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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). We can make a couple of approximations in the limit of large and small loop transmission.

Vertical force, vertical position, and magnet current are given as the sum of a DC component and an incremental component: fZ ϭ FZ ϩ fz z ϭ Zo ϩ z iM ϭ I M ϩ im [1-39] Putting this into the force equation results in: 2 Z Ϫ CI 2 z Ϫ 2 CI Z i Ϫ CI M o M M o m fZ [1-40] where second-order and higher terms have been neglected. At equilibrium, there is a resultant magnetic force that balances the force of gravity: 2 Z FZ ϭ Mg ϭ ϪCI M o [1-41] Newton’s law applied to the magnet results in: M d2z 2 z Ϫ 2CI Z i ϭ fZ Ϫ Mg ϭ ϪCI M M o dt 2 [1-42] resulting in: M d2z 2 Mg im ϩz ϭ 2 2 km I M CI M dt [1-43] Using the spring constant k and converting the equation to the frequency domain results in: ⎞ ⎛M 2Mg ⎜⎜ 2 ϩ 1⎟ ⎟⎟ z(s) ϭ im (s) s ⎟⎠ ⎜⎜⎝ km km I M [1-44] resulting in the transfer function between magnet position and magnet control current: z( s) ϭ im (s) 2Mg ⎛M ⎞ s2 ϩ 1⎟⎟⎟ km I m ⎜⎜⎜ ⎟⎠ ⎜⎝ k m w ww.

Stability So far, we haven’t discussed the issue regarding the stability of closed-loop systems. 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.