By Robert Pease
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There are many deﬁnitions of stability in the literature, but we’ll consider BIBO stability. In other words, we’ll consider the stability problem given that we’ll only excite our system with bounded inputs. The system is BIBO stable if bounded inputs generate bounded outputs, a condition that is met if all poles are in the left-half plane (Figure 1-6). w ww. c om Review of Feedback Systems 7 jv Poles in left-half plane for BIBO stability s Figure 1-6: Closed-loop pole locations in the left-half plane for bounded input, bounded output (BIBO) stability.
Com 6 Chapter 1 a(s) 1 f log (v) Figure 1-5: Plot for estimating closed-loop transfer function graphically. The curve a(s) depicts the frequency dependence of the forward-path gain. The line 1/f is the inverse of the feedback gain, shown here for resistive feedback. 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).
0 GHz Frequency Figure 1-36: Frequency response results for photodiode ampliﬁer with Cf :2, 4, 6, 8, 10 pF, with Rf ؍1k and Cp ؍60 pF. VCC IL RL Vin VL ϩ a(s) Ϫ va Vsense Rsense Figure 1-37: MOSFET current source. The voltage Vsense senses MOSFET drain current. , not oscillating). A small-signal model is shown in Figure 1-38. Following are the parameters: h Time constant of op-amp high-frequency pole rout Output resistance of op-amp Cgs MOSFET gate-source capacitance Cgd MOSFET gate-drain capacitance gm MOSFET transconductance RL Load resistance The transfer function of the MOSFET source follower is estimated using the method of open-circuit time constants.