Closed-loop Gain and Sensitivity

For example, if a 1% change in amplifier gain produces approximately a 1% change in overall gain, the sensitivity is 1. If the overall gain changes by only about 0.1%, the sensitivity is 0.1. To compare changes in quantities of different magnitudes, use the change relative to each original value rather than the change alone.

The relative change in $K_a$ is $\Delta K_a/K_a$, and that in $T$ is $\Delta T/T$. For very small changes, the normalized sensitivity relating these two ratios is

$$S_{K_a}^T=\frac{K_a}{T}\frac{dT}{dK_a}$$

. The derivative calculated below shows how much the overall gain responds to a small change in amplifier gain.

Holding $\beta$ fixed and differentiating the $T$ just obtained, the quotient rule gives

$$\frac{dT}{dK_a}=\frac{(1+\beta K_a)-\beta K_a}{(1+\beta K_a)^2}=\frac1{(1+\beta K_a)^2}.$$

Multiplying this by the previously obtained $K_a/T=1+\beta K_a$ gives

$$S_{K_a}^T=\frac{K_a}{T}\frac{dT}{dK_a}=\frac1{1+\beta K_a}.$$

The full overall-gain expression is used to calculate normalized sensitivity. As $\beta K_a$ increases, this ratio decreases, so the overall gain changes less for the same amplifier-gain change.

The inverting amplifier from the lecture is represented by $T=-K_a/(1+\beta K_a)$. Here, both $T$ and its derivative have negative signs, so normalized sensitivity is the same as above.

This amplifier model omits dynamics and noise. In a real controller, when increasing the gain, also check oscillation and measurement-noise effects.

Let the value before the change be $K_0$ and the corresponding overall gain be $T_0=T(K_0)$. As with the linear approximation in Week 3, estimating the change from the slope at the reference point gives

$$\Delta T\approx T'(K_0)\Delta K_a.$$

Divide both sides by $T_0$, then multiply and divide by $K_0$ to obtain the sensitivity found earlier.

$$\frac{\Delta T}{T_0}\approx\left[\frac{K_0}{T_0}T'(K_0)\right]\frac{\Delta K_a}{K_0}=S_0\frac{\Delta K_a}{K_0}.$$

For a +10% change, therefore, use $S_0\times0.1$. This estimate uses the slope at the reference point. For the exact value, insert the new gain into the original $T(K_a)$ expression to obtain $T_1$, then calculate $(T_1-T_0)/T_0$. Both methods divide by the unchanged $T_0$. Multiply by 100 only at the end when expressing the result as a percentage.

In this exercise, compare relative changes of 1%, 10%, and 20%. As the change grows, check how far an estimate using only the slope at the baseline point deviates from the actual change.

Changing β also changes the target gain. State whether the experiment compares the same target or different feedback strengths.