The non inverting op amp stage boosts a small sensor signal without flipping its polarity or loading the source. Here you will learn its gain equation, its limits and how to pick resistors that work on a real board.
The gain of a non inverting op amp stage depends only on two resistors and is always one or more. This guide covers the formula, its derivation, the follower case, bandwidth and practical resistor choice with solved examples.
What Is Non Inverting Amplifier Gain?
Non inverting amplifier gain is the ratio of output voltage to input voltage of an op amp stage where the signal enters the plus input, and it equals Av = 1 + Rf/R1. The output stays in phase with the input, and the gain can never fall below one.
Av = 1 + Rf ÷ R1
Gain in dB = 20 × log10(Av)
Rf = feedback resistor, R1 = resistor to ground
The circuit is the twin of the stage described in our inverting amplifier gain formula article, with the input and ground swapped. If you are new to op amps, start with op amp basics to learn the two golden rules used below.
In the lower circuit the input goes straight to the plus pin, while Rf and R1 form a divider from the output back to the minus pin. That divider is the whole secret of the gain.
Analog Devices notes that a precision op amp can have an open loop DC gain of about 160 dB, or 100 million. Negative feedback trades almost all of that for a stable gain set only by resistors.
Non Inverting Amplifier Formula Derivation
With negative feedback, the op amp drives its output until both inputs sit at nearly the same voltage, so the minus pin also equals Vin. No current flows into the inputs, so Rf and R1 carry the same current.
V minus pin = Vin (virtual short)
Vout ÷ Vin = (R1 + Rf) ÷ R1
Av = 1 + Rf ÷ R1
The first line is just the voltage divider rule applied to the feedback network. Because the divider ratio is always less than one, the gain is always at least one, which is why this stage cannot attenuate.
- Av = 1 + Rf/R1, always 1 or more
- Output is in phase with the input
- Input impedance is very high, close to that of the op amp itself
- Bandwidth ≈ GBW ÷ noise gain, and noise gain equals signal gain here
- Rf = 0 or R1 open gives a voltage follower with gain 1
| Rf | R1 | Gain Av | Gain in dB |
|---|---|---|---|
| 0 Ω (short) | open | 1 | 0 |
| 10 kΩ | 10 kΩ | 2 | 6.02 |
| 9.09 kΩ | 1.01 kΩ | 10 | 20 |
| 47 kΩ | 4.7 kΩ | 11 | 20.83 |
| 99 kΩ | 1 kΩ | 100 | 40 |
The 9.09 kΩ and 1.01 kΩ pair comes from a Texas Instruments design note for a gain of 10, using standard 1 % values. Choosing values from the E96 series lets you hit an exact gain without trimming.
Note that TI calls the feedback resistor R1 and the ground resistor R2, so its formula reads 1 + R1/R2. Always check which resistor is which before you copy a gain equation from a datasheet.
Difference Between Inverting and Non Inverting Amplifier
Both stages use the same parts, but the input pin and the gain formula differ. The table compares them for a quick choice.
| Feature | Non inverting | Inverting |
|---|---|---|
| Input pin | Plus (+) | Minus through R1 |
| Gain | 1 + Rf/R1 | minus Rf/R1 |
| Phase | In phase | 180° shift |
| Lowest gain | 1 | Any, even below 1 |
| Input impedance | Very high | Equal to R1 |
| Noise gain | Same as signal gain | 1 + Rf/R1 |
- Very high input impedance, no loading of sensors
- No phase inversion, easy to follow
- Good for buffering high impedance sources
- Simple two resistor gain setting
- Gain cannot be below 1
- Input common mode voltage changes with the signal
- Not suited to summing many inputs
- Common mode limits of the op amp still apply
For signals that sit on a common mode voltage, a differential amplifier or an instrumentation amplifier is the better choice. Bridge sensors and thermocouples usually need that kind of input.
Input Impedance and the Voltage Follower Case
Texas Instruments states that the input impedance of this circuit equals the input impedance of the op amp, so high impedance sources are acceptable. For a CMOS op amp with picoamp bias current, this can be many gigaohms.
That is why this stage is the natural first block after a pH probe, a photodiode amplifier or a high value divider. The output impedance stays very low because the feedback holds the output where it should be.
Av = 1 + 0 = 1
Vout = Vin, voltage follower
The follower is the limiting case of non inverting amplifier gain, and it is covered in detail in voltage follower unity gain buffer. It isolates a weak source from a heavy load such as an ADC input or a long cable.
Non Inverting Amplifier Gain and Bandwidth
Analog Devices explains in tutorial MT 033 that the closed loop bandwidth equals the gain bandwidth product divided by the noise gain. For a non inverting stage the noise gain is the same as the signal gain, 1 + Rf/R1.
GBW 1 MHz, gain 11
f ≈ 1,000 kHz ÷ 11 ≈ 90.9 kHz
The TI design shows this clearly: its OPA171 has a 3 MHz unity gain bandwidth, and at a gain of 10 the simulated bandwidth is 323.2 kHz. The response is already 3 dB down at that point, so you need more headroom than the formula alone suggests.
Analog Devices advises a safety factor of at least 5 on GBW. For a gain of 10 with 100 kHz bandwidth the bare minimum is 1 MHz, but a 5 MHz op amp is the sensible choice.
Phase margin and stability at high gain are best understood with a Bode plot. If you also need to remove noise above the signal band, follow the stage with a Sallen Key active filter.
Solved Examples of Non Inverting Amplifier Gain
Example 1: a sensor gives 0 to 0.5 V and the ADC needs about 5.5 V at full scale. With Rf = 47 kΩ, R1 = 4.7 kΩ and an op amp of 1 MHz GBW, find the gain, output and bandwidth.
Vout = 11 × 0.5 = 5.5 V
Gain in dB = 20 × log10(11) = 20.83 dB
Bandwidth ≈ 1000 ÷ 11 = 90.91 kHz
Example 2: for a gain of exactly 10, pick R1 = 1.01 kΩ and solve Rf = R1 × (Av minus 1) = 9.09 kΩ, the same pair TI uses. With a 1 V input the output is 10 V, so the supply must be at least plus and minus 12 V or the op amp must be rail to rail.
The calculator ignores supply limits, so always compare Vout with the output swing in the datasheet. Negative inputs are shown with the word minus in the result.
Resistor Selection for a Non Inverting Amplifier
TI warns that high value resistors can reduce phase margin and add noise, while very low values load the output and waste current. A range of about 1 kΩ to 100 kΩ for Rf suits most general purpose op amps.
Resistor tolerance adds directly to the error in non inverting amplifier gain, so 1 % or 0.1 % parts are common in instrument circuits. Their colour bands or SMD markings are explained in the resistor color code guide.
These blocks often sit inside the front end described in signal conditioning in instrumentation. The scaled output then feeds the converter covered in analog to digital conversion.
- Gain worked out from input and output ranges
- Rf and R1 picked from E96 values
- GBW at least 5 times gain times signal bandwidth
- Output swing checked against the supply
- Input common mode range checked
- Small capacitor across Rf if needed for stability
If the output sticks at a rail, check the input common mode range first. Many single supply op amps cannot accept inputs close to the positive rail even in a follower.
After the gain is set, use the SNR calculator to see whether the amplified signal stands well above the noise floor. A higher gain raises noise too, so non inverting amplifier gain is only one part of the design.
Pair this tutorial with the transimpedance amplifier article to see how noise gain behaves in another common op amp circuit.
Non Inverting Amplifier Gain FAQ
The non inverting amplifier gain is Av = 1 + Rf/R1, where Rf is the feedback resistor and R1 goes from the minus input to ground. With Rf of 47 kΩ and R1 of 4.7 kΩ the gain is 11.
In decibels this is 20 times the log of the gain, so a gain of 11 is about 20.83 decibels. The output stays in phase with the input at all gains.
The feedback network is a divider that returns only part of the output to the minus input. Since the op amp makes that fraction equal to the input, the output must be at least as large as the input.
When Rf is zero or R1 is removed, the full output goes back and the gain becomes exactly 1. This special case is the voltage follower used as a buffer.
In a non inverting amplifier the signal enters the plus input and the output stays in phase. In an inverting amplifier the signal enters the minus input through R1 and the output is turned upside down.
The non inverting gain is 1 + Rf/R1 and its input impedance is very high. The inverting gain is minus Rf/R1 and its input impedance is only the value of R1.
Texas Instruments states that the input impedance equals the input impedance of the op amp itself. For modern CMOS and JFET parts this is very high, often in the gigaohm range.
This makes the stage ideal for weak, high impedance sources such as pH probes and piezo sensors. The feedback resistors do not load the source at all, unlike in the inverting stage.
Divide the gain bandwidth product of the op amp by the noise gain, which equals 1 + Rf/R1 in this stage. A 1 MHz op amp at a gain of 11 gives roughly 90.9 kHz of bandwidth.
Analog Devices recommends a safety factor of at least 5 on the gain bandwidth product. The response is already 3 dB down at the calculated frequency, so leave some margin.
No, the lowest possible gain of this stage is exactly 1, reached in the voltage follower. Any added Rf only increases the gain above that value.
If you need a gain below one without inversion, place a resistor divider before a follower. You can also use an inverting stage followed by a second inverting stage to restore the phase.
Related Articles
- Inverting Amplifier Gain Formula
- Voltage Follower Unity Gain Buffer
- Operational Amplifier Basics
- Differential Amplifier and CMRR
- Instrumentation Amplifier Working
External References
- Analog Devices MT 033: Voltage Feedback Op Amp Gain and Bandwidth (PDF)
- Texas Instruments: Non inverting amplifier circuit (SBOA271)
- Wikipedia: Operational amplifier applications
What We Learn Today
- The non inverting amplifier gain is Av = 1 + Rf/R1, so the stage always amplifies by at least one and keeps the output in phase.
- Input impedance equals that of the op amp, which makes the stage ideal for buffering high impedance sensors, with the follower as the gain of one case.
- Closed loop bandwidth is roughly GBW divided by the gain, and Analog Devices advises a safety factor of five when choosing the op amp.
