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Appendix С

Middiebrooks Extra Element Theorem

The Exmi ESfmeist Thenreiii of R. D. Middlebrook [1-3] shows how a transfer iunetion is changed by the addition of an impedance to the network. The theorem allows one to determine the effects of this extra element on any transfer ftmction of interest, without solving the system ail t)ver again. The Extra Element Theorem is a powerful technique of design-oriented analysis. It leads to intpedance inequalities which guarantee that an element does not substantially alter a transfer function. The Extra Element Theorem is employed in Chapter Ш, where it leads to a relatively simple methodology for designing input filters that do not degrade the \oop gains of switching regulators. It is also employed in Section 19.4, to determine how the load resistance affects the properties of a resonant inverter. In this appendix. Middle-brooks Extra Element Theorem is derived, based on the principle of superposition. Its application is illustrated via examples.

C.1 BASIC RKSULT

Consider the linear circuit of Fig, C, 1(a). This network contains an input V; (i) and an output v, ,j(j). In addition, il contains a port whose terminals are open-circuited. It is assumed that the transfer function from V; (i) to IoufWis known, and is givenby

The Extra Element Theorem tells us how the transfer function G{s) is modified when an impedance Z(s) is connected between the terminals at the port, as in Fig. C.l(b). The result is