Advanced Calculator
Enter practical integrator values. The calculator estimates feedback factor, open loop gain, loop gain, margins, and sweep response.
Formula Used
Integrator impedance: Zc = 1 / (jωC)
Optional feedback path: Zf = Rf || Zc
Feedback factor: β(jω) = Zin / (Zin + Zf)
Amplifier open loop model: A(jω) = A0 / [(1 + jf/fp1)(1 + jf/fp2)]
Loop gain: L(jω) = A(jω) × β(jω)
Gain in decibels: LdB = 20 log10(|L|)
Phase margin: PM = 180° + ∠L at |L| = 1
Integrator corner: fi = 1 / (2πRC)
How to Use This Calculator
- Enter the input resistor and feedback capacitor values.
- Add the selected test frequency for direct loop gain output.
- Enter the amplifier DC open loop gain in decibels.
- Add the dominant pole frequency from the amplifier model.
- Use a second pole when high frequency roll-off is important.
- Add leakage resistance when the capacitor has a parallel resistor.
- Press the calculate button and review the results above the form.
- Download the result table as CSV or PDF for reporting.
Example Data Table
| Case | R | C | Frequency | A0 | fp1 | Use |
|---|---|---|---|---|---|---|
| Lab integrator | 10 kΩ | 10 nF | 1 kHz | 100 dB | 10 Hz | General stability check |
| Fast signal stage | 4.7 kΩ | 1 nF | 20 kHz | 95 dB | 20 Hz | Higher bandwidth design |
| Slow sensor stage | 100 kΩ | 100 nF | 10 Hz | 110 dB | 5 Hz | Low frequency physics measurement |
Physics Background and Design Notes
Why Loop Gain Matters
An active integrator is more than a simple resistor and capacitor network. It also depends on amplifier behavior. Loop gain shows how strongly the amplifier corrects error at a selected frequency. A high value means feedback is strong. A low value means the output follows the ideal rule less closely.
What the Model Measures
This calculator uses complex impedance. It treats the feedback capacitor as a frequency dependent path. It also includes the input resistor, finite open loop gain, amplifier poles, and optional phase lag. These choices make the estimate useful for real physics circuits, sensors, filters, charge amplifiers, and lab integrators.
Reading the Gain Result
Loop gain is shown as a ratio and in decibels. Positive decibels mean feedback correction is greater than one. Zero decibels marks the gain crossover. The phase at that point helps decide stability. A phase margin near sixty degrees is often comfortable. A small margin can cause ringing or oscillation.
Integrator Behavior
The ideal closed loop integrator has gain equal to one divided by angular frequency, resistance, and capacitance. Its phase is near minus ninety degrees. Real amplifiers add extra phase shift. At high frequency, poles reduce gain and add delay. This is why a circuit that works at low frequency may fail at higher bandwidth.
Practical Use
Start with measured or datasheet amplifier values. Then enter the target resistor and capacitor. Compare the test frequency with the integrator corner. Review the Bode style plot. If phase margin is low, reduce bandwidth, change component values, or choose a faster amplifier. Always verify critical circuits with simulation and bench testing.
FAQs
1. What is loop gain in an integrator?
Loop gain is the product of open loop amplifier gain and feedback factor. It shows how much correction feedback applies at one frequency.
2. Why is phase margin important?
Phase margin estimates how far the loop is from oscillation at gain crossover. Higher margin usually means less ringing and better stability.
3. What does gain crossover mean?
Gain crossover is the frequency where loop gain equals one, or zero decibels. Phase at this point is used for phase margin.
4. Can I use a leakage resistor?
Yes. Enter a feedback leakage resistor when the capacitor has a parallel resistance. Use zero when the feedback path is only capacitive.
5. What is a good phase margin?
Many designs aim for about 45 to 60 degrees or more. Sensitive physics instruments may need more conservative margins.
6. Why add a second pole?
A second pole models extra high frequency roll-off. It can reduce phase margin and reveal stability problems missed by a one-pole model.
7. Is this equal to circuit simulation?
No. It is an analytical estimate. Use it for design guidance, then verify demanding circuits with a simulator and real measurements.
8. Why is the ideal integrator gain included?
It gives a reference for the expected closed loop response. Comparing it with loop gain helps judge real circuit accuracy.