The two most critical op-amp formulas for linear circuit design dictate the closed-loop voltage gain of the non-inverting and inverting amplifier configurations. For a non-inverting amplifier, the gain formula is Av = 1 + (Rf / Ri). For an inverting amplifier, the formula is Av = -(Rf / Ri). These equations allow you to set precise amplification using external passive components, bypassing the massive, unpredictable open-loop gain of the bare silicon.
Core Op-Amp Formulas and Symbol Definitions
Before deriving or rearranging these equations, we must strictly define the variables. In both configurations, the operational amplifier uses negative feedback to force the voltage difference between its inverting (-) and non-inverting (+) inputs to virtually zero. This virtual short is the foundation of the formulas below.
| Symbol | Parameter | Standard Unit | Description |
|---|---|---|---|
| Av | Voltage Gain | V/V (dimensionless) | The ratio of output voltage to input voltage. Often converted to decibels (dB) in audio/RF. |
| Rf | Feedback Resistor | Ohms (Ω) | The resistor connecting the op-amp output pin back to the inverting (-) input pin. |
| Ri | Input / Ground Resistor | Ohms (Ω) | Non-inverting: connects inverting input to ground. Inverting: connects signal source to inverting input. |
| Vin | Input Voltage | Volts (V) | The AC or DC signal applied to the amplifying input terminal. |
| Vout | Output Voltage | Volts (V) | The resulting amplified signal at the op-amp output pin. |
While the ideal formulas assume perfect components, real-world op-amps introduce limitations. The table below provides benchmark data for four common op-amps to contextualize where ideal formulas hold up and where real-world physics takes over.
| Op-Amp Model | Open-Loop Gain (AOL) | GBWP | Slew Rate | Primary Use Case |
|---|---|---|---|---|
| TI TLV9002 | 120 dB (1,000,000 V/V) | 1.0 MHz | 2.0 V/μs | Modern low-voltage (1.8V-5.5V) sensor interfaces and IoT battery devices. |
| TI TL072 | 106 dB (200,000 V/V) | 3.0 MHz | 13.0 V/μs | JFET-input audio mixers, guitar pedals, and high-impedance sensor buffers. |
| TI OPA1678 | 114 dB (500,000 V/V) | 16 MHz | 9.0 V/μs | High-fidelity audio DAC output stages and studio equipment. |
| TI LM358 | 100 dB (100,000 V/V) | 1.0 MHz | 0.6 V/μs | Legacy single-supply industrial control and basic DC signal conditioning. |
Assumptions, Applicability, and Unit Traps
The closed-loop op-amp formulas apply strictly when the circuit operates in its linear region and the op-amp meets ideal assumptions. According to foundational semiconductor theory outlined by resources like All About Circuits, the ideal model assumes infinite open-loop gain (AOL = ∞), infinite input impedance, and zero output impedance. Because AOL is functionally massive (e.g., 1,000,000 V/V in the TLV9002), the 1+AOL term in the exact derivation simplifies to just AOL, collapsing the complex fraction into the clean Rf/Ri ratios we use on the bench.
Realistic Answer Magnitudes
A common mistake for beginners is calculating a closed-loop gain of 50,000 V/V because they used a 5 MΩ feedback resistor and a 100 Ω ground resistor. This is physically impossible in a linear circuit. A realistic closed-loop gain magnitude for standard sensor and audio applications ranges from 1 V/V to 100 V/V. If your formula outputs a gain higher than your op-amp's open-loop gain, or if the resulting Vout exceeds the power supply rails, the op-amp saturates and acts as a comparator, not an amplifier.
Unit Mistakes That Break the Math
- The kΩ Ratio Trap: The gain formula relies on the ratio of Rf to Ri. You can use ohms, kilohms, or megaohms, but you must use the same unit for both. Mixing 10 kΩ and 1,000 Ω without converting yields a 10x error.
- V/V vs. dB Confusion: The formula outputs linear gain (V/V). If a spec sheet calls for 20 dB of gain, you cannot plug '20' into the Av variable. You must convert decibels to linear gain first using Av = 10(dB/20). (20 dB = 10 V/V).
Rearranged Forms for Component Selection
On the workbench, you rarely solve for Av. Usually, you know your sensor's output voltage, your ADC's required input voltage, and the resistors you have in your drawer. Here are the non-inverting formulas rearranged to solve for every variable, assuming Av is expressed in linear V/V:
- Solve for Feedback Resistor: Rf = Ri × (Av - 1)
- Solve for Ground Resistor: Ri = Rf / (Av - 1)
- Solve for Required Input: Vin = Vout / Av
- Solve for Expected Output: Vout = Vin × [1 + (Rf / Ri)]
Note: For the inverting configuration, drop the '1 +' and apply a negative sign to the gain, keeping in mind the input impedance of the circuit becomes exactly Ri, unlike the non-inverting topology which boasts near-infinite input impedance.
Worked Examples with Unit Tracking
Let's apply these formulas to two real-world scenarios, tracking units at every step to ensure dimensional consistency.
Problem 1: Non-Inverting Sensor Amplifier
Scenario: You are reading a thermocouple amplifier that outputs a maximum of 120 mV (0.120 V). You need to scale this to exactly 3.0 V to maximize the resolution of a 3.3V microcontroller ADC. You have a 1.0 kΩ precision resistor available for Ri. What value must Rf be?
Step 1: Determine required linear gain (Av).
Av = Vout / Vin
Av = 3.0 V / 0.120 V
Av = 25 V/V
Step 2: Rearrange the non-inverting formula to solve for Rf.
Rf = Ri × (Av - 1)
Step 3: Substitute values and track units.
Rf = 1.0 kΩ × (25 V/V - 1 V/V)
Rf = 1.0 kΩ × 24
Rf = 24 kΩ
Result: You need a 24 kΩ feedback resistor. Since 24 kΩ is not a standard 1% E96 series value, you would use a 23.7 kΩ resistor in series with a 300 Ω resistor, or select a 24.3 kΩ and accept a 1.25% gain error, which is easily calibrated out in firmware.
Problem 2: Inverting Audio Stage (The Rail Trap)
Scenario: You are building an inverting preamp using a TLV9002 powered by a single 5.0 V supply (V+ = 5V, V- = GND). The circuit uses Rf = 47 kΩ and Ri = 10 kΩ. An AC audio signal peaks at Vin = -0.8 V. What is the theoretical Vout, and what actually happens on the bench?
Step 1: Calculate theoretical gain.
Av = -(Rf / Ri)
Av = -(47 kΩ / 10 kΩ)
Av = -4.7 V/V
Step 2: Calculate theoretical Vout.
Vout = Vin × Av
Vout = -0.8 V × -4.7 V/V
Vout = +3.76 V
Step 3: Evaluate against real-world supply rails.
The theoretical output is +3.76 V. Because the TLV9002 is a rail-to-rail output op-amp on a 5V single supply, its maximum output swing is roughly 4.9 V under light loads. Since 3.76 V is well below the 4.9 V ceiling and above the 0.1 V floor, the signal will pass cleanly without clipping.
What if Vin peaked at -1.5 V?
Theoretical Vout = -1.5 V × -4.7 = +7.05 V. Because the op-amp is powered by a 5V supply, it cannot output 7.05 V. The output will hard-clip at approximately 4.9 V, flattening the top of your audio waveform and introducing severe harmonic distortion. The formula holds mathematically, but fails physically. For deep dives into avoiding these saturation instabilities, Texas Instruments' Precision Op-Amp resources provide extensive application notes on headroom and common-mode limits.
Real-World Deviations: When the Formula Fails
The closed-loop gain formulas assume DC or low-frequency operation. As signal frequency increases, the op-amp's internal compensation capacitor causes the open-loop gain to roll off at -20 dB/decade. This is defined by the Gain Bandwidth Product (GBWP).
If you design a non-inverting amplifier with a closed-loop gain of 100 V/V (40 dB) using an LM358 (GBWP = 1 MHz), your maximum usable bandwidth is only 10 kHz (1,000,000 / 100). If you attempt to amplify a 50 kHz ultrasonic sensor signal with this circuit, the actual gain will be drastically lower than the Rf/Ri formula predicts. Always divide the op-amp's GBWP by your required closed-loop gain to verify the circuit will support your target signal frequency.
