State-Space Averaging

• approximates the switching converter as a continuous linear system

• requires that the effective output filter corner frequency to be much smaller than the switching frequency

Power switching converters Dynamic analysis of switching converters

State-Space Averaging

• Step 1: Identify switched models over a switching cycle. Draw the linear switched circuit model for each state of the switching converter (e.g., currents through inductors and voltages across capacitors).

• Step 2: Identify state variables of the switching converter. Write state equations for each switched circuit model using Kirchoff's voltage and current laws.

• Step 3: Perform state-space averaging using the duty cycle as a weighting factor and combine state equations into a single averaged state equation. The state-space averaged equation is

Procedures for state-space averaging

1 21 2x = [ A d + A (1- d)] x + [ B d + B (1- d)] u .

Power switching converters Dynamic analysis of switching converters 4

State-Space Averaging

• Step 4: Perturb the averaged state equation to yield steady-state (DC) and dynamic (AC) terms and eliminate the product of any AC terms.

• Step 5: Draw the linearized equivalent circuit model.

• Step 6: Perform hybrid modeling using a DC transformer, if desired.

Power switching converters Dynamic analysis of switching converters 5

State-Space Averaged Model for an Ideal Buck Converter

x

1

Qs 1

2

L

-

xu R

+

CDfw

Power switching converters Dynamic analysis of switching converters 6

State-Space Averaged Model for an Ideal Buck Converter

2

x

C

1

1

x

2

L

R

x

x L

-

+

1

(b) (1-d)T interval

-

u

+

(a) dT interval

C R

11 2u L x x

212xx = C x +

R

1 20 = L x + x

212xx = C x +

R

1

2

11

2

0 1 1-xx L = + [ ]L u1 1 xx 0-

C RC

1

2

11

2

0 1- 0xx L = + [ ]u1 1 0xx -C RC

Power switching converters Dynamic analysis of switching converters 7

State-Space Averaged Model for an Ideal Buck Converter

0 1 0 1- -L LA = d + (1- d)

1 1 1 1- -C RC C RC

0 1-LA = .

1 1-C RC

1 d0B = d + (1- d) = .L L

00 0

1

2

11

2

0 1 d-xx L = + [ ] .L u1 1 xx 0-

C RC

Power switching converters Dynamic analysis of switching converters 8

State-Space Averaged Model for an Ideal Boost Converter

C

x

u

Dfw

RQs

+

x2

-

2

1L

u1

Power switching converters Dynamic analysis of switching converters 9

State-Space Averaged Model for an Ideal Boost Converter

+

L

C

-

C

L

R

(b) (1-d) T interval

R

+

-

(a) dT interval

u1

u1

1x

1x

x2

x2

2u

2u

11 = L xu

22

2x = C x + uR

1

2

1 1

2 2

1 00 0x ux L = + .0 1 0 1- x ux RC C

11 2 = L x + u x

22

1 2x + = C x + x uR

1

2

1 1

2 2

0 1 1 o-x ux L L = + .

1 1 0 1x ux -C RC C

Power switching converters Dynamic analysis of switching converters 10

State-Space Averaged Model for an Ideal Boost Converter

1 2 (1 )

0 10 0 -LA = A d A d = d + (1- d)0 1 1 1- -RC C RC

0 -(1- d)LA = .

(1- d) 1-C RC

1 2 (1 )

1 0 1 0L LB = B d B d = d + (1- d)0 1 0 1

C C

1 0LB = .0 1

C

1

2

1 1

2 2

0 -(1- d) 1 0x ux L L = +

(1- d) 1 0 1x ux -C RC C

1

2

2 1

1 2 2

-(1- d)x ux L L = + .

(1- d)x x ux -C RC C

Buck Boost Converter

Power switching converters Dynamic analysis of switching converters 11