0= ax + BU V = CX + EU

where the averaged matrices are

A = DA, +DA,

С = 0С, + яс2

The eqtiilibrium dc components are

X = equilibrium (dc) stiile vector

U = e4uilibrium(di;) input ve (7.94)

Y = equili brium (dc) output vettor D = equilibrium (dc) duty cycle

Quantitiesdefined in Eq. (7.94) represent the equilibrium values ofthe averaged vectors. Equation (7.92) can be solved to find the equilibrium state and output vectors:

7* ° ,

The state equations of the small-signal ac model are

к = as(f) + B0(/) + (a , - л i) x + [b. - b,j и j <?{0 y{r) = cm + E0{/) + J(C, - Cj) x + (e. - Ej) uj j(0

(7.96)

The quantities u(t), J(f), and c/(r) in Eq. (7.96) are small ac variations about the equilibrium solution,

or quiescent operating point, defined by Eqs. (7.92) to (7.95).

So if we can write the converter state equations, Eqs. (7.90) and (7.91), then we can always find the averaged dc and small-signal ac models, by eva.luation of Eqs. (7.92) to (7.96).

7.3.3 Dicussiun of the State-Space Averaging Result

As in Sections 7.1 and 7.2, the low-frequency components ofthe inductor currents and capacitt>r voltages are inodeled by averaging over an interval of length T. Hence, we can define the average ofthe state vector as

that describes the converter in equilihrium is

a v d¥1

(7.92)

The Inw-Frcquency components of the input and output vectors are modeled in a similar manner. By averaging the inductor voltages and capacitor currents, one then obtains the following low-frequency state equation:

(7.98)

This result is equivalent to Eq. (7.2).

For example, let us consider how the elements of the state vector x(r) change over one switching period. During the first siihinterval, with the switches in position 1, the converter state equations are given hy Eq. (7.90). Therefore, the elements of x(/) change with the slopes K (AX(r) + BjU(r)), ff we make the small ripple approximation, that x(f) and u(f) do not change much over one switching period, then the slopes are essentially constant and are approximately equal to

A.{:t(;)) + B,(uC0),

(7,99)

This assumption coincides with the requirements for small switching ripple in all eletnents nf x(r) and that variations in u(f) be slow compared to the switching frequency. If we assume that the state vector is initially equal to x(0), then we can write

(7.100)

final inldal interval value value length

ilope

Similar arguments apply during the second subinterval. With the switch in position 2, the state equations are given by Eq. (7.91). With the assumption of small ripple during this subinterval, the state vector now changes with slope

Aj{x(()), + B, {u[t)) The state vector at the end of the switching period is

x(7-,) = x(rfr,) + {d%) K- [Aj (x(0),-bBj{u(0),

final initial interval value value length

Slope

Substitution of Eq. (7.100) into Eq. (7.102) allows us to determine х(7) in terms of x(t)):

х(7-,) = х(0) + 7Д-

Л, {xO)) + B, {ti(/))j JT.K- A, {x(0),+ Bj {u(())

(7.101)

(7.102)

(7.103)

Upon collecting terms, one obtains

73 Siaie-Space Агегащ

xCO)

(rfA, + d-A i) (x)+ (Л, + В2) (u)

Fig. 7,29 ilovi an element of the state vector, and its average, evolve over one switching period.

C,(x(0), + E,{ (/)),

0 л; г, r

Fig. 7.JO Averaging an element of the output vector y(r).

Ti[TJ = xCO) + 7:,K [d(r)A, + dXt)\i] {xO))., + r,K- [</C()B, + d(OBi} {u(0) С-*)

Next, we approximate the derivative of(x(()), using the net change over one switching period:

( W)., х(Г.)х(0) (7,105)

Suhstitution of Eq. (7.104) into (7.105) leads to

К y = ((J(0 л, + di,) Aj) (x(/)) + [d{t) В, + rf(t) Bj] (u(()),.

(7.106)

which is identical to Eq. (7.99). This is the basic averaged model which describes the convertsr dynamics. It is nonlinear because the control inptit 40multiplied by (x(t)).. and (u(t))r,-Variation of atypical element of x(r) and its average are illustrated in Fig, 7,29,

It is also desired to find the low-frequency components ofthe output vector y(f) hy averaging. The vector y(f) is described by Eq. (7.90) for the first subinterval, and by Eq. (7.91) for the second sub-interval. Hence, the elements of y(f) may be discontinuous at the switching transitions, as illustrated in Fig. 7.30. We can again remove the switching harmonics by averaging over one switching period; the