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
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 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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Model-Limited Control
• Any type of PWM implies switching
that takes place much faster than
system dynamics.
• Dc-dc converters use controllers
designed based on averaging.
• We often learn that bandwidths are
limited 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
7
9
11
13
15
-9 dB
-42 dB
-70 dB
-110 dB
-154 dB
-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 smallsignal analysis.
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Buck Converter
V o ltag e V
( ) , cu rren t A
( )
• In this example, a “feedforward”
compensation is used to eliminate changes
caused by line variation.
iIN #1
L
IOUT
+
+
VIN
vOUT RLOAD
VOUT
#2 i ( t)
v ( t)
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tm
i e
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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
trip ( j  k)
s3lev( j  k  m)
ref ( j  m)
- 1.1
0
j
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Compensated PWM Output
• Filter time constant about 1/10 of switching.
0.5
Cur r ent
0
0.5
0
500
1000
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1500
2000
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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
trip ( j  k)
s3lev( j  k  m)
ref ( j  m)
- 1.1
0
j
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Output Ripple
10
Cur r ent
5
s3iiii
0
5
10
0
500
1000
1500
2000
2500
3000
3500
4000
iii
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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.
12
10
Voltage
Line input
8
6
4
2
0
10
20
30
Time (us)
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50
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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.
iIN #1
L
IOUT
+
+
RLOAD
VIN
vOUT
VOUT
#2 • 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 are
specific times at which a
device 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.
t1
t2 t3
t4 t5
t6
t7
t8
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t10 t11
t12 t13
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The Ultimate Formulation
• Example: boost dc-dc converter.
L
C
VIN
R
+
VOUT
-
• 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 i uH, C = 10 uF, 100LkHz Iswitching.
in
ou t
+
+
#1
V in_
#2
+
v ou t
R
v ou t
load
_
_
v ou t ( t)
in
V o ltag e
V
v ou t
tmi e
0
0
T
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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.
1.1
i L (t)
1.05
1
v out (t)
expanded
0.95
0.9
0
5
10
15
20
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30
35
40
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Result in State Space
• Same data plotted in state space.
1.1
Inductor current
1.05
1
Steady state
0.95
0.9
4.99
4.995
5
Capacitor voltage
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Hysteresis Control
• Alternative: simply switch based on whether
the output is above or below 5 V.
• No frequency constraint.
1.1
Hysteresis control on output voltage.
1.05
1
0.95
0.9
0
20
40
60
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100
120
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Hysteresis Control
• Same result, in state space.
• These controls need timing constraints to
prevent chattering.
1.1
State space.
1.05
Y
1
0.95
0.9
4.99
4.995
5
Y2
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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.
1.1
1
0.9
0
50
100
150
200
250
300
350
400
Time (us)
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State Space
• State space plot shows how much the
behavior deviates.
1.1
State space
1.05
iL i
1
0.95
0.9
4.98
4.99
5
5.01
vc i
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5.02
5.03
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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.
1.1
1.05
1
0.95
0.9
0
20
40
60
80
100
Time (us)
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State Space
• The step is cancelled perfectly – essentially in
zero time.
1.1
State space
1.05
iL i
1
0.95
0.9
4.99
4.995
5
5.005
vc i
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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
L
IIN
+
VIN
vL
iOUT
-
+
iC
+
-
ILOAD
vin
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C
R
VOUT
-
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Steady State Behavior
2.5
i L (t)
2
v out (t)
expanded
1.5
1
0.5
0
5
10
15
20
25
30
35
40
20
25
30
35
40
2.5
2.5
2.45
I ndu cto r cur r en t
2
1.5
2.4
State space
1
2.35
0.5
0
5
10
2.3
11.85
15
11.9
11.95
12
12.05
12.1
12.15
Capac itor voltage
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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.
3
Current
2
1
Voltage
0
0
200
400
600
800
1000
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1400
1600
1800
2000
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State Space
• Suggests a faster transition is possible.
State space
I ndu cto r cur r en t
2.4
2.2
2
1.8
1.6
11.4
11.6
11.8
12
12.2
12.4
12.6
12.8
13
13.2
Capac itor voltage
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Ad Hoc Control
• Short-term overshoot can be used to
dramatically speed the response.
2.5
2
1.5
1
0.5
0
100
200
300
400
500
600
700
800
900
1000
Time (us)
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State Space
• Rapid move toward final desired result.
2.4
I ndu cto r cur r en t
State space
2.2
2
1.8
1.6
1.4
1.2
11.8
12
12.2
12.4
12.6
12.8
13
13.2
Capac itor voltage
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Augmented Boost
• Now alter the boost to achieve timing targets.
• This control eliminates the transient.
2.5
i L (t)
2
v out (t)
expanded
1.5
1
0.5
0
50
100
150
200
250
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350
400
450
500
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State Space
• The response never goes outside ripple limits.
Start
Inductor current
2.4
2.3
2.2
2.1
2
End
1.9
11.85
11.9
11.95
12
12.05
Capacitor voltage
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12.1
12.15
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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.
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