How does amplifier feedback help?
Consider again our classical feedback amplifier in Fig. 1. We have the feedback amplifier equation:
Gcl = A / (1 + A * beta)
where Gcl is the closed loop gain of the amplifier with a feedback loop, A is the intrinsic gain of the amplifier (without the feedback loop), beta is the reciprocal of the external loop gain elements for example in the case of the inverting op-amp beta is
beta = Rin / Rf
One effect of feedback that we can calculate is the effect on gain accuracy. We can show that if we differentiate the gain vs. the intrinsic amplifier gain A, we have the equation
dGcl /dA = 1/ (1 + 2A*beta +A^2 * beta^2)
Let’s say that a particular op-amp had an intrinsic gain of 80db. This corresponds to a gain value of 10,000 ! And let’s say that the external gain of the amplifier is 10, or beta is 0.1. Then the above equation evaluates with this data to
dGcl/A = 1/ (1 + 2*10000*0.1 + 100000000 * 0.01) = 1/(1+ 2000 + 1000000) = 0.000001
We see two things from the above calculation. First the closed loop gain sensitivity to the amplifier intrinsic gain is only one part in a million. Second, we can approximate the result with a simple equation:
dGcl/A ~ 1 / ( A*beta)^2
So the larger we can make the product A*beta the less gain error sensitivity we will have.
For another example, in the case of the buffer amplifier with a beta of 1.0 and an intrinsic gain of 10,000, the gain error would only be 1 part in 100,000,000 ! In actual practice, this accuracy is not really achieved because of input offset voltage and bias or offset current errors. But the calculation shows that the great value of the operational amplifier is its insensitivity to amplifier intrinsic gain and that its gain is accurately determined by the external gain elements, e.g., Rin and Rf.
Another effect of feedback is the reduction of some of the distortion that may be generated by the amplifier as long as the distortion is inside the loop of the amplifier and its feedback path. Assume that there is some change in the output of the amplifier that is not the result of the input signal and is therefore an unwanted distortion product of some kind. We want to find out how much the feedback loop will do to correct this output variation, or what is the delta change applied to the input to correct or partially correct the variation? A way to answer this question is to find the derivative of Vin with respect to Vout using our gain equation from analysis of the circuit of Fig. 1. We have that
Vout = Vin * A / (1+A*beta)
or
Vin = Vout * (1+A*beta) / A
and then
dVin/dVout = (1+A*beta) / A
which we can re-write as
dVin = ((1/A) + beta) * dVo
For the large values of intrinsic gain A, we can approximate dVin as
dVin ~ beta * dVo = dVout / (1/beta) = (1/Gcl)* dVout
Here we see that if we have a closed loop gain Gcl of 10, our feedback correction will be about 10% of the output error. So if we had a distortion of 1.0% due to the intrinsic amplifier our net output distortion would be reduced to 0.9% (maybe enough to get our design to meet specifications.) If we have a unity gain buffer, we would have a correction of approximately 100% meaning that we should have almost zero distortion out of a unity gain buffer amplifier with the feedback loop.
Consider again our classical feedback amplifier in Fig. 1. We have the feedback amplifier equation:
Gcl = A / (1 + A * beta)
where Gcl is the closed loop gain of the amplifier with a feedback loop, A is the intrinsic gain of the amplifier (without the feedback loop), beta is the reciprocal of the external loop gain elements for example in the case of the inverting op-amp beta is
beta = Rin / Rf
One effect of feedback that we can calculate is the effect on gain accuracy. We can show that if we differentiate the gain vs. the intrinsic amplifier gain A, we have the equation
dGcl /dA = 1/ (1 + 2A*beta +A^2 * beta^2)
Let’s say that a particular op-amp had an intrinsic gain of 80db. This corresponds to a gain value of 10,000 ! And let’s say that the external gain of the amplifier is 10, or beta is 0.1. Then the above equation evaluates with this data to
dGcl/A = 1/ (1 + 2*10000*0.1 + 100000000 * 0.01) = 1/(1+ 2000 + 1000000) = 0.000001
We see two things from the above calculation. First the closed loop gain sensitivity to the amplifier intrinsic gain is only one part in a million. Second, we can approximate the result with a simple equation:
dGcl/A ~ 1 / ( A*beta)^2
So the larger we can make the product A*beta the less gain error sensitivity we will have.
For another example, in the case of the buffer amplifier with a beta of 1.0 and an intrinsic gain of 10,000, the gain error would only be 1 part in 100,000,000 ! In actual practice, this accuracy is not really achieved because of input offset voltage and bias or offset current errors. But the calculation shows that the great value of the operational amplifier is its insensitivity to amplifier intrinsic gain and that its gain is accurately determined by the external gain elements, e.g., Rin and Rf.
Another effect of feedback is the reduction of some of the distortion that may be generated by the amplifier as long as the distortion is inside the loop of the amplifier and its feedback path. Assume that there is some change in the output of the amplifier that is not the result of the input signal and is therefore an unwanted distortion product of some kind. We want to find out how much the feedback loop will do to correct this output variation, or what is the delta change applied to the input to correct or partially correct the variation? A way to answer this question is to find the derivative of Vin with respect to Vout using our gain equation from analysis of the circuit of Fig. 1. We have that
Vout = Vin * A / (1+A*beta)
or
Vin = Vout * (1+A*beta) / A
and then
dVin/dVout = (1+A*beta) / A
which we can re-write as
dVin = ((1/A) + beta) * dVo
For the large values of intrinsic gain A, we can approximate dVin as
dVin ~ beta * dVo = dVout / (1/beta) = (1/Gcl)* dVout
Here we see that if we have a closed loop gain Gcl of 10, our feedback correction will be about 10% of the output error. So if we had a distortion of 1.0% due to the intrinsic amplifier our net output distortion would be reduced to 0.9% (maybe enough to get our design to meet specifications.) If we have a unity gain buffer, we would have a correction of approximately 100% meaning that we should have almost zero distortion out of a unity gain buffer amplifier with the feedback loop.
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