p. t. krein, director grainger center for electric...

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

Electronics and Drives

P. T. Krein, DirectorGrainger Center for Electric Machinery

and ElectromechanicsDept. of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign

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

2

Outline

• Fundamentals: power electronics control at its basic level

• Motivation• False starts and model-limited control• Small-signal examples• Ultimate formulation• Geometric control examples


3

Fundamentals

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

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

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

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


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Fundamentals

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

based only on present conditions)– Other approaches

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


5

Motivation

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

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

for audio amplifiers, provides nearly unlimited fidelity.

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

that is unreachable with averaging approaches.


6

False Starts

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

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

• Small-signal analysis methods are even less direct.

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


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Space Vectors in Time Domain

• Space vector modulation

• Third-harmonic injection sine-triangle PWM

1

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0 60 120 180 240 300 36001234567

Time (angle)Sector boundaries

Switch periodboundaries

Samplinginterval

Ideal phase a

Sw

itch

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1

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0

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0 60 120 180 240 300 36001234567

Time (angle)

Sw

itch

sta

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Model-Limited Control

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

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


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

• Dc-dc converters use controllersdesigned based on averaging.

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

• We finally have the tools to interpret this rigorously.


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

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

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

5 -9 dB7 -42 dB9 -70 dB11 -110 dB13 -154 dB15 -201 dB 10-10

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


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

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

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

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

large-signals circuits and disturbances.


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Small-Signal Response Examples

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

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

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

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


13

Buck Converter

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

VIN

iIN

vOUT

L IOUT

VOUT

#2

#1

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v (t)

Volta

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), cu

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(A)

(t)i


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Example Dc-Dc Converter Problem

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

1.2

1.1

trip j k( )

s3lev j k m( )

ref j m( )

20480 j


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Compensated PWM Output

• Filter time constant about 1/10 of switching.

0 500 1000 1500 20000.5

0

0.5

Current


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Result?

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

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

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

preferred.• Now, ignore small signal limits.


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Example Dc-Dc Converter Problem

• 10% disturbance around 80% reference value.• 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 Ripple

0 500 1000 1500 2000 2500 3000 3500 400010

5

0

5

10

s3iiii

iii

Current


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Result?

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

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

• The disturbance frequency does not appear in the output.


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Quick Performance Check

• Hysteresis control instead, 150 kHz disturbance.

0 10 20 30 40 502

4

6

8

10

12

Time (us)

Vol

tage

Line input


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Hysteresis Method

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

• The disturbance has no noticeable influence on the output.

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

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


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Comments

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

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

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

change?• Hysteresis controls respond rapidly. This is

an issue of timing flexibility more than of switching frequency.


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Consideration of Disturbance Timing

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

• No line disturbance will have any effect if it occurs while the active switch is off.

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

VIN

iIN

vOUT

L IOUT

VOUT

#2

#1

RLOAD


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Consideration of Disturbance Timing

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

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

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

responses and others.


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The Ultimate Formulation

• A converter has some number of switches.• For each switch, there are

specific times at which adevice should turn on or off.

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

• For each switch i, find a sequence of times ti,j that produce the desired operation of the converter.

This image cannot currently be displayed.


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• A converter with ten switches.

• Time sequences t1,j through t10,j.


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

• Example: for a dc-dc converter with one active switch, find the sequence of times ti that yields an output voltage close to a desired reference value.


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


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• Example: boost dc-dc converter.

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

VIN

L

VOUT

C R


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• Still too generic – no unique solution.• Also limited in utility.• The proposed constraint deals with

steady-state output and only one specific dynamic disturbance.

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



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

• That is, find a set of times tk that minimizes an objective function.



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

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

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

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

The Ultimate FormulationThis image cannot currently be displayed.


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Implications

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

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

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

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


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Geometric Control Examples

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

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#1

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Volta

ge


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Fixed Duty Ratio

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

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

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


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Result 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


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Hysteresis Control

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

• No frequency constraint.

0 20 40 60 80 100 1200.9

0.95

1

1.05

1.1

Hysteresis control on output voltage.


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Hysteresis Control

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

prevent chattering.

4.99 4.995 5 5.0050.9

0.95

1

1.05

1.1

Y

Y2

State space.


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Response to Step Line Input

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

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

0 50 100 150 200 250 300 350 4000.9

1

1.1

Time (us)


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State Space

• State space plot shows how much the 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


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Same Step – Different Control

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

0 20 40 60 80 1000.9

0.95

1

1.05

1.1

Time (us)


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State Space

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

4.99 4.995 5 5.0050.9

0.95

1

1.05

1.1

iLi

vc i

State space


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Boost Converter – A Harder Test

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

V input, 12 V output, 100 kHz switching

VIN

IIN

vin

L iOUT

C R

ILOAD

iC vL

VOUT


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Steady State Behavior

0 5 10 15 20 25 30 35 400.5

1

1.5

2

2.5

iL(t)

vout(t) expanded

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

ctor

curr

ent

State space


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Step Change Behavior

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

ratio values are set to cancel the change.

0 200 400 600 800 1000 1200 1400 1600 1800 20000

1

2

3

Current

Voltage


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State Space

• 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

ctor

curr

ent

State space


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Ad Hoc Control

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

0 100 200 300 400 500 600 700 800 900 10000.5

1

1.5

2

2.5

Time (us)


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State Space

• 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

ctor

curr

ent

State space


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Augmented Boost

• Now alter the boost to achieve timing targets.• 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


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State Space

• 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


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More General Result


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More General Result


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More General Result


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Research Topics

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

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

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

factors.• Seek simplifications that reduce costs with

little (or no) sacrifice in performance.


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Conclusion

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

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

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

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

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

p. t. krein, director grainger center for electric...

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