Transferfunction

Transfer function

Gis)

Open-circuit

Fig. C.l How an added element changes a transfer function Gis); (a) (irlglnal ciinditions, before addition of the flew element; (b) addition uf element having impedanee Z(jf).

Transfer function

Transfer function

Linear circuit Input Output Port

Short-circuit

Fig. C.2 The dual form of the Extra Element Theorem, iir which the extra eleutetit replaces a short circuit: (a) original conditions, (b) additioti of element having impedance Z(.v),

ад Z[s)

Z(s) j

(C,2)

The right-hand side terms involving Z{s) account for the influence of Z{s) on G(s), and are known as the correction factor.

The Extra Element Theorem also apphes to the dual form illustrated in Fig. C.l. In this form, the transfer functitm is initially known under the conditions that the port is short-circuited. In Fig. C.2(b), the short-circuit is replaced by the impedance Z(.t). In this case, the addition of the impedance Z(.!) causes the transfer function to become

vsl

z(.v) - 0

ZM j

(C,3)

Short-circuit

Linear circuit

input Output

Port

+ vis) ~

Linear circuit Input Ouiput

Port

Fig. С J Determination of the quantities Zt) and Zfj): (a) Z,(i) is the Thevenin-equivalent iinpedanue at the poit, and is measured with the input Vj {s) set to zero; (b) Zi) is the impedance seen at the port under the condition that the output is nulled.

The Zy{j) and Z(j) terms in Eqs. (C.2) and (C.3) are identical. By equating the G(s) expressions of Eqs. (C.2) and (C.3), one can show that

HA - Zj)(i)

(C,4)

This is known as the reciprocity relationship.

The quantities ZJ,s) and Zq(,s) can be found by measuring impediinces at the port. The term Zg(s) is the Thevenin equivalent impedance seen looking into the port, also known as the driving-point impedance. As illustrated in Fig. C.3(a), this impedance is found by setting the independent source vO?) to zero, and then measuring the impedance between the terminals of the port:

Zy(j) =

(C.5)

Thus, Z{.v) is the impedance between the port terminals when the input v(i) is set to zero.

Determination of the impedance Zs) is illustrated in Fig. C.3(b). The term 2(s) is found under the conditions that the output v ,(.t) is nulled to zero. A current source s{s) is connected to the terminals ofthe port. In the presence of the input signal Vf (.r), the current i{s) is adjusted so that the output

nulled to zero. Under these conditions, the quantity Zfs) is given by

(.CA)

Note that mtlling the output is not the same as .horlin tlie output. If one simply shorted the output, then a euiient would flow through the short, which would induce voltage drops and currents in other elements of fhe network. These voltage drops and currents ;ue not present when the output is nulled. The null condition of Fig. C.3(b) does not employ any connections to the output of the circuit. Rather, the null condition employs the adjustment of the independent .sources ) t<) and i(s) in a special way that causes the output V ,(.t) to be zero. By superposition, can be expressed as a linear combinatitm of У,- -) and

i(j); therefore, for a given v-,Jis), it is always possible to choose an i(i) that will cause v (.v) to be zero. Under these null conditions, Zfs) is measured as the ratio of vj) to i(s). In practice, the circuit analysis to find Zfjs) is simpler than analysis of Zp(s), because the null condition cau.ses many of the signals within the circuit to be zero. Several examples ate given in Section C.4.

The input and output quantities need not be voltages, but could also be currents or other signals that can be set or nulled to zero. The next section contains a derivation of the Extra Element Tlieoreni with a general input u[s) and output y(s).

C.2 DERIVATION

Figure C.4(a) illustrates a general linear system having an input ii(.s) and an output y(.0- In addition, the system contains an electrical port having voltage v(.t) and current i(.i), widi the polarities illustiated. Initially, the port is open-circuited; ((.t) = 0. Tlie transfer function of this system, with the port open-circuited, is

(C.7)

The objective of the extra element theorem is to determine the new transfer function G(s) that is obtained when an impedance Z{s) is connected to the pott:

<C,8)

The situation is illustrated in Fig. C.4(b). It can be seen that the conditions at the port are now given by

Inptit

Linear network

Port

Open-circuit

Oittput

Ф) -j

Fig. C.4 Modifioatiou of a hnear network by addinon of an extra element: (a) origiitEil .system, (b) modified system, with impedance Z(,s) connected at an eltscUical port.