state space averaging
DESCRIPTION
PWM class notes on State Space Averaging. B. Tech final year or M. TechTRANSCRIPT
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