canonical_circuit_model

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3. Converter must contain an effective lowpass filter characteristic • necessary to filter switching ripple • also filters ac variations
He(s) 1 : M(D) +
Fundamentals of Power Electronics
127
Chapter 7: AC equivalent circuit modeling
Steps in the development of the canonical circuit model
e(s) d(s)
He(s) 1 : M(D) + Zei(s) Effective low-pass filter – Zeo(s)
Fundamentals of Power Electronics
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Chapter 7: AC equivalent circuit modeling
Transfer functions predicted by canonical model
e(s) d(s)
He(s) 1 : M(D) + Zei(s) Effective low-pass filter – Zeo(s)
node B
Vg – V d D
+ –
1:D
sLI d D'
L
D' : 1 + C
V + v(s)
+ –
DI d D'
Vg + vg(s)
+ –
Id
DI d D'
R

Fundamentals of Power Electronics
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Chapter 7: AC equivalent circuit modeling
V + v(s)
+ –
Vg + vg(s) + –
j(s) d(s)
R
D + d(s)
Power input
Control input
Load
Line-to-output transfer function: Control-to-output transfer function:
Gvg(s) =
Gvd(s) =
Vg – V d D
node A
1:D
L
D' : 1 +
+ –
Vg + vg(s)
+ –
Id
I d D'
I d D'
C
V + v(s)
R

Fundamentals of Power Electronics
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Chapter 7: AC equivalent circuit modeling
Step 3
1 : M(D) + + –
V + v(s)
R

D Power input Control input Load
Fundamentals of Power Electronics
126
Chapter 7: AC equivalent circuit modeling
Steps in the development of the canonical circuit model
Small-signal ac model of the buck-boost converter
L
Vg – V d
1:D
D' : 1 +
Id
+ –
Vg + vg(s)
+ –
Id
C
V + v(s)
R

• Push independent sources to input side of transformers • Push inductor to output side of transformers • Combine transformers
Vg + vg(s)
+ –
Id
I d D'
R

Fundamentals of Power Electronics
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Chapter 7: AC equivalent circuit modeling
Step 4
Step 4
Now push current source through 1:D transformer. Push current source past voltage source, again by: Breaking ground connection of current source, and connecting to node B instead. Connecting an identical current source from node B to ground, so that the node equations are unchanged. Note that the resulting parallel-connected voltage and current sources are equivalent to a single voltage source.
V + v(s)
+ –
Vg + vg(s) + –
j(s) d(s)
R
D + d(s)
Power input
Control input
Load
4. Control input variations also induce ac variations in converter waveforms • Independent sources represent effects of variations in duty cycle • Can push all sources to input side as shown. Sources may then become frequency-dependent
Vg – V – s LI d(s) D DD' L D' 2
D' : D
Vg + vg(s) + –
I d(s) D'
+ –
C Effective low-pass filter
135
+
V + v(s)
R

Fundamentals of Power Electronics
Chapter 7: AC equivalent circuit modeling
Vg – V d D
1:D
L
D' : 1 +
I d D'
+ –
Vg + vg(s)
+ –
Id
C
V + v(s)
R

Fundamentals of Power Electronics
131
Chapter 7: AC equivalen move the current source past the inductor: Break ground connection of current source, and connect to node A instead. Connect an identical current source from node A to ground, so that the node equations are unchanged.
The parallel-connected current source and inductor can now be replaced by a Thevenin-equivalent network:
Vg – V d D sLI d D'
+ –
1:D
L
D' : 1 + C
V + v(s)
+ –
+ –
Converter model 1 : M(D) + Vg V – R
D Power input Control input Load
Fundamentals of Power Electronics
125
Chapter 7: AC equivalent circuit modeling
Steps in the development of the canonical circuit model
Coefficient of control-input voltage generator
Voltage source coefficient is:
e(s) =
Vg + V s LI – D D D'
Simplification, using dc relations, leads to
Step 5: final result
Push voltage source through 1:D transformer, and combine with existing input-side transformer. Combine series-connected transformers.
e.g. 100 Hz variations
2. Ac variations in vg(t) induce ac variations in v(t) • these variations are also transformed by the conversion ratio M(D)