The terms theoretical gain, and maximum frequency or bandwidth, are often used in conjunction with gain-bandwidth product (GBWP), which is the frequency at which the gain of the operational amplifier is unity. Operational amplifiers usually exhibit this value at 1 MHz, which is when the gain falls to unity.

The blue graph shows the how the gain of an operational amplifier falls as we increase the frequency. Usually, at 1 MHz, the gain tends to be unity otherwise known as 1. When scientists calculate any theoretical gain (G_t) of an op amp, they also need to know the maximum theoretical frequency (F_t) over which that gain will be available. We therefore use the GBWP value to calculate the theoretical gain, or the maximum frequency (also known simply as bandwidth). The simplest way to remember the formula is that GBWP is exactly as it describes, that it is the product of gain and bandwidth, when the gain is 1 and the bandwidth is 1 MHz (in this example).

• When the gain is 1, the bandwidth will 1 MHz.
• When the gain is 10, the bandwidth will be 100 kHz.
• When the gain is 100, the bandwidth will be 10 kHz.
• When the gain is 1000, the bandwidth will be 1 kHz.

The products of gain and bandwidth values above are always the same, which is 1000000 Hz.

Therefore, unity × 1 MHz = Gt × Bt as shown in the graph.

Hence,

GBWP = Theoretical Gain × Theoretical Bandwidth

The theoretical gain is the maximum gain, and the theoretical bandwidth is the maximum frequency over which this gain occurs. hence, by transposition we get the following.

Theoretical gain = GBWP / Theoretical Bandwidth

Alternatively, if your question has provided GBWP and theoretical gain, then you can calculate the theoretical bandwidth frequency.

Theoretical Bandwidth = GBWP / Theoretical Gain

Example

Here is a simple example where we have an operational amplifier with negative feedback at the inverting input. In this type of circuit, current passes through the input resistor and through the feedback resistor, but never through the input terminal of the operational amplifier because that junction point forms a virtual earth (VE).

If this amplifier had gain bandwidth product (GBWP) of 1 MHz, calculate the maximum frequency over which this amplifier will have maximum gain.

Since we have the resistor values, we can calculate the theoretical gain using those and the simple gain formula for inverting amplifiers.

Gain = - feedback resistance / input resistance

Gain = 100 kΩ / 10 kΩ

Therefore, this amplifier circuit has theoretical gain of 10.

GBWP = Theoretical Gain × Theoretical Bandwidth

Hence,

Theoretical Bandwidth = GBWP / Theoretical Gain

Bandwidth = 1000000 Hz / 10

Bandwidth = 100 kHz

Therefore, you can expect to get a gain of 10, up to 100 kHz, after which gain starts to fall.