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# Gain-bandwidth product

The gain bandwidth product (GBW or GB) for an amplifier is the product of the open loop gain (constant for a given amplifier) and its 3 dB bandwidth.

Relevance to design

This quantity is commonly specified for operational amplifiers, and allows circuit designers to determine the maximum gain that can be extracted from the device for a given frequency (or bandwidth) and vice versa.

When adding LC circuits to the input and output of an amplifier the gain raises and the bandwidth decreases, but the product remains constant.

Theory

The gain-bandwidth product may be understood from a conservation-of-power viewpoint. The difference between the output signal power and the input signal power can never be greater than the DC power supplied to the amplifier through its bias circuitry. Stated mathematically, if "P"out is the total output signal power from the amplifier, "P"in is the total signal power input to the amplifier, and "P"DC is the total DC power supplied to the amplifier, then

$P_\left\{out\right\} - P_\left\{in\right\} le P_\left\{DC\right\}$.

In practice, equality in the above expression is never achieved since the DC bias circuitry supplies DC current as well as DC voltage, and the DC current flows through resistors which convert some of the available electrical energy into heat energy. So, this expression represents a theoretical upper limit on how much amplification ("i"."e"., gain) can be obtained from the device. We may now express the total input and output signal power as an integral over their respective power spectral density functions. If "s"("t") is the input signal as a function of time, and "S"(ω) is the signal as a function of frequency ("i"."e"., the Fourier transform of "s"("t"), or the power spectral density of the input signal), then if "G"(ω) is the gain of the amplifier as a function of frequency, the equation above can be rewritten as:

$int G\left(omega \right) S\left(omega\right) ,domega - int S\left(omega\right) ,domega le P_\left\{DC\right\}$

or,

$int \left[G\left(omega \right)-1\right] S\left(omega\right) ,domega le P_\left\{DC\right\}$

If the amplifier gain is much greater than unity over its operational bandwidth, this expression can be approximated as:

$int G\left(omega \right) S\left(omega\right) ,domega le P_\left\{DC\right\}$

If we denote the amplifier bandwidth as BW, then if both the gain and the signal spectra are constant over the bandwidth BW (at "G" and S respectively), then the equation above becomes:

$G imes BW le P_\left\{DC\right\}/S$

and we see the origin of the gain-bandwidth product limit for amplifiers. The available DC power to the amplifier can either be put to use as high signal gain over a limited bandwidth or limited gain over a wide bandwidth. We also note that for fixed DC input power, the greatest signal gains are achieved with weak input signals. To get high gains in already amplified signals (as in output stages), increased amounts of DC power must be used.

Examples

If the GBWP of an op-amp is 1 MHz, it means that the gain of the device falls to unity at 1 MHz. Hence, when the device is wired for unity gain, it will work up to 1 MHz (GBW product = gain x bandwidth, therefore if BW = 1 MHz, gain = 1) without excessively distorting the signal. The same device when wired for a gain of 10 will work only up to 100 kHz, in accordance with the GBW product formula. Further, if the maximum frequency of operation is 1 Hz, then the maximum gain that can be extracted from the device is 1 x 106.

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