Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ... 1 Chapter 3. Steady-State Equivalent Circuit Modeling, Losses, and Efficiency 3.1. The dc transformer model 3.2. Inclusion of inductor copper loss 3.3. Construction of equivalent circuit model 3.4. How to obtain the input port of the model 3.5. Example: inclusion of semiconductor conduction losses in the boost converter model 3.6. Summary of key points
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Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...1
Chapter 3. Steady-State Equivalent CircuitModeling, Losses, and Efficiency
3.1. The dc transformer model
3.2. Inclusion of inductor copper loss
3.3. Construction of equivalent circuit model3.4. How to obtain the input port of the model
3.5. Example: inclusion of semiconductor conductionlosses in the boost converter model
3.6. Summary of key points
Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...2
3.1. The dc transformer model
Basic equations of an idealdc-dc converter:
Pin = Pout
Vg Ig = V I(η = 100%)
V = M(D) Vg (ideal conversion ratio)Ig = M(D) I
These equations are valid in steady-state. Duringtransients, energy storage within filter elements may causePin ≠ Pout
Switching
dc-dc
converter
D
Control input
Power
input
Power
output
Ig I
+
V
–
+
Vg
–
Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...3
Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...28
Construction of equivalent circuits
Vg – IRL – IDRon – D'VD – ID'RD – D'V = 0
D'I – V/R = 0
RL
+–
D'RD
+ –
D'VDDRon
+ IRL – + IDRon – + ID'RD –
+–
D'VI
Vg
R
+
V
–
V/R
D'I
Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...29
Complete equivalent circuit
RL
+–
D'RD
+ –
D'VDDRon
+–
D'V R
+
V
–
D'II
Vg
RL
+–
D'RD
+ –D'VDDRon
R
+
V
–
I
D' : 1
Vg
Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...30
Solution for output voltage
V = 1D'
Vg – D'VDD'2R
D'2R + RL + DRon + D'RD
VVg
= 1D'
1 –D'VD
Vg
1
1 +RL + DRon + D'RD
D'2R
RL
+–
D'RD
+ –
D'VDDRon
R
+
V
–
I
D' : 1
Vg
Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...31
Solution for converter efficiency
Pin = (Vg) (I)
Pout = (V) (D'I)
η = D' VVg
=
1 –D'VD
Vg
1 +RL + DRon + D'RD
D'2R
Conditions for high efficiency:
RL
+–
D'RD
+ –
D'VDDRon
R
+
V
–
I
D' : 1
Vg
Vg/D' > VD
D'2R > RL + DRon + D'RD
Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...32
Accuracy of the averaged equivalent circuitin prediction of losses
• Model uses averagecurrents and voltages
• To correctly predict powerloss in a resistor, use rmsvalues
• Result is the same,provided ripple is small
MOSFET current waveforms, for variousripple magnitudes:
Inductor current ripple MOSFET rms current Average power loss in Ron
(a) ∆i = 0 I D D I2 Ron
(b) ∆i = 0.1 I (1.00167) I D (1.0033) D I2 Ron
(c) ∆i = I (1.155) I D (1.3333) D I2 Ron
i(t)
t
0
DTs Ts0
I
2 I
1.1 I(a)
(c)
(b)
Fundamentals of Power Electronics Chapter 3: Steady-state equivalent circuit modeling, ...33
Summary of chapter 3
1. The dc transformer model represents the primary functions of any dc-dcconverter: transformation of dc voltage and current levels, ideally with100% efficiency, and control of the conversion ratio M via the duty cycle D.This model can be easily manipulated and solved using familiar techniquesof conventional circuit analysis.
2. The model can be refined to account for loss elements such as inductorwinding resistance and semiconductor on-resistances and forward voltagedrops. The refined model predicts the voltages, currents, and efficiency ofpractical nonideal converters.
3. In general, the dc equivalent circuit for a converter can be derived from theinductor volt-second balance and capacitor charge balance equations.Equivalent circuits are constructed whose loop and node equationscoincide with the volt-second and charge balance equations. In convertershaving a pulsating input current, an additional equation is needed to modelthe converter input port; this equation may be obtained by averaging theconverter input current.