Op-Amp Basics: Inverting and Non-Inverting Amplifier Gain

The operational amplifier is a high-gain differential amplifier in an IC package. On its own its gain is so high, often 100 000 or more, that it is useless as a direct amplifier. With negative feedback from a pair of resistors, though, it becomes a precise and predictable building block whose gain depends only on the resistor ratio. This tutorial covers the ideal op-amp rules, the two basic amplifier configurations, and the real-world limits you must check before your circuit works on the bench.

The Two Golden Rules

An op-amp has a non-inverting input (+), an inverting input (-) and an output. Its output is Vout = Aol x (V+ - V-), where Aol is the open-loop gain. When negative feedback is applied (output connected back to the inverting input through a network), analysis becomes simple using two rules for an ideal op-amp:

The output does whatever is necessary to make the voltage difference between the two inputs zero (V+ = V-).
The inputs draw no current.

These rules hold only while the op-amp is operating within its limits: output not saturated at a supply rail, and signal frequency well below the bandwidth limit.

The Inverting Amplifier

Vin ---[ Rin ]---+---[ Rf ]---+
                 |            |
                (-)           |
                   op-amp ----+---- Vout
                (+)
                 |
                GND

Gain:  Av = Vout / Vin = -Rf / Rin
Input resistance = Rin

Derivation: the + input is grounded, so by rule 1 the - input is also at 0 V, a 'virtual ground'. The current through Rin is Vin / Rin. By rule 2 none of it enters the op-amp, so it all flows through Rf, giving Vout = -(Vin / Rin) x Rf. The minus sign means the output is inverted, 180 degrees out of phase.

Worked example

Design an amplifier with a gain of -10 and an input resistance of 10 kΩ. Rin = 10 kΩ, Rf = 10 x 10 kΩ = 100 kΩ. A 0.2 V peak input gives a 2 V peak output, inverted. A useful variant, the summing amplifier, adds more input resistors to the same virtual-ground node: with three 10 kΩ inputs and Rf = 10 kΩ, Vout = -(V1 + V2 + V3).

The Non-Inverting Amplifier

Vin ------------(+)
                   op-amp ----+---- Vout
          +-------(-)         |
          |                   |
          +------[ Rf ]-------+
          |
        [ Rg ]
          |
         GND

Gain:  Av = 1 + Rf / Rg
Input resistance: very high (the op-amp input itself)

Derivation: by rule 1, the - input equals Vin. Rf and Rg form a voltage divider from Vout, so Vin = Vout x Rg / (Rf + Rg). Rearranging gives Vout / Vin = 1 + Rf / Rg. The output is in phase with the input, and the gain can never be less than 1.

With Rf = 0 and Rg removed, the gain is exactly 1: the voltage follower or buffer. It draws almost no current from its source and can drive a much heavier load, which makes it the standard fix for the voltage-divider loading problem.

Worked example

Amplify a sensor output of 0 to 0.3 V to 0 to about 3 V. Required gain is 10. Choose Rg = 1 kΩ, then Rf = 9 kΩ; the nearest E24 value is 9.1 kΩ, giving Av = 1 + 9.1 = 10.1 and a maximum output of 3.03 V.

Real Op-Amp Limits

Gain-bandwidth product

For a voltage-feedback op-amp, gain times bandwidth is roughly constant. The LM358 and the 741 both have a gain-bandwidth product of about 1 MHz.

Bandwidth ~ GBW / noise gain
Non-inverting gain 10.1 with GBW = 1 MHz:
   f(-3 dB) ~ 1 000 000 / 10.1 ~ 99 kHz
Inverting gain -10: noise gain = 1 + Rf/Rin = 11
   f(-3 dB) ~ 1 000 000 / 11 ~ 91 kHz

Note that for the inverting amplifier the bandwidth is set by 1 + Rf / Rin, not by Rf / Rin. Slew rate imposes a second limit on large, fast signals: the 741 is specified at about 0.5 V/µs, so a 10 V peak sine wave above roughly 8 kHz will distort.

Supply and output swing

Op-amp	Supply	Input range includes ground?	Output swing
741	Dual, typically ±15 V	No	About 1 to 2 V from each rail
LM358	Single 3 to 32 V or dual	Yes	Down to near 0 V, up to about 1.5 V below V+
Rail-to-rail CMOS types (e.g. MCP6002)	Low voltage single supply	Yes	Close to both rails

In the worked non-inverting example, an LM358 running from 5 V cannot reach 3.03 V comfortably if the load is heavy, and cannot exceed about 3.5 V at all. Running it from a higher supply, or using a rail-to-rail part, solves this.

Common Mistakes

Connecting feedback to the + input; this is positive feedback and the output slams to a rail.
Using an inverting amplifier on a single supply without biasing the + input to mid-supply; negative outputs are impossible.
Leaving out 100 nF decoupling capacitors on the supply pins, leading to oscillation.
Choosing very large resistors (megohms), which increases noise and offset errors from input bias current.
Expecting a 741 to work from a single 5 V supply.

Summary

With negative feedback, an op-amp holds its inputs at equal voltage and draws no input current. That gives a gain of -Rf / Rin for the inverting amplifier and 1 + Rf / Rg for the non-inverting one. Before building, check bandwidth with the gain-bandwidth product, check slew rate for large signals, and confirm the op-amp's input and output ranges suit your supply.