Mastering the Non Inverting Op Amp Gain Equation: Derivation and Worked Examples

The Core Non Inverting Op Amp Gain Equation

The closed-loop voltage gain of a non inverting operational amplifier configuration is determined entirely by the ratio of two external feedback resistors. The direct answer for the voltage gain ($A_v$) is expressed by the following formula:

A_v = 1 + (R_f / R_i)

Unlike the inverting configuration, which introduces a 180-degree phase shift and can yield fractional gains, the non inverting topology preserves the input signal phase and enforces a minimum gain of 1 (unity). Below is the complete definition of every symbol used in this equation.

Derivation and Core Assumptions

To understand when the non inverting op amp gain equation applies, we must derive it using Kirchhoff’s laws and the ideal op-amp model. According to All About Circuits, the derivation relies on two critical assumptions: infinite open-loop gain and infinite input impedance.

Step 1: The Virtual Short
Because the open-loop gain is assumed infinite, the op-amp drives its output to force the voltage difference between the non inverting (+) and inverting (-) inputs to zero. Therefore, V_- = V₊. Since V₊ is tied directly to V_in, we know that V_- = V_in.

Step 2: The Voltage Divider
No current flows into the ideal inverting input pin (infinite input impedance). Therefore, R_f and R_i act as a simple series voltage divider between V_out and ground. The voltage at the inverting pin (V_-) is:

V_- = V_out × [ R_i / (R_i + R_f) ]

Step 3: Equating and Solving
Substitute V_in for V_- and rearrange to solve for the gain (V_out / V_in):

V_in = V_out × [ R_i / (R_i + R_f) ]
V_out / V_in = (R_i + R_f) / R_i
A_v = (R_i / R_i) + (R_f / R_i)
A_v = 1 + (R_f / R_i)

When this formula breaks down: This equation assumes the op-amp is operating in its linear region. If V_out attempts to exceed the DC supply rails (e.g., trying to output 14V on a ±12V supply), the op-amp saturates, and the equation no longer predicts the actual output. Furthermore, at high frequencies, the finite Gain-Bandwidth Product (GBWP) of real silicon causes the actual gain to roll off below this calculated DC value.

Rearranged Forms for Circuit Design

On the bench, you rarely calculate gain from scratch; you usually have a target gain and need to find a missing resistor value. Here are the algebraically rearranged forms of the non inverting op amp gain equation for practical design:

• Solve for Feedback Resistor (R_f):
  R_f = R_i × (A_v - 1)
• Solve for Input Resistor (R_i):
  R_i = R_f / (A_v - 1)
• Solve for Required Input Voltage (V_in):
  V_in = V_out / [ 1 + (R_f / R_i) ]
• Solve for Expected Output Voltage (V_out):
  V_out = V_in × [ 1 + (R_f / R_i) ]

Worked Examples with Unit Tracking

Let’s apply these formulas to real bench scenarios. A common trap for hobbyists is mixing base units (Ohms) with prefix units (kilo-Ohms) without conversion, leading to catastrophic design errors. Always track your units through the calculation.

Problem 1: Calculating Output Voltage

Scenario: You are building a sensor preamplifier using an LM358. The non inverting input receives a 250 mV (0.25 V) DC signal from a thermocouple. Your feedback network uses R_f = 47 kΩ and R_i = 10 kΩ. The op-amp is powered by a single +12V supply and ground.

Goal: Find V_out.

Solution Steps:

1. Convert all resistances to base Ohms to ensure unit cancellation: R_f = 47,000 Ω, R_i = 10,000 Ω.
2. Calculate Gain: A_v = 1 + (47,000 Ω / 10,000 Ω) = 1 + 4.7 = 5.7 V/V.
3. Calculate Output: V_out = V_in × A_v = 0.25 V × 5.7 = 1.425 V.
4. Verify against rails: 1.425 V is well within the 0V to ~10.5V linear output swing of the LM358 on a 12V supply. The answer is valid.

Problem 2: Sizing the Feedback Resistor

Scenario: You need a precise gain of 15 V/V to amplify an audio signal using a TL072. You have a 2.2 kΩ resistor in your parts bin and want to use it as R_i.

Goal: Find the required R_f value.

Solution Steps:

1. Use the rearranged formula: R_f = R_i × (A_v - 1).
2. Substitute values: R_f = 2,200 Ω × (15 - 1).
3. Calculate: R_f = 2,200 Ω × 14 = 30,800 Ω (or 30.8 kΩ).
4. Real-world adjustment: 30.8 kΩ is not a standard E24 resistor value. You must select the closest 1% E96 value (30.9 kΩ) or series two standard resistors (e.g., 27 kΩ + 3.9 kΩ) to achieve the exact gain.

Realistic Answer Magnitudes: In practical DC and audio designs, closed-loop gains typically range from 1 to 100. If your calculation yields a gain of 10,000, you are likely designing a comparator, not a linear amplifier. Attempting to force a real op-amp into a gain of 10,000 will result in severe bandwidth restriction (due to GBWP limits) and massive amplification of input offset voltage.

Real-World Component Selection and Edge Cases

The non inverting op amp gain equation assumes ideal components. According to Texas Instruments, real silicon introduces errors that scale with your resistor choices. Here is how to bridge the gap between theory and the breadboard:

• Resistor Impedance Limits: Keep R_f and R_i between 1 kΩ and 100 kΩ. Values below 1 kΩ force the op-amp to source excessive current, increasing THD (Total Harmonic Distortion). Values above 100 kΩ make the circuit highly susceptible to electromagnetic interference (EMI) and exacerbate errors from the op-amp’s input bias current.
• Input Bias Current (I_b): In bipolar op-amps like the NE5532, input bias currents flow through R_f and R_i, creating an unintended DC offset voltage at the output. To minimize this, place a compensation resistor (R_comp = R_f || R_i) in series with the non inverting input. For FET-input op-amps like the OPA2134, I_b is in the picoamp range, making R_comp unnecessary.
• Tolerance Stacking: A gain of 10 using two 5% resistors can vary between 9.05 and 11.05. For precision instrumentation, always specify 1% or 0.1% metal film resistors (e.g., Yageo MFR series) to ensure the physical circuit matches the equation.

Frequently Asked Questions

Why is the non inverting op amp gain equation always greater than or equal to 1?

The mathematical floor of the equation is A_v = 1, which occurs when R_f = 0 Ω (a direct short from output to inverting input) or when R_i approaches infinity (an open circuit to ground). This specific configuration is called a voltage follower or unity-gain buffer. Because the input signal is applied directly to the non inverting (+) terminal, the output must at least match the input voltage to satisfy the virtual short condition. It cannot attenuate the signal below the input magnitude; for attenuation, you must use a passive voltage divider before the buffer or switch to an inverting topology.

How does the non inverting op amp gain equation apply to AC signals and impedance?

For AC signals, the scalar resistors R_f and R_i are replaced by complex impedances Z_f and Z_i. The equation becomes A_v = 1 + (Z_f / Z_i). If you place a capacitor in series with R_i, the impedance of that branch increases at low frequencies, effectively rolling off the DC gain to 1 (acting as an AC-coupled amplifier). Conversely, placing a capacitor in parallel with R_f decreases the feedback impedance at high frequencies, creating an active low-pass filter. In all AC applications, the magnitude and phase of the gain will vary with frequency, bounded ultimately by the op-amp's Gain-Bandwidth Product.

What happens to the non inverting op amp gain equation if the feedback resistor is zero?

If R_f = 0 Ω, the equation evaluates to A_v = 1 + (0 / R_i), which simplifies to A_v = 1. The circuit becomes a unity-gain buffer. In this state, the output pin is wired directly to the inverting input pin, providing 100% negative feedback. The op-amp will source or sink whatever current is necessary to make V_out exactly equal to V_in. This is highly useful for isolating high-impedance sensors (like piezo pickups or pH probes) from low-impedance loads (like ADC inputs or long cable runs) without altering the signal amplitude.