Fig, 20.14 Input current waveforin t,Ct), and the areas and if during subintervals 1 and 2 respectively.

(20.32)

The charge quantttie.4 and are the area.s under the ij(f) waveform during the first and second sub-intervals, respectively. The charge ijj is given by the triangle area forrauln

The tinte (Х/Щ is the length of subinterval 1, The charge (/j is

<7i =

Ц + Р

(20.33)

(20.34)

Accttfding to Fig. 20.12, during subinterval 2 the current ([(/) can be related to the tanlt capacittir current /(-(() and the switch output current by the node equatitin

( ,(0 = i(.(ti + h. Substitution ofEq. (20.35) into Eq. (20.34) leads to

a+ [1 Ц+ [1

h=\2 cm-

l.clt

(20.35)

(20.3(3)

Both integrals in Eq. (20.36) can easily 1эе evaljated, as follows. Since the .second term involves the integral ofthe constant current this terra is

!2 -4

(20.37)

The first terra in Eq. (20.36) involves the integral tif the capacitor current over subinterval 2. Hence, this term is equal to the change in capacittw charge t)ver the second subinterval:

-5!Г

(recall that 4 = Civ in a capacitor). During the second .subinterval, the tank capacitor voltage is initially zero, and has a ftnal value of V.,. Hence, Eq. (20.38) reduces to

U+Ji

jo i,<f)rff = c(V -0]=CV

(20.39)

Substitution of Eqs. (20.37) and (20.39) into Eq. (20.36) leads to the following expressioti for q.

Equations (20.33) and (20.40) can now be inserted into Eq. (20.32), to obtain the following expression for the switch input current:

, . cv fib.

(20.41)

Substitution ofEq. (20.41) into (20.8) leads to the following expression for the switch conver-

sion ratio:

- icQj, /,7-. + tu J

(20.42)

2 CQflT .J,

Finally, the quantities a, p,and V.jcan he eliminated, using Eqs. (20.13), (20.20), (20.23). The result is

p. = fi Ay, + ir + sin-V) +j[l+VTT7f

where

Equation (20.43) is of the form

where

(20,43)

(20.44)

(20.45)

y, + 7l+sin-(/,)-l-j (i

(20.46)

Thus, the switch conversion ratio is directly controllable by variation of the switching frequency, through F. The switch conversion ratio is also a function of the applied terminal voltage V, and current /j, via 7j. The function P{{J) is sketched in Fig. 20.15. The switch conversion ratio )i. is sketched in Fig. 20.16, for various values of F and These characteristics are similar in shape to the function PiJ), and are simply scaled by the factor F. It can be seen that the conversion ratio Д is a strong function

10 t

Fig. 2вЛ5 Tlie function (У,).

S

6

4 -

2

ZCS boundary

Fig. 20.16 Charaeterlsiics of the half-wave ZCS quasi-resottant switch. j

of the current /j, via J, Tlie cliaracteristics end at J, - \ according to Eq, (20.31), the zero current switching property is lost when J> l.The characteristics also end at the maximum switching frequency limit given by Eq. (20.31). This expression can be simplified by use ofEq. (20.43), to express the limit in terms of as follows: