# State Space Averaging of Buck-Boost Converter

In on state, the circuit is the same of the boost in on state, then its equations are the Eqs. 7.65 and 7.67, recalled in Sect. 7.12 and rewritten here for clarity. rL 1

TiL + L V

Vc    dt RC

As a consequence, the coefficient matrices in on state are the same as those obtained for the boost in on state, i. e., the matrices in Eq. 7.133, rewritten here for clarity

 f rL Aon = L 0 z о fiQ і i ° pr i_________ i CON = [ 0 1 Don = 0
 0 CR

In off state, the circuit can be deduced by the buck circuit in on state, setting Vs equal to zero. The corresponding equations are Eq. 7.150, obtained from Eq. 7.41a with the position Vs = 0, and Eq. 7.42a, recalled in Sect. 7.12 and rewritten here.

(7.150)    The coefficient matrices in off state are deduced by the buck in the same state.

Comparing Eq. 7.151 with Eq. 7.134, it can be observed that only the BOFF of the buck-boost is different from the corresponding boost matrix, having all its elements null.    As for matrices A, B and C, they can be calculated according to Eq. 7.100 and Eq. 7.102.

C = ConD + Coff(1 – D) = [rc(1 – D) 1 ]

It should be noted that the matrices A and C coincide with the corresponding matrices obtained for the boost.

On the other hand, the matrix B is equal to the corresponding one of the boost multiplied by the duty cycle D.     The static gain is obtained from Eq. 7.113

the obtained expression is the same of that obtained for the boost converter multiplied by D; comparing Eq. 7.136 with Eq. 7.153 it follows that:  where the approximation R + rc & R is done.

The output voltage to duty cycle transfer function is obtained by Eq. 7.118 that becomes:

( ) = C[sl — A] 1 [(AoN — Aoff )x0 + Bon Vs] + (Con — Coff)*0 a(s)  The term (Aon — AOFF)xo is given by: rCD2 + RD(1 – D) + R(1 – D)2

(7.158)

where the term rcD has been neglected respect to the term R. Table 7.2 Outline of the static gain including parasitic parameters and of the output voltage versus duty cycle and input voltage for buck, boost, and buck-boost converters Buck

Vo _______________ R(1 – D)_____________

Vs (1 – D)2R + rL + rc(1 – D) Vo(s) @-s2RLCrc + s[rcR2C(1 – D) – RL – rcrLRC] + |r2(1 – D)2 — rLR – r2(1 – D)

d(s) у s2RLC + s{L + RC[rc(1 – D) + rL]} + [rc(1 – D) + rL] + R(1 – D)2

do(s) _____________________________ R(srcC + 1)(1 – D)_________________________________

Vs(s) s2RLC + s{L + RC[rc(1 – D) + rL]} + [rc(1 – D) + rL] + R(1 – D)2

Buck-boost RD(1-D)

(1-D) 2R+rL+rc(1-D)   s[Crc2 (1 – D)D2 + CrcR(1 – D)2 – LD2]+ R(1 – D)2 – rtD2
(1 – D) D(S2RLC + s{L + RC[rc(1 – D) + rL]} + [rc(1 – D) + rL]+ R(1 – D)) +

Therefore, the final expression of the output voltage to duty cycle transfer function is given by:

Updated: August 18, 2015 — 4:04 am