Non Inverting Amplifier Gain Formula: Derivation & Worked Examples

The Non Inverting Amplifier Gain Formula at a Glance

When designing signal conditioning circuits, the non inverting amplifier gain formula dictates the closed-loop voltage amplification of your operational amplifier (op-amp). The direct answer for the ideal closed-loop voltage gain ($A_v$) is:

A_v = 1 + (R₂ / R₁)

Unlike the inverting configuration, this topology preserves the phase of the input signal and provides a minimum gain of exactly 1 (unity). Below is the definitive symbol reference for every variable in the equation.

Table 1: Symbol Definitions for the Non Inverting Amplifier Gain Formula
Symbol	Parameter	Standard Unit	Practical Notes
`A_v`	Closed-Loop Voltage Gain	V/V (Dimensionless)	Often expressed in decibels (dB) as 20·log₁₀(A_v).
`V_out`	Output Voltage	Volts (V)	Constrained by the op-amp's supply rails (typically ~1.5V below V+ for non-rail-to-rail parts like the LM358).
`V_in`	Input Voltage	Volts (V)	Applied directly to the non-inverting (+) terminal.
`R₁`	Grounded Feedback Resistor	Ohms (Ω)	Connects from the inverting (-) terminal to ground. Typical values: 1kΩ to 100kΩ.
`R₂`	Feedback Resistor	Ohms (Ω)	Connects from the output terminal back to the inverting (-) terminal.

Derivation and Core Assumptions

To understand when the non inverting amplifier gain formula applies, we must derive it using Kirchhoff’s laws and the golden rules of ideal op-amps. This derivation assumes an ideal op-amp model, which holds true for DC and low-frequency AC signals in general-purpose parts like the TL072 or precision parts like the OPA2134.

The Ideal Op-Amp Assumptions

Infinite Input Impedance: No current flows into the non-inverting (+) or inverting (-) input pins ($I_+ = I_- = 0$).
Infinite Open-Loop Gain ($A_{OL}$): Forces a 'virtual short' between the inputs, meaning $V_- = V_+$.
Linear Operation: The output is not saturated against the positive or negative supply rails.

Step-by-Step Derivation

Establish Input Voltage: The signal is applied to the non-inverting pin, so $V_+ = V_{in}$.
Apply Virtual Short: Because of infinite open-loop gain, the op-amp drives its output to force $V_- = V_+$. Therefore, $V_- = V_{in}$.
Analyze the Feedback Network: Resistors $R_1$ and $R_2$ form a voltage divider between $V_{out}$ and ground. Since no current enters the inverting pin, the voltage at $V_-$ is determined strictly by the divider:
V_- = V_out × [R₁ / (R₁ + R₂)]
Equate and Solve: Substitute $V_{in}$ for $V_-$:
V_in = V_out × [R₁ / (R₁ + R₂)]
Rearranging for the gain ratio (V_out / V_in):
A_v = (R₁ + R₂) / R₁
Splitting the fraction yields the final non inverting amplifier gain formula:
A_v = 1 + (R₂ / R₁)

Rearranged Forms for Component Selection

On the bench, you rarely solve for $A_v$ blindly; you usually have a target gain and need to pick E24 series resistors. Here are the rearranged forms of the non inverting amplifier gain formula solving for each variable:

Solve for Feedback Resistor (R₂): R₂ = R₁ × (A_v - 1)
Solve for Grounded Resistor (R₁): R₁ = R₂ / (A_v - 1)
Solve for Required Input (V_in): V_in = V_out / A_v
Solve for Expected Output (V_out): V_out = V_in × A_v

Worked Examples with Unit Tracking

Let’s apply the formula to real-world design scenarios. A common unit mistake that breaks the formula is failing to cancel prefixes properly (e.g., dividing $k\Omega$ by $\Omega$ without converting). Because $R_2 / R_1$ is a ratio, as long as both resistors are in the same unit prefix, the ratio is dimensionless. Furthermore, gain ($A_v$) has no units; it is strictly Volts per Volt (V/V).

Problem 1: Audio Preamplifier Output Calculation

Scenario: You are building a microphone preamp using an NE5532. The input signal is $V_{in} = 150 \text{ mV}_{RMS}$. Your feedback network uses $R_1 = 10 \text{ k}\Omega$ and $R_2 = 47 \text{ k}\Omega$. What is $V_{out}$?

Calculate Gain: A_v = 1 + (47 kΩ / 10 kΩ) = 1 + 4.7 = 5.7 V/V.
Calculate Output: V_out = V_in × A_v = 150 mV × 5.7 = 855 mV (or 0.855 V).
Realistic Magnitude Check: 855 mV RMS translates to roughly 2.4 V peak-to-peak. If your NE5532 is powered by a standard ±9V dual supply, this output is well within the linear range. If you attempted this on a single 3.3V supply without proper DC biasing, the signal would clip severely.

Problem 2: Sensor Scaling Resistor Selection

Scenario: You need to amplify a 50 mV thermocouple signal to exactly 5.0 V to maximize the resolution of an Arduino Uno's 10-bit ADC. You have a precision $R_1 = 4.7 \text{ k}\Omega$ (1% tolerance) in your kit. What value do you need for $R_2$?

Determine Target Gain: A_v = V_out / V_in = 5.0 V / 0.050 V = 100 V/V.
Rearrange for R₂: R₂ = R₁ × (A_v - 1).
Substitute and Solve: R₂ = 4.7 kΩ × (100 - 1) = 4.7 kΩ × 99 = 465.3 kΩ.
Practical Implementation: 465.3 kΩ is not a standard E24 value. You would use a 464 kΩ (E96 series) 1% resistor, or combine a 470 kΩ potentiometer in series with a fixed resistor to trim the gain exactly, compensating for the thermocouple's non-linearities.

⚠️ Unit Mistake Warning: In Problem 2, if you accidentally used $R_1 = 4700 \Omega$ but left $R_2$ in $k\Omega$ during intermediate steps without tracking units, you would calculate an impossible resistor value. Always convert both resistors to base Ohms (Ω) or ensure both share the exact same prefix (kΩ, MΩ) before dividing.

Frequently Asked Questions

What happens to the non inverting amplifier gain formula if R1 is zero?

If $R_1 = 0 \Omega$ (a direct short to ground) and $R_2$ is any finite value, the ratio $R_2 / R_1$ approaches infinity, which theoretically implies infinite gain. However, in practice, if $R_1$ is zero, the inverting pin is tied directly to ground, breaking the feedback loop and turning the circuit into an open-loop comparator.

Conversely, if $R_2 = 0 \Omega$ (shorting the output directly to the inverting pin) and $R_1 = \infty$ (open circuit), the formula yields $A_v = 1 + (0 / \infty) = 1$. This specific configuration is called a voltage follower or unity-gain buffer, widely used for impedance matching.

How does the non inverting amplifier gain formula change for high-frequency AC signals?

At high frequencies, the ideal DC formula fails because the op-amp's internal open-loop gain ($A_{OL}$) rolls off. You must replace the resistors with complex impedance ($Z$) and factor in the Gain-Bandwidth Product (GBWP). For example, an LM358 has a GBWP of roughly 1 MHz. If you design a non-inverting amplifier with a closed-loop gain of 100 V/V (40 dB), your maximum usable bandwidth drops to 1 MHz / 100 = 10 kHz. Beyond 10 kHz, the actual measured gain will be significantly lower than the formula predicts, and phase shift will occur.

Why is my measured gain lower than the non inverting amplifier gain formula predicts?

If your bench multimeter shows a lower $V_{out}$ than calculated, check these three real-world limitations:
1. Saturation: Your calculated $V_{out}$ exceeds the op-amp's supply rails. A standard LM358 on a 5V single supply will clip at roughly 3.5V to 4V, not 5V.
2. Loading Effects: Your load resistance (e.g., a 100 Ω speaker or heavy cable capacitance) is too low, drawing more current than the op-amp's output stage can source (typically limited to 20-30 mA for general-purpose parts).
3. Slew Rate Limiting: For fast AC transients, the op-amp cannot change its output voltage fast enough. The OPA2134 has a slew rate of 14 V/µs; if your signal demands a faster voltage change, the waveform will distort into a triangle wave, lowering the measured RMS voltage.

Can the non inverting amplifier gain formula yield a negative number?

No. Because resistor values are strictly positive, the ratio $R_2 / R_1$ is always positive. Therefore, the minimum possible gain is $1 + 0 = +1$. The non-inverting topology can never attenuate a signal below unity, nor can it invert the phase (which would require a negative gain like -5 V/V). If your design requires signal attenuation or a 180-degree phase inversion, you must use the inverting amplifier topology instead, where the gain formula is simply $A_v = - (R_2 / R_1)$.