1 sea_uniovi_cc1_00 lección 4 teoría básica de los convertidores cc/cc (i) (convertidores con un...
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1SEA_uniovi_CC1_00
Lección 4
Teoría básica de los convertidores CC/CC (I) (convertidores con un único transistor)
Diseño de Sistemas Electrónicos de Potencia
4º Curso. Grado en Ingeniería en Tecnologías y Servicios de Telecomunicación
Universidad de Oviedo
2
Introducing switching regulators
Basis of their analysis in steady state
Detailed study of the basic DC/DC converters in continuous conduction mode Buck, Boost and Buck-Boost converters Common and different properties Introduction to the synchronous rectification Four-order converters
Outline (I)
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Study of the basic DC/DC converters in discontinuous conduction modeDC/DC converters with galvanic isolation How and where to place a transformer in a DC/DC converter The Forward and Flyback converters
Outline (II)
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Linear DC/DC conversion (analog circuitry)
First idea
= (vOiO)/(vgig)
iO ig
vO/vg
Actual implementation
-
Vref
Av
Feedback loop
vE
RLvgvO
Q iOig
Only a few components Robust No EMI generation Only lower output voltage Efficiency depends on input/output voltages Low efficiency Bulky
-
Vref
Av
Feedback loop
vE
RLvgvO
RV iOig
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Linear versus switching DC/DC conversion
Linear
-
Vref
Av
Feedback loop
vE
RLvgvO
Q iOig
Switching (provisional)
-
Vref
Av
Feedback loop
vE
RLvgvO
SiOig
PWM
vOvg
t
Features:
100% efficiency
Undesirable output voltage waveform
vO_avg
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Introducing the switching DC/DC conversion (I)
Basic switching DC/DC converter (provisional)
-
Vref
Av
Feedback loop
vE
RLvgvO
SiOig
PWM
vOvg
t
vO_avg
The AC component must be removed!!
-
Vref
Av
Feedback loop
vE
RLvgvO
SiOig
PWM
Filter
VOVgt
RLvgvO
SiOig
C filter
C filter
It doesn’t work!!!
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Introducing the switching DC/DC conversion (II)
Basic switching DC/DC converter
vDVg
t
VO
-
Vref
Av
Feedback loop
vE
RLvgvO
SiOig
PWM
FilterLC filter
Infinite voltage across L when S1 is opened
It doesn’t work either!!!
RLvgvO
SiOig
LC filter
iL
C
L
RLvgvO
SiOig
LC filter
iL
C
L
iD DvD
+
-
+
-
Including a diode
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Introducing the switching DC/DC conversion (III)
Buck converter
RLvgvO
SiOig
LC filter
iL
C
L
iD DvD
+
-
+
-RLvg
vO
S
iOig
iL
CL
iD DvD
+
-
+
-
iS
Starting the analysis of the Buck converter in steady state:
L & C designed for negligible output voltage ripple (we are designing a DC/DC converter)
iL never reaches zero (Continuous Conduction Mode, CCM)
The study of the Discontinuous Conduction Mode (DCM) will done later
t
iLCCM
tiL
DCM
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First analysis of the Buck converter in CCM
- RLvgvO
S
iOig
iL
CL
iD DvD
+ +
-
iS
(In steady-state)
-RL
vO
iO
iL
CLvD
+ +
LC filter
vDvg
t
vD_avg
The AC component is removed by the filter
Analysis based on the specific topology of the Buck converter
= vO
vO = vD_avg = d·vg
T
dT
t
vD
vO
vg
d: “duty cycle”
This procedure is only valid for converter with explicit LC filter
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Introducing another analysis method (I)
Obviously, there is not an explicit LC filter
Therefore, we must use another method
R
VgVO
+
-
ig
iS
iDL1
C2S
D
iL2
L2
C1
+ -
Could we use the aforementioned analysis in the case of this converter (SEPIC)?
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Introducing another analysis method (II)
Powerful tools to analyze DC/DC converters in steady-state
Step 1- To obtain the main waveforms (with no quantity values) using Faraday’s law and Kirchhoff’s current and voltage laws
Step 2- To take into account the average value of the voltage across inductors and of the current through capacitors in steady-state
Step 2 (bis)- To use the volt·second balance
Step 3- To apply Kirchhoff’s current and voltage laws in average values
Step 4- Input-output power balance
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Introducing another analysis method (III)
Any electrical circuit that operates in steady-state satisfies:
The average voltage across an inductor is zero. Else, the net current through the inductor always increases and, therefore, steady-state is not achieved
The average current through a capacitor is zero. Else, the net voltage across the capacitor always increases and, therefore, steady-state is not achieved
+
-vL_avg = 0
iC_avg = 0
Vg
Circuit in steady-state
L
C
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Introducing another analysis method (IV)
Particular case of many DC/DC converters in steady-state:
Voltage across the inductors are rectangular waveforms
Current through the capacitors are triangular waveforms
+
-vL
iC
Vg
Circuit in steady-state
L
C
vL_avg = 0 iC_avg = 0
TdT
vL
t-
+v1
-v2
tiC
-+
Volt·second balance:V1dT – V2(1-d)T = 0
Same areas
Same areas
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Vg
iL1
iS
L1
S L2
C1
+ -
iC1
vL1+ -
vL2
+
-
vC1
Example
Introducing another analysis method (V)
Any electrical circuit of small dimensions (compared with the wavelength associated to the frequency variations) satisfies:
Kirchhoff’s current law (KCL) is not only satisfied for instantaneous current values, but also for average current values
Kirchhoff’s voltage law (KVL) is not only satisfied for instantaneous voltage values, but also for average voltage values
KVL applied to Loop1 yields:vg - vL1 - vC1 - vL2 = 0
vg - vL1_avg - vC1_avg - vL2_avg = 0
Therefore: vC1_avg = vg
KCL applied to Node1 yields:iL1 - iC1 - iS = 0
iL1_avg - iC1_avg - iS_avg = 0
Therefore: iS_avg = iL1_avg Loop1
Node1
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Introducing another analysis method (VI)
A switching converter is (ideally) a lossless system
RLvgvO
iOig
+
-
Switching-mode DC/DC converter
Input power:Pg = vgig_avg
Output power:PO = vOiO = vO
2/RL
Power balance:Pg = PO
DC Transformer
vg
iOig_avg
RLvO
+
-1:N
A switching-mode DC/DC converter as an ideal DC transformer
being N = vO/vg
Important concept!!
ig_avg = iOvO/vg = N·iO
Therefore: vgig_avg = vO2/RL
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Steady-state analysis of the Buck converter in CCM (I)
Step 1: Main waveforms. Remember that the output voltage remains constant during a switching cycle if the converter has been properly designed
RLvgvO
S
iOig iL
CLiD
DvD
+
-
+
-
iSvS+ -
iOiL
RLvgvO
CL +
-
During dT
S on, D off
iOiL
RLvO
CL +
-
During (1-d)T
S off, D on
t
t
t
t
iS
iD
iL
Driving signal
dT
T
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Step 1: Main waveforms (cont’)
RLvgvO
S
iOiL
CL
DvD
+
-
+
-
vS+ -
Steady-state analysis of the Buck converter in CCM (II)
vL+ -
dT
vg-vO
S off, D on,
(1-d)T
iOiL
RLvO
CL +
-
vL+ -
S on, D off,dT
iOiL
RLvg
vO
CL +
-
vL+ -
T
- vO
Driving signal
t
t
t
vL
iL
iL_avg
DiL
From Faraday’s law: DiL = vO(1-d)T/L
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+-
Step 2 and 2 (bis): Average inductor voltage and capacitor current
Steady-state analysis of the Buck converter in CCM (III)
dT
vg-vO
T
- vO
Driving signal
t
t
t
vL
iL
iL_avg
KCL applied to Node1 yields:iL - iC - iO = 0
iL_avg - iC_avg - iO = 0
Therefore: iL_avg = iO = vO/RL
Volt·second balance over L:(vg - vO)dT - vO(1-d)T = 0
Therefore: vO = d·vg (always vO < vg)
RLvO
iOiL
C
L +
-
vL+ -
iCNode1vg
S
ig
iD
D
iS Average value of iC:
iC_avg = 0
Step 3: Average KCL and KVL:
Step 4: Power balance:ig_avg = iS_avg = iOvO/vg = d·iO
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Summary
Steady-state analysis of the Buck converter in CCM (IV)
RLvgvO
S
iOig iL
CLiD
DvD
+
-
+
-
iSvS+ -
iO
t
t
tiS
iD
iL
dT
T
t
DiL
t
Driving signal
vDvg
iL_avg = iO = vo/RL
vO = d·vg (always vO < vg)
ig_avg = iS_avg = d·iO
iD_avg = iL_avg - iS_avg = (1-d)·iO
DiL = vO(1-d)T/L
iL_peak = iL_avg + DiL/2 = iO + vO(1-d)T/(2L)
vSmax = vDmax = vg
iS_peak = iD_peak = iL_peak
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Steady-state analysis of the Boost converter in CCM (I)
t
t
t
t
iS
iD
iL
Driving signal
dT
T
Can we obtain vO > vg? Þ Boost converter
S on, D off,during dT
iL
vg
L
vL+ -
Step 1: Main waveforms
+
-C vg
iL
iSL
S
iD
D
RLvO
iO
+
-
vL+ -ig
S off, D on,during (1-d)T
iOiL
RLvO
CL +
-
vL+ -
DiL
From Faraday’s law: DiL = vgdT/L
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Steady-state analysis of the Boost converter in CCM (II)
Step 2 and 2 (bis): Average values
KCL applied to Node1 yields:iD - iC - iO = 0
iD_avg - iC_avg - iO = 0
Therefore: iD_avg = iL_avg(1-d) = iO = vO/RL
Volt·second balance over L:vgdT - (vO - vg)(1-d)T = 0
Therefore: vO = vg/(1-d) (always vO > vg)
Average value of iC:
iC_avg = 0
Step 3: Average KCL and KVL:
Step 4: Power balance:ig_avg = iL_avg = iOvO/vg = iO/(1-d)
+
-C vg
iL
iSL
S
iD
D
RLvO
iO
+
-
vL+ -ig
iC
Node1
dT
vg
T
Driving signal
t
t
t
vL
iD iD_avgDiL
-(vO-vg)
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Steady-state analysis of the Boost converter in CCM (III)
Summary
iO
t
t
tiS
iD
iL
dT
T
t
DiL
t
Driving signal
vDvO
iL_avg = ig_avg = iO/(1-d) = vo/[RL(1-d)]
vO = vg/(1-d) (always vO > vg)
iS_avg = d·iL_avg = d·vo/[RL(1-d)]
iD_avg = iO DiL = vgdT/L
iL_peak = iL_avg + DiL/2 = iL_avg + vgdT/(2L)
vSmax = vDmax = vO
iS_peak = iD_peak = iL_peak
vS
+
-
vD +-
C vg
iL
iSL
S
iDD
RLvO
iO
+
-
vL+ -ig
iC
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Steady-state analysis of the Buck-Boost converter in CCM (I)
t
t
t
t
iS
iD
iL
Driving signal
dT
T
Can we obtain either vO < vg or vO > vg? Þ Buck-Boost converter
DiL
From Faraday’s law: DiL = vgdT/L
+
-
C
D
vg
iL
iS
LS
iD
RL
iO
vO
-
+
ig
vL
+
-
S on, D off,during dT
Charging stage
iL
vg LvL
+
-
ig
S off, D on,during (1-d)T
iO
RLvO
C -
+
iL
LvL
+
-
Discharging stage
+-
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Steady-state analysis of the Buck-Boost converter in CCM (II)
Step 2 and 2 (bis): Average values
KCL applied to Node1 yields:iD - iC - iO = 0
iD_avg - iC_avg - iO = 0
Therefore: iD_avg = iL_avg(1-d) = iO = vO/RL
Volt·second balance over L:vgdT - vO(1-d)T = 0
Therefore: vO = vgd/(1-d)
Average value of iC:
iC_avg = 0
Step 3: Average KCL and KVL:
Step 4: Power balance:ig_avg = iS_avg = iOvO/vg = iOd/(1-d)
Node1
+
-
C
D
vg
iL
iS
LS
iD
RL
iO
vO
-
+
ig
vL
+
-
iC
dT
vg
T
Driving signal
t
t
t
vL
iD iD_avgDiL
-vO
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Steady-state analysis of the Buck-Boost converter in CCM (III)
Summary
iL_avg = iD_avg/(1-d) = iO/(1-d) = vo/[RL(1-d)]
vO = vgd/(1-d) (both vO < vg and vO > vg)
iS_avg = ig_avg = d·iL_avg = d·vo/[RL(1-d)]
iD_avg = iO DiL = vgdT/L
iL_peak = iL_avg + DiL/2 = iL_avg + vgdT/(2L)
vSmax = vDmax = vO + vg
iS_peak = iD_peak = iL_peak
iO
t
t
tiS
iD
iL
dT
T
t
DiL
t
Driving signal
vDvO + vg
+
-
C
D
vg
iL
iS
LS
iD
RL
iO
vO
-
+
ig
vL
+
-
vD -+vS -+
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Common issues in basic DC/DC converters (I)
RLvgvO
SC
L
D
+
-
+
-
Buck
+
-C vg
L
S
D
RLvO
+
-
Boost
+
-
C
D
vg LS
RL
vO
+
-
Buck-Boost
Complementary switches + inductor
vgRL
vO
+
-
+-C
L
DS
d 1-d
Voltage source
The inductor is an energy buffer to connect two voltage sources
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Common issues in basic DC/DC converters (II)
+
-
C vg
L
S
D
RLvO
+
-Boost
vO
RLvgvOS C
L
D
+
-
+
-Buck
vg
+
-CD
vg LS RL
vO
+
-Buck-Boost
vO + vg
Diode turn-off
The diode turns off when the transistor turns on
The diode reverse recovery time is of primary concern evaluating switching losses
Schottky diodes are desired from this point of view
In the range of line voltages, SiC diodes are very appreciated
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Comparing basic DC/DC converters (I)
Generalized study as DC transformer (I)
DC Transformer
vg
iOig_avg
RLvO
+
-1:N
+
-C vg
L
S
D
RLvO
iO
+
-
ig
Boost
+
-
C
D
vg LS
RL
iO
vO
+
-
ig
Buck-Boost
RLvgvO
S
iOig
C
L
D
+
-
+
-
Buck
Buck: N= d (only vO < vg)
Boost: N= 1/(1-d) (only vO > vg)
Buck-Boost: N= -d/(1-d) (both vO < vg and vO > vg)
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Comparing basic DC/DC converters (II)
Generalized study as DC transformer (II)
Buck: ig_avg = iON = iOd
Boost: ig_avg = iON = iO/(1-d)
Buck-Boost: ig_avg = iON = - iOd/(1-d)
DC Transformer
vg
iOig_avg
RLvO
+
-1:N
ig_avg = iON = iOd/(1-d)
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Comparing basic DC/DC converters (III)
Electrical stress on components (I)
Buck:
vSmax = vDmax = vg
iS_avg = ig_avg
iL_avg = iO
iD_avg = iL_avg - iS_avg
vg
ig
iD
DvD
+
-S
iS vS+ -
RLvO
iO
+
-
DC/DC converter
Boost:
vSmax = vDmax = vO
iL_avg = ig_avg
iD_avg = iO
iS_avg = iL_avg - iD_avg
Buck-Boost:
vSmax = vDmax = vO + vg
iS_avg = ig_avg
iD_avg = iO
iL_avg = iS_avg + iD_avg
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Comparing basic DC/DC converters (IV)
Example of electrical stress on components (I)
vS_max = vD_max = 100 V
iS_avg = iD_avg = 1 A
iL_avg = 2 A
FOMVA_S = FOMVA_D = 100 VA
RL
SC
L
D
+
-
+
-50 V
100 V
2 A1 A (avg)
100 W Buck, 100% efficiency
+
-
C
D
LS
RL
-
+50 V
100 V
2 A1 A (avg)
100 W Buck-Boost, 100% efficiency
vS_max = vD_max = 150 V
iS_avg = 1 A
iD_avg = 2 A
iL_avg = 3 A
FOMVA_S = 150 VA
FOMVA_D = 300 VA Higher electrical stress in the case of Buck-Boost converter
Therefore, lower actual efficiencySEA_uniovi_CC1_30
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Comparing basic DC/DC converters (V)
Example of electrical stress on components (II)
vS_max = vD_max = 50 V
iS_avg = iD_avg = 2 A
iL_avg = 4 A
FOMVA_S = FOMVA_D = 100 VA
+-C
L
S
D
RL
+
-50 V25 V
2 A4 A (avg)
100 W Boost, 100% efficiency
+
-
C
D
LS
RL
-
+50 V
25 V
2 A4 A (avg)
100 W Buck-Boost, 100% efficiency
vS_max = vD_max = 75 V
iS_avg = 4 A
iD_avg = 2 A
iL_avg = 6 A
FOMVA_S = 300 VA
FOMVA_D = 150 VA Higher electrical stress in the case of Buck-
Boost converter Therefore, lower actual efficiency
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Comparing basic DC/DC converters (VI)
Price to pay for simultaneous step-down and step-up capability:
Higher electrical stress on components and, therefore, lower actual efficiency
Converters with limited either step-down or step-up capability:
Lower electrical stress on components and, therefore, higher actual efficiency
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Comparing basic DC/DC converters (VII)
300 W Boost, 98% efficiency
+-C
L
S
D
RL
+
-60 V50 V
5 A6.12 A (avg)
1.12 A (avg)
Example of power conversion between similar voltage levels based on a Boost converter
Very high efficiency can be achieved!!!
vS_max = vD_max = 60 V
iS_avg = 1.12 A
iD_avg = 5 A
iL_avg = 6.12 A
FOMVA_S = 67.2 VA
FOMVA_D = 300 VA
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Comparing basic DC/DC converters (VIII)
The opposite case: Example of power conversion between very different and variable voltage levels based on a Buck-
Boost converter
High efficiency cannot be achieved!!!
300 W Buck-Boost, 75% efficiency
+
-
C
D
LS
RL
-
+60 V
20 - 200 V
5 A20 - 2 A (avg)
vS_max = vD_max = 260 V
iS_avg_max = 20 A
iD_avg_max = 5 A
iL_avg = 25 A
FOMVA_S_max = 5200 VA
FOMVA_D = 1300 VA
Remember previous example:FOMVA_S = 67.2 VA
FOMVA_D = 300 VA
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Comparing basic DC/DC converters (IX)
One disadvantage exhibited by the Boost converter:
The input current has a “direct path” from the input voltage source to the load. No switch is placed in this path. As a consequence, two problems arise:
Large peak input current in start-up
No over current or short-circuit protection can be easily implemented (additional switch needed)
Buck and Buck-Boost do not exhibit these problems
+
-C vg
L
S
D
RLvO
+
-Boost
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Synchronous rectification (I)
To use controlled transistors (MOSFETs) instead of diodes to achieve high efficiency in low output-voltage applications
This is due to the fact that the voltage drop across the device can be lower if a transistor is used instead a diode
The conduction takes place from source terminal to drain terminal
In practice, the diode (Schottky) is not removed
SL
D
S1
L
S2
S1
L
S2
idevice
vdevice
Diode
MOSFET
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Synchronous rectification (II)
In converters without a transformer, the control circuitry must provide proper driving signals
In converters with a transformer, the driving signals can be obtained from the transformer (self-driving synchronous rectification)
Nowadays, very common technique with low output-voltage Buck converters
S1
L
S2
Feedback loop
-Vref
Av
vO
PWMQ
Q’
RLvgvOC
L
+
-
+
-
Synchronous Buck
S1
S2
D
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Input current and current injected into the output RC cell (I)
vg
ig
iD
DvD
+
-S
iS vS+ -
RLvO
+
-
DC/DC converter
+-C
iRC
t
Desired current
ig
t
iRC
Desired current
If a DC/DC converter were an ideal DC transformer, the input and output currents should also be DC currents
As a consequence, no pulsating current is desired in the input and output ports and even in the current injected into the RC output cell
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40
Input current and current injected into the output RC cell (II)
t
ig
Noisy
RLvgvO
S
iRCig
C
L
D
+
-
+
-
Buck
tLow noise
iRC
+
-C vg
L
S
D
RLvO
+
-
Boost
ig iRC
Low noiset
ig
t
Noisy
iRC
vO
+
-+
-
C
D
vg LS
RL
Buck-Boost
ig iRC
t
Noisy
ig
t
Noisy
iRC
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Input current and current injected into the output RC cell (III)
RLvg
vOS
iRCig
C
L
D
+
-
+-
Buck
CF
LF +-
+-CF vg
L
S
D
Boost
ig iRC
RLvO
+
-C
LF+-
iRCig
+
-CF
D
LS
Buck-Boost
RLvO
-
+C
LF-+vg
CF
LF +-
Filter Filter
Filter
Filter
Adding EMI filters
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Four-order converters (converters with integrated filters)
RL
vgvO
+
-
ig
iS
iDL1
C2S
D
iL2
L2
C1
+ -vC1
SEPIC
RL
vgvO+
-
ig
iSiD
L1
C2SD
iL2 L2C1
+ -vC1
Cuk
RL
vgvO
+
-
iS
iD
L1 C2
SD
iL2
L2
C1
+-
iL1
vC1
Zeta
Same vO/vg as Buck-Boost
Same stress as Buck-Boost
vC1 = vg
Filtered input
Same vO/vg as Buck-Boost
Same stress as Buck-Boost
vC1 = vg + vO
Filtered input and output
Same vO/vg as Buck-Boost
Same stress as Buck-Boost
vC1 = vO
Filtered output
SEA_uniovi_CC1_41
43
DC/DC converters operating in DCM (I)
Only one inductor in basic DC/DC converters
The current passing through the inductor decreases when the load current decreases (load resistance increases)
vg
ig
DS RL
vO
iO
+
-DC/DC converter
L
iL
TdT
t
t
iL
Driving signal
iL_avg
Buck:
iL_avg = iO
Boost:
iL_avg = iO/(1-d)
Buck-Boost:
iL_avg = iS_avg + iD_avg = diO/(1-d) + iO
= iO/(1-d)
SEA_uniovi_CC1_42
44
When the load decreases, the converter goes toward Discontinuous Conduction Mode (DCM)
iL_avg
t
iL
RL_1
t
iL
RL_2 > RL_1
iL_avg
iL
t
RL_crit > RL_2 iL_avg
Dec
reas
ing
load
It corresponds to RL = R L_crit
Boundary between CCM and DCM
Operation in CCM
DC/DC converters operating in DCM (II)
SEA_uniovi_CC1_43
45
What happens when the load decreases below the critical value?
iL
t
RL_crit iL_avg
Dec
reas
ing
load
DCM starts if a diode is used as rectifier
If a synchronous rectifier (SR) is used, the operation depends on the driving signal
CCM operation is possible with synchronous rectifier with a proper driving signal (synchronous rectifier with signal almost complementary to the main transistor)
iL
t
RL_3 > RL_critiL_avg
CCM w. SR
iL
t
RL_3 > RL_critiL_avg
DCM w. diode
DC/DC converters operating in DCM (III)
SEA_uniovi_CC1_44
46
Remember:
iL_avg = iO (Buck) or iL_avg = iO/(1-d) (Boost and Buck-Boost)
For a given duty cycle, lower average value (due to the negative area) Þ lower output current for a given load Þ lower output voltage
iL
t
iL_avg
RL > RL_crit
DCM w. diode
For a given duty cycle, higher average value (no negative area) Þ higher output current for a given load Þ higher output voltage
iL
t
RL > RL_crit
CCM w. SRiL_avg
The voltage conversion ratio vO/vg is always higher in DCM
than in CCM (for a given load and duty cycle)
DC/DC converters operating in DCM (IV)
SEA_uniovi_CC1_45
47
How can we get DCM (of course, with a diode as rectifier) ?
t
iL
t
iL
t
iL
After decreasing the inductor inductance
After decreasing the switching frequency
After decreasing the load (increasing the load resistance)
DC/DC converters operating in DCM (V)
SEA_uniovi_CC1_46
48
DC/DC converters operating in DCM (VI)
Three sub-circuits instead of two:
The transistor is on. During d·T
The diode is on. During d’·T
Both the transistor and the diode are off. During (1-d-d’)T
tiL
t
iL_avg
vL
T
d·Tt
d’·T
+-
iD
t
iD_avg
-vO
vg
Driving signal
+
-
C
D
vg
iL
iS
LS
iD
RL
iO
vO
-
+
ig
vL
+
-
iL
vg LvL
+
-
ig
During d·T
iO
RLvO
C -
+
iL
LvL
+
-+-
During d’·T
iL
LvL
+
-
During (1-d-d’)T
Example: Buck-Boost converter
SEA_uniovi_CC1_47
49
DC/DC converters operating in DCM (VII)
Voltage conversion ratio vO/vg for the Buck-Boost converter in DCM
iL
vg LvL
+
-
ig
During d·T
iO
RLvO
C -
+
iL
LvL
+
-+-
During d’·T
From Faraday’s law:vg = LiL_max/(dT)
And also: vO = LiL_max/(d’T)
Also:
iD_avg = iL_maxd’/2, iD_avg = vO/R
And finally calling M = vO/vg we obtain:
M =d/(k)1/2 where k =2L/(RT)
tiL
t
iL_avg
vL
T
d·Tt
d’·T
+-
iD
t
iD_avg
-vO
vg
Driving signal
iL_max
iL_max
SEA_uniovi_CC1_48
50
Due to being in DCM: M = vO/vg = d/(k)1/2, where: k = 2L/(RT)
Due to being in CCM: N = vO/vg = d/(1-d)
Just on the boundary: M = N, R = Rcrit, k = kcrit
Therefore: kcrit = (1-d)2
The converter operates in CCM if: k > kcrit
The converter operates in DCM if: k < kcrit
DC/DC converters operating in DCM (VIII)
The Buck-Boost converter just on the boundary between DCM and CCM
iLt
RL = RL_crit
iL_avg
SEA_uniovi_CC1_49
51
N = d
2M =
1 + 1 + 4kd2
kcrit = (1-d)
kcrit_max = 1
Buck
dM =
k
dN =
1-d
kcrit = (1-d)2
kcrit_max = 1
Buck-Boost
2M =
1 + 1 + 4d2
k
1N =
1-d
kcrit = d(1-d)2
kcrit_max = 4/27
Boost
DC/DC converters operating in DCM (IX)
Summary for the basic DC/DC converter
k = 2L/(RT)
SEA_uniovi_CC1_50
52
CCM versus DCM
DC/DC converters operating in DCM (X)
t
t
tiS
iD
iL
dT
T
t
t
Driving signal
vD
iL_avg
t
t
tiS
iD
iL
dT
T
t
t
Driving signal
vD
iL_avg
- Lower conduction losses in CCM (lower rms values)
- Lower losses in DCM when S turns on and D turns off
- Lower losses in CCM when S turns off
- Lower inductance values in DCM (size?)
SEA_uniovi_CC1_51
53
vi = ni d/dt
= B - A = (vi/ni)·dtB
A
From Faraday’s law:
In steady-state:
()in a period= 0
Achieving galvanic isolation in DC/DC converters (I)
(vi /ni)avg = 0
And therefore:
Volt·second balance: If all the voltages are DC voltages, then: CCM: dT(V1/n1) – (1-d)T(V2/n2) = 0
DCM: dT(V1/n1) –d’T(V2/n2) = 0
vg Circuit in steady-state
n1:n2
v1
+
-v2
+
-
SEA_uniovi_CC1_52
- A two-winding magnetic device is needed
- The volt·second balance in the case of magnetic devices with two windings must be used
54
Achieving galvanic isolation in DC/DC converters (II)
n1:n2
Model 1:Circuit Theory
element
n1:n2
Lm1
Model 2:Magnetic transformer with perfect coupling
n1:n2
Lm1
Ll1 Ll2
Model 3:Magnetic transformer
with real coupling
Model 1 Model 2
Transformer models
At least the magnetizing inductance must be taken into account analyzing DC/DC converters
SEA_uniovi_CC1_53
55
Achieving galvanic isolation in DC/DC converters (III)
n1:n2
Lm1
Where must we place the transformer?
vg
ig
vDD
+
-
vS
S
+ -
RLvO
iO
+
-
DC/DC converter
In a place where the average voltage is zero
SEA_uniovi_CC1_54
56
Achieving a Buck converter with galvanic isolation (I)
RLvgvO
SC
L
D
+
-
+
-Buck
n1:n2
Lm1
No place with average voltage equal to zero
RLvgvOS C
L
D
+
-
+-
New node with possible zero average voltage
vgS RL
vOC
L
D1
+
-
+
-
D2S off
It does not work!!
S on
SEA_uniovi_CC1_55
57
Achieving a Buck converter with galvanic isolation (II)
vextra
n3D2
n1:n2
Lm1vg
RLvOC
L +
-
+
-D1
S onS off
S
n1:n1:n2
Lm1
vg
RLvOC
L +
-
+
-D1
D2
D3Final implementation: the
Forward converter
Standard design:
vextra = vg
n3 = n1
A circuit to apply a given DC voltage across Lm1 when S is off
SEA_uniovi_CC1_56
58
The Forward converter
S
n1:n1:n2
Lm1
vg
RLvOC
L +
-
+
-D1
D2
D3
As the Buck converter replacing vg with vgn2/n1
Transformer magnetizing stage
vg Lm1
vL
+
-
im1
iOiL
RLvgn2/n1vO
CL +
-
Inductor magnetizing stageS & D2 on, D1
& D3 off, during dT
D3 on, during d’T
Transformer reset stage
vg
Lm1
vL
+
-
im1iOiL
RLvO
CL +
-
Inductor demagnetizing stage
during (1-d)T
S & D2 off, D1 on,
vO = dvgn2/n1
vSmax = 2 vg
dmax = 0.5 (reset transformer)SEA_uniovi_CC1_57
59
Achieving a Buck-Boost converter with galvanic isolation (I)
n1:n2
Lm1
There is a place with average voltage equal to
zero: the inductor
RLvOC-
+
-
+
D
vO
-
++
-CD
vg LS
RLBuck-Boost
vgS
L
S offS on
n1:n2
L RLvOC-
+
-
+
Dvg
S
Inductor and transformer integrated into only one
magnetic device (two-winding inductor)
SEA_uniovi_CC1_58
60
Achieving a Buck-Boost converter with galvanic isolation (II)
n1:n2
L RLvOC-
+
-
+
Dvg
S
Final implementation: the Flyback converter
S
n1:n2
vg
RLvOC
+
-
+
-
D
L1 L2
S off, D on,during (1-d)T
iO
RLvO
C -
+vLn2/n1
+
-
Discharging stage
+-
L2
S on, D off,during dT
Charging stage
vg L1
vL
+
-
ig
Two-winding inductor
SEA_uniovi_CC1_59
61
The Flyback converter
S
n1:n2
vg
RLvOC
+
-
+
-
D
L1 L2
Analysis in steady-state in CCM
Volt·second balance:
dTvg/n1 - (1-d)TvO/n2 = 0
Þ vO = vg(n2/n1)·d/(1-d)
Therefore, the result is the same as Buck-Boost converter replacing vg with
vgn2/n1
vSmax = vg + vOn1/n2
vDmax = vgn2/n1 + vO Very simple topology
Useful for low-power, low-cost converters
Critical “false transformer” (two-winding inductor) design
SEA_uniovi_CC1_60
62
Achieving other converters with galvanic isolation (I)
+
-C vg
L
S
D RLvO
+
-Boost
It is not possible with only one transistor!!
RLvg
vO
+
-
L1
C2S
DL2
C1
+ -
SEPIC
n1:n2
n1:n2
RLVgVO+
-
L1
C3S D
L2C1
+ -C2
+ -
Cuk
Zeta converter is also possible
vO = vg(n2/n1)d/(1-d)
vSmax = vg + vOn1/n2
vDmax = vgn2/n1 + vO
Like the Flyback converter
SEA_uniovi_CC1_61