The Non-Inverting Op-Amp Formula: Direct Answer & Symbol Map

The closed-loop output voltage of a non-inverting operational amplifier is defined by the following formula:

Vout = Vin × (1 + (Rf / Ri))

The closed-loop voltage gain (Av) is therefore strictly 1 + (Rf / Ri). Unlike the inverting configuration, the non-inverting topology cannot yield a gain of less than 1 (unity gain), and the output signal maintains the exact same phase polarity as the input signal.

Non-Inverting Op-Amp Symbol Definition
Symbol Parameter Standard Unit Practical Bench Notes
Vout Output Voltage Volts (V) Hard-limited by the op-amp's supply rails (VCC and VEE or GND).
Vin Input Voltage Volts (V) Applied directly to the non-inverting (+) terminal.
Rf Feedback Resistor Ohms (Ω) Connects Vout to the inverting (-) terminal. Keep under 100kΩ to minimize noise and bias current errors.
Ri Input/Ground Resistor Ohms (Ω) Connects the inverting (-) terminal to ground (or a reference voltage).
Av Closed-Loop Voltage Gain Unitless (V/V) Minimum value is 1.0. Often expressed in decibels (dB) in audio/RF applications.

When the Formula Applies and Core Assumptions

This formula assumes an ideal op-amp operating in its linear region. The core assumptions are:

  • Infinite open-loop gain (AOL): Ensures the virtual short between the (+) and (-) inputs is exact.
  • Infinite input impedance: Assumes zero current flows into the (+) and (-) terminals, meaning no voltage drop across the source impedance.
  • Zero output impedance: Assumes Vout does not sag under load.
  • Linear operation: The calculated Vout must fall within the op-amp's output voltage swing limits (the supply rails).

Realistic Answer Magnitudes

On the bench, a realistic Av for a single DC-coupled non-inverting stage ranges from 2 V/V to 100 V/V. If your math yields an Av of 10,000, you are designing a comparator, not a linear amplifier, and the formula no longer applies. Furthermore, a realistic Vout magnitude is typically 1V to 3V below the absolute supply rails for standard parts (like the LM358 or TL072), or within 50mV of the rails for true rail-to-rail output (RRIO) devices like the MCP6002.

Deriving the Gain: Why the "+1" Matters

To understand why the non-inverting op-amp formula includes the "+1", we apply Kirchhoff's laws and the concept of the virtual short. For a deeper dive into the underlying semiconductor physics, the All About Circuits Op-Amp Basics guide provides excellent foundational context.

  1. The Virtual Short: Because the open-loop gain (AOL) is theoretically infinite, the op-amp drives its output to force the voltage at the inverting terminal (V-) to equal the non-inverting terminal (V+). Therefore, V- = V+ = Vin.
  2. The Voltage Divider: The feedback network (Rf and Ri) acts as a voltage divider between Vout and Ground. The voltage at the inverting terminal is:
    V- = Vout × [ Ri / (Ri + Rf) ]
  3. Equating the Nodes: Substitute Vin for V-:
    Vin = Vout × [ Ri / (Ri + Rf) ]
  4. Solving for Vout: Multiply both sides by the inverse of the divider ratio:
    Vout = Vin × [ (Ri + Rf) / Ri ]
  5. Simplifying: Separate the fraction:
    Vout = Vin × [ (Ri / Ri) + (Rf / Ri) ]
    Vout = Vin × (1 + Rf / Ri)

The "+1" exists because the input signal is applied directly to the output-driving side of the feedback network, meaning the baseline signal always passes through even if Rf is zero.

Rearranged Forms: Solving for Any Variable

When designing a circuit, you rarely start with all resistor values. Use these rearranged forms to solve for the missing component based on your target gain (Av) or voltage constraints.

  • Solve for Feedback Resistor (Rf):
    Rf = Ri × (Av - 1)
  • Solve for Ground Resistor (Ri):
    Ri = Rf / (Av - 1)
  • Solve for Required Input Voltage (Vin):
    Vin = Vout / (1 + (Rf / Ri))
  • Solve for Target Gain (Av):
    Av = Vout / Vin

Solved Problems: Tracking Units from Millivolts to Volts

Abstract math hides real-world scaling errors. Here are two bench-realistic problems with explicit unit tracking.

Problem 1: Thermistor Bridge Amplification

Scenario: A Wheatstone bridge outputs a small differential DC voltage representing a temperature change. We need to amplify this for a 3.3V microcontroller ADC.

  • Given: Vin = 14.2 mV, Rf = 47 kΩ, Ri = 2.2 kΩ
  • Find: Vout

Step-by-step Solution:

  1. Calculate the resistor ratio: Rf / Ri = 47,000 Ω / 2,200 Ω = 21.36 (unitless).
  2. Calculate total gain (Av): 1 + 21.36 = 22.36 V/V.
  3. Multiply Vin by Av: 14.2 mV × 22.36 = 317.5 mV.
  4. Convert to standard volts for the final answer: Vout = 0.3175 V.

Problem 2: Audio DAC Line Driver

Scenario: You are buffering a DAC output to drive a line-level audio input. The DAC outputs 200 mVRMS, and the target line level is 2.0 VRMS.

  • Given: Vin = 200 mVRMS, Vout = 2.0 VRMS, Ri = 10 kΩ
  • Find: Rf

Step-by-step Solution:

  1. Convert Vin to Volts to match Vout: 200 mV = 0.2 V.
  2. Calculate required gain (Av): Vout / Vin = 2.0 V / 0.2 V = 10 V/V.
  3. Apply the rearranged formula for Rf: Rf = Ri × (Av - 1).
  4. Substitute values: Rf = 10,000 Ω × (10 - 1) = 10,000 Ω × 9.
  5. Final answer: Rf = 90,000 Ω (or 90 kΩ). Use a standard 91kΩ 1% resistor in practice.

Bench War Story: When the Math Meets the Rails

Formulas assume ideal components; reality enforces physics. Here is a documented failure from a recent prototype build.

Setup: Designing a single-supply piezo vibration sensor preamp using an LM358 op-amp. The system runs on a 5V USB supply (VCC = 5V, VEE = GND).
Numbers: The piezo generates a Vin of 0.8V peak. We need a gain of 10 to drive an ADC. We select Rf = 90 kΩ and Ri = 10 kΩ. The non-inverting op-amp formula dictates Vout = 0.8V × (1 + 9) = 8.0V peak.
Outcome: The oscilloscope showed a flat-topped, heavily distorted waveform clipping at exactly 3.4V. The ADC readings were garbage.
What Went Wrong: The math was correct, but the assumptions failed. First, the LM358 is not a rail-to-rail output device. According to the Texas Instruments LM358 Datasheet, the maximum output high swing (VOH) on a 5V supply is typically VCC - 1.5V, meaning it physically cannot exceed 3.5V. Second, the required 8V output was impossible on a 5V single supply anyway. The fix was twofold: switch to an MCP6002 (true RRIO op-amp) and lower the gain to 4 (Rf = 30k, Ri = 10k) to keep the 3.2V peak signal safely inside the 5V rail.

Unit Traps and Assumption Failures

When the non-inverting op-amp formula yields a circuit that fails on the bench, it is almost always due to one of these specific unit or assumption traps:

1. The Missing "+1" (Inverting vs. Non-Inverting Confusion)

The most common math error is calculating gain as simply Rf / Ri. This is the formula for the inverting amplifier magnitude. If you design for a gain of 10 using Rf=90k and Ri=10k, a non-inverting stage will actually yield a gain of 11. Always add 1 to the resistor ratio.

2. Mixing Peak, Peak-to-Peak, and RMS Units

If your Vin is specified in VRMS (common in audio) but your supply rail limits are defined in Peak or Peak-to-Peak, the formula will lie to you. A 2.0 VRMS sine wave has a peak voltage of 2.83V (2.0 × √2). If you amplify this by a gain of 5, the math says 10 VRMS, but the physical peaks will hit 14.1V, instantly clipping a standard ±12V supply. Always convert AC units to Peak before checking against supply rails.

3. Ignoring Input Bias Current with Megaohm Resistors

The formula assumes infinite input impedance. In reality, op-amps require a small input bias current (Ib). If you use massive resistors to save power (e.g., Rf = 10 MΩ, Ri = 1 MΩ for a gain of 11), the bias current flowing through the feedback network creates a severe DC offset voltage. For a standard TL072 with an Ib of 60 pA, the offset might be negligible, but for a bipolar part like the LM358 with an Ib of 45 nA, a 10 MΩ feedback resistor will generate a 450 mV DC error at the output. Keep resistor values in the 1kΩ to 100kΩ range unless using specialized CMOS op-amps.

For comprehensive guidance on avoiding these real-world errors, the TI Precision Labs Op-Amp Training series provides excellent video modules on input bias currents and output swing limitations.

4. Exceeding the Gain-Bandwidth Product (GBP)

The non-inverting formula is a DC and low-frequency construct. Every op-amp has a Gain-Bandwidth Product limit. If you use a TL072 (GBP ≈ 3 MHz) and set your resistor ratio for a gain of 100 (40 dB), your maximum usable frequency drops to 30 kHz. If you attempt to amplify a 100 kHz signal with that same gain, the formula will predict 100x amplification, but the physical chip will act as a low-pass filter, severely attenuating the signal. Always verify that your target Av multiplied by your maximum signal frequency is less than the op-amp's GBP.