Ultimate%20Switching:%20Toward%20a%20Deeper%20Understanding%20of%20Switch%20Timing%20Control%20in%20Power%20Electronics%20and%20Drives

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Ultimate Switching Toward a Deeper Understanding
of Switch Timing Control in Power Electronics and
Drives

• P. T. Krein, Director
• Grainger Center for Electric Machinery and
  Electromechanics
• Dept. of Electrical and Computer Engineering
• University of Illinois at Urbana-Champaign

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Outline

• Fundamentals power electronics control at its
  basic level
• Motivation
• False starts and model-limited control
• Small-signal examples
• Ultimate formulation
• Geometric control examples

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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.

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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.

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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

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Model-Limited Control

• Many control methods used in todays 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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Model-Limited Control

• 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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Model-Limited Control

• 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 dB 7
  -42 dB 9 -70 dB 11 -110 dB 13
  -154 dB 15 -201 dB ? 10-10
• This is consistent with signal arguments that
  yield 2? as the minimum ratio and rules of
  thumb about a ratio of 10 for best results.

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Model-Limited Control

• 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.

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Buck Converter

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

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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).

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Compensated PWM Output

• Filter time constant about 1/10 of switching.

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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.

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Output Ripple
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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.

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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.

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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 arespecific 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.

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The Ultimate Formulation

• A converter with ten switches.
• Time sequences t1,j through t10,j.

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The Ultimate Formulation

• 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.

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The Ultimate Formulation

• Example boost dc-dc converter.
• Find the best time sequence to correct a step
  load change and maintain fixed output voltage.

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The Ultimate Formulation

• 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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The Ultimate Formulation

• 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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The Ultimate Formulation

• 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.

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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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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.

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Result in State Space

• Same data plotted in state space.

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Hysteresis Control

• Alternative simply switch based on whether the
  output is above or below 5 V.
• No frequency constraint.

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Hysteresis Control

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

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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.

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State Space

• State space plot shows how much the behavior
  deviates.

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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.

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State Space

• The step is cancelled perfectly essentially in
  zero time.

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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

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Steady State Behavior
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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.

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State Space

• Suggests a faster transition is possible.

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Ad Hoc Control

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

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State Space

• Rapid move toward final desired result.

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Augmented Boost

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

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State Space

• The response never goes outside ripple limits.

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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.