Operational amplifier frequency response

An operational amplifier’s frequency response describes how its voltage gain changes with the frequency of an input signal. Voltage gain is the ratio of the output voltage amplitude to the input voltage amplitude.

For a real amplifier, the gain is typically approximately constant across a range of low frequencies. As the frequency increases beyond this range, the gain falls. Thus, for input signals of the same amplitude, higher-frequency signals in the falling-gain region produce smaller output amplitudes.

Frequency response curve and cutoff

A frequency response curve plots gain against frequency. It shows the low-frequency region where gain is approximately constant and the region where gain decreases as frequency rises.

The upper cutoff frequency is the frequency at which the gain falls to

1/\sqrt{2}

, or approximately

$$0.707$$

, of its low-frequency value. This is equivalent to a decrease of approximately

3,\mathrm{dB}

. It is a reference point on the falling part of the curve, not a frequency at which the output suddenly stops.

Bandwidth

The bandwidth is the range of frequencies over which the response is considered to extend. If the response extends down to very low frequencies, the bandwidth is approximately the range from zero to the upper cutoff frequency. Its numerical value is then approximately equal to the upper cutoff frequency.

Operational amplifier gain–bandwidth product

An operational amplifier can be configured to give different voltage gains, but a higher gain comes with a narrower range of frequencies over which that gain is maintained. Under the usual gain–bandwidth model, the product of gain and bandwidth is approximately constant for a given operational amplifier.

\text{gain} \times \text{bandwidth} \approx \text{gain–bandwidth product}

Here, gain means the magnitude of the closed-loop voltage gain in the frequency range where it is approximately constant. Using the magnitude means that an inverting gain is treated as positive for this calculation. Bandwidth is the corresponding frequency range up to the cutoff, above which the gain falls further.

The relationship is inverse: if the gain is doubled, the available bandwidth is approximately halved. If the gain is reduced, the available bandwidth increases.

Calculating gain or bandwidth

When the device’s gain–bandwidth product is known, the relationship can be rearranged to find either quantity:

\text{bandwidth} \approx \frac{\text{gain–bandwidth product}}{\text{gain}} \qquad \text{gain} \approx \frac{\text{gain–bandwidth product}}{\text{bandwidth}}

For example, an amplifier has a gain–bandwidth product of $1,\text{MHz}$ and a closed-loop gain magnitude of $$$10$$$ . Its approximate bandwidth is:

\text{bandwidth} \approx \frac{1\,\text{MHz}}{10} = 0.1\,\text{MHz} = 100\,\text{kHz}

If the gain magnitude is increased to $$$100$$$ , the same model gives a bandwidth of $10,\text{kHz}$ . Both results are approximate because the gain–bandwidth product is only approximately constant under this model.