two resistors. Choose precise resistors and you get a precise gain. The virtual short is the first key to solving countless circuits, such as inverting amplifiers, summers, and integrators; from circuit to circuit, only where the input goes and how the feedback is wired change.

ObserveV− = V+

Feedback ties the two inputs to the same voltage.

Choosei_in = ? (R_in = ∞)

The ideal input resistance is infinite.

Fill inG = 1 + ?

The non-inverting gain is 1 plus the resistor ratio.

On your ownG = 3 → RfRg = ?

G = 3 means Rf/Rg = 2.

Back to the first screen

As you changed the feedback ratio, the gold arrow lengthened and the output grew to (1 + Rf/Rg) times the input, reaching exactly 3 times, and the dashed mark, at Rf/Rg = 2. The op-amp's own colossal gain appeared nowhere in the result. Because negative feedback tied the two inputs together and kept the input current at 0, the gain depended on nothing but the ratio of two resistors.

An ideal operational amplifier has infinite gain and input resistance and 0 output resistance. With negative feedback, its two input voltages become equal (virtual short, V+ = V−) and the input current is 0. By these two rules the gain of a non-inverting amplifier is G = 1 + Rf/Rg, set by the external resistor ratio alone and independent of the op-amp's own gain.