Fig. C.S CLirronl injection at the electrical port, by addition of independent CLiricnt source i(.v).

v(s)-HsyAs)

(C.9)

To express the new transfer function G(.r) in Eq. (C.S) in terms ofthe original transfer function Gj ;(s) of Eq. (C.7), we use current injection at tire port, as illustrated in Fig. C.5. There are now two independent inputs: the input и(.г) and the independent current source /(.!). The dependent quantities yis} and \is} can be expressed as functions of these independent inputs using the principle of superposition:

where

y{s)=G,(.s)u(s) + Gs)i() v(s)G,{s)u(s)-Zds)i(s)

G,.js) =

4(s)

(CIO) (C.ll)

(C.12)

ttrt-O

G/J) =

i(.s)

(C.13)

Zs).

(C.14)

(CI 5)

are the transfer fmictions from the independent inputs to the respective dependent quantities y(s) and v(s).

The transfer function Gis) can be found by elimination of ф) and /(,!) from the system of equations (C.9) to (C. 11), and solution for y(s) as a function of u{s). The result is

(С. 16)

This intermediate result expresses the new transfer function G(s) as a function of the original transfer function Cijjfjis) and the extra element Z(.i), as well as the quantities Zj(i), and G(s).

Equation (C.]4) gives a direct way to find tiiequatitity 7.{s). Z(s) is tiie driving-point impedatice at tlie port, wlien tlie input u(s) is set to zero. This quantity can be found eitlier by conventional circuit analysis or simulation, or by laboratory measurement.

Aldiough Crfs) and G-(s) could also be determined from the definitions (C. 13) and (C. 15), it is preferable to eliminate these quantities, and instead express G{s) as a function of the impedances at the given port. This can be accomplished via the following thought experiment. In the pre.sence of the input u(s), we adjust the independent current source i{s) in the special way that causes the output y(.!) to be nulled to zero. The impedance Zs) is defined as the ratio of v(s) to i{s) under these null conditions:

(С. 17)

The value of i(s) that achieves the null condition y(.v) jj 0 can be ftHmd by setting y(s) = 0 in Eq. (C.IO), as follows:

Gm(s)u(.s) + G,(s)Ks)

Hence, the output y{s) is nulled when the inputs u{s) and iis) are related as follows: Under this null condititm, the voltage v(s) is given by

null M

(CIS)

(Ci9)

GMG,U)

(C.20)

which follows from Eqs. (C.ll) and (C.19). Substitution of Eq. (C.17) into Eq. (C.20) yields

Gf.s)G,is)

Hence,

Solution for thequantity GJsyGs) yields

Gfs)G;{s) = (z (,o - ги))а,ф)

(C.2])

(C.22)

(C.23)

Thus, the unltnown quantities Gfs) and G,(i) can be related to Z;(i) and Zj(s), which are properties of the port at which the new impedance Z(s) will be connected, and to the original transfer function GjJs). The final step is to substitute Eq. (C.23) into Eq. (C.lfi), leading to

This expression can be simplified as follows:

1 +

(C,24)

(C.26)

This is the desireil restilt. It states how the transfer function G{s) is modified by aildition of the extra element Z(s). The right-raost terra in Eq. (C.26) is calleil the correction factor; this term gives a quantitative measure of the ehange in G{s) arising from the introduction of Z(.r).

Derivation of the dual resuh, Eq. (C.3), foiiows similar steps.

DISCUSSION

The general form of the extra element theorem makes it useful for designing a system such that tinwanted circuit element.s do not degrade the de.sirable .system performance already obtained. Forexample, suppose that we already know some transfer function or similar quantity G(i), under simplified or ideal conditions, and have designed the system such that this quantity meets specifications. We can then use the extra element theorem to answer the following questions:

What is the effect of a parasitic element Z(,!) that was not included in the original analysis?

What happens if we later decide to add some additional components having impedance Z{s) to the system?

Can we establish some conditions on Z(.!) that ensure that Gis) is not substantially changed? A common application ofthe extra element theorem is the determination of conditions on the extra element that guarantee that the transfer function G{s) is not significantly altered. According to Eqs. (C.2) and (C.26), this will occur when the correction factor is approximately equal to unity. The conditions are:

\z{m\-.

-2,vCj )

(C.27)

This gives a formal way to show when an impedance can be ignored: one can plot the impedances II f/JlO) II and 11 ZJj(3)} II, and compare tbe results with a plot of il Z(ftD) [, The impediuice Z(.9) cim be igntffed over the range of frequencies where the inequalities (C.27) are satisfied.

For the dual case in which the new impedance is inserted where there was previously a short circuit, Eq. (C.3), the inequalities are reversed: