By Robert Pease
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Extra resources for Analog Circuits (World Class Designs)
The thick line indicates our estimate for closed-loop transfer function. For a(s)f Ͼ Ͼ1, the closed-loop gain is approximately 1/f. For a(s)f Ͻ Ͻ1, the closed-loop gain is approximately a(s). a 1 ϩ af ⎛ 1 (1 ϩ af ) Ϫ af ϭ = ⎜⎜ 2 ⎜⎝ 1 ϩ af (1 ϩ af ) da ⎛⎜ 1 ⎞⎟ ⎟ ϭ ⎜ a ⎜⎝ 1 ϩ af ⎟⎟⎠ Aϭ dA da ) dA A ⎞⎟ ⎛ 1 ⎟⎟ ⎜⎜ ⎠⎟ ⎜⎝ 1 ϩ af ⎟⎟⎞ ϭ A ⎛⎜⎜ 1 ⎟⎟ a ⎜ 1 + af ⎝ ⎠ ⎞⎟ ⎟⎟ ⎟⎠ [1-6] This result means that if af Ͼ Ͼ 1, then the fractional change in closed-loop gain (dA/A) is much smaller than the fractional change in forward-path gain (da/a).
This is very useful, since in some cases the feedback function f can be implemented with a simple resistive divider . . which can be cheap and accurate. T. T. by doing a thought experiment: Cut the feedback loop in one place, inject a signal, and ﬁnd out what returns where you cut. The gain around the loop is the loop transmission. T . [1-5] Note that if the loop transmission is large at frequencies of interest, then the output due to the disturbance will be small. The term (1 ϩaf ) is called the desensitivity of the system.
M. ( jv) 0 Ϫ45 Ϫ90 Ϫ135 Ϫ180 Ϫ225 Ϫ270 fm log (v) vc Figure 1-10: Plot of gain margin and phase margin. Relationship Between Damping Ratio and Phase Margin The damping ratio and phase margin are directly related. For a second-order system, a low phase margin in general implies a low damping ratio. 6 corresponds to a phase margin of 60°. The actual relationship over the range of damping ratios 0 Ͻ Ͻ 2 is shown in Figure 1-11. 8 2 Damping ratio Figure 1-11: Relationship between phase margin and damping ratio.
Analog Circuits (World Class Designs) by Robert Pease