High Gain Feedback for Reduction of Systematic Errors

The benefit of adding high-gain feedback to many measurement systems is illustrated by considering the case of the voltage-measuring instrument whose block diagram is shown in Figure.

In this system, the unknown voltage Ei is applied to a motor of torque constant Km and the induced torque turns a pointer against the restraining action of a spring with spring constant Ks. The effect of environmental inputs on the motor and spring constants is represented by variables Dm and Ds. In the absence of environmental inputs, the displacement of the pointer X0 is given by: X0 D KmKsEi. However, in the presence of environmental inputs, both Km and Ks change, and the relationship between X0 and Ei can be affected greatly.

Therefore, it becomes difficult or impossible to calculate Ei from the measured value of X0. Consider now what happens if the system is converted into a high-gain, closed-loop one, as shown in Figure, by adding an amplifier of gain constant Ka and a feedback device with gain constant Kf. Assume also that the effect of environmental inputs on the values of Ka and Kf are represented by Da and Df. The feedback device feeds back a voltage E0 proportional to the pointer displacement X0. This is compared with the unknown voltage Ei by a comparator and the error is amplified. Writing down the equations of the system, we have:

[pmath]E0=KfXo;   Xo=(Ei-E0)KaKmKs=(Ei-KfXo)KaKmKs[/pmath]

Thus:

[pmath]EiKaKmKs=(1+KfKaKmKs)Xo[/pmath]

i.e.

[pmath]Xo=KaKmKs/(1+KfKaKmKs) Ei[/pmath]

Because Ka is very large (it is a high-gain amplifier), Kf. Ka. Km. Ks >> 1, and equation reduces to:

[pmath]Xo=Ei/Kf[/pmath]

This is a highly important result because we have reduced the relationship between X0 and Ei to one that involves only Kf. The sensitivity of the gain constants Ka, Km and Ks to the environmental inputs Da, Dm and Ds has thereby been rendered irrelevant, and we only have to be concerned with one environmental input Df. Conveniently, it is usually easy to design a feedback device that is insensitive to environmental inputs: this is much easier than trying to make a motor or spring insensitive. Thus, highgain feedback techniques are often a very effective way of reducing a measurement system’s sensitivity to environmental inputs. However, one potential problem that must be mentioned is that there is a possibility that high-gain feedback will cause instability in the system. Therefore, any application of this method must include careful stability analysis of the system.