nonlinear phenomena in power...
TRANSCRIPT
![Page 1: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/1.jpg)
NONLINEAR PHENOMENAIN POWER ELECTRONICS
Soumitro Banerjee
Department of Electrical EngineeringIndian Institute of TechnologyKharagpur — 721302, India
NONLINEAR PHENOMENA. . . 1
![Page 2: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/2.jpg)
All power electronic circuits share the following properties:
• Switches make the circuit toggle between two or more differenttopologies (different sets of differential equations) at differenttimes.
• Storage elements (inductors and capacitors) absorb energyfrom a circuit, store it and return it.
• The switching times are nonlinear functions of the variables tobe controlled (mostly the output voltage).
The basic source of nonlinearity:
feedback controlled switching
NONLINEAR PHENOMENA. . . 2
![Page 3: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/3.jpg)
In addition there are “parasitic” nonlinearities —
1. the nonlinear v − i characteristics of switches,
2. nonlinear inductances and capacitances,
3. electromagnetic couplings between components.
However, the main source of nonlinearity is the ubiquitousswitching element — which makes all power electronic systemsstrongly nonlinear even if all components are assumed to be ideal.
Therefore
Power electronics engineers/researchers are invariably dealingwith nonlinear problems.
NONLINEAR PHENOMENA. . . 3
![Page 4: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/4.jpg)
Example 1: The voltage-mode controlled buck converter
inV Vo
NONLINEAR PHENOMENA. . . 4
![Page 5: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/5.jpg)
Example 1: The voltage-mode controlled buck converter
inV Vo
Vramp
refVVcon
NONLINEAR PHENOMENA. . . 5
![Page 6: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/6.jpg)
Example 1: The voltage-mode controlled buck converter
inV Vo
Vramp
refVVcon
NONLINEAR PHENOMENA. . . 6
![Page 7: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/7.jpg)
Nominal periodic behavior
(a)
4
5
6
7
8
0 0.001 0.002 0.003 0.004
But such a behavior occurs only within certain limits of the externalparameters.
NONLINEAR PHENOMENA. . . 7
![Page 8: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/8.jpg)
Change of behaviour due to fluctuation of input voltage:
(b)
0 0.001 0.002 0.003 0.004
4
5
6
7
8
The change in behavior occurred when the input voltage changedfrom Vin =24 V to Vin =25 V.
NONLINEAR PHENOMENA. . . 8
![Page 9: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/9.jpg)
When Vin is increased to 35V, the behavior becomes aperiodic, orchaotic.
NONLINEAR PHENOMENA. . . 9
![Page 10: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/10.jpg)
(a) (b)
(a) The experimental period-2 attractorand (b) the chaotic attractor.
NONLINEAR PHENOMENA. . . 10
![Page 11: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/11.jpg)
CHAOS
• Aperiodic waveform
• Seemingly random, noise-like behavior
• Completely deterministic
• The orbit is sensitively dependent on the initial condition
• Statistical behaviour (average values of state variables, powerspectrum etc.) completely predictable.
• Unstable at every equilibrium point, but globally stable.Waveform bounded.
NONLINEAR PHENOMENA. . . 11
![Page 12: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/12.jpg)
Example 2: The current mode controlled boost converter
i=Iref
in
ref
i
L D
R
SI
V
Q
RS C
+
Clock next clock
Switch off when
Switch on at the
Instability and transition to period-2 subharmonic at the critical dutyratio of 0.5.
NONLINEAR PHENOMENA. . . 12
![Page 13: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/13.jpg)
How to probe such phenomena?
• Averaged modeldx
dt= f(x, µ, t)
(t)
x(t)
Simple, but details destroyed. Eliminates nonlinear effects.
NONLINEAR PHENOMENA. . . 13
![Page 14: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/14.jpg)
• Sampled-data (discrete) model
xn+1 = f(xn, µ)
time orbit ObservationsDiscreteContinuous−
0 T 2T 3T 4T
x
y
stat
esp
ace
t
Relatively complex, but accurate. Captures nonlinearity.
NONLINEAR PHENOMENA. . . 14
![Page 15: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/15.jpg)
off on onoff
v vcon ramp
n(v ,i n) v ,in+1 n+1( ) v ,in+2 n+2( )
Sampled data model: (vn, in) 7→ (vn+1, in+1)
NONLINEAR PHENOMENA. . . 15
![Page 16: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/16.jpg)
The procedure: stacking of solutions
• Start from an initial condition (xn, yn) at a clock instant.
• Using the on-time equation and the value of Iref , obtain thelength of the on-period.
• Obtain the state vector at the end of the on-period.
• Use this state vector as the initial condition in the off-timeequations, and evolve for (T − Ton). This gives (xn+1, yn+1) atthe next clock instant.
Thus we obtain the map
(xn+1, yn+1) = f(xn, yn)
NONLINEAR PHENOMENA. . . 16
![Page 17: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/17.jpg)
Dynamics in discrete time
Iterate the map starting from any initial condition.Obtain a sequence of points in the discrete state space.Plot the discrete-time evolution, called the “phase-portrait”.
n n(x ,y )
n n(x ,y ) n+1n+1(x ,y )
statespace
f
(x ,y )
n+3(x ,y )n+3
n+2n+2
NONLINEAR PHENOMENA. . . 17
![Page 18: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/18.jpg)
For periodic systems, after some initial transient, all the iterates fallon the same point in the discrete state space. The fixed point ofthe map is stable −→ period-1 attractor.
y
xtime
cont
rol v
olta
ge
Continuous-time Discrete-time
NONLINEAR PHENOMENA. . . 18
![Page 19: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/19.jpg)
If the system is period-2 (the same state repeats after 2 clocks),there will be 2 points in the discrete-time state space−→ period-2 attractor.
y
xtime
cont
rol v
olta
ge
Continuous-time Discrete-time
NONLINEAR PHENOMENA. . . 19
![Page 20: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/20.jpg)
If the system is period-n (the same state repeats after n clocks),there will be n points −→ period-n attractor.
If a system is chaotic, there will be an infinite number of points inthe phase portrait. −→ chaotic attractor.
NONLINEAR PHENOMENA. . . 20
![Page 21: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/21.jpg)
The phase portrait for the buck converter in the chaotic mode.
NONLINEAR PHENOMENA. . . 21
![Page 22: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/22.jpg)
Question: Why, and in what ways, does the system behaviourchange with the change in a parameter?
Studied through Bifurcation diagrams(panoramic view of stability status).
• Sampled variable at steady state versus parameter, e.g.,iL(nT ) vs. R.
• Bifurcation diagrams can be plotted against the variation of anyof the system parameters, e.g., Vin, R, Iref etc.
NONLINEAR PHENOMENA. . . 22
![Page 23: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/23.jpg)
An experimental bifurcation diagram of the buck converter.x-coordinate: input voltage Vin,y-coordinate: sampled value of the control voltage.Parameter values are: R = 86Ω, C = 5µF, L = 2.96mH, VU = 8.5V,VL = 3.6V, clock speed 11.14 kHz.
NONLINEAR PHENOMENA. . . 23
![Page 24: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/24.jpg)
What is a bifurcation?
• Nothing but loss of stability.
• In a linear system, a loss of stability means the systemcollapses.
• In a nonlinear system, when a periodic orbit loses stability,some other orbit may become stable.
• Thus, at a specific parameter value, one may observe aqualitative change in the character of the orbit. This is abifurcation.
NONLINEAR PHENOMENA. . . 24
![Page 25: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/25.jpg)
Bifurcation: qualitative change in system behaviour with thechange of a parameter
Study of bifurcations:
Derive a discrete time map xn+1 = f(xn, µ).
Obtain the fixed point xn+1 = xn.
Examine the Jacobian at the fixed point and find the loci of the
eigenvalues when a bifurcation parameter is varied.
Identify the condition for the eigenvalue(s) moving out the unit circle in
the complex plane.
NONLINEAR PHENOMENA. . . 25
![Page 26: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/26.jpg)
Two types of bifurcation seen in power electronics
➀ Smooth bifurcations (found in other systems as well)
➁ Border collision (characteristic of power electronics)Abrupt change of behavior due to a structural change
Structural change in switching converters = Alteration in topologicalsequence e.g., change of operating mode, reaching a saturationboundary.
NONLINEAR PHENOMENA. . . 26
![Page 27: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/27.jpg)
time
cont
rol v
olta
ge
time
cont
rol v
olta
ge
time
cont
rol v
olta
ge
Loss of stability,
No structural change
For voltage
mode control:
(smooth bifurcation)
Structural change
(border collision bifurcation)
NONLINEAR PHENOMENA. . . 27
![Page 28: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/28.jpg)
on on off on off off offon offoff onon
on on off on off off offon offoff onon
Loss of stability, No structural change(Smooth bifurcation)
NONLINEAR PHENOMENA. . . 28
![Page 29: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/29.jpg)
on on off on off off offon offoff onon
on on off on on off on offon
Change of topological sequence(involves Border collision bifurcation)
NONLINEAR PHENOMENA. . . 29
![Page 30: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/30.jpg)
The two classes of bifurcations
Smooth bifurcations
• Loss of stability without
structural change
• Standard appearance of bi-
furcation diagrams, e.g., pe-
riod doubling cascade, peri-
odic windows etc.
Border collision
• Loss of stability due to struc-
tural change
• Non-standard appearance in
bifurcation diagrams, e.g.,
bendings, sudden transition
to chaos.
NONLINEAR PHENOMENA. . . 30
![Page 31: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/31.jpg)
How to analyse such loss of stability of power converters?
Bifurcation theory can help.
A bifurcation occurs when a fixed point of the discrete map losesstability.
=⇒ At least one eigenvalue crosses the unit circle.
NONLINEAR PHENOMENA. . . 31
![Page 32: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/32.jpg)
If the map is smooth, there are three possibilities:
** **
* *
***
*
(c)(b)(a)
(a) A period doubling bifurcation:eigenvalue crosses the unit circle on the negative real line,
(b) A saddle-node or fold bifurcation:an eigenvalue touches the unit circle on the positive real line,
(c) A Hopf or Naimark-Sacker bifurcation:a complex conjugate pair of eigenvalues cross the unit circle.
NONLINEAR PHENOMENA. . . 32
![Page 33: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/33.jpg)
on on off on off off offon offoff onon
on on off on off off offon offoff onon
time
cont
rol v
olta
ge
time
cont
rol v
olta
ge
Examples of period-doubling bifurcation, associated with oneeigenvalue becoming −1.
Common in all feedback-controlled dc-dc converters.
NONLINEAR PHENOMENA. . . 33
![Page 34: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/34.jpg)
A saddle-node bifurcation results in the creation of a new orbit (orthe destruction of an existing orbit).
A new periodic orbit coming into being may imply
• Destruction of chaos for a range of parameters (periodicwindow)
• Creation of coexisting attractors (multistability).
NONLINEAR PHENOMENA. . . 34
![Page 35: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/35.jpg)
A Naimark-Sacker bifurcation causes a periodic orbit change into aquasiperiodic orbit.
x
xx
x
t
1
23
1
(Combination of two incommensurate frequencies.)
In discrete time,
NONLINEAR PHENOMENA. . . 35
![Page 36: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/36.jpg)
0.6 1.0 1.4 1.8 2.20.2
0
2
4
6C
ontr
ol V
olta
ge (
V)
Time (msec)
Con
trol
Vol
tage
(V
)
0
2
4
6
0.2 0.6 1.0 1.4 1.8 2.2Time (msec)
(V)conv−60
0
80
n(m
A)
i
−9 3 15con (V)v
i n
−60
0
80
−9 3 15
(mA
)Quasiperiodic and mode-locked periodic behavior in a PWM buckconverter. Slow scale instability. Can also be predicted with theaveraged model.
NONLINEAR PHENOMENA. . . 36
![Page 37: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/37.jpg)
These are the three possible ways that a converter can losestability without structural change, i.e., without alteration oftopological sequence.
How can we analyse the loss of stability caused by structuralchange?
NONLINEAR PHENOMENA. . . 37
![Page 38: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/38.jpg)
It has been found that sampled data modeling of all powerelectronic circuits yield piecewise smooth maps.
• The discrete state space is divided into two or morecompartments with different functional forms of the map.
• The compartments are separated by borderlines. TheJacobian changes discontinuously across the borderlines.
x 2
x1
x x=fn+1 1( n)
x x=fn+1 ( n)2
Borderline
NONLINEAR PHENOMENA. . . 38
![Page 39: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/39.jpg)
The current mode controlled boost converter
i=Iref
in
ref
i
L D
R
SI
V
Q
RS C
+
Clock next clock
Switch off when
Switch on at the
refI
i
NONLINEAR PHENOMENA. . . 39
![Page 40: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/40.jpg)
Structure of discrete state space inin current mode controlled converters
Iref
T
Iref
T T
ii i
(a) (b) (c)
(a) and (b): The two possible types of evolution between twoconsecutive clock instants yielding two different functional forms inthe sampled-data model and (c): the borderline case.7→ Piecewise smooth map.
NONLINEAR PHENOMENA. . . 40
![Page 41: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/41.jpg)
Iref
T
Iref
T T
ii i
(a) (b) (c)
Case (a): in+1 = f1(in) =(
1 + m2
m1
)
Iref − m2T −m2
m1
in.
Case (b): in+1 = f2(in) = in + m1T
Boderline: Ib =Iref−m1T
Slope
1Slope
−m
2 /m1
in
in+1
NONLINEAR PHENOMENA. . . 41
![Page 42: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/42.jpg)
Structure of discrete state spacein voltage mode controlled converters
(a) (b) (c)
(e)(d)
(a), (b) and (c): The three possible types of evolution between twoconsecutive clock instants, and (d) and (e): the “grazing” situationsthat create the borderlines.7→ Piecewise smooth map.
NONLINEAR PHENOMENA. . . 42
![Page 43: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/43.jpg)
Dynamics of Piecewise Smooth Maps
• If a fixed point loses stability while in either side, the resultingbifurcations can be categorized under the generic classes forsmooth bifurcations.
• But what if a fixed point crosses the borderline as someparameter is varied?
The Jacobian elements discretelychange at this point
NONLINEAR PHENOMENA. . . 43
![Page 44: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/44.jpg)
• The eigenvalues may jump from any value to any other valueacross the unit circle.
• The resulting bifurcations are calledBorder Collision Bifurcations.
Continuous movement ofeigenvalues in a smoothbifurcation
Discontinuous jump ofeigenvalues in a bordercollision bifurcation
NONLINEAR PHENOMENA. . . 44
![Page 45: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/45.jpg)
Theories have been developed that can predict the outcome of a border
collision bifurcation (which orbit will become stable) depending on the
eigenvalues of the Jacobian matrix at the two sides of the border. These
transitions are usually sudden and abrupt.
µ
x
µ
x
µ
x
µ
x
µ
x
µ
x
NONLINEAR PHENOMENA. . . 45
![Page 46: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/46.jpg)
As a system parameter is varied, the change of dynamicalbehavior of power converters exhibit a succession of smooth andborder collision bifurcations.
• Generally the first bifurcation from a normal period-1 operationis of smooth type.
• The development of a smooth bifurcation sequence (e.g., aperiod-doubling cascade) is interrupted by border collisionbifurcations.
NONLINEAR PHENOMENA. . . 46
![Page 47: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/47.jpg)
Sam
pled
indu
ctor
cur
rent
Load resistance
Smooth Bifurcation
Border Collision
Bifurcation diagram of the boost converter with load resistance asvariable parameter.
NONLINEAR PHENOMENA. . . 47
![Page 48: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/48.jpg)
Which type of instability is likely in which converter?
1. Voltage mode controlled buck converter (PWM-2): The firstbifurcation is always period doubling. Subsequently perioddoubling cascade interrupted by border collision, leading todirect transition to chaos. Hamill and Deane (1990,1992),Fossas and Olivar (1996), Chakrabarty and Banerjee (1996)
2. Voltage mode controlled boost converter (PWM-2): Hopfbifurcation leading to quasiperiodicity. Tse (1997)
3. Voltage mode controlled converters (PWM-1): Hopf bifurcationleading to quasiperiodic and mode-locked periodic orbits.Period doubling also occurs. El Aroudi et al. (2000), Maity andBanerjee (2007)
NONLINEAR PHENOMENA. . . 48
![Page 49: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/49.jpg)
4. Current mode controlled boost converter: The first bifurcationis always period doubling. Subsequently period doublingcascade interrupted by border collision, leading to directtransition to chaos. Hamill and Deane (1990,1992,1995),Chakrabarty and Banerjee (1997), Chan and Tse (1997),
5. Current mode controlled Cuk converter: Period doubling routeto chaos, truncated by border collision. Tse and Chan (1995)
6. Converters operated in DCM: Period doubling. Tse (1994)
7. Converters undergoing transition from DCM to CCM: bordercollision bifurcation leading to quasiperiodicity. Maity andBanerjee (2007)
NONLINEAR PHENOMENA. . . 49
![Page 50: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/50.jpg)
In many applications the operating point constantly changes withtime:
• Power factor correction power supplies;
• Dc-AC inverters;
• Audio amplifiers using dc-dc converters.
Bifurcations are expected even under normal operating condition.
0.08 0.085 0.09 0.095 0.1-2
0
2
4
6
Time [s]
Cur
rent
[A]
NONLINEAR PHENOMENA. . . 50
![Page 51: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/51.jpg)
How to analyze the stability of power electronic circuits?
If the initial condition is perturbed and the solution converges backto the orbit, then the orbit is stable. The stability margin can beassessed from the rate of convergence.
Suppose the initial condition is given a perturbation δx(t0). If theoriginal trajectory and the perturbed trajectory evolve for a time t,
δx(t) = Φ δx(t0).
Here Φ is the state transition matrix.
NONLINEAR PHENOMENA. . . 51
![Page 52: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/52.jpg)
The properties of Φ :
x x xAB C
If xB = ΦAB xA, xC = ΦBC xB
then δxB = ΦAB δxA, δxC = ΦBC δxB
and δxC = ΦAC δxA, ΦAC = ΦBCΦAB
NONLINEAR PHENOMENA. . . 52
![Page 53: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/53.jpg)
For LTI systems, the state transition matrix can be obtained as thematrix exponential
Φ(t, t0) = eA(t−t0)
so that the perturbation at time t can be written as
δx(t) = eA(t−t0)δx0.
NONLINEAR PHENOMENA. . . 53
![Page 54: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/54.jpg)
Suppose we are able to find the state transition matrices during theON period and the OFF period:
δx(dT ) = A1 δx(0)
δx(T ) = A2 δx(dT )
The product A2 · A1 does not give the state transition matrix overthe whole cycle. One has to take into account how theperturbations change when they cross the border.
M. A. Aizerman and F. R. Gantmakher, “On the stability of periodic motions,” Journal
of Applied Mathematics and Mechanics (translated from Russian), 1958, pages
1065-1078.
NONLINEAR PHENOMENA. . . 54
![Page 55: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/55.jpg)
∆x(tB+) = S ∆x(tB−),
where S, the “jump matrix” or “saltation matrix”, can be expressedas
S = I +(f+ − f−)nT
nT f− + ∂h/∂t.
f− : RHS of the differential equations before switching
f+ : RHS of the differential equations after switching
h(x, t) = 0: the switching condition (a surface in the state space)
I : the identity matrix
n : the vector normal to the switching surface.
NONLINEAR PHENOMENA. . . 55
![Page 56: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/56.jpg)
The state transition matrix over the complete clock cycle, called themonodromy matrix is expressed as
Φcycle(T, 0) = S2 × Φoff(T, dT ) × S1 × Φon(dT, 0),
where S1 is the saltation matrix related to the first switching event,and S2 is that related to the second switching event.
NONLINEAR PHENOMENA. . . 56
![Page 57: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/57.jpg)
The monodromy matrix then relates the perturbation at the end ofthe clock period to that at the beginning:
∆x(T ) = Φ(T, 0)∆x(0).
The eigenvalues of the monodromy matrix are also called theFloquet multipliers. If all the eigenvalues are inside the unit circle,perturbations will die down and the system is stable.
NONLINEAR PHENOMENA. . . 57
![Page 58: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/58.jpg)
The voltage mode controlled buck converter:
closeds
vv rampcon
= <
di(t)
dt=
vin − v(t)
L, S is conducting
−v(t)
L, S is blocking.
,dv(t)
dt=
i(t) −v(t)
RC
NONLINEAR PHENOMENA. . . 58
![Page 59: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/59.jpg)
time
cont
rol v
olta
ge
11.4 11.5 11.6 11.7 11.8 11.9 120.44
0.46
0.48
0.5
0.52
0.54
0.56
0.58
0.6
0.62
voltage, V
curr
ent,
A
1,9
X(dT)
X(0)
2
3
4
5
6
7
8
1 2 3 4
5
6 7 8 9
NONLINEAR PHENOMENA. . . 59
![Page 60: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/60.jpg)
The switching hypersurface (h) is given by
h(x(t), t) = x1(t) − Vref −vramp(t)
A= 0,
vramp(t) = VL + (VU − VL)
(
t
Tmod 1
)
The normal to the hypersurface is:
n = ∇h(x(t), t) =
∂h(x(t), t)
∂x1(t)∂h(x(t), t)
∂x2(t)
=
1
0
NONLINEAR PHENOMENA. . . 60
![Page 61: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/61.jpg)
By defining x1(t) = v(t) and x2(t) = i(t), the system equations are
x =
Asx + Bu, A(x1(t) − Vref) < vramp(t),
Asx, A(x1(t) − Vref) > vramp(t).
Where,
As =
−1/RC 1/C
−1/L 0
, Bu =
0
1/L
Vin
NONLINEAR PHENOMENA. . . 61
![Page 62: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/62.jpg)
When the state goes from the off state to the on state,
fp− = limt↑tΣ
f−(x(t)) =
x2(tΣ)/C − x1(tΣ)/RC
−x1(tΣ)/L
,
fp+ = limt↓tΣ
f+(x(t)) =
x2(tΣ)/C − x1(tΣ)/RC
(Vin − x1(tΣ))/L
.
where tΣ is the switching instant.
NONLINEAR PHENOMENA. . . 62
![Page 63: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/63.jpg)
Thus
fp+ − fp− =
0Vin
L
,
(
fp+ − fp−
)
nT =
0 0Vin
L0
,
nT fp− =x2(tΣ)
C−
x1(tΣ)
RC.
NONLINEAR PHENOMENA. . . 63
![Page 64: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/64.jpg)
∂h(x(t), t)
∂t=
∂
(
x1(t) − Vref −TVL + (VU − VL)t
AT
)
∂t
= −VU − VL
AT.
Hence the saltation matrix is calculated as
S =
1 0
Vin/L
x2(tΣ) − x1(tΣ)/R
C−
VU − VL
AT
1
NONLINEAR PHENOMENA. . . 64
![Page 65: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/65.jpg)
For a buck converter with the parameters
Vin = 24V , Vref = 11.3V , L = 20mH, R = 22Ω, C = 47µF,
A = 8.4, T = 1/2500s, VL = 3.8V and VU = 8.2V
the switching instant was calculated to be 0.4993 × T .
The state at the switching instants are
x(0) =
12.0222
0.6065
and x(d′T ) =
12.0139
0.4861
.
NONLINEAR PHENOMENA. . . 65
![Page 66: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/66.jpg)
The saltation matrix is calculated as
S =
1 0
−0.4639 1
The state matrix is
As =
−1/RC 1/C
−1/L 0
=
967.12 21276.6
−50 0
.
NONLINEAR PHENOMENA. . . 66
![Page 67: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/67.jpg)
The state transition matrices for the two pieces of the orbit are:
1. Off period:
Φ(d′T, 0) = eAsd′T =
0.8058 3.8366
−0.0090 0.9802
2. On period:
Φ(T, d′T ) = eAsdT =
0.8052 3.8468
−0.0090 0.9800
.
NONLINEAR PHENOMENA. . . 67
![Page 68: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/68.jpg)
Hence the monodromy matrix is
Φ(T, 0, x(0)) = Φ(T, d′T ) · S · Φ(d′T, 0)
=
−0.8238 0.0131
−0.3825 −0.8184
The eigenvalues are −0.8211 ± 0.0708j implying that at the aboveparameter values the system is stable.
NONLINEAR PHENOMENA. . . 68
![Page 69: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/69.jpg)
An idea for increasing the stability margin:
Φ(T, 0, x(0)) = Φ(T, d′T ) · S · Φ(d′T, 0)
The state transition matrices for the ON and OFF periods are givenby the parameters and the duty ratio. Set by the user’s specs.
There is another handle: the saltation matrix.
S = I +(f+ − f−)nT
nT f− + ∂h/∂t.
NONLINEAR PHENOMENA. . . 69
![Page 70: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/70.jpg)
• ∂h/∂t can be manipulated by varying the slope of the ramp.
• nT can be manipulated by using current feedback along withvoltage feedback.
22 24 26 28 30 32 34
2
6
−2
10
14
Con
trol
Vol
tage
(V
)
Input Voltage (V) (a)
22 24 26 28 30 32 34
2
6
−2
10
14
Con
trol
Vol
tage
(V
)Input Voltage (V) (b)
The experimentally obtained bifurcation diagrams (a) under normalPWM-2 control, and (b) when the secondary control loop is added,with δVU = 0.7 V.
NONLINEAR PHENOMENA. . . 70
![Page 71: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/71.jpg)
Take home message:
• All power electronic circuits are strongly nonlinear systems.
• Power converters may exhibit subharmonics and chaos forspecific parameter ranges. Linear analysis does not predictthese instabilities.
• Smooth as well as nonsmooth (or border collision) bifurcationsoccur in such converters.
• To analyse the smooth bifurcations, one has to obtain theeigenvalues of the monodromy matrix. To analyse the bordercollision bifurcations, one has to obtain the eigenvalues of thesame periodic orbit before and after border collision.
• Bifurcation theory helps in delimiting the parameter space andin devising better controllers to avoid instability,.
NONLINEAR PHENOMENA. . . 71
![Page 72: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/72.jpg)
For further reading:
1. Soumitro Banerjee and George C. Verghese, (Editors)“Nonlinear Phenomena in Power Electronics: Attractors,Bifurcations, Chaos, and Nonlinear Control”, IEEE Press,2001.
2. C. K. Tse, “Complex Behavior of Switching Power Converters,”CRC Press, USA, 2003.
3. Z. T. Zhusubaliyev and E. Mosekilde, “Bifurcations and Chaosin Piecewise-Smooth Dynamical Systems”, World Scientific,Singapur, 2003.
NONLINEAR PHENOMENA. . . 72
![Page 73: NONLINEAR PHENOMENA IN POWER ELECTRONICSpublish.illinois.edu/grainger-ceme/files/2014/06/CEME508India.pdf · 2. nonlinear inductances and capacitances, 3. electromagnetic couplings](https://reader036.vdocuments.us/reader036/viewer/2022081521/5ed50f1aab6e6a47610358e1/html5/thumbnails/73.jpg)
THANK YOU
NONLINEAR PHENOMENA. . . 73