is the itnpedance seen looking into the power inpnt port of the eonverter wlien is set to zero, and

(C.46)

is the impedance seen looking into the power input port ofthe eonverter when the converteroutput С is nulled. The null condition is achieved by injecting a test current source Г., at the converter input port, in the presence of г/ variations, and adjusting such that \ is nulled. Derivation of expressions forZ,(.!) and Zp(,i) for a buck converter example is described in Section 11X3.1.

According to Eq. (C.28), the input filter does not significantly affect G,{s) provided that

; I zM) \

(C.47}

These inequalities can provide an effective set of criteria for designing the input filter Bode pk)ts of 11 ZfJJii)) II and II Zsijiii) II are constructed, and then the filter element values are chosen to satisfy (C.47). Several examples of this procedure arc explained in Chapter 10.

C.4.4 Dependence of Tran.si.stor Current un Load in a Resunant Inverter

The conduction loss caused by circulating tank ctirrents is a major problem in resonant converter design. These currents are independent of, or only weakly dependent on, the load current, and lead to poor efficiency at light load. The origin of this probletn is the weak dependence ofthe tank network input impedance on the load resistance. Forexample, Fig. C.19 illustrates the model ofthe ac portion of a resonant inverter, derived using the sinusoidal approximation of Section 19,1. The resonant network contains the tank inductors and capacitors of the converter, and the load is the resistance R. The current ijf) flowing in the effective sinusoidal source is equal to the switch current. This model predicts that the switch current !,(s) is equal to ij,](.?)/Z,(j), where Zj(j) is the input impedance ofthe resonant tank network. If we want the switch current to track the load current, then at the switching frequency Z. should be dominated by, or at least strongly influenced by, the load resistance R. Unfortunately, this is often not consistent with other requirements, in which Z is dominated by the impedances ofthe tank elements. To design a resonant converter that exhibits good properties, the engineer must develop physical insight into how the load resistance Я affects the tank input impedance and otitput voltage.

Transfer funclion

Kg. c.19 Resonant lEivetter model.

Effeclive .vinu.soidat source

.,(0

Resonant network.

Purely reactive

Effective resistive load R

Inpul

Extra element R

Fig. C.20 ApplicBiioti of the Extra Element Theorem lo the system ot Fig. C. 19, to expose the dependence of Z/i) on Д.

To expose the dependence of Zj(.l) on the load resi.stance R, we can treat R as the extra element as in Fig. C.20. The inptit impedance Z((.v) 1.4 viewed as the transferfunction from the current i, to the voltage У.,; in thi.s sense, i is the input and ij is the otitput. Equations (C.2) and (С.Э) then imply that Zj(i) can be expressed as follows:

l. Z(,v)

1 +-

Z (.)

Z,j(.v)

(C.4S)

(C.49)

Here, the impedance Z/fjIs) is

i.e., the input impedance Z-(s} when the load terminals ate shorted. Likewise, the impedance Z.J,s} is

Z, (J)= Z,(.t)L

which is the input itnpedance Z{s) when the load is disconnected (open circuited),

Determination of Zj,(.?) aitd Z(j(,?) is illustrated in Fig. C.21. The quantity Zi) is found by nulling the output vj to zero, and then solving for v(j)(.s). Thequantity 7(л) coincides with the conventional output impedance Z (.0 illustrated in Fig. C.19. In Fig. C.21(a), the act of nulling v, iseqiiivalent to shorting the source v, of Fig. C.19. In Section 19.4, thequantity Z,(j) is denoted Z,(j), because il coincides with the converter output impedance with the switch network shorted.

The quantity Z,(i) is found by setting the input !\. to zero, and then solving for v(syi{s). The quantity Z(,?) coincides with the output impedance Z/j) illustrated in Fig. C.19, under the conditions that the .source is open-circuited. In Section 19.4, thequantity Z(,t) is denoted Z(.v), because it coincides with the converter output impedance with the switch network open-circuited. The reciprocity relationship, Eq. (C.4), becomes

Zjjs)

(C.51)

The above results are used in Section 19.4 to expose how conduction losses and the zero-voltage switching boundary depend on the loading ofa resonant converter.

Resonant netH-ork

Fig. C.21 Derermination of the quantities Zf/.s) and Zp(s-) for tiie neiwork of Fig. C,20: (a) finding Zis), (b) finding ZpW.

References

[1] R. D. Middlebrook, Nnll Double Injection and the Extra Element Theorem, IEEE Transactions on Education, vol. 32. No. 3. Aug. 167-ISO.

[2] R. D. MiDDLEBROOK, Thc Two Extra Element Theorem, IEEE Frontiers in Education Conference Pre-

ceedings, Sept. 1991, pp. 702-708.

[3] R. D, Middlebrook, V. Vorperi.as, and J. Lindal, The N Extra Element Thcoiciu, IEEE Transac lions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 45, No. 9, Sept. 199S, pp. 919-935.