Ultimate Switching: Toward a Deeper Understanding of Switch Timing Control in Power Electronics and Drives P. T. Krein, Director Grainger Center for Electric

Ultimate Switching: Ultimate Switching: Toward a Deeper Toward a Deeper Understanding of Switch Timing Control in Power Understanding of Switch Timing Control in Power

Electronics and DrivesElectronics and Drives

P. T. Krein, DirectorP. T. Krein, DirectorGrainger Center for Electric MachineryGrainger Center for Electric Machinery

and Electromechanics and ElectromechanicsDept. of Electrical and Computer EngineeringDept. of Electrical and Computer Engineering

University of Illinois at Urbana-ChampaignUniversity of Illinois at Urbana-Champaign

Grainger Center for Electric Machines and Electromechanics University of Illinois at Urbana-ChampaignGrainger Center for Electric Machines and Electromechanics University of Illinois at Urbana-Champaign

22

OutlineOutline

• Fundamentals: power electronics control at Fundamentals: power electronics control at its basic levelits basic level

• MotivationMotivation• False starts and model-limited controlFalse starts and model-limited control• Small-signal examplesSmall-signal examples• Ultimate formulationUltimate formulation• Geometric control examplesGeometric control examples


33

FundamentalsFundamentals

• In any power electronic circuit or system, In any power electronic circuit or system, control can be expressed in terms of the control can be expressed in terms of the times at which switches operate.times at which switches operate.

• The fundamental challenge is to find The fundamental challenge is to find switching times for each device.switching times for each device.

• Example:Example:– For each switch in a converter, find switching For each switch in a converter, find switching

times that best address a set of constraints.times that best address a set of constraints.– This is an optimal control problem of a sort.This is an optimal control problem of a sort.– Might represent this with a switching function q(t).Might represent this with a switching function q(t).


44

FundamentalsFundamentals

• The general problem is daunting, so we The general problem is daunting, so we simplify and address switch timing indirectly.simplify and address switch timing indirectly.– Averaging (address duty ratio rather than q)Averaging (address duty ratio rather than q)– PWM (use d as the actuation, not just the control)PWM (use d as the actuation, not just the control)– Sigma-delta (make one decision each period Sigma-delta (make one decision each period

based only on present conditions)based only on present conditions)– Other approachesOther approaches

• We are researching to try and identify ways to We are researching to try and identify ways to address the timing questions more directly.address the timing questions more directly.


55

MotivationMotivation

• We believe that a new and more fundamental We believe that a new and more fundamental consideration of a switch timing framework consideration of a switch timing framework has strong potential benefits.has strong potential benefits.

• Motivated by our work on switching audioMotivated by our work on switching audio– Showed that sine-triangle PWM, used as a basis Showed that sine-triangle PWM, used as a basis

for audio amplifiers, provides nearly unlimited for audio amplifiers, provides nearly unlimited fidelity.fidelity.

• Motivated by past work on geometric and Motivated by past work on geometric and nonlinear controlnonlinear control– Performance can be achieved in power converters Performance can be achieved in power converters

that is unreachable with averaging approaches.that is unreachable with averaging approaches.


66

False StartsFalse Starts

• Many argue that space-vector modulation Many argue that space-vector modulation (SVM) gets more directly at switch timing.(SVM) gets more directly at switch timing.

• In fact, SVM addresses duty ratios and yields In fact, SVM addresses duty ratios and yields (at best) (at best) exactlyexactly the same result as a PWM the same result as a PWM process. It is usually worse because uniform process. It is usually worse because uniform sampling is involved.sampling is involved.

• Small-signal analysis methods are even less Small-signal analysis methods are even less direct.direct.

• Sliding-mode controls “confine” the switching Sliding-mode controls “confine” the switching without getting to the timing challenge.without getting to the timing challenge.


77

Space Vectors in Time DomainSpace Vectors in Time Domain

• Space vector Space vector modulationmodulation

• Third-harmonic injection Third-harmonic injection sine-triangle PWMsine-triangle PWM


88

Model-Limited ControlModel-Limited Control

• Many control methods used in today’s Many control methods used in today’s switching power converters are limited by switching power converters are limited by the the modelsmodels of the systems. of the systems.

• ““Model-limited control” is an important Model-limited control” is an important barrier to improvement of converters.barrier to improvement of converters.


99


• Any type of PWM implies switchingAny type of PWM implies switchingthat takes place much faster thanthat takes place much faster thansystem dynamics.system dynamics.

• Dc-dc converters use controllersDc-dc converters use controllersdesigned based on averaging.designed based on averaging.

• We often learn that bandwidths areWe often learn that bandwidths arelimited to a fraction of the switching rate.limited to a fraction of the switching rate.

• We finally have the tools to interpret this We finally have the tools to interpret this rigorously.rigorously.


1010


• Distortion in the low-frequency band can be Distortion in the low-frequency band can be computed as a function of switching frequency ratio.computed as a function of switching frequency ratio.

• Distortion must be at least -40 dB (better -60 dB) to Distortion must be at least -40 dB (better -60 dB) to justify control loop design.justify control loop design.

• Based on natural sampling:Based on natural sampling:Frequency ratioFrequency ratio In-band distortionIn-band distortion

55 -9 dB -9 dB 7 7 -42 dB -42 dB 9 9 -70 dB -70 dB1111 -110 dB -110 dB1313 -154 dB -154 dB1515 -201 dB -201 dB 10 10-10-10

• This is consistent with signal arguments that yield 2This is consistent with signal arguments that yield 2 as the minimum ratio and “rules of thumb” about a as the minimum ratio and “rules of thumb” about a ratio of 10 for best results.ratio of 10 for best results.


1111


• These models are convenient and useful, but These models are convenient and useful, but do not use the full capability of a conversion do not use the full capability of a conversion circuit.circuit.

• We gave up a factor of 10 on dynamic We gave up a factor of 10 on dynamic performance in exchange for precision.performance in exchange for precision.

• Consider an example:Consider an example:– Small-signal methods and models are powerful Small-signal methods and models are powerful

tools for analysis and design.tools for analysis and design.– They can only go so far toward the analysis of They can only go so far toward the analysis of

large-signals circuits and disturbances. large-signals circuits and disturbances.


1212

Small-Signal Response ExamplesSmall-Signal Response Examples

• Take a dc-dc converter, with a well-designed Take a dc-dc converter, with a well-designed feedback control. Explore its response.feedback control. Explore its response.

• In this case, a known sinusoidal disturbance In this case, a known sinusoidal disturbance is applied at the line input.is applied at the line input.

• Its frequency is 5% of the switching rate.Its frequency is 5% of the switching rate.• Its magnitude is 10%.Its magnitude is 10%.• The controller is adjusted to The controller is adjusted to cancel line cancel line

variation completelyvariation completely – the duty ratio tracks – the duty ratio tracks and cancels the disturbance based on small-and cancels the disturbance based on small-signal analysis.signal analysis.


1313

Buck ConverterBuck Converter

• In this example, a “feedforward” In this example, a “feedforward” compensation is used to eliminate changes compensation is used to eliminate changes caused by line variation.caused by line variation.

VIN

iIN

vOUT

L IOUT

VOUT

#2

#1

RLOAD

time

v (t)

Volta

ge (V

), cu

rrent

(A) (t)i


1414

Example Dc-Dc Converter ProblemExample Dc-Dc Converter Problem

• 10% disturbance around 80% reference value.10% disturbance around 80% reference value.• Frequency is 1/20 of switching (e.g. 5 kHz on 100 kHz).Frequency is 1/20 of switching (e.g. 5 kHz on 100 kHz).


1515

Compensated PWM OutputCompensated PWM Output

• Filter time constant about 1/10 of switching.Filter time constant about 1/10 of switching.


1616

Result?Result?

• Is the disturbance rejected or not?Is the disturbance rejected or not?– Yes and no.Yes and no.

• Does this controller achieve the requested Does this controller achieve the requested bandwidth?bandwidth?– In fact, the controller is In fact, the controller is completelycompletely eliminating eliminating

linearlinear aspects of the disturbance. aspects of the disturbance.– But the output ripple has features that may not be But the output ripple has features that may not be

preferred.preferred.

• Now, ignore small signal limits.Now, ignore small signal limits.


1717

Example Dc-Dc Converter ProblemExample Dc-Dc Converter Problem

• 10% disturbance around 80% reference value.10% disturbance around 80% reference value.• Frequency is 3/4 of switching.Frequency is 3/4 of switching.

1.2

1.1

trip j k( )

s3lev j k m( )

ref j m( )

40960 j


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Output RippleOutput Ripple

0 500 1000 1500 2000 2500 3000 3500 400010

5

0

5

10

s3iiii

iii

Current


1919

Result?Result?

• In several ways, the result is the same, In several ways, the result is the same, although filtering is less effective because of although filtering is less effective because of the higher frequency.the higher frequency.

• There is an aliasing effect (but there was There is an aliasing effect (but there was previously as well).previously as well).

• The disturbance frequency does not appear The disturbance frequency does not appear in the output.in the output.


2020

Quick Performance CheckQuick Performance Check

• Hysteresis control instead, 150 kHz disturbance.Hysteresis control instead, 150 kHz disturbance.


2121

Hysteresis MethodHysteresis Method

• Now the ripple is tied only to the switching Now the ripple is tied only to the switching rate.rate.

• The disturbance has no noticeable influence The disturbance has no noticeable influence on the output.on the output.

• This is true even though the disturbance is This is true even though the disturbance is faster than the switching frequency!faster than the switching frequency!

• Does this mean the converter has a Does this mean the converter has a “bandwidth” greater than its switching “bandwidth” greater than its switching frequency?frequency?


2222

CommentsComments

• ““Frequency response” and “bandwidth” imply Frequency response” and “bandwidth” imply certain converter models.certain converter models.

• Physical limits are more fundamental:Physical limits are more fundamental:– When should the active switch operate to provide When should the active switch operate to provide

the best response?the best response?– How soon can the next operation take place?How soon can the next operation take place?– How fast can the converter slew to make a How fast can the converter slew to make a

change?change?

• Hysteresis controls respond rapidly. This is Hysteresis controls respond rapidly. This is an issue of timing flexibility more than of an issue of timing flexibility more than of switching frequency.switching frequency.


2323

Consideration of Disturbance TimingConsideration of Disturbance Timing

• In a buck converter, In a buck converter, anyany line disturbance while line disturbance while the active switch is on will have a direct and the active switch is on will have a direct and immediate effect at the output.immediate effect at the output.

• NoNo line disturbance will have line disturbance will have anyany effect if it effect if it occurs while the active switch is off.occurs while the active switch is off.

• This means an impulse response cannot be This means an impulse response cannot be written without a switching function.written without a switching function.

VIN

iIN

vOUT

L IOUT

VOUT

#2

#1

RLOAD


2424

Consideration of Disturbance TimingConsideration of Disturbance Timing

• This indicates that the nonlinearity cannot be This indicates that the nonlinearity cannot be removed for impulse response.removed for impulse response.

• ““Impulse” is not adequate information to Impulse” is not adequate information to determine the response.determine the response.

• Average models cannot capture timing issues.Average models cannot capture timing issues.• Notice that similar arguments apply to step Notice that similar arguments apply to step

responses and others.responses and others.


2525

The Ultimate FormulationThe Ultimate Formulation

• A converter has some number of switches.A converter has some number of switches.

• For each switch, there areFor each switch, there arespecific times at which aspecific times at which adevice should turn on or off.device should turn on or off.

• The times represent the control action. The times represent the control action. Selection of the times is the control Selection of the times is the control principle.principle.

• For each switch For each switch ii, find a sequence of times , find a sequence of times tti,ji,j that produce the desired operation of the that produce the desired operation of the

converter. converter.


2626


• A converter with ten switches.A converter with ten switches.

• Time sequences Time sequences tt1,j1,j through through tt10,j10,j..


2727


• This is too generic -- there must be constraints This is too generic -- there must be constraints and objectives.and objectives.

• Example: for a dc-dc converter with one active Example: for a dc-dc converter with one active switch, find the sequence of times switch, find the sequence of times ttii that yields an that yields an

output voltage close to a desired reference value.output voltage close to a desired reference value.

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13


2828


• Example: boost dc-dc converter.Example: boost dc-dc converter.

• Find the best time sequence to correct a step Find the best time sequence to correct a step load change and maintain fixed output load change and maintain fixed output voltage.voltage.

VIN

L

VOUT

C R


2929


• Still too generic – no unique solution.Still too generic – no unique solution.

• Also limited in utility.Also limited in utility.

• The proposed constraint deals with The proposed constraint deals with steady-state output and only one specific steady-state output and only one specific dynamic disturbance.dynamic disturbance.

• There were no constraints on switching There were no constraints on switching rates or other factors.rates or other factors.


3030


• More practical: Given an objective that More practical: Given an objective that takes into account power loss, output takes into account power loss, output steady-state accuracy, dynamic accuracy, steady-state accuracy, dynamic accuracy, response times, and other desired factors, response times, and other desired factors, find a sequence of times that yield an find a sequence of times that yield an optimum result.optimum result.

• That is, find a set of times That is, find a set of times ttkk that minimizes that minimizes

an an objective functionobjective function..


3131

• This is a general formulation in terms of a This is a general formulation in terms of a hybrid control problemhybrid control problem..

• Unfortunately, with results framed this way Unfortunately, with results framed this way there are very limited results about there are very limited results about existence of solutions, uniqueness, existence of solutions, uniqueness, stability, and other attributes.stability, and other attributes.

• Still very general, but with a well-formed Still very general, but with a well-formed cost function it might even have a solution.cost function it might even have a solution.

• There is a control opportunity every time a There is a control opportunity every time a switch operates.switch operates.



3232

ImplicationsImplications

• For steady-state analysis, this must yield For steady-state analysis, this must yield familiar results.familiar results.

• A dc-dc converter with loss constraints must A dc-dc converter with loss constraints must act at a specific switching frequency with act at a specific switching frequency with readily calculated duty ratio.readily calculated duty ratio.

• For dynamic situations, the implications are For dynamic situations, the implications are deeper.deeper.– Should a converter operate for a short time at Should a converter operate for a short time at

higher frequency when disturbed?higher frequency when disturbed?– How do EMI considerations affect times?How do EMI considerations affect times?– Are our models accurate and complete enough?Are our models accurate and complete enough?


3333

Geometric Control ExamplesGeometric Control Examples

• Dc-dc buck converter, 12 V to 5 V nominal.Dc-dc buck converter, 12 V to 5 V nominal.• L = 200 uH, C = 10 uF, 100 kHz switching.L = 200 uH, C = 10 uF, 100 kHz switching.

#2

+

_

+

_R

+

_

#1

L

V vin

v

i in Iout

outout load

02TT0

v (t)out

vout

V in

time

Volta

ge


3434

Fixed Duty RatioFixed Duty Ratio

• Steady state, fixed duty ratio.Steady state, fixed duty ratio.• This shows the inductor current and ten times This shows the inductor current and ten times

the normalized capacitor voltage.the normalized capacitor voltage.• The “best” solution given fixed 100 kHz The “best” solution given fixed 100 kHz

switching.switching.

0 5 10 15 20 25 30 35 400.9

0.95

1

1.05

1.1

0 5 10 15 20 25 30 35 400.9

0.95

1

1.05

1.1

iL(t)

vout(t) expanded


3535

Result in State SpaceResult in State Space

• Same data plotted in state space.Same data plotted in state space.

4.99 4.995 5 5.0050.9

0.95

1

1.05

1.1

Capacitor voltage

Indu

ctor

cur

rent

Steady state


3636

Hysteresis ControlHysteresis Control

• Alternative: simply switch based on whether Alternative: simply switch based on whether the output is above or below 5 V.the output is above or below 5 V.

• No frequency constraint.No frequency constraint.

0 20 40 60 80 100 1200.9

0.95

1

1.05

1.1

Hysteresis control on output voltage.


3737

Hysteresis ControlHysteresis Control

• Same result, in state space.Same result, in state space.• These controls need timing constraints to These controls need timing constraints to

prevent chattering.prevent chattering.

4.99 4.995 5 5.0050.9

0.95

1

1.05

1.1

Y

Y2

State space.


3838

Response to Step Line InputResponse to Step Line Input

• Line step from 12 V to 15 V at 42 us.Line step from 12 V to 15 V at 42 us.• Duty ratio adjusts instantly to the right values. Duty ratio adjusts instantly to the right values.

(This would happen in open-loop SCM.) (This would happen in open-loop SCM.)• Transient in voltage occurs.Transient in voltage occurs.

0 50 100 150 200 250 300 350 4000.9

1

1.1

Time (us)


3939

State SpaceState Space

• State space plot shows how much the State space plot shows how much the behavior deviates.behavior deviates.

4.98 4.99 5 5.01 5.02 5.030.9

0.95

1

1.05

1.1

iLi

vci

State space


4040

Same Step – Different ControlSame Step – Different Control

• This is a current hysteresis control, with the This is a current hysteresis control, with the switch set to turn off at a defined peak and on switch set to turn off at a defined peak and on at a defined valley. Same line step.at a defined valley. Same line step.

0 20 40 60 80 1000.9

0.95

1

1.05

1.1

Time (us)


4141


• The step is cancelled perfectly – essentially in The step is cancelled perfectly – essentially in zero time.zero time.

4.99 4.995 5 5.0050.9

0.95

1

1.05

1.1

iLi

vc i

State space


4242

Boost Converter – A Harder TestBoost Converter – A Harder Test

• What about a boost converter step?What about a boost converter step?• Example converter: L = 200 uH, C = 20 uF, 5 Example converter: L = 200 uH, C = 20 uF, 5

V input, 12 V output, 100 kHz switchingV input, 12 V output, 100 kHz switching

VIN

IIN

vin

L iOUT

C R

ILOAD

iC

vLVOUT


4343

Steady State BehaviorSteady State Behavior

0 5 10 15 20 25 30 35 400.5

1

1.5

2

2.5

iL(t)

vout(t) expanded

0 5 10 15 20 25 30 35 400.5

1

1.5

2

2.5

11.85 11.9 11.95 12 12.05 12.1 12.152.3

2.35

2.4

2.45

2.5

Capacitor voltage

Indu

cto

r cu

rren

t

State space


4444

Step Change BehaviorStep Change Behavior

• Step input from 5 V to 6 V at 42 us.Step input from 5 V to 6 V at 42 us.• Very slow transient – even though the duty Very slow transient – even though the duty

ratio values are set to cancel the change.ratio values are set to cancel the change.


4545


• Suggests a faster transition is possible.Suggests a faster transition is possible.

11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 13.21.6

1.8

2

2.2

2.4

Capacitor voltage

Indu

cto

r cu

rren

t

State space


4646

Ad Hoc ControlAd Hoc Control

• Short-term overshoot can be used to Short-term overshoot can be used to dramatically speed the response.dramatically speed the response.

0 100 200 300 400 500 600 700 800 900 10000.5

1

1.5

2

2.5

Time (us)


4747


• Rapid move toward final desired result.Rapid move toward final desired result.

11.8 12 12.2 12.4 12.6 12.8 13 13.21.2

1.4

1.6

1.8

2

2.2

2.4

Capacitor voltage

Indu

cto

r cu

rren

t State space


4848

Augmented BoostAugmented Boost

• Now alter the boost to achieve timing targets.Now alter the boost to achieve timing targets.• This control eliminates the transient.This control eliminates the transient.

0 50 100 150 200 250 300 350 400 450 5000.5

1

1.5

2

2.5iL(t)

vout(t) expanded


4949


• The response never goes outside ripple limits.The response never goes outside ripple limits.

11.85 11.9 11.95 12 12.05 12.1 12.151.9

2

2.1

2.2

2.3

2.4

Capacitor voltage

Indu

ctor

cur

rent

Start

End


5050

More General ResultMore General Result


5151



5252



5353

Research TopicsResearch Topics

• Find examples of high-performance converter Find examples of high-performance converter controls, based on a timing control controls, based on a timing control perspective.perspective.

• Develop design methodologies for them.Develop design methodologies for them.• Formulate sample optimization problems that Formulate sample optimization problems that

address timing control directly.address timing control directly.• Seek controls that address system-level Seek controls that address system-level

factors.factors.• Seek simplifications that reduce costs with Seek simplifications that reduce costs with

little (or no) sacrifice in performance.little (or no) sacrifice in performance.


5454

ConclusionConclusion

• The ultimate in power electronics control is to The ultimate in power electronics control is to find a sequence of switching times that find a sequence of switching times that optimizes a specific objective function.optimizes a specific objective function.

• Some test cases show that performance far Some test cases show that performance far outside the accepted range can be obtained.outside the accepted range can be obtained.

• Good ways to specify constraints, quantify the Good ways to specify constraints, quantify the problem, and optimize are issues for research.problem, and optimize are issues for research.

• Examples show Examples show existenceexistence of such solutions. of such solutions.• The objective is to identify and develop The objective is to identify and develop control control

concepts and methods that use the full physical concepts and methods that use the full physical capability of power electronics.capability of power electronics.

Documents

Ultimate Switching: Toward a Deeper Understanding of Switch Timing Control in Power Electronics and Drives P. T. Krein, Director Grainger Center for Electric