с Ф R k viO v/t) Q i с ф я i v(r)

(a) (b)

Fig. 7.8 Buct-boo.st converter circuit; (a) when the switch is in position 1, (h) when the switch is itr posititMi 2.

(7.7)

(7.8)

Hence, during the first subinterval, the inductor current j(/) and the capacitor voltage ifr) change with the essentially constant slopes given by Eqs. (7.7) and (7.8). With the switch in position 2, the circuit of Fig. 7.8(b) is obtained. Its inductor voltage and capacitor current are:

(,) = C = -i(/)- Use ofthe small-ripple approximation, to replace ((r) and v(f) with their averaged values, yields

,(r) = C.-(i(r)),-

t))r.

(7.У) (7.10)

(7.11) (7.12)

During the second subinterval, the inductor current and capacitor voltage change with the essentially constant slopes given by Eqs. (7.11) and (7.12).

7.2.1 Averaging the Inductor Waielorms

The inductor voltage and current waveforms are sketched in Fig. 7.9. The low-frequency average of the inductor voltage is found by evaluation of Eq. (7.3)-the inductor voltage during the first and second subintervals, given by Eqs. (7.7) and (7.11), are averaged:

(v,(f))=J] \,(T)dT=rf(o (vfi)) + di,) (v(o), o.m

where d(j) = -(,1). The right-hand side [)f Eq. (7.13) contains no switching harmonics, and models

..........Ж- T /

<<0)

( <>)r

Fif;. 7.9 Buck-boost converter witveforms: (a) inductor voitage, (b) inductor currenl.

only the low-frequency components of the indnctor voltage waveform. Insertion of this equation into Eq. (7.2) leads to

diiit)).,.

(7.14)

This equation describes how the low-frequency components of the inductor current vary with time.

7,2,2 DisciLSsiun of the Averaging Approximation

In steady-state, die actual inductor current waveform i(t) is periodic with period equal to the switching period T,. i(t + Г,) = i{t). During transients, there is a net change in ((f) over one switching period. This net change in inductor current is correctly predicted by use of the average inductor voltage. We can show that this is true, based on the inductor equation

Divide by L, and integrate both sides from t to (-i- T:

Cms:

Viii)dT:

(7.15)

(7.1щ

The left-hand side of Eq. (7.16) is iit + - iit), while the right-hand side can be expressed in terms of the definition nf (i;(OX;. Eq. (7.3), by multiplying and dividing by obtain

H, + T,)-iit)i-T,{v,it})j.

(7.17)

The left-hand side of Eq. (7.17) is the net change in inductor current over one complete switching period. Equation (7.17) states that this change is exactly equal to the switching period multiplied by the aver-

age slope {v(t)y./L.

Equation (7.17) ean be rearranged to obtain

i(/ + r)-i(f)

Let us now find the derivative of {i(t)\j d{i{i))

J-. d di

i(< + T,) - iji)

~~T,-

Substitution ofEq. (7.19) into (7.18) leads to

(7.18)

(7.19)

(7.20)

which c[)iucides with Eq. (7.2).

It us next compute how the inductor current changes over one switching period in nnr buck-boo.st example. The inductor current waveform is sketched in Fig. 7.9(b). Assume that the inductor current begins at some arbitrary value i(0). During the first subinterval, the inductor current changes with the essentially constant value given by Eq. (7.7). The value at the end ofthe first subinterval is

iidT,) =

!(0)

(7.21)

(final value) = (inidal value) + (length of interval) (average slope)

During the secotid subinterval. the inductor curretit chatiges with the essentially cotistatit value given by Eq. (7.11). Hence, the value at the end ofthe .second subinterval is

(7,22)

(final value) = (mitial value) + (length of interval) (average .sk)pe) By substitution of Eq. (7.21) intt) Eq. (7.22), we can express i(TJ in terms of ((0),

;(T;) = i(0) +J

i(f){v,(0)+0(v(t))r

(7.23)

Equations (7.21) to (7.23) are illustrated in Fig. 7.10, Equation (7.23) expresses the final value i(Tj directiy in terms of / (0). without the intermediate step of calculating r(D7). This equation can be iiiter-p-eted in the same manner as Eqs, (7.21) and (7,22): the final value i(TJ is equal to the initial value ((0), plus the letigth ofthe interval T,. multiplied by the average slope (i((r)).yL. But tiote that the interval length is chosen to coincide with the switching period, such that the switching ripple is effectively