Editing Step response (section)

====Results====
[[Image:Step response for two-pole feedback amplifier.PNG|thumbnail|350px|Figure 3: Step-response of a linear two-pole feedback amplifier; time is in units of&nbsp;1/''ρ'', that is, in terms of the time constants of ''A''<sub>OL</sub>; curves are plotted for three values of ''mu''&nbsp;=&nbsp;''μ'', which is controlled by&nbsp;''β''.]]
Figure 3 shows the time response to a unit step input for three values of the parameter μ. It can be seen that the frequency of oscillation increases with μ, but the oscillations are contained between the two asymptotes set by the exponentials [&nbsp;1&nbsp;−&nbsp;exp(−''ρt'')&nbsp;] and [&nbsp;1&nbsp;+&nbsp;exp(−ρt)&nbsp;]. These asymptotes are determined by ρ and therefore by the time constants of the open-loop amplifier, independent of feedback.

The phenomenon of oscillation about the final value is called '''[[ringing (signal)|ringing]]'''. The '''[[overshoot (signal)|overshoot]]''' is the maximum swing above final value, and clearly increases with μ. Likewise, the '''undershoot''' is the minimum swing below final value, again increasing with μ. The '''[[settling time]]''' is the time for departures from final value to sink below some specified level, say 10% of final value.

The dependence of settling time upon μ is not obvious, and the approximation of a two-pole system probably is not accurate enough to make any real-world conclusions about feedback dependence of settling time. However, the asymptotes [&nbsp;1&nbsp;−&nbsp;exp(−''ρt'')&nbsp;] and [&nbsp;1&nbsp;+&nbsp;exp&nbsp;(−''ρt'')&nbsp;] clearly impact settling time, and they are controlled by the time constants of the open-loop amplifier, particularly the shorter of the two time constants. That suggests that a specification on settling time must be met by appropriate design of the open-loop amplifier.

The two major conclusions from this analysis are: 
#Feedback controls the amplitude of oscillation about final value for a given open-loop amplifier and given values of open-loop time constants,  τ<sub>1</sub> and τ<sub>2</sub>.
#The open-loop amplifier decides settling time. It sets the time scale of Figure 3, and the faster the open-loop amplifier, the faster this time scale.

As an aside, it may be noted that real-world departures from this linear two-pole model occur due to two major complications: first, real amplifiers have more than two poles, as well as zeros; and second, real amplifiers are nonlinear, so their step response changes with signal amplitude.
[[Image:Overshoot control.PNG|thumbnail|300px|Figure 4: Step response for three values of&nbsp;α. Top: α &nbsp;=&nbsp;4; Center: α&nbsp;=&nbsp;2; Bottom: α&nbsp;=&nbsp;0.5. As α is reduced the pole separation reduces, and the overshoot increases.]]