1. Power Electronics Conversion 2
ECE 3795
Power Electronics Conversion 2
Fall 2017
© A. Kwasinski, 2017

2. Power electronic applications
Dynamic:
Variable speed drives
Arguably for wind generation
Electric and hybrid electric cars
Stationary:
UPS
Energy storage integration
Information and communication technologies power plants
Power supplies
Solar power
Micro-grids

3. Power electronics basics
Types of interfaces:
dc-dc: dc-dc converter
ac-dc: rectifier
dc-ac: inverter
ac-ac: cycloconverter (used less often)
Power electronic converters components:
Semiconductor switches:
Diodes
MOSFETs
IGBTs
SCRs
Energy storage elements
Inductors
Capacitors
Other components:
Transformer
Control circuit
Diode
MOSFET
SCR
IGBT
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4. Power electronics basics
dc-dc converters
Buck converter
Boost converter
Buck-boost converter
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5. Power electronics basics
Rectifiers v v v t t t
Rectifier
Filter
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6. Power electronics basic concepts
Inverters
dc to ac conversion
Several control techniques. The simplest technique is square wave modulation (seen below).
The most widespread control technique is Pulse-Width-Modulation (PWM).
© A. Kwasinski, 2017

7. Power electronics basic concepts
Inverters
dc to ac conversion
Several control techniques. The simplest technique is square wave modulation (seen below).
The most widespread control technique is Pulse-Width-Modulation (PWM).
© A. Kwasinski, 2017

8. Power electronics basic concepts
Inverters
dc to ac conversion
Several control techniques. The simplest technique is square wave modulation (seen below).
The most widespread control technique is Pulse-Width-Modulation (PWM).
© A. Kwasinski, 2017

9. Power electronics basic concepts
Energy storage
When analyzing the circuit, the state of each energy storage element contributes to the overall system’s state. Hence, there is one state variable associated to each energy storage element.
In an electric circuit, energy is stored in two fields:
Electric fields (created by charges or variable magnetic fields and related with a voltage difference between two points in the space)
Magnetic fields (created by magnetic dipoles or electric currents)
Energy storage elements:
Capacitors: Inductors: L C
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10. Power electronics basic concepts
Capacitors:
state variable: voltage
Fundamental circuit equation:
The capacitance gives an indication of electric inertia. Compare the above equation with Newton’s
Capacitors will tend to hold its voltage fixed.
For a finite current with an infinite capacitance, the voltage must be constant. Hence, capacitors tend to behave like voltage sources (the larger the capacitance, the closer they resemble a voltage source)
A capacitor’s energy is
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11. Power electronics basic concepts
Inductors
state variable: current
Fundamental circuit equation:
The inductance gives an indication of electric inertia. Inductors will tend to hold its current fixed.
Any attempt to change the current in an inductor will be answered with an opposing voltage by the inductor. If the current tends to drop, the voltage generated will tend to act as an electromotive force. If the current tends to increase, the voltage across the inductor will drop, like a resistance.
For a finite voltage with an infinite inductance, the current must be constant. Hence, inductors tend to behave like current sources (the larger the inductance, the closer they resemble a current source)
An inductor’s energy is
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12. Power electronics basic concepts
Since capacitors behave like constant voltage sources you shall never connect a switch in parallel with a capacitor. Any attempt to violate this load will lead to high currents. Likewise, you shall never connect a switch in series with an inductor. Any attempt to violate this rule will lead to high voltages.
Steady state:
In between steady states there are transient periods,
In steady state:
That is, in steady state the energy in each of the energy storage elements is the same at the beginning and end of the cycle T.
Of course, during the transient periods (if they could be called “periods”) there is a difference between the initial and final energy. or
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13. Power electronics basic concepts
The average voltage across an inductor operating in periodic steady state is zero.
Likewise, the average current through a capacitor operating in periodic steady state is zero.
Hence, Both KCL and KVL apply in the average sense.
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14. Power electronics basic concepts
Switch matrix
It is a very useful tool to represent a power electronics circuit operation and to related (input) variables and (output) signals.
Analysis with a switch matrix involves:
1) Identify and define all possible states. States are defined based on all possible combinations of the switches in the matrix. Switches have two possible states: ON (1) or OFF (0).
2) For each possible state relate (output) signals to (input) variables by taken into consideration the time at each state (i.e. the portion of the time with respect to the switching period).
3) Combine the previous relationship in order to calculate average values for the (output) signals.
© A. Kwasinski, 2017

15. Switching function and duty cycle
The logic signal used to represent the control signal in power electronic switches is called switching signal q(t).
When q(t) =1 the switch (e.g. a MOSFET or an IGBT) is closed
When q(t) =0 the switch is open.
When power electronic switches are operated at a constant frequency in steady state q(t) is usually a periodic signal that may look like this:
Then the duty cycle D is the portion of the time the switch is conducting current so, mathematically it equals the average of the switching function.

16. Power electronics basic concepts
Switch selection
There are two criteria:
Current conduction direction
There are two possible directions:
Forward – Usually from source to load
Bi directional – Both directions
(if current only circulates in the reverse direction, just reverse the switch and make it a forward conducting switch).
Voltage present at the switch when it is blocking the current flow.
The definition relies on the voltage polarity off the switch when it is blocking current flow and with respect to the forward current direction convention.
Can be reverse blocking (RB - diode), forward blocking (FB – BJT or MOSFET), or bi-directional blocking (BB - GTO).
Switches power rating is significantly higher than their losses. + -
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17. Average value of periodic instantaneous power p(t)

18. Two-wire sinusoidal case
zero average
Displacement power factor
Average power

19. Root-mean squared value of a periodic waveform with period T
Compare to the average power expression
compare
The average value of the squared voltage
Apply v(t) to a resistor
rms is based on a power concept, describing the equivalent voltage that will produce a given average power to a resistor

20. Root-mean squared value of a periodic waveform with period T
For the sinusoidal case

21. RMS of some common periodic waveforms
Duty cycle controller DT T V
0 < D < 1
By inspection, this is the average value of the squared waveform

22. RMS of common periodic waveforms, cont.
Sawtooth V T

23. RMS of common periodic waveforms, cont.
Using the power concept, it is easy to reason that the following waveforms would all produce the same average power to a resistor, and thus their rms values are identical and equal to the previous example V V -V V V V V

24. RMS of common periodic waveforms, cont.
Now, consider a useful example, based upon a waveform that is often seen in DC-DC converter currents. Decompose the waveform into its ripple, plus its minimum value.
the ripple + =
the minimum value

25. RMS of common periodic waveforms, cont.
Define

26. RMS of common periodic waveforms, cont.
Recognize that

27. RMS of segmented waveforms
Consider a modification of the previous example. A constant value exists during D of the cycle, and a sawtooth exists during (1-D) of the cycle.
In this example, is defined as the average value of the sawtooth portion DT
(1-D)T

28. RMS of segmented waveforms, cont.
a weighted average
So, the squared rms value of a segmented waveform can be computed by finding the squared rms values of each segment, weighting each by its fraction of T, and adding

29. Practice Problem
The periodic waveform shown is applied to a 100Ω resistor. What value of α yields 50W average power to the resistor?
Since α represents the duty cycle and the rms value of a triangular waveform is the peak value divided by the square root of 3:

30. Fourier Analysis Harmonics
Concept: periodic functions can be represented by combining sinusoidal functions
Underlying assumption: the system is linear (superposition principle is valid.)
e.g. square-wave generation
Due to the presence of harmonics power electronic signals usually have components in the order of MHz.

31. Fourier Analysis
Additional definitions related with Fourier analysis

32. Fourier series for any physically realizable periodic waveform with period T
When using arctan, be careful to get the correct quadrant

33. Square wave V –V T
T/2

34. Triangle wave V –V T
T/2

35. Half-wave rectified cosine wave

36. Power electronics basic concepts
In power electronic circuits, signals usually have harmonics added to the desired (fundamental) signal.
Energy storage elements are used to
Provide intermediate energy transfer buffers.
Filter undesired harmonics
There are two approaches:
Linear approximation (based on time constants considerations). I.e., current and voltage ripples)
Harmonic superposition
© A. Kwasinski, 2017

37. RMS in terms of Fourier Coefficients
which means that
and that
for any k

38. More on power in ac circuits
Instantaneous power
Average power
Power in linear single frequency ac single phase circuits operating in steady state.
if and
Energy exchanges with energy storage components
Displacement power factor
Average power

39. More on power in (single phase) ac circuits
if and
Their phasors are
So complex power is defined as
where
is the reactive power representing steady state energy exchanges with energy storage components.

40. Power in steady state balanced three phase ac circuits
Now,
if phase voltages and their respective phasors are
Then, line voltage phasors are
and

41. Power in steady state balanced three phase ac circuits
Since the phase currents are:
In 3-phase circuits with Y configuration
Then, the power in this type of circuits is

42. Power in steady state balanced three phase ac circuits
Since,
Other definitions applicable to 3-phase power in balanced circuits are

43. Average power in terms of Fourier coefficients
Messy!

44. Average power in terms of Fourier coefficients, cont.
Cross products disappear because the product of unlike harmonics are themselves harmonics whose averages are zero over T!
Not wanted in an AC system
Harmonic power – usually small wrt. P1
Due to the DC
Due to the 1st harmonic
Due to the 2nd harmonic
Due to the 3rd harmonic

45. Power Factor
In the restricted case in which we have a linear circuit operating at a single frequency, the power factor is the displacement power factor:
In a more general definition when a circuit operates with multiple frequencies, power factor is
In both cases the power factor provides an idea related with efficiency because the power factor is a ratio of real power (usually the useful component of power) divided by the total power which includes the power associated to the energy necessary to build fields or to conversion processes.

46. Power Factor
In a linear circuit with a single frequency excitation the power factor is not 1 when the reactance of the equivalent circuit is not zero (so the circuit is inductive or capacitive). The fact that the power factor is not 1, represents the fact that we are storing energy in magnetic and/or electric fields.
In a circuit in which the voltage and/or the current signal has multiple harmonics, the power factor is not 1. In a power electronic circuit the fact that the power factor is not 1 represents the fact that power conversion processes usually create harmonic content in the signals.

47. Total harmonic distortion − THD (for voltage or current)

48. Some measured current waveforms
Refrigerator
THDi = 6.3%
240V residential air conditioner
THDi = 10.5%
277V fluorescent light (electronic ballast)
THDi = 11.6%
277V fluorescent light (magnetic ballast)
THDi = 18.5%

49. Some measured current waveforms, cont.
Vacuum cleaner
THDi = 25.9%
Microwave oven
THDi = 31.9% PC
THDi = 134%

50. Resulting voltage waveform at the service panel for a room filled with PCs
THDV = 5.1%
(2.2% of 3rd, 3.9% of 5th, 1.4% of 7th)
-200
-150
-100
-50 50
100
150
200
Volts
THDV = 5% considered to be the upper limit before problems are noticed
THDV = 10% considered to be terrible

51. Some measured current waveforms, cont.
Bad enough to cause many power electronic loads to malfunction
5000HP, three-phase, motor drive
(locomotive-size)

52. Consider a special case where one single harmonic is superimposed on a fundamental frequency sine wave
Fund. freq
Combined +
Harmonic
Using the combined waveform,
Determine the order of the harmonic
Estimate the magnitude of the harmonic
From the above, estimate the RMS value of the waveform,
and the THD of the waveform

53. Single harmonic case, cont. Determine the order of the harmonic
Count the number of cycles of the harmonic, or the number of peaks of the harmonic 17 T

54. Single harmonic case, cont. Estimate the magnitude of the harmonic
Estimate the peak-to-peak value of the harmonic where the fundamental is approximately constant
Viewed near the peak of the underlying fundamental (where the fundamental is reasonably constant), the peak-to-peak value of the harmonic appears to be about 30
Imagining the underlying fundamental, the peak value of the fundamental appears to be about 100
Thus, the peak value of the harmonic is about 15

55. Single harmonic case, cont. Estimate the RMS value of the waveform
Note – without the harmonic, the rms value would have been 70.7V (almost as large!)

56. Single harmonic case, cont. Estimate the THD of the waveform

57. KVL and KCL in periodic steady-state
Since KVL and KCL apply at any instance, then they must also be valid in averages. Consider KVL,
KVL applies in the average sense
The same reasoning applies to KCL
KCL applies in the average sense

58. KVL and KCL in the average sense
Consider the circuit shown that has a constant duty cycle switch
+ VSavg −
+ VRavg −
0 A R1
Iavg +
VLavg = 0 − V L R2
Iavg
A DC multimeter (i.e., averaging) would show
and would show V = VSavg + VRavg

59. KVL and KCL in the average sense, cont.
Consider the circuit shown that has a constant duty cycle switch
+ VSavg −
+ VRavg − R1
Iavg
Iavg +
VCavg − V C R2
A DC multimeter (i.e., averaging) would show
and would show V = VSavg + VRavg + VCavg

60. Energy in steady state conditions
Consider:
Then,

61. Energy in steady state conditions
Also
Then, for any inductor, including one with infinitely large L:
and since
Then, if L is infinitely large but vL , the current is constant so…..

62. Energy in steady state conditions
and,
Then,
Notice that VL,discharge is negative
“Green” areas = “Orange” areas
So the average inductor voltage is zero

63. Energy in steady state conditions
So for an inductor
Since capacitors are duals of inductors:
“Green” areas = “Orange” areas
So the average inductor voltage is zero
“Green” areas = “Orange” areas
So the average capacitor current is zero

64. Other figures of merit used in power electronics
Efficiency:
% Line regulation:
% Load regulation
System

65. Filtering
In most power electronic circuits filtering is necessary in order to eliminate undesirable harmonics that are a result of the power conversion process.
High-pass filters are rarely used because they tend to amplify high frequency components that are the result of the switching action, thus, increasing the noise in the circuit.
Low pass filters are used both in ac and dc circuits.

66. Effects of “fast” switching on waveforms
Switching is “fast” when the period of the switching function is much shorter than the circuit time constants.
As a result, charging or discharging times are much shorter than the circuit time constants
In these conditions, exponential and other waveforms representing the actual response of circuit variables can be approximated to lines.
Remember the Taylor Series Expansion of a function f(x) about a point xo:

67. Power electronics basic concepts
Time constants: In power electronics we tend to work in many circuits with “large” capacitances and inductances which leads to “large” time constants.
What does “large” means? Large means time constants much larger than the period (whatever the period is. For example, a switching period.
If you look close and for a short time interval, exponentials look like lines
Time constant time scale
Period time scale
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68. Power electronics basics
Additional definitions
Average
RMS value
Instantaneous power
(Average) power
Total harmonic distortion
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69. Power electronics basics
Additional definitions
Power factor
Line regulation
Load regulation
© A. Kwasinski, 2017