The closed-loop voltage gain of an ideal inverting operational amplifier is defined strictly by the ratio of the feedback resistor to the input resistor. The direct answer for the gain equation for inverting op amp circuits is: Av = -Rf / Rin. This formula assumes the op-amp is operating in its linear region with negative feedback, relying on the principle of a virtual ground at the inverting input terminal.

The Gain Equation for Inverting Op Amp: Symbols and Core Assumptions

Before soldering any components, you must understand the boundaries of the math. The derivation of this equation relies on Kirchhoff's Current Law (KCL) at the inverting node, assuming no current flows into the op-amp's input pins. According to Analog Devices' design guidelines, this ideal model holds true for DC and mid-band AC signals, provided the op-amp's open-loop gain is sufficiently high (typically >100,000 V/V).

Table 1: Symbol Definitions for the Inverting Amplifier
Symbol Parameter Standard Unit Practical Bench Range
Av Closed-Loop Voltage Gain V/V (Dimensionless) -0.1 to -1000
Rf Feedback Resistor Ohms (Ω) 1 kΩ to 1 MΩ
Rin Input Resistor Ohms (Ω) 1 kΩ to 100 kΩ
Vout Output Voltage Volts (V) Constrained by supply rails
Vin Input Voltage Volts (V) Within common-mode range
Core Assumption Check: The formula assumes an ideal op-amp with infinite input impedance (zero input bias current) and infinite open-loop gain. In reality, input bias currents flowing through Rf and Rin create an offset voltage. For high-impedance circuits (>100 kΩ), you must select a JFET/CMOS op-amp (like the TL072 or OPA1642) or add a compensation resistor to the non-inverting pin.

Rearranged Forms and the Unit Mistakes That Break Circuits

On the bench, you rarely calculate gain from known resistors; you usually know the required gain and need to find the missing resistor. Here is the complete list of rearranged forms solving for every variable in the system:

  • Solve for Gain: Av = -(Rf / Rin)
  • Solve for Feedback Resistor: Rf = -Av × Rin
  • Solve for Input Resistor: Rin = -Rf / Av
  • Solve for Output Voltage: Vout = Vin × -(Rf / Rin)
  • Solve for Input Voltage: Vin = Vout / -(Rf / Rin)

Unit Traps That Will Destroy Your Prototype

When plugging numbers into these rearranged forms, three specific unit mistakes routinely ruin builds:

  1. The kΩ/Ω Mismatch: If Rf is 100 kΩ and Rin is 10 Ω, the gain is -10,000, not -10. Always convert both resistors to base Ohms (or both to kΩ) before dividing.
  2. Dropping the Negative Sign: The negative sign is not a mathematical artifact; it represents a 180-degree phase inversion. If you are summing audio signals or feeding an ADC expecting a 0-3.3V positive swing, ignoring the phase inversion will result in destructive interference or negative voltage rail saturation.
  3. Ignoring Rail Limits: The equation will happily tell you that a 1V input with a gain of -20 yields -20V. If your op-amp is powered by a single 5V supply (like an LM358), the output will hard-clip at roughly -0.1V (ground), rendering the math useless for that operating point.

Worked Examples: Tracking Units from Input to Output

Let's run through two realistic bench scenarios, tracking units at every step to ensure dimensional consistency.

Problem 1: Audio Line-Level Scaling

Scenario: You have a microphone preamp outputting a peak signal of 50 mV. You need to invert and scale this to a -500 mV peak signal to feed a mixing console. You have chosen a standard 1 kΩ input resistor. What feedback resistor do you need?

  1. Identify knowns: Vin = 50 mV (0.05 V), Vout = -500 mV (-0.5 V), Rin = 1 kΩ (1000 Ω).
  2. Calculate required gain: Av = Vout / Vin = -0.5 V / 0.05 V = -10 V/V.
  3. Rearrange for Rf: Rf = -Av × Rin.
  4. Substitute and solve: Rf = -(-10) × 1000 Ω = 10,000 Ω.
  5. Result: You need a 10 kΩ feedback resistor.

Problem 2: Sensor Interface with High Impedance

Scenario: A piezo vibration sensor outputs 20 mV RMS. You need to amplify this to 2.0 V RMS for a microcontroller ADC. To avoid loading the high-impedance piezo element, you select a 100 kΩ input resistor. Calculate Rf and verify the output magnitude.

  1. Identify knowns: Vin = 20 mV, Target Vout magnitude = 2.0 V (2000 mV), Rin = 100 kΩ.
  2. Calculate required gain magnitude: |Av| = 2000 mV / 20 mV = 100. (Actual Av = -100).
  3. Rearrange for Rf: Rf = |-100| × 100 kΩ = 10,000 kΩ.
  4. Result: Rf = 10 MΩ.
Bench Warning: While the math yields 10 MΩ, using a 10 MΩ feedback resistor with a standard BJT-input op-amp (like the LM741) will generate massive thermal noise and DC offset due to input bias currents. For this specific 10 MΩ requirement, you must switch to a CMOS op-amp like the LMC6482 which features femtoamp-level bias currents.

Resistor Selection Decision Tree

Calculating the ratio is only half the battle. The absolute values of Rf and Rin dictate the circuit's noise floor, power consumption, and bandwidth. Use this decision path to terminate on a concrete component pick.

Table 2: Resistor Value Decision Matrix
Design Constraint If your priority is... Then choose this Rin range... Concrete Pick (1% Metal Film)
Low Noise (Audio/RF) Minimizing Johnson-Nyquist thermal noise 1 kΩ to 10 kΩ Vishay MRS25 series (e.g., 2.2 kΩ)
Low Power (Battery/IoT) Minimizing current draw from the op-amp output 100 kΩ to 1 MΩ Vishay MCT0603 series (e.g., 220 kΩ)
High Speed (Video/Fast ADC) Minimizing parasitic capacitance RC time constants 500 Ω to 2 kΩ Panasonic ERA-6A (e.g., 1 kΩ)
High Impedance Source Preventing signal attenuation at the input > 100 kΩ Yageo RC1206 (e.g., 470 kΩ)
The Parasitic Capacitance Trap: If you choose Rf values above 100 kΩ for high-speed signals, the parasitic capacitance of the resistor and PCB traces (typically 1-2 pF) creates a low-pass filter with the feedback node. This introduces a pole in the feedback loop, causing phase margin degradation and high-frequency ringing. Keep Rf under 10 kΩ for signals above 1 MHz.

Realistic Magnitudes and the Default Bench Recommendation

What does a realistic answer magnitude look like when you step back from the breadboard? In 90% of general-purpose analog designs—whether you are scaling sensor data, inverting a control voltage, or summing audio—closed-loop gains fall between -1 and -100. Gains higher than -100 in a single inverting stage are generally avoided because the required high resistor ratios exacerbate offset voltage errors and reduce the closed-loop bandwidth (due to the op-amp's constant Gain-Bandwidth Product).

If you need a gain of -500, the professional approach is to cascade two inverting stages (e.g., a gain of -22 followed by a gain of -22.7) rather than forcing a 5 MΩ feedback resistor onto a single stage.

The Default Recommendation

If you are starting a new prototype and need a reliable, low-noise, general-purpose inverting amplifier without doing a deep-dive noise analysis, use this exact bill of materials:

  • Op-Amp: Texas Instruments TL072 (Dual JFET input, low noise, wide supply range up to ±18V).
  • Rin: 10 kΩ (Vishay MRS25, 1%, 50 ppm/°C).
  • Rf: 100 kΩ (Vishay MRS25, 1%, 50 ppm/°C).
  • Resulting Gain: Exactly -10 V/V.

This specific 10k/100k combination provides an input impedance high enough not to load most microcontroller DACs or sensor outputs, while keeping the feedback resistor low enough to minimize thermal noise and avoid stray capacitance issues up to roughly 100 kHz. It is the gold-standard starting point for analog bench work.