Op-Amp Differentiator 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, Feedback Resistor 10 kΩ, Input Capacitor 100 nF, the result is 0.1659 kV. 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 differentiator uses a capacitor at the input and a resistor in the feedback path. The output magnitude increases linearly with frequency at 20 dB/decade (like a first-order high-pass filter). At high frequencies, the gain can become very large, amplifying noise and potentially causing oscillation. Therefore, practical differentiators always include a small series resistor with the input capacitor or a small feedback capacitor to limit the high-frequency gain. Differentiators are used in rate-of-change detection, edge detection, frequency demodulation, and the D term of PID controllers.

Variables

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

Practical Notes

The unity-gain frequency occurs at f = 1/(2πRC), the same as the integrator but with gain increasing instead of decreasing. To stabilize the circuit, add a series resistor Rs with the input capacitor: Rs ≈ R/(10-100) limits the maximum gain to R/Rs. Alternatively, add a feedback capacitor Cf in parallel with R, creating a rolloff at f = 1/(2πRCf). The noise gain of a differentiator increases without bound, making it inherently noisier than an integrator.

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 Feedback Resistor kΩ mΩ, Ω, kΩ, MΩ
C Input Capacitor nF pF, nF, µF, mF, F

Step by step

  1. 1. What you know: Input Voltage 26.4 V, Frequency 1 kHz, Feedback Resistor 10 kΩ, Input 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 = 0.1659 kV

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, Feedback Resistor 100 Ω, Input Capacitor 100 pF

Result: Output Voltage = 0.2281 µV

Low range

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

Result: Output Voltage = 2.281 mV

Typical

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

Result: Output Voltage = 0.1659 kV

High range

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

Result: Output Voltage = 6359 kV

Maximum range

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

Result: Output Voltage = 6.36 × 10⁷ kV

Solving for each variable

Solve for Output Voltage

Input Voltage 0.726 V, Frequency 5 Hz, Feedback Resistor 100 Ω, Input Capacitor 100 pF → 0.2281 µV

Solve for Input Voltage

Output Voltage 0.33 V, Frequency 5 Hz, Feedback Resistor 100 Ω, Input Capacitor 100 pF → 1050 kV

How Output Voltage changes with Input Voltage

Input Voltage (input) Output Voltage (result)
5.81 V 0.3887 kV
14.53 V 0.9718 kV
29.05 V 1.944 kV
43.58 V 2.915 kV
58.1 V 3.887 kV
87.15 V 5.831 kV
116.2 V 7.774 kV
0.2324 kV 15.55 kV
0.4067 kV 27.21 kV
0.581 kV 38.87 kV

The same result in other units

Unit Value
mV 165900
V 165.9
kV 0.1659

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Frequently asked questions

How do you calculate Op-Amp Differentiator Gain?

Use |V_out| = V_in × 2πfRC. With Input Voltage 26.4 V, Frequency 1 kHz, Feedback Resistor 10 kΩ, Input Capacitor 100 nF, Output Voltage works out to 0.1659 kV.

What is the formula for Op-Amp Differentiator Gain?

The formula is |V_out| = V_in × 2πfRC. An op-amp differentiator uses a capacitor at the input and a resistor in the feedback path.

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, Feedback Resistor 100 Ω, Input Capacitor 100 pF, Input Voltage is 1050 kV.

What units does Op-Amp Differentiator Gain use?

Output Voltage is shown in 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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