Input offset voltage; proof of constant GBP

Input offset voltage

Input offset voltage (Vio) is the dc differential voltage that must be applied between the inverting and non-inverting inputs of an op-amp to make its output voltage zero. It is caused by mismatch between the input transistors. With feedback it appears at the output as Voo = (1 + Rf/R1)·Vio. For the 741, Vio is 1 mV typical and 6 mV maximum.

Proof that GBP is constant

The open-loop gain of an internally compensated op-amp (e.g. 741) has a single dominant pole at fo:

A(f) = A0 / (1 + j f/fo)
A0 = dc open-loop gain,  fo = open-loop break frequency

Apply negative feedback with feedback factor β (for a non-inverting amplifier β = R1/(R1 + Rf)):

Af(f) = A(f) / (1 + β A(f))
      = [A0/(1 + jf/fo)] / [1 + βA0/(1 + jf/fo)]
      = A0 / (1 + βA0 + jf/fo)
      = [A0/(1 + βA0)] / [1 + j f / (fo(1 + βA0))]

This is again a single-pole response, with

Closed-loop dc gain : Af0 = A0 / (1 + βA0)
Closed-loop BW      : fF  = fo (1 + βA0)
Product             : Af0 × fF = A0 × fo = constant

The factor (1 + βA0) that divides the gain multiplies the bandwidth, so gain × bandwidth = A0·fo = ft, the unity-gain bandwidth, which is fixed for a given op-amp. For the 741, A0 ≈ 2 × 10⁵ and fo ≈ 5 Hz, so GBP ≈ 1 MHz: with a gain of 10 the bandwidth is 100 kHz, with a gain of 100 it is 10 kHz.

This shows that for an op-amp, gain can be traded for bandwidth, but their product cannot be increased.