Math Bridge: Op-Amp Gain and Bandwidth

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Math BridgeElectronicsStruggle-friendly runway

Why does asking for more op-amp gain cost bandwidth?

Connect feedback factor, closed-loop gain, gain-bandwidth product, and finite open-loop gain.

Eddie, the electronics guideEddie guides
The one targetTurn a requested non-inverting gain into its ideal bandwidth ceiling.
The chapter caseA 1 MHz GBW op-amp at closed-loop gain 100.
What it buys youA front-end gain choice checked against sensor bandwidth and accuracy.

A field team faces an unresolved physical question: Why does asking for more op-amp gain cost bandwidth? They must answer it before changing requested gain on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is requested gain. The middle card applies this page's relationship. The green card is feedback factor. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Requested gain changes feedback factor An input card leads through the page relationship to the feedback factor result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Feedback accuracy and bandwidth both depend on beta times the op-amp's available open-loop gain.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for requested gain is 100.

  2. 2

    Name the relationship. Av=100, so beta=1/100=0.010 f-3dB=(1,000,000 Hz)/100=10,000 Hz=10.0 kHz unity gain keeps about 1.00 MHz with AOL,DC=100,000, ACL,DC=99.900 and error≈0.100%

  3. 3

    Substitute the chapter fixture. Set requested gain to 100. The page ledger gives feedback factor as 0.010.

  4. 4

    Read the result. Keep the stated output unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change requested gain

Try Predict the direction of feedback factor. Move one control, calculate, then check your prediction.

100
Chapter baseline
Feedback factor

Observe Feedback accuracy and bandwidth both depend on beta times the op-amp's available open-loop gain. Reset the control to 100 and compare feedback factor.

Explain Only requested gain moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only requested gain moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only requested gain. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

An op-amp starts with enormous open-loop gain, but that gain falls with frequency. Negative feedback spends some gain to make a predictable closed-loop stage. Asking for more closed-loop gain leaves less loop gain at high frequency.

Eddie: GBW is a budget: the same device can provide much gain or much bandwidth, but not both without limit.

2. Name every algebra move

1

Set feedbackFor a non-inverting stage, Av=1+Rf/Rg and beta=1/Av.

2

Use the falling open-loop gainAbove its dominant pole, AOL≈GBW/f.

3

Find the boundarySet beta AOL=1 at the approximate -3 dB point.

4

Rearrangef-3dB=beta GBW=GBW/Av.

5

Check finite DC gainUse ACL=AOL/(1+beta AOL) rather than assuming infinity.

3. Reproduce the chapter case

Av=100, so beta=1/100=0.010
f-3dB=(1,000,000 Hz)/100=10,000 Hz=10.0 kHz
unity gain keeps about 1.00 MHz
with AOL,DC=100,000, ACL,DC=99.900 and error≈0.100%

The 10 kHz result is a first-order bandwidth ceiling, not proof that the whole sensor path is accurate to 10 kHz.

4. Try one real input

TryChange closed-loop gain and predict bandwidth and finite-gain error.

Requested gain
Feedback factor
Rf/Rg
Bandwidth
Bandwidth
Unity/bandwidth ratio
DC loop gain
Finite DC gain
Finite-gain error

ObserveDoubling requested gain halves the ideal bandwidth and also reduces DC loop gain.

ExplainFeedback accuracy and bandwidth both depend on beta times the op-amp's available open-loop gain.

Technical boundaries.

This is a single-dominant-pole, small-signal teaching ledger.

Frequency
Extra poles, zeros, phase margin, capacitive load, and compensation can lower usable bandwidth.
Signal
Slew rate, output current, input common mode, output swing, and settling may dominate large signals.
Error
Offset, bias current, resistor tolerance, noise, CMRR, PSRR, and drift remain outside finite-gain error.

Correct, not complete: this ledger does not select, stabilise, or qualify an op-amp sensor front end.

5. Use the result in the design

Require bandwidth and loop-gain margin beyond the sensor band, then check slew, stability, noise, input/output range, and worst-case datasheet curves.

6. Record the evidence state

Record op-amp variant, supply, gain network, source impedance, load, GBW, phase margin, slew rate, signal amplitude, frequency, temperature, and measured settling.

7. Check yourself

Why does gain 100 give about 10 kHz from a 1 MHz part?
Answer: The single-pole gain-bandwidth approximation divides 1,000,000 Hz by the closed-loop noise gain 100.
Does unity gain guarantee stability?
Answer: No. The exact part, compensation, phase margin, load capacitance, and layout still decide stability.
Why is ideal resistor gain not exact?
Answer: Open-loop gain is finite, so the feedback error is nonzero even before resistor tolerance, offset, and noise.
Honesty boundary.

The arithmetic reproduces the named gain-100 case; it is not a front-end approval.

Frequency
Extra poles, zeros, phase margin, capacitive load, and compensation can lower usable bandwidth.
Signal
Slew rate, output current, input common mode, output swing, and settling may dominate large signals.
Error
Offset, bias current, resistor tolerance, noise, CMRR, PSRR, and drift remain outside finite-gain error.

Correct, not complete: this ledger does not select, stabilise, or qualify an op-amp sensor front end.

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