Op-Amp Integrator Gain

|Vout| = Vin / (2πfRC)

Quick answer

This calculator solves |V_out| = V_in / (2πfRC) for Output Voltage. With Input Voltage 26.4 V, Frequency 1 kHz, Input Resistor 10 kΩ, Feedback Capacitor 100 nF, the result is 4.202 V. It solves in 2 directions, so you can enter the values you have and get the one you need.

Calculator

Output Voltage (V)
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Formula

|V_out| = V_in / (2πfRC)

Description

An op-amp integrator uses a capacitor in the feedback path instead of a resistor. For a sinusoidal input, the output magnitude decreases with frequency at 20 dB/decade (like a first-order low-pass filter), with a gain of 1/(2πfRC). At DC (f = 0), the ideal gain is infinite, causing the output to drift to the supply rail. In practice, a large feedback resistor is added in parallel with the capacitor to limit DC gain and prevent saturation. Integrators are fundamental building blocks in analog computers, PID controllers (the I term), active filters, and waveform generators (triangle from square wave).

Variables

  • V_out — Output voltage magnitude (V)
  • V_in — Input voltage magnitude (V)
  • f — Signal frequency (Hz)
  • R — Input resistor (Ω)
  • C — Feedback capacitor (F)

Practical Notes

The unity-gain frequency (where |Vout| = |Vin|) occurs at f = 1/(2πRC). Below this frequency, the integrator amplifies; above it, the integrator attenuates. For a triangle wave generator, apply a square wave to the input: the output triangle amplitude is Vsq × T/(4RC) where T is the square wave period. The feedback resistor Rf for DC stability should satisfy Rf >> 1/(2πf_min×C) where f_min is the lowest signal frequency of interest.

Symbols and units

Symbol Meaning Default unit Accepted units
Vout Output Voltage V pV, nV, µV, mV, V, kV
Vin Input Voltage V pV, nV, µV, mV, V, kV
f Frequency Hz Hz, kHz, MHz, GHz
R Input Resistor kΩ mΩ, Ω, kΩ, MΩ
C Feedback Capacitor nF pF, nF, µF, mF, F

Step by step

  1. 1. What you know: Input Voltage 26.4 V, Frequency 1 kHz, Input Resistor 10 kΩ, Feedback Capacitor 100 nF
  2. 2. The relation: |V_out| = V_in / (2πfRC)
  3. 3. Substitute: Vin = 26.4 V, f = 1 kHz, R = 10 kΩ, C = 100 nF
  4. 4. Result: Output Voltage = 4.202 V

Related quantities from the same inputs

  • P = V² / R = 69.7 mW
  • I = V / R = 2.64 mA

Worked examples

Minimum range

Given: Input Voltage 0.726 V, Frequency 5 Hz, Input Resistor 100 Ω, Feedback Capacitor 100 pF

Result: Output Voltage = 2311 kV

Low range

Given: Input Voltage 7.26 V, Frequency 50 Hz, Input Resistor 1 kΩ, Feedback Capacitor 1 nF

Result: Output Voltage = 23.11 kV

Typical

Given: Input Voltage 26.4 V, Frequency 1 kHz, Input Resistor 10 kΩ, Feedback Capacitor 100 nF

Result: Output Voltage = 4.202 V

High range

Given: Input Voltage 0.506 kV, Frequency 20 kHz, Input Resistor 100 kΩ, Feedback Capacitor 1 µF

Result: Output Voltage = 40.27 mV

Maximum range

Given: Input Voltage 5.06 kV, Frequency 0.2 MHz, Input Resistor 1 MΩ, Feedback Capacitor 10 µF

Result: Output Voltage = 0.4027 mV

Solving for each variable

Solve for Output Voltage

Input Voltage 0.726 V, Frequency 5 Hz, Input Resistor 100 Ω, Feedback Capacitor 100 pF → 2311 kV

Solve for Input Voltage

Output Voltage 0.33 V, Frequency 5 Hz, Input Resistor 100 Ω, Feedback Capacitor 100 pF → 103.7 nV

How Output Voltage changes with Input Voltage

Input Voltage (input) Output Voltage (result)
5.81 V 86.84 mV
14.53 V 0.2171 V
29.05 V 0.4342 V
43.58 V 0.6513 V
58.1 V 0.8684 V
87.15 V 1.303 V
116.2 V 1.737 V
0.2324 kV 3.474 V
0.4067 kV 6.079 V
0.581 kV 8.684 V

The same result in other units

Unit Value
µV 4202000
mV 4202
V 4.202
kV 0.004202

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Frequently asked questions

How do you calculate Op-Amp Integrator Gain?

Use |V_out| = V_in / (2πfRC). With Input Voltage 26.4 V, Frequency 1 kHz, Input Resistor 10 kΩ, Feedback Capacitor 100 nF, Output Voltage works out to 4.202 V.

What is the formula for Op-Amp Integrator Gain?

The formula is |V_out| = V_in / (2πfRC). An op-amp integrator uses a capacitor in the feedback path instead of a resistor.

How do you find Input Voltage when you know the other values?

Rearrange for Input Voltage: with Output Voltage 0.33 V, Frequency 5 Hz, Input Resistor 100 Ω, Feedback Capacitor 100 pF, Input Voltage is 103.7 nV.

What units does Op-Amp Integrator Gain use?

Output Voltage is shown in µV, mV, V, kV. Values are held in SI base units internally, so you can mix prefixes without converting by hand.

Related Concepts

All Op-Amps formulas →
Non-Inverting Amplifier Inverting Amplifier Summing Amplifier Difference Amplifier Gain-Bandwidth Product Common-Mode Rejection Ratio Op-Amp Integrator Gain Op-Amp Differentiator Gain

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