The Core Inverting Amplifier Formula and Symbol Definitions
If you are designing an analog front-end or scaling a sensor signal, the inverting amplifier formula is your baseline tool. The direct answer for calculating the output voltage of an ideal inverting operational amplifier is:
Vout = -Vin × (Rf / Rin)
Alternatively, if you are solving for the closed-loop voltage gain (Av), the formula is expressed as:
Av = - (Rf / Rin)
The negative sign indicates phase inversion: a positive input voltage yields a negative output voltage, assuming a dual-polarity power supply. Below is the definitive spec-sheet-table defining every variable in the equation.
| Symbol | Parameter | Standard Unit | Practical Bench Range |
|---|---|---|---|
| Vout | Output Voltage | Volts (V) | Bounded by supply rails (e.g., ±12V) |
| Vin | Input Voltage | Volts (V) | mV to low single-digit V |
| Rf | Feedback Resistor | Ohms (Ω) | 1kΩ to 1MΩ |
| Rin | Input Resistor | Ohms (Ω) | 1kΩ to 100kΩ |
| Av | Closed-Loop Voltage Gain | Dimensionless (V/V) | -1 to -100 (per stage) |
Rearranged Forms: Solving for Resistors and Input Voltage
On the bench, you rarely know all variables upfront. Usually, you have a target gain and a fixed input signal, and you need to select standard E24 resistor values. Here are the algebraically rearranged forms of the inverting amplifier formula, solving for each individual variable:
- Solving for Feedback Resistor (Rf): Rf = - (Vout × Rin) / Vin or Rf = -Av × Rin
- Solving for Input Resistor (Rin): Rin = - (Vin × Rf) / Vout or Rin = -Rf / Av
- Solving for Input Voltage (Vin): Vin = - Vout / (Rf / Rin)
When selecting resistors, always calculate the exact theoretical value first, then map it to the nearest standard 1% tolerance resistor (E96 series) or 5% tolerance (E24 series) available in your kit. Keep resistor values between 1kΩ and 100kΩ to avoid excessive thermal noise (too high) or overloading the op-amp's output stage (too low).
Worked Examples with Strict Unit Tracking
Abstract math fails on the breadboard when units are mixed. Here are two solved problems demonstrating strict unit tracking.
Problem 1: Calculating Output Voltage from Sensor Data
Given: A thermocouple amplifier outputs a Vin = 45 mV DC signal. Your circuit uses Rin = 2.2 kΩ and Rf = 33 kΩ. Find Vout.
- Convert units to base SI: Vin = 45 mV = 0.045 V. Rin = 2200 Ω. Rf = 33000 Ω.
- Calculate Gain (Av): Av = - (33000 / 2200) = -15 V/V.
- Apply Formula: Vout = - (0.045 V) × 15 = -0.675 V.
- Sanity Check: The output is negative (inverted) and well within standard ±12V supply rails. The magnitude is realistic.
Problem 2: Sizing Resistors for a Target Gain
Given: You need to invert and amplify a 1V peak audio signal to a -10V peak output. You have a 4.7 kΩ input resistor in your bin. Find the required Rf.
- Determine Target Gain: Av = Vout / Vin = -10V / 1V = -10 V/V.
- Rearrange Formula for Rf: Rf = -Av × Rin.
- Substitute Values: Rf = -(-10) × 4.7 kΩ = 10 × 4700 Ω = 47,000 Ω (47 kΩ).
- Component Selection: 47 kΩ is a standard E24 value. You can use a single 47kΩ 1% metal film resistor. No series/parallel combinations required.
Real-World Bench Scenario: When the Math Meets the Rails
The inverting amplifier formula assumes an ideal op-amp operating in its linear region. Reality introduces power supply constraints. Here is a narrative walkthrough of a common bench failure.
The Setup
You are breadboarding an inverting preamp for an electret microphone using a standard LM358 dual op-amp. To keep the circuit simple, you power the LM358 with a single +5V supply (VCC = 5V, GND = 0V). The microphone provides a 0.2V peak AC signal centered around 0V. You choose Rin = 1 kΩ and Rf = 10 kΩ to achieve a gain of -10.
The Numbers
According to the formula: Vout = - (0.2V) × (10k / 1k) = -2.0V peak. You expect to see a 2V peak inverted sine wave on your oscilloscope.
The Outcome
You probe pin 1 (Output) with the oscilloscope. The trace is a flatline at 0.02V. There is no amplified signal, just a tiny amount of high-frequency noise.
What Went Wrong (and the Fix)
The formula is mathematically correct, but physically impossible in this configuration. The LM358 cannot output a negative voltage when its negative supply pin (VEE) is tied to 0V ground. The output transistor saturated at the lower rail (~0V). Furthermore, the LM358 is not a rail-to-rail output op-amp; even on a dual supply, its output swing typically stops 1.5V short of the positive rail.
The Fix: You must establish a virtual ground. By adding a voltage divider (two 10kΩ resistors) from the 5V rail to ground, you create a 2.5V bias. Connect this 2.5V to the non-inverting input (Pin 3) instead of grounding it. AC-couple the input and output with 10μF capacitors. Now, the op-amp amplifies the signal around the 2.5V midpoint, swinging from 0.5V to 4.5V, safely avoiding the supply rails. For a cleaner alternative, swap the LM358 for a true rail-to-rail CMOS op-amp like the TLV2462.
Assumptions, Unit Traps, and Realistic Magnitudes
To use the inverting amplifier formula reliably, you must understand the boundaries of the math.
When the Formula Applies (and Its Assumptions)
The formula Vout = -Vin × (Rf / Rin) relies on the ideal op-amp model. It assumes:
- Infinite Open-Loop Gain: The op-amp has enough internal gain to force the voltage difference between the inverting and non-inverting pins to exactly zero (virtual short).
- Infinite Input Impedance: No current flows into the op-amp's input pins; all current from Vin flows through Rin and Rf.
- Linear Operation: The calculated Vout must fall strictly between the op-amp's positive and negative supply rails (minus the headroom voltage specified in the component datasheet).
Unit Mistakes That Break the Math
The most common bench error is mixing milli-Volts and Volts without conversion. If Vin is 50mV and you mistakenly plug '50' into the formula while expecting Vout in Volts, your calculated output will be 1000x larger than reality.
Conversely, mixing resistor units (e.g., Rf in kΩ and Rin in Ω) does not break the gain calculation, provided you are only calculating the ratio (Av). The units cancel out in the division. However, if you are calculating current (I = Vin / Rin), failing to convert kΩ to Ω will yield a current reading off by a factor of 1000.
What a Realistic Answer Magnitude Looks Like
If your formula spits out a gain of -5,000, your design is flawed. While the math allows it, real-world op-amps suffer from input offset voltage and noise gain. An input offset voltage of just 2mV, amplified by a gain of 5,000, results in a 10V DC error at the output, instantly saturating the amplifier.
For stable, low-noise analog design, keep single-stage inverting gain magnitudes between 1 and 100. If you need a gain of 1,000, cascade two inverting stages (e.g., a gain of -31.6 followed by another gain of -31.6) and AC-couple them to block DC offset accumulation. For deep-dive design rules on multi-stage cascading and bandwidth limitations, refer to Analog Devices' Op-Amp Basics guide.
