The non-inverting operational amplifier configuration is a foundational building block in analog circuit design. It provides high input impedance, low output impedance, and a positive voltage gain that never drops below unity. Whether you are scaling a 50 mV thermocouple signal for a 3.3V microcontroller ADC or buffering an audio line, the math governing this topology is exact, provided you respect its boundary conditions. The closed-loop voltage gain (Av) of a non-inverting op-amp is defined as 1 + (Rf / Ri).
The Core Non Inverting Op Amp Equations
To calculate the behavior of the circuit in its linear region, we rely on two primary mathematical expressions. The first defines the closed-loop voltage gain (Av), and the second calculates the absolute output voltage (Vout) based on a known input voltage (Vin).
Gain Equation:
Av = 1 + (Rf / Ri)
Output Voltage Equation:
Vout = Vin × [1 + (Rf / Ri)]
Symbol Definition and Specification Table
| Symbol | Parameter | Standard Unit | Typical Practical Range |
|---|---|---|---|
| Vout | Output Voltage | Volts (V) | Bounded by supply rails (e.g., 0V to 5V, or ±15V) |
| Vin | Input Voltage (at non-inverting pin) | Volts (V) | µV to V range (must remain within common-mode limits) |
| Rf | Feedback Resistor (Output to Inverting Pin) | Ohms (Ω) | 1 kΩ to 1 MΩ (keeps bias current errors low) |
| Ri | Input/Ground Resistor (Inverting Pin to Ground) | Ohms (Ω) | 1 kΩ to 100 kΩ |
| Av | Closed-Loop Voltage Gain | V/V (Dimensionless) | ≥ 1 (Unity to ~100 for stable bandwidth) |
When the Formula Applies and Its Assumptions
These non inverting op amp equations assume an ideal operational amplifier operating in its linear region. According to Texas Instruments design fundamentals, the ideal model relies on four core assumptions:
- Infinite Open-Loop Gain (AOL): Forces a "virtual short" between the inverting (-) and non-inverting (+) inputs, meaning V- = V+.
- Infinite Input Impedance: Zero current flows into the op-amp input pins, meaning all current through Ri must flow through Rf.
- Zero Output Impedance: The op-amp can drive the calculated Vout regardless of the load attached.
- Linear Operation: The calculated Vout must not exceed the physical voltage supply rails (VCC and VEE/GND).
In reality, a legacy part like the LM358 has an open-loop gain of ~100 dB and an input offset voltage of up to 3 mV, which introduces slight errors at high gains. Modern precision op-amps like the OPA2134 or chopper-stabilized variants mitigate these errors, making the ideal equations highly accurate for 95% of bench and production designs.
Rearranged Forms: Solving for Any Variable
On the workbench, you rarely have the luxury of picking arbitrary resistor values. You usually have a target gain and a limited inventory of E24 or E96 series resistors. Here are the algebraically rearranged forms of the core equation to solve for any single variable:
- Solving for Output Voltage (Vout):
Vout = Vin × [1 + (Rf / Ri)] - Solving for Input Voltage (Vin):
Vin = Vout / [1 + (Rf / Ri)] - Solving for Feedback Resistor (Rf):
Rf = Ri × [(Vout / Vin) - 1] or Rf = Ri × (Av - 1) - Solving for Ground Resistor (Ri):
Ri = Rf / [(Vout / Vin) - 1] or Ri = Rf / (Av - 1)
Worked Examples with Unit Tracking
Abstract formulas are useless without rigorous unit tracking. Below are two real-world scenarios demonstrating how to apply the non inverting op amp equations step-by-step.
Problem 1: Audio Preamp Output Calculation
Scenario: You are designing a microphone preamp stage using a TL072 op-amp powered by ±15V rails. The input signal from the electret capsule is 200 mV (peak). Your feedback network consists of Rf = 47 kΩ and Ri = 10 kΩ. What is the peak output voltage?
Step-by-Step Solution:
- Identify knowns: Vin = 200 mV; Rf = 47 kΩ; Ri = 10 kΩ.
- Calculate Gain (Av):
Av = 1 + (Rf / Ri)
Av = 1 + (47 kΩ / 10 kΩ)
Note: The 'k' (kilo) prefix cancels out in the ratio.
Av = 1 + 4.7 = 5.7 V/V - Calculate Vout:
Vout = Vin × Av
Vout = 200 mV × 5.7 V/V
Vout = 1140 mV - Convert to standard units:
1140 mV × (1 V / 1000 mV) = 1.14 V (peak) - Verify against rails: 1.14 V is well within the ±15V supply limits (and the TL072's typical ±13V linear swing limit). The equation holds.
Problem 2: Sensor Scaling Resistor Selection
Scenario: You need to amplify a 0–100 mV shunt resistor signal to fully utilize a 0–3.3V microcontroller ADC. You choose an MCP6002 rail-to-rail op-amp on a 3.3V single supply. You have a 2.2 kΩ precision resistor for Ri. What value of Rf is required?
Step-by-Step Solution:
- Identify knowns: Vin(max) = 100 mV; Vout(max) = 3.3 V; Ri = 2.2 kΩ.
- Determine required Gain (Av):
Av = Vout / Vin
Av = 3.3 V / 0.1 V (converting 100 mV to Volts for consistent units)
Av = 33 V/V - Rearrange formula to solve for Rf:
Rf = Ri × (Av - 1) - Substitute and solve:
Rf = 2.2 kΩ × (33 - 1)
Rf = 2.2 kΩ × 32
Rf = 70.4 kΩ - Practical Implementation: 70.4 kΩ is not a standard value. Selecting the nearest 1% E96 series resistor yields 71.5 kΩ. This results in an actual gain of 33.5, pushing the 100 mV input to 3.35 V. Because the MCP6002 cannot swing perfectly to the 3.3V rail under load, you may need to slightly reduce Vin(max) or add a software scaling factor in your ADC code.
Real-World Magnitudes and Unit Mistakes That Break the Math
Understanding what a realistic answer magnitude looks like prevents silent design failures. In a non-inverting topology, the gain magnitude can never be less than 1. If your math yields a gain of 0.5, you have either wired an inverting amplifier, created a passive voltage divider, or made an algebraic error. If you require attenuation (gain < 1) before amplification, you must place a passive resistive divider at the non-inverting input, or switch to an inverting op-amp topology.
Furthermore, the calculated Vout magnitude is strictly bounded by the op-amp's output swing limits. A standard LM358 on a 5V single supply will typically saturate at ~3.5V. If your equation predicts 4.8V, the physical circuit will clip at 3.5V, and the linear equation no longer applies.
Unit Mistakes That Break the Math
While the ratio of Rf / Ri is dimensionless (meaning you can mix kΩ and kΩ, or Ω and Ω), sloppy unit tracking causes catastrophic errors in two specific areas:
- Mixing mV and V in the Gain Ratio: If Vout is 3.3V and Vin is 50mV, calculating gain as 3.3 / 50 yields 0.066. You must convert 50mV to 0.05V first, yielding the correct gain of 66.
- Current Calculations in the Feedback Network: The current flowing through the feedback network is I = Vout / (Rf + Ri). If Vout is in Volts and your resistors are in kΩ, the resulting current is in milliamps, not amps. Failing to track this leads to selecting op-amps that cannot source the required output current, causing thermal shutdown or voltage sag.
Frequently Asked Questions
Why is the minimum gain of a non-inverting op amp 1?
The gain equation is Av = 1 + (Rf / Ri). Because physical resistors cannot have negative resistance, the lowest possible value for the ratio (Rf / Ri) is zero. This occurs when Rf is a short circuit (0 Ω) or Ri is an open circuit (∞ Ω). In both cases, the equation resolves to 1 + 0 = 1. This specific configuration is known as a unity-gain buffer or voltage follower, used exclusively for impedance matching rather than voltage amplification.
How do non inverting op amp equations change for a single-supply circuit?
The core DC equations do not change, but the reference point does. In a dual-supply circuit (±15V), ground (0V) is the center reference. In a single-supply circuit (e.g., 0V to 5V), the op-amp cannot output negative voltages. Therefore, you must bias the non-inverting input to a virtual ground, typically VCC/2 (2.5V), using a voltage divider. The AC signal is then capacitively coupled. The gain equation still applies to the AC signal component, but the absolute DC output voltage will be Vbias + (Vin(ac) × Av).
What happens to the non inverting op amp equations when the op-amp saturates?
When the calculated Vout exceeds the op-amp's maximum output swing (dictated by the supply rails and the internal transistor saturation voltages), the op-amp enters non-linear saturation. The "virtual short" assumption breaks down because the open-loop gain is no longer sufficient to force the inverting pin to match the non-inverting pin. At this point, the linear equations become invalid, and Vout simply clips at the maximum or minimum rail voltage limit.
Can I use non inverting op amp equations for AC signals?
Yes, the equations apply perfectly to AC signals, provided you use consistent magnitude metrics (e.g., both Vin and Vout in RMS, or both in Peak-to-Peak). However, you must verify that the signal frequency does not exceed the op-amp's Gain-Bandwidth Product (GBWP). For example, if an op-amp has a GBWP of 1 MHz and you configure it for a gain of 100 V/V, the maximum frequency it can amplify before the gain begins to roll off is 10 kHz (1 MHz / 100). For high-frequency AC designs, consult the manufacturer's Bode plot in the datasheet.
