Inverting Amplifier Feedback

Fig. 1 shows an ideal feedback amplifier that represents the inverting amplifier circuit configuration of Fig. 2. The inverting amplifier has similar feedback calculations to the non-inverting amplifier. As a matter of fact the loop gains are identical (A*beta) and this is the quantity that is important to stability.

To calculate the loop gain we need to adjust the amplifier gain with the factor R2/(R1+R2). Ref. Intersil Application Note AN9415.3. Referring to Fig. 1, we calculate A as

A = Ai * R2 / (R1 + R2)

where Ai is the intrinsic gain of the op-amp, R1 is the inverting input resistance or impedance, and R2 is the feedback resistance or impedance. The feedback beta is R1/ R2 for the inverting amplifier so our loop gain calculation is

A*beta = Ai * (R2 / (R1+R2)) * (R1/R2)

Canceling out R2 we have

A*beta = Ai * R1/(R1+R2)

Now we have a similar equation for the inverting amplifier as:



G = Ai / (1 +A*beta)

The stability of the amplifier is governed again by the denominator 1+A*beta. When the denominator is 0, i.e., when A*beta = -1, then we have an unstable amplifier, if the signal is greater than unity (within the bandwidth) of the amplifier. Note that in this case the feedback is shifted 180 degrees in phase so that the feedback is now positive instead of negative. This is why it is necessary to “roll-off” loop gain to less than unity before the frequency is reached where the phase has naturally shifted to the positive feedback condition (as will usually happen if there is more than one pole in the amplifier response.)

Also, when there is more than one pole in the frequency response, the roll-off of loop gain to the unity gain threshold (also called the "cross-over frequency") or less, should begin before the second pole of the response of the amplifier is reached. In the Bode plot of the loop gain vs. frequency this is evidenced by a slope of -20db per decade of frequency at the unity gain cross-over frequency. If this is achieved the amplifier is usually stable.

The closed loop gain for the inverting amplifier is different from the non-inverting amplifier as we noted before. The closed loop gain is

Gcl = -R2/R1