User:CraigNGC/subpage GBW

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 applied to the amplifier through its bias circuitry. Stated mathematically, if Pout is the total output signal power from the amplifier, Pin is the total signal power input to the amplifier, and PDC is the total DC power supplied to the amplifier, then

Pout - Pin ≤ PDC

and 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 the electrical energy to 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:

$$ \int G(\omega ) S(\omega) \,dx - \int S(\omega)  \,dx \le  P_{DC} $$

or,

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

If the amplifier gain is much greater than unity, then, this becomes

$$ \int G(\omega ) S(\omega) \,dx   \le  P_{DC} $$

If we denote the amplifier bandwidth as BW, then if both the gain and the signal spectrum are constant over the bandwidth BW (at G and 1 respectively, then the equation above becomes:

$$ G \times BW   \le  P_{DC} $$

and we see the origin of the gain-bandwidth product limit for amplifiers.