Switch Mode Power Supplies:From Circuit Theory to the Workbench

Michael TsePower Electronics Research CentreDepartment of Electronic & Information EngineeringHong Kong Polytechnic University

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Prelude

How much can we trust the theory?What can we do if the theory doesn’t match the outcome?Can we live without heuristics?Are complex models always better?What makes the engineers lose faith in the theory?Can theory really be used in practice?

How can we get the most of circuit theory forpractical design and analysis?

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The theorists at fault!!My EMI filter isn’t quite doing what it is supposed to do. Letme change the order and a different filter configuration, andtry again.

I need to derive a more accurate model for my powerconverter, preferrably up to 10 times the switchingfrequency.

Oh! I have found the answer from my algebra. A duty cycleof 1.2 for this theoretical power factor control!! How could Iachieve this?

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The engineers at fault!!The snubber gets pretty hot. Let me choose a smallerresistance.

I would tend to think that the transformer has splitcharacters. The flyback transformer has never convinced methat it is related to the forward transformer.

Air gap stores more energy.

The output voltage of my constant-power-controlled circuitisn’t high enough. Let me get a few more turns in thesecondary.

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Objectives

To show how one can arrive at a practical circuit fromconsideration of basic circuit theory.

To show how circuit theory can be used to explainphenomena observed in practical circuits.

To show how a switch mode power supply can besystematically cons-tructed, starting from the simplestconverter topology.

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Contents

Basic topologies and practical requirementsHow theory solves problems:

First problem: practical transformerSecond problem: real device switchingThird problem: closed-loop controlFourth problem: isolationFifth problem: input filter

Conclusion

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Genesis of convertersAim: To convert controllable powerfrom a voltage source to a load,with NO LOSS.

Source Load

?

Lossless

Controllable

Simple

Kirchhoff’s laws restrict terminalconditions.

Source Load

?

Lossless

Controllable

Simple

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An old solutionLossy linear regulator

Source Load

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Elements wantedLossless requirements

current sinking for inputcurrent sourcing foroutput

IdeasAn inductor switchingbetween source and loadRelative sourcing andsinking durations wouldcontrol the energy flow

Bear in mind whatKirchhoff’s laws say:

Inductors must not be leftopen.

-> An inductor switchingbetween source and load.

-> At least one current pathmust be available at alltimes.

-> At least two switches areneeded to divert theinductor current

-> THREE POSSIBILITIES

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Three possibilities

Boost

Buck-boost

Buck

Source LoadiL

Source Load

iL

Source Load

L

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The buck converterSwitch S is turned on and off very quickly, at a rate muchgreater than the output filter natural frequency

Control parameter is duty cycle

iLLS

D

ON

OFF+

–E

+

–U

dS t

T

Duration when is on

Periodon= =

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Steady-state operationSuppose we fix the duty cycle and wait until a steady state isreached.Inductor current goes up during the ON time, and goes down duringthe OFF time.

Periodic operation forces

Increment during ON = Decrement during OFF

⇒−

=−

⇒ = ×

( ) ( )E U DT

L

U D T

L

U D E

1

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Putting it to practiceWe need

transformer isolationclosed-loop controldrivers for MOSFETsself start-upsnubbers as switching aidsprotectioninput EMI filterProper component selectionmechanical design: heat sink, layout, packaging, etc

DirectMandatoryRequirements

IndirectMandatoryRequirements

RegulatoryRequirements

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Practical circuit requirementForward converter—transformer isolatedbuck converter

?

Does it work?

?

driver

EMI filter needed

ControlIC andcircuits

?

isolation?

?

start-up

fromaux

output

? ?

EMI

protection (over I / V)

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First problem: practical transformer

The previous forward converter worked only if thetransformer were ideal.However, practical transformers have magnetisinginductance.

L m

Ideal transformer

1:n

v nv

in

i

2 =

=

1

2 11

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What the theory says?

If we have to use a transformer for the forwardconverter, the transformer should be idea.That means INDEFINITELY LARGE magnetizinginductance.

⇓Either an infinitely permeable core

OR an infinite number of turns

THAT’S IMPOSSIBLE!!

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Probing further into the theory

First, consider the ON time.

Ideal transformer

1:n

Practical transformer

What happens when the FET is off?

Secondary of T/F hasno current.

↓Primary has no currenteither.

↓Current in magnetisinginductance can gonowhere!!

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Deriving solution: core reset

A path must exist during OFF time to bring themagnetizing current back to zero

Circuit theory works and explains everything!

Ideal transformer

1:n

Practical transformer

+

–Vz

0A0A

+

– –

+

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Waveforms (with core reset)Secondary current of ideal t/f

same asabove;scaled by factor n

0

0

Primary current of practical t/f

Primary current of ideal t/f

Magntizing current

slopecontrolledby outputinductance

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Requirements for core reset

Negative voltage polarity applied to the windingduring OFF time.This voltage must be large enough to bring themagnetising current back to zero.

If d = 0.5, then .

If d = 0.8, then .

Technique: clamping the voltage during OFF time.

V Vz in≥

V Vz in≥ 4

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Some core reset possibilities1:n

Vz

1:n

1:n

1:n

nt

Tertiary winding as voltage clamp

n controls clamped voltage for re-sett

Advantage:

Two-wheeler forward converter

Duty cycle restricted below 0.5

Advantage: simple transformerDisadvantage:

Disadvantage: bulky transformer

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Recapitulation

Although we cannot make an ideal transformer, wesolve the problem with a reset circuit.

We now care much less how large Lm is, since wehave a way to get around it.

QUESTION: Can the magnetizing inductance be usedto advantage?

YES, in a flyback converter!

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Flyback converterIn this case, we don’t needan ideal transformer. That’sgood. We don’t have itanyway.The magnetizing inductancebecomes crucial as part ofthe circuit element.Requirement:

Linear inductor!Air-gap to augment BHcurve (then more turns toobtain inductance)

1:n

0A0A

1:n

During ON-time, magnetizing inductance charges up.

+

–

reverse polarity, current can't go!

ideal t/f primary current = 0

During OFF-time, magnetizing current forces its way outthrough the ideal t/f primary.

+

–

di/dt negative,hence reversepolarity

+

–

+

–

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Second problem: real switchingConsider turning theswitch in a buckconverter.The result of this realdevice switching is

POWER LOSS(switching loss)

+

–E

+

–U

0

5

idsvds+ –

turn-off

ids

vds

diode won't conduct unless forward biased

ON OFF

diode turns on

ids can't go downunless diode turns on

i.e., vds reaches the input voltage

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Deriving solution: snubberGive the switch current achance to go down before thediode turns on.

⇓Set up a PARALLEL CURRENTPATH right after the turn-offinstant.

⇓Place a capacitor across theswitch at turn-off to supplycurrent for the output inductor.

+

–E

+

–U

0

5

ids

turn-off

idsvds

ON OFF

ids goes down immediately

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Completing the solutionWhat happens when the switch is turned on again in the next cycle?The current will rush through the switch!!

⇓We must protect the switch from such huge in-rush.The complete snubber is:

Energy loss per cycle

=+

–E

+

during turn-on

during turn-off

Snubber

1

22C vssnubber

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Other examples (same principle)

+

–E

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Third problem: closed-loop controlWhy?Because the system is dynamic.What is a dynamical system?A simplified definition: a system that does not assume an operating point instantlywhen an input parameter is changed.

0.2

0.4

4.8V

9.6Voutput voltage

duty cycle

Consider a buck converter with input 24V.

Suppose d is forced to step up from 0.2 to 0.4.

Suppose d stays constant, but the loadresistance steps up.

4.8Voutput voltage

load resistance

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The need for controlObviously we need to controlthe duty cycle if we want thesystem to have a dynamicbehaviour different from thenatural behaviour.

2 common approachesVoltage mode controlCurrent mode control

+–

voltage-mode fb

current-mode fb

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Voltage mode controlA general feedback circuit representation is:

Z f

Vref

–+

Vout

–+

Vm

vcon R1

R2

vR R

Z R Rv

R

R R

Z

Z R Rvcon

fref

f

fout

1 2

1 2

2

1 2 1 2

||

( || ) ( || )+

= −+ +

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Small-signal analysis

We can separate the AC from theDC component.Let’s not worry about the steady-state operating point.The small-signal AC equation is:

Taking into account the PWM, wehave

vR R

Z R Rv

R

R R

Z

Z R Rvcon

fref

f

fout

1 2

1 2

2

1 2 1 2

||

( || ) ( || )+

= −+ +

∆ ∆vZ

Rvcon

fout= −

1

∆ ∆dV

Z

Rv

m= −

1

1

fout

If we know the duty-cycle-to-output small-signal transferfunction, then we can find theloop gain and hence be able todesign the required compensatorto give sufficient bandwidth andstability.

HERE, we need small-signalmodels from circuit theory.

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Design criteria

Fast response — high gain and wide bandwidth of loop gainStability — phase shift must be well below 180deg at 0dBcrossover.

The converter is a second-order system which can becomeunstable under closed-loop condition, especially when the gain ishigh causing the phase shift of the loop gain to get close to180deg at 0dB crossover.

We must limit the bandwidth somehow if we allow a high DCgain.

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Workbench construction

The concept of POLE — frequency location where the gain beginsto roll off.

E.g., if the circuit has a capacitor to ground forming an RC networkwith the rest of the equivalent resistance, then a pole exists at1/2 CR Hz.

A simple educated trial-and-error:Select an RC combination (in the compensation circuit associated withthe control IC) for a deep lag compensation — narrow band first.Then, relax the time constant (widen the band) until the circuit beginsto oscillate!Finally, reduce the capacitor value down 10 times to restore stability.

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Current mode control

Essential concept — The systemreduces to first order, more orless!The system is therefore faster,with less chance for instability.

IDEA:Make the inductor currentdependent on the output voltageby forcing the current peak tofollow the output voltage analog.Disqualify the inductor current asa state variable.The converter becomes firstorder.

Z f

Vref

–+

Vout

–+

v con

R1

R2

currentsensor

R

SQ

driver

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Fourth problem: isolation

Need for isolating the load fromthe mains.But the control circuit connectsthe two!Put control in the primary-side

Mainsrectifier

driver

control IC

Vout

+V

aux. supply

bootstrapstart-up

Mainsrectifier

driver

control IC

Vout

+V

aux. supply

Put control in the secondary-side

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Fifth problem: filter

An input filter is always needed toprevent differential-mode andcommon-mode noise from getting intothe mains.

L

N

E

~ differential

commoncommon

SMPS

Basic requirement:Let 50Hz gets in, butprevent highfrequencies fromgetting out!

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Basic filter theory

Voltage filter (low-passleft-to-right)

Current filter (low-passright-to-left)

Voltage filter (low-passright-to-left)

noise currentat switchingfrequencyand above

50Hz SMPS load

x

noise voltageat switchingfrequencyand above

x+–

Rs

Rs

Rs

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Conceptual placement

Voltage filter (low-passleft-to-right)

Current filter (low-passright-to-left)

Voltage filter (low-passright-to-left)

SMPS

L

N

E

SMPS

L

N

E

Differential-mode filter

Common-mode filter

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Practical placement

NOTE:

The EMI filter oftenfails to do what itis supposed to do.Does the theoryfall short ofanything?Or have we missedout someimportant things?!

SMPS

L

N

E

common-modeand differential-mode

capacitor

common-mode chokedifferential-mode choke(may be provided by leakageof common-mode choke)

differential-modecapacitor

differential-modecapacitor

LcLd

Cc1 Cc2Cm1

Cc1Cm

2+

Cm2,

1 Cc2Cm

2+ 2

2LdLc+

Ld2

2Cm1

2Cm2

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Conclusion

The curses to the theory:A capacitor may not behave as a capacitor.An inductor may not behave as an inductor.Parasitics and nonlinearity strike in.Signals get around the filter, instead of being filtered.

The theory does not fail. The engineers fail to identify theright ingredients for constructing theories that are viable.

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There is nothing more practical than a good theory.

— James C. Maxwell