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Universidad de Oviedo

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

SEA_uniovi_CC1_001

Outline (I)

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  Common and

Buck-Boost converters

different properties

 Introduction to

the synchronous rectification

 Four-order converters

SEA_uniovi_CC1_012

Outline (II)

Study of the basic DC/DC converters in discontinuous conduction mode DC/DC converters with galvanic isolation  How and  The

where to place a transformer in a DC/DC converter

Forward and Flyback converters

SEA_uniovi_CC1_023

Linear DC/DC conversion (analog circuitry) RV

ig

iO RL

vg vE

Av

Feedback loop

First idea

= (vOiO)/(vgig)

vO

iO  ig

Vref

Q

ig

  vO/vg

iO RL

vg vE

Av

Feedback loop

vO

Vref

Actual implementation

 Only a few components  Robust  No EMI generation  Only lower output voltage  Efficiency depends on input/output voltages  Low efficiency  Bulky SEA_uniovi_CC1_034

Linear versus switching DC/DC conversion Q

ig

iO RL

vg vE

Av

Feedback loop

ig

vO

vg

iO RL

vg

PWM Vref

vE

vO

Av

Feedback loop

Linear

vO

S

Vref

Switching (provisional)

Features:

vO_avg

100% efficiency  Undesirable output voltage waveform

t SEA_uniovi_CC1_045

Introducing the switching DC/DC conversion (I) S

ig

iO RL

vg PWM

vE

vg

t The AC component must be removed!!

-

Av

Vref

S RL C filter

Vg

vO

vg

iO Filter

VO

RL

vO

C filter PWM

t It doesn’t work!!!

S

ig

iO

vg

vO_avg

vO

Feedback loop

ig

vO

vE

Av

Feedback loop

Vref

Basic switching DC/DC converter (provisional) SEA_uniovi_CC1_056

Introducing the switching DC/DC conversion (II) S

ig

iO Filter

vg

PWM

RL

ig

iL

vg

iD

+ vD D -

C

iO

L C

RL

vO

Infinite voltage across L when S1 is opened It doesn’t work either!!!

Including a diode S

vg LC filter

Vref

Feedback loop

iO

L

LC filter

-

Av

vE

ig

vO

iL

S

RL

+ vO -

Vg

vD

VO t

LC filter

Basic switching DC/DC converter SEA_uniovi_CC1_067

Introducing the switching DC/DC conversion (III) iL

S

ig

iD

vg

+ vD D -

iO

L C

RL

+ vO -

iL

iS

ig S vg

LC filter

iD

+ L vD D -

iO C RL

+ vO -

Buck converter

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 iL

DCM

CCM

iL t

t SEA_uniovi_CC1_078

First analysis of the Buck converter in CCM (In steady-state) Analysis based on the specific topology of the Buck converter iL

iS

ig S vg

vg

iD

+ L vD D -

iL

iO C RL

+

+ vO -

L

iO C

vD

RL

+ vO -

LC filter

vD vg

vO

vD

vD_avg = vO

t

t dT T

d: “duty cycle”

vO = vD_avg = d·vg

The AC component is removed by the filter

 This procedure is only valid for converter with explicit LC filter SEA_uniovi_CC1_089

Introducing another analysis method (I) Could we use the aforementioned analysis in the case of this converter (SEPIC)?

ig

L1

C1 + iS

Vg S

iD

D

iL2 L2

+

C2 -

R VO

 Obviously, there is not an explicit LC filter  Therefore, we must use another method 10 SEA_uniovi_CC1_09

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 11 SEA_uniovi_CC1_10

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

Vg

L

Circuit in steady-state

+

vL_avg = 0

C

iC_avg = 0 12 SEA_uniovi_CC1_11

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

Vg

L

Circuit in steady-state

+

vL

+

Same areas v1 t

-

-

dT C

vL_avg = 0

vL

iC

-v2

Volt·second balance: V1dT – V2(1-d)T = 0

T

iC_avg = 0

+

iC

-

t

Same areas 13 SEA_uniovi_CC1_12

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:

Example iL1

Node1

L1 + vL1 -

Vg

S

iS

vg - vL1 - vC1 - vL2 = 0

vg - vL1_avg - vC1_avg - vL2_avg = 0 iC1

Therefore: vC1_avg = vg

C1 + vC1

 KCL applied to Node1 yields: +

vL2

Loop1 -

iL1 - iC1 - iS = 0 L2

iL1_avg - iC1_avg - iS_avg = 0 Therefore: iS_avg = iL1_avg 14 SEA_uniovi_CC1_13

Introducing another analysis method (VI) A switching converter is (ideally) a lossless system ig

 Input power:

iO vg

Switching-mode DC/DC converter

RL

Therefore: vgig_avg = vO2/RL

Pg = vgig_avg + vO -

 Output power: PO = vOiO = vO2/RL  Power balance: Pg = PO

 A switching-mode DC/DC converter as an ideal DC transformer ig_avg

iO RL

vg

+ vO -

being N = vO/vg ig_avg = iOvO/vg = N·iO

1:N DC Transformer

Important concept!! 15 SEA_uniovi_CC1_14

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 ig

+

vS

iS

-

vg

S

iO

iL

iD

+ vD D -

L

Driving signal

C RL

L

C RL

vg

t iL

iO

iL

S on, D off

+ vO -

+ vO -

t iS t

During dT iO

iL

S off, D on

L

C RL

iD + vO -

t dT T

During (1-d)T 16 SEA_uniovi_CC1_15

Steady-state analysis of the Buck converter in CCM (II) Step 1: Main waveforms (cont’) +

vS

vg

iL

S

+ vL L

+ vD D -

C RL

t

+ vO -

iL

L

iO C RL

vg

+ vO -

vg-vO

t - vO T

iL + vL L

t

vL

dT

S off, D on, (1-d)T

DiL

iL_avg

i L + vL -

S on, D off, dT

Driving signal

iO

iO C RL

+ vO -

 From Faraday’s law: DiL = vO(1-d)T/L 17 SEA_uniovi_CC1_16

Steady-state analysis of the Buck converter in CCM (III) Step 2 and 2 (bis): Average inductor voltage and capacitor current  Average value of iC:

ig

+ vL - iL

iS

iC_avg = 0

iD

 Volt·second balance over L:

vg

L

Node1

S D

iO

iC C

RL

+ vO -

(vg - vO)dT - vO(1-d)T = 0

Therefore:

vO = d·vg (always vO < vg) Driving signal

Step 3: Average KCL and KVL:

t

 KCL applied to Node1 yields:

iL

iL - iC - iO = 0

iL_avg

iL_avg - iC_avg - iO = 0 Therefore: iL_avg = iO = vO/RL Step 4: Power balance: ig_avg = iS_avg = iOvO/vg = d·iO

vg-vO

t

vL

+

-

dT

t - vO

T 18 SEA_uniovi_CC1_17

Steady-state analysis of the Buck converter in CCM (IV) ig

+ vg

vS

iS

S

iD

+ vD D -

Summary

iO

iL L

C RL

Driving signal

+ vO -

vO = d·vg (always vO < vg) vSmax = vDmax = vg

t vg

vD t

iL

iO

iL_avg = iO = vo/RL ig_avg = iS_avg = d·iO

t iS

DiL t

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) iS_peak = iD_peak = iL_peak

iD

t dT T 19 SEA_uniovi_CC1_18

Steady-state analysis of the Boost converter in CCM (I)

Can we obtain vO > vg?  Boost converter Step 1: Main waveforms ig

+ vL - i L L

vg

Driving signal

iD

iO

D

iS

+

- RL

C

S

t + vO -

iL t iS

iL + vL -

S on, D off, during dT

DiL

t

L

iD

vg

t i L + vL -

S off, D on, during (1-d)T

L

dT

iO C RL

+ vO -

T

 From Faraday’s law: DiL = vgdT/L 20 SEA_uniovi_CC1_19

Steady-state analysis of the Boost converter in CCM (II) Step 2 and 2 (bis): Average values

ig

 Average value of iC:

+ vL - i L L

iC_avg = 0  Volt·second balance over L:

iD

vg

iC

D

iS

Node1

C

S

iO

+ - RL

+ vO -

vgdT - (vO - vg)(1-d)T = 0 Therefore:

vO = vg/(1-d) (always vO > vg) Driving signal

Step 3: Average KCL and KVL:

t

 KCL applied to Node1 yields:

iD

iD - iC - iO = 0

t

iD_avg - iC_avg - iO = 0 Therefore: iD_avg = iL_avg(1-d) = iO = vO/RL Step 4: Power balance: ig_avg = iL_avg = iOvO/vg = iO/(1-d)

DiL

iD_avg

vg

vL t dT

-(vO-vg) T 21 SEA_uniovi_CC1_20

Steady-state analysis of the Boost converter in CCM (III) ig

+ v L - i L - vD + L

vg

iS S

D iD + vS -

Summary

iO iC

C

RL

Driving signal

+ vO -

vO = vg/(1-d) (always vO > vg)

t vO

vD t

iL

vSmax = vDmax = vO iL_avg = ig_avg = iO/(1-d) = vo/[RL(1-d)]

t iS

DiL t

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) iS_peak = iD_peak = iL_peak

iD

iO

t

dT T

22 SEA_uniovi_CC1_21

Steady-state analysis of the Buck-Boost converter in CCM (I)

Can we obtain either vO < vg or vO > vg?  Buck-Boost converter ig

iD

iS iL S

vg

L

+ vL -

iO D

Driving signal vO +

C

+ RL

t iL

DiL

ig

S on, D off, during dT

t iL

vg

L

iS

+ vL

t

-

iD

Charging stage

t

iO

S off, D on, during (1-d)T

+ C vL + L -

iL

RL

vO +

Discharging stage

dT T

 From Faraday’s law: DiL = vgdT/L 23 SEA_uniovi_CC1_22

Steady-state analysis of the Buck-Boost converter in CCM (II) Step 2 and 2 (bis): Average values  Average value of iC:

ig

vgdT - vO(1-d)T = 0

iS iL

iC_avg = 0  Volt·second balance over L:

Node1 iD iO + vL -

S vg

L

D iC C

+ RL

vO +

Therefore: vO = vgd/(1-d) Driving signal

Step 3: Average KCL and KVL:

t

 KCL applied to Node1 yields:

iD

iD - iC - iO = 0 iD_avg - iC_avg - iO = 0 Therefore: iD_avg = iL_avg(1-d) = iO = vO/RL Step 4: Power balance: ig_avg = iS_avg = iOvO/vg = iOd/(1-d)

DiL

iD_avg t

vg

vL

t -vO

dT T

24 SEA_uniovi_CC1_23

Steady-state analysis of the Buck-Boost converter in CCM (III) ig

i + vS - S iL S vg

L

i D + vD -

+ vL -

D

-

C

Summary

iO

+ RL

Driving signal

vO +

vO = vgd/(1-d) (both vO < vg and vO > vg)

t vO + vg

vD t

iL

vSmax = vDmax = vO + vg iD_avg = iO

DiL = vgdT/L

t iS

DiL t

iL_avg = iD_avg/(1-d) = iO/(1-d) = vo/[RL(1-d)] iS_avg = ig_avg = d·iL_avg = d·vo/[RL(1-d)]

iL_peak = iL_avg + DiL/2 = iL_avg + vgdT/(2L) iS_peak = iD_peak = iL_peak

iD

iO

t

dT T

25 SEA_uniovi_CC1_24

Common issues in basic DC/DC converters (I) L

S

vg

+ C

D

RL

-

+ vO -

Complementary switches + inductor

Buck

L

S

D

vg

+ C

S

- RL

d + vO -

vg

D

1-d L

C

+ - RL

+ vO -

Boost

D S vg

L

Buck-Boost

C

+ RL

+ vO -

Voltage source

The inductor is an energy buffer to connect two voltage sources 26 SEA_uniovi_CC1_25

Common issues in basic DC/DC converters (II) Diode turn-off vg L vg

S

D

Buck

L vg

D

S

vO

Boost

+ C

RL

-

C + - RL

+ vO -

+ vO -

 The diode reverse recovery time is of primary concern evaluating switching losses  Schottky diodes are desired from this point of view

vO + vg S vg

 The diode turns off when the transistor turns on

L

D C Buck- + RL Boost

+ vO -

 In the range of line voltages, SiC diodes are very appreciated

27 SEA_uniovi_CC1_26

Comparing basic DC/DC converters (I) Generalized study as DC transformer (I) ig

iO L

S

vg

C -

D

L S

iO

iO RL

+

+ vO -

- RL

C

+ vO -

1:N DC Transformer

 Buck: N= d (only vO < vg)

Boost

ig

ig_avg

vg

D

vg

iO D

-

S vg

RL

Buck

ig

+ vO -

+

L

C

+ RL

 Boost: N= 1/(1-d) (only vO > vg) + vO -

 Buck-Boost: N= -d/(1-d) (both vO < vg and vO > vg)

Buck-Boost 28 SEA_uniovi_CC1_27

Comparing basic DC/DC converters (II) Generalized study as DC transformer (II) ig_avg

iO RL

vg

+ vO -

1:N DC Transformer

ig_avg = iON = iOd/(1-d)

 Buck: ig_avg = iON = iOd  Boost: ig_avg = iON = iO/(1-d) Buck-Boost: ig_avg = iON = - iOd/(1-d)

29 SEA_uniovi_CC1_28

Comparing basic DC/DC converters (III) Electrical stress on components (I) iS

ig

vg

+

vS

iO

-

S

iD

+ vD D -

RL

+ vO -

DC/DC converter

 Buck:

 Boost:

 Buck-Boost:

vSmax = vDmax = vg

vSmax = vDmax = vO

vSmax = vDmax = vO + vg

iS_avg = ig_avg

iL_avg = ig_avg

iS_avg = ig_avg

iL_avg = iO

iD_avg = iO

iD_avg = iO

iD_avg = iL_avg - iS_avg

iS_avg = iL_avg - iD_avg

iL_avg = iS_avg + iD_avg 30 SEA_uniovi_CC1_29

Comparing basic DC/DC converters (IV)

Example of electrical stress on components (I) 1 A (avg)

2A L

S

100 V

C

D

+

+ -

RL

50 V

-

100 W Buck, 100% efficiency 2A

1 A (avg) D S 100 V

L

C

-

+ RL +

50 V

100 W Buck-Boost, 100% efficiency

vS_max = vD_max = 100 V iS_avg = iD_avg = 1 A iL_avg = 2 A FOMVA_S = FOMVA_D = 100 VA 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 BuckBoost converter  Therefore, lower actual efficiency 31 SEA_uniovi_CC1_30

Comparing basic DC/DC converters (V)

Example of electrical stress on components (II) 4 A (avg)

2A L

25 V

D S

C

vS_max = vD_max = 50 V iS_avg = iD_avg = 2 A iL_avg = 4 A FOMVA_S = FOMVA_D = 100 VA

+

+ - RL

-

50 V

100 W Boost, 100% efficiency 2A

4 A (avg) D S 25 V

L

C

-

+ RL +

50 V

100 W Buck-Boost, 100% efficiency

 Higher electrical stress in the case of BuckBoost converter  Therefore, lower actual 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

32 SEA_uniovi_CC1_31

Comparing basic DC/DC converters (VI)

 Price to pay for simultaneous step-down and stepup 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 33 SEA_uniovi_CC1_32

Comparing basic DC/DC converters (VII) Example of power conversion between similar voltage levels based on a Boost converter 6.12 A (avg)

50 V

5A

L

1.12 A (avg)

D S

C

+ - RL

300 W Boost, 98% efficiency

+ -

60 V

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

Very high efficiency can be achieved!!! 34 SEA_uniovi_CC1_33

Comparing basic DC/DC converters (VIII) The opposite case: Example of power conversion between very different and variable voltage levels based on a BuckBoost converter 20 - 2 A (avg)

5A D

S 20 - 200 V

L

C

-

+ RL +

60 V

300 W Buck-Boost, 75% efficiency Remember previous example: FOMVA_S = 67.2 VA FOMVA_D = 300 VA

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

High efficiency cannot be achieved!!! 35 SEA_uniovi_CC1_34

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)

L vg

S

D

+

C Boost

- RL

+ vO -

Buck and Buck-Boost do not exhibit these problems 36 SEA_uniovi_CC1_35

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

L

S

idevice MOSFET

D

Diode

L S1 S2

L S1 S2

vdevice 37 SEA_uniovi_CC1_36

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 L L

S2

S1 vO

S2

S1 D

C -

RL

+ vO -

Synchronous Buck

Q’ PWM

Q

vg

+

Av

Vref

Feedback loop 38 SEA_uniovi_CC1_37

Input current and current injected into the output RC cell (I)

 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

ig

iS

vg

+

vS S

iRC

iD

+ vD D -

DC/DC converter

Desired current

ig

C

+ + v - RL - O

Desired current

iRC t

t 39 SEA_uniovi_CC1_38

Input current and current injected into the output RC cell (II) iRC

ig

ig S

vg

t

L

+ C

D

Noisy

-

iRC

ig

L

D

vg

t

t Low noise

Buck

ig

S

Low noise

+ - RL

C

+ vO -

iRC t Noisy

Boost

ig

iRC D

ig

S

t Noisy

RL

+ vO -

iRC

vg

L Buck-Boost

C

+ RL

+ vO -

iRC t

Noisy 40 SEA_uniovi_CC1_39

Input current and current injected into the output RC cell (III) Adding EMI filters

iRC

ig LF CF

vg

+ -

L

S D

+ C -

+ vO -

RL

Buck

Filter

iRC

ig L

D

vg

S

CF

Boost

vg

CF Filter

+ -

C

Filter

ig LF

LF

+ -

D S L Buck-Boost

+

LF CF

+ - RL

+ vO -

R + L

vO +

iRC

C

Filter 41 SEA_uniovi_CC1_40

Four-order converters (converters with integrated filters) L1

ig

iD

C1 + iS

vg

 Same vO/vg as Buck-Boost  Same stress as Buck-Boost  vC1 = vg  Filtered input

D

-

vC1

iL2 L2

S

+ C2 -

vO

RL

ig

SEPIC

 Same vO/vg as Buck-Boost  Same stress as Buck-Boost  vC1 = vg + vO  Filtered input and output iS

C1

vg

iS

vg

- +

L1

+ C2

iL1

iD

Zeta

-

vC1

S

iD D

-

RL

vO C2 +

Cuk

L2 D

L2

iL2

C1 +

iL2

vC1

S

L1

-

RL

vO

 Same vO/vg as Buck-Boost  Same stress as Buck-Boost  vC1 = vO  Filtered output 42 SEA_uniovi_CC1_41

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) iL L

ig

vg

S

D DC/DC converter

iL

iL_avg

iO RL

t + vO -

Driving signal

t dT T

 Buck:

 Boost:

 Buck-Boost:

iL_avg = iO

iL_avg = iO/(1-d)

iL_avg = iS_avg + iD_avg = diO/(1-d) + iO = iO/(1-d) 43 SEA_uniovi_CC1_42

DC/DC converters operating in DCM (II)

 When the load decreases, the converter goes toward Discontinuous Conduction Mode (DCM)

Decreasing load

iL

RL_1 iL_avg

t iL

Operation in CCM

RL_2 > RL_1 iL_avg

t iL

RL_crit > RL_2 iL_avg t

Boundary between CCM and DCM

It corresponds to RL = R L_crit 44 SEA_uniovi_CC1_43

DC/DC converters operating in DCM (III)

Decreasing load

What happens when the load decreases below the critical value?

iL

iL

 DCM

RL_crit iL_avg

 If a synchronous rectifier (SR) is used,

iL_avg

the operation depends on the driving signal

RL_3 > RL_crit

t

RL_3 > RL_crit

iL_avg

DCM w. diode

rectifier

t

CCM w. SR

iL

starts if a diode is used as

 CCM

operation is possible with synchronous rectifier with a proper driving signal (synchronous rectifier with signal almost complementary to the main transistor)

t

45 SEA_uniovi_CC1_44

DC/DC converters operating in DCM (IV) Remember: iL_avg = iO (Buck) or iL_avg = iO/(1-d) (Boost and Buck-Boost)

iL

 For a given duty cycle, lower average

RL > RL_crit CCM w. SR

iL_avg t

 For a given duty cycle, higher average

RL > RL_crit

iL

value (due to the negative area)  lower output current for a given load  lower output voltage

DCM w. diode

iL_avg t

value (no negative area)  higher output current for a given load  higher output voltage

The voltage conversion ratio vO/vg is always higher in DCM than in CCM (for a given load and duty cycle) 46 SEA_uniovi_CC1_45

DC/DC converters operating in DCM (V) How can we get DCM (of course, with a diode as rectifier) ? iL After decreasing the inductor inductance t iL

iL

After decreasing the switching frequency t After decreasing the load (increasing the load resistance) t 47 SEA_uniovi_CC1_46

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-

Driving signal

t

iL

iL_avg t

iD

vg

d-d’)T Example: Buck-Boost converter ig

iD_avg

S

+

vg

- -vO d’·T

t

iD

iL

t

vL

d·T

iS

L

ig

vg

D

C

+ RL

vO +

iO iL

T

+ vL -

iO

L

+ vL -

During d·T

+ C vL + RL L -

iL

During d’·T

vO +

iL

+ vL

L

-

During (1-d-d’)T 48 SEA_uniovi_CC1_47

DC/DC converters operating in DCM (VII) Voltage conversion ratio vO/vg for the Buck-Boost converter in DCM ig

iL

Driving signal

iL

vg

iL_max

t iL_avg

iD

vg

t

iL_max iD_avg

+ d·T

- -vO d’·T T

vL

From Faraday’s law: vg = LiL_max/(dT)

-

During d·T + C vL + RL L -

iL

iO vO +

And also: vO = LiL_max/(d’T)

During d’·T

t

vL

L

+

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) 49 SEA_uniovi_CC1_48

DC/DC converters operating in DCM (VIII) The Buck-Boost converter just on the boundary between DCM and CCM

iL

RL = RL_crit iL_avg

t

 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 50 SEA_uniovi_CC1_49

DC/DC converters operating in DCM (IX) Summary for the basic DC/DC converter

Buck

Boost

N=d

1

N=

1-d

Buck-Boost N=

2

M= 1+

4k 1+ 2 d

kcrit = (1-d)

kcrit_max = 1

1+ M=

4d2 1+ k 2

kcrit = d(1-d)2 kcrit_max = 4/27

M=

d 1-d d k

kcrit = (1-d)2 kcrit_max = 1 k = 2L/(RT) 51 SEA_uniovi_CC1_50

DC/DC converters operating in DCM (X) CCM versus DCM Driving signal

Driving signal

t t

vD

t iL iS

iD

T

- Lower conduction losses in CCM (lower rms values)

iL_avg

- Lower losses in DCM when S turns on and D turns off

t

- Lower losses in CCM when S turns off

t

t

dT

vD

- Lower inductance values in DCM (size?)

t iL

iL_avg

iS

t

iD

t

t

dT T

52 SEA_uniovi_CC1_51

Achieving galvanic isolation in DC/DC converters (I)

- A two-winding magnetic device is needed - The volt·second balance in the case of magnetic devices with two windings must be used From Faraday’s law:

+ v1 -

+ v2 n1:n2

vg

Circuit in steadystate

vi = ni d/dt



B

D= B - A = (vi/ni)·dt A

In steady-state: (D)in a period = 0 And therefore: (vi /ni)avg = 0

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 53 SEA_uniovi_CC1_52

Achieving galvanic isolation in DC/DC converters (II) Transformer models Model 1

Lm1 n1:n2

Ll1

Model 2

Ll2

Lm1 n1:n2

n1:n2

Model 1:

Model 2:

Model 3:

Circuit Theory element

Magnetic transformer with perfect coupling

Magnetic transformer with real coupling

At least the magnetizing inductance must be taken into account analyzing DC/DC converters

54 SEA_uniovi_CC1_53

Achieving galvanic isolation in DC/DC converters (III) Where must we place the transformer?

Lm1 n1:n2

In a place where the average voltage is zero

ig

iO + vg

vS S

-

+ vD D -

RL

+ vO -

DC/DC converter

55 SEA_uniovi_CC1_54

Achieving a Buck converter with galvanic isolation (I) L

S

vg

C Buck

D

+ - RL

No place with average voltage equal to zero

+ vO -

New node with possible zero average voltage L vg

S D

S on

S off

C

+ vO -

+ - RL

D2 L

vg

S

Lm1

D1

+

C

-

RL

+ vO -

n1:n2

It does not work!! 56 SEA_uniovi_CC1_55

Achieving a Buck converter with galvanic isolation (II)

vextra

A circuit to apply a given DC voltage across Lm1 when S is off

S off

S on

D2

n3

L vg

D1

Lm1

+

C

-

RL

+ vO -

n1:n2

n1:n1:n2

D2 L

Lm1

vg S

D1

+ C

-

RL

+ vO -

D3 Final implementation: the

Standard design: vextra = vg n3 = n1

Forward converter 57 SEA_uniovi_CC1_56

The Forward converter As the Buck converter replacing vg with vgn2/n1 n1:n1:n2

L D1

Lm1

+

C

L

+ vO -

S & D2 on, D1 & D3 off, during dT

S & D2 off, D1 on,

S

RL

-

D3

vg

iO

iL

D2

C RL

vgn2/n1

+ vO -

Inductor magnetizing stage im1 +

D3 on, during d’T im1

during (1-d)T iO

iL L

C RL

vg

+ vO -

Inductor demagnetizing stage

Lm1

vL -

Transformer reset stage

-

Transformer magnetizing stage

+

vg

Lm1

vL

vO = dvgn2/n1

vSmax = 2 vg dmax = 0.5 (reset transformer) 58 SEA_uniovi_CC1_57

Achieving a Buck-Boost converter with galvanic isolation (I) D S vg

L

BuckBoost

S

vg

C

+ RL

vO +

There is a place with average voltage equal to zero: the inductor

D

Inductor and transformer integrated into only one - magnetic device (two-winding inductor) vO +

C

Lm1

L

+

RL

RL

vO +

n1:n2

S on

vg

S off S

D

C

L

+

n1:n2 59 SEA_uniovi_CC1_58

Achieving a Buck-Boost converter with galvanic isolation (II)

vg

D

S

C

L

+

RL

vO +

n1:n2

Two-winding inductor D

n1:n2

L1

L2

+ C

-

RL

+ vO -

S on, D off, during dT

ig

+ vg

L1

vL -

Charging stage

iO vg

+ S

Final implementation: the Flyback converter

S off, D on, during (1-d)T

vLn2/n1 -

C L2 +

RL

vO +

Discharging stage 60 SEA_uniovi_CC1_59

The Flyback converter Analysis in steady-state in CCM  Volt·second balance:

L1 vg S

dTvg/n1 - (1-d)TvO/n2 = 0

D

n1:n2

L2

+ C

-

RL

+ vO -

 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 61 SEA_uniovi_CC1_60

Achieving other converters with galvanic isolation (I) L1

L

D C Boost

S

vg

RL

+ -

+

+ vO -

vg

S

It is not possible with only one transistor!! L1

C1 +

n1:n2 -

L2

C2 + -

Vg D

S

Cuk

C3

C1 L2

n1:n2 D

+ C2 -

RL vO

SEPIC

 Zeta converter is also possible RL VO +

 vO = vg(n2/n1)d/(1-d)  vSmax = vg + vOn1/n2  vDmax = vgn2/n1 + vO Like the Flyback converter 62 SEA_uniovi_CC1_61

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