PREFACE
The technology of power electronics and drives has gone through intense technological evolution during the last 30 years, although its history dates back for nearly a century. Many inventions in devices, components, circuits, controls, and systems have caused power electronics to emerge as a major technology in recent years. This course note makes an introduction to an emerging technologyPower Electronics, it addresses the development history of early semiconductor devices and makes an introduction of the recently developed high-performance power semiconductor switching devices. This course note gives a brief and broad spectrum review of the present power electronics. Special emphasis has been placed on the development of modern power electronics. The listed references provide some further readings on this emerging technology.
1 HISTORY
Power electronics originated at the beginning of this century. Many technical articles and several books on the subject were published during the period from 1930-1947. These dealt primarily with the application of grid-controlled gas-filled tubes. Because of the limitations of the mercury-arcrectifier and gas-filled thyratrons, only a relatively limited number of equipments were manufactured.
It was in 1930 when the field-effect principle was first disclosed in U.S. patent by Julius Lilienfeld, a former professor of physics at the University of Leipzig who has recently immigrated to the United States.
On December 16, 1947, the point-contact transistor was demonstrated by Walter H. Brattain and John Bardeen (Shockley 1972 & 1976), with William Shockley as an intensely interested observer. After an additional week of further experimentation and polishing of the demonstration, it was repeated for several key Bell Laboratories managers on December 23, 1947, the date that has come to be taken as the "official" date of reduction to practice. Walter H. Brattain, John Bardeen and William Shockley shared a Nobel prize for the transistor in 1956.
The invention of the bipolar junction transistor in 1948 was the beginning of semiconductor electronics. This device and semiconductor diodes spawned a revolution electronics. Drastic reduction in size, cost, and power consumption were achieved simultaneously with greatly increased equipment complexity and capability.
At the same time, in 1948, Shockley and Pearson tried fabricating a rudimentary FET using evaporated layers of germanium on dieletric. However, it was not until Bardeen theoried on the surface state phenomenon and Shockley published his theoretical analysis of the unipolar field-effect transistor (Shockley 1952).
Semiconductor power diodes become available shortly after 1950. However, it was not until late 1957, when the most popular member of the thyristor family - the SCR - was announced by General Electric (1957), that semiconductor power electronics really began. Starting from a single 16-A device, the thyristor family has grown tremendously. Hundreds of thyristor devices are available from numerous manufacturers throughout the world. In addition, there have been continuing improvements in ratings and characteristics of power transistors so that a wide variety of these components are also now available.
The development of power FET has also an important impact on the power semicondctor industry. In 1964, when two significant papers were published, one by Zuleeg, the other by a French physicist, Teszner (Zuleeg 1967, Teszner & Giquel 1964). It was in these papers that FETs were first considered suitable for handling power. These papers reported to have solved two mutually exclusive problems: the "high-frequency" problem and means to achieve "high saturation currents". These solutions have significant effects, because it was customary at that time to achieve higher gain and higher current-handling capacity simply by increasing the physical size of the FET geometry. But increasing the size also increase the Parasitic capacitances. Additionally, an increase in the channel resistance occurred. Together they limit the high-frequency performance. The higher parasitic capacitance reduced the frequency response while the increased channel resistance lowered the forward transconductance and limited the current.
It appeared that both Zuleeg and Teszner had solved the problem. Zuleeg called his device a multichannel field-effect transistor, which he nicknamed the MUCH-FET. Teszner, in his coauthored paper, called his gridistor, as he felt that it closely resembled the triode vacuum tube except, of course, it was solid state. Actually, triode called the SCL (space-charge-limited) triode. Zuleeg, however, had made another major contribution, for his MUCH-FET was fabricated vertically, as is practically every power FET that we find in the market today.
During this activity the Japanese were quitely active. In 1950, professor Nishizawa and Watanabe at Tokyo University patented in Japan what they called an analog transistor. It was a FET. About the time that patents were being issued to both Zuleeg and Teszner for their inventions, the U.S. Army and later the department of the Navy released a research contract to RCA to develop a high frequency power FET. As it turned out when the final report, underwritten by the Army, was issued in 1968, it did not use either Zuleeg's nor Teszner's ideas. Had they used the solutions proposed by these two researchers, the money spent might have resulted in a more worthwhile contribution.
Simultaneous with the work of Zuleeg and Teszner, the Japanese were developing their analog transistor field-effect transistor, a work that had remarkable parallels to the work of Zuleeg. This analog transistor was later to be named the SIT (Static induction transistor), capable of remarkable power-handling abilities at high frequency.
Concurrent with this work by the Japanese research team of Nishazawa and Watanabe, the Japanese Electrotechnical Laboratory, in 1969, reported what we were able to read as a major technological breakthrough. This we discovered was what has now become the well-known vertical V-groove MOSFET structure that no longer relied upon the limited dimensional accuracy of the photolithographical process.
As the decade of the 1970s closed, power FET development had spread worldwide with each year announcing a new technology. The VMOSFET was the first commercially available power FET we saw followed by a vertical DMOS (double-diffused MOS), the most noteworthy being the HEXFET. (HEXFET is the trademark of International Rectifier, Inc., Semiconductor Division, EI Segundo, Calif., U.S.A.) In this decade the Japanese introduced their SIT, where application was soon to settle in high-fidelity audio power amplifiers.
2 POWER SEMICONDUCTOR DEVICES AND DEVELOPMENT TRENDS
Table 2.1 is a list of a number of thyristor devices presently available. A wide range of power switching transistors are also available, with current rating up to several hundred amperes and voltage ratings well over 1000V. However, the combination of the highest current and voltage rating is not available in a single device. One of the highest power devices presently available is the Westinghouse D60T/D62T. These are npn power-switching transistors with a maximum collect current of 200A, a maximum of VCEO(sus) of 500V and with a current gain of 10 at 50A collect current.
There are many additional special thyristor and transistor devices. These include multiple bipolar transistors fabricated in a single package, chip-type (unpackaged) thyristors, power FETs, and Schottky diodes. In addition, the continued development of IC components and microprocessors is having a most important impact on control circuits for power electronic systems.
Device trends of most importance at this time include the following. There is continuing growth in the number of switching power transistors with a combination of high voltage and current ratings. This is causing an increase in the rating level at which power transistor equipment are preferable to thyristor hardware. For the past several years, there has been considerable activity on the development of high-power gate turn-off devices with improved characteristics. These components could have great application potential. Presently, at least four Japanese companies manufacture GTOs with ratings up to 2500V, 1000A - Hitachi, Mitsubishi, Toshiba, and IR-Japan. A third area are of interest is the activity on improved cooling techniques for power semiconductors. New types of heat sinks integral with the device (power module) have significant potential for reducing size and cost per unit of equipment rating. More power FETs are certain to become available, a great number of power integrated devices (smart power) is anticipated, and there is considerable opportunity for device improvement with clever new packaging techniques. Table 2.2 is a list of manufacturers of power semiconductor components in U.S.A.
| Asymmetrical silicon bilateral switch (ASBS) Asymmetrical silicon-controlled rectifier (ASCR) Bidirectional diode thyrisitor (DIAC) Bidirectional triode thyrisitor (TRIAC) Gate-assisted turn-off thyristor (DIAC) Gate turn-off thyristor (GTO) Light-activated programmable unijunction transistor (LAPUT) Light-activated reverse-blocking diode thyrisitor (LAS-light-activated switch) Light-activated reverse-blocking tetrode thyrisitor (LASCS) Light-activated reverse-blocking triode thyrisitor (LASCR) Programmable unijuction transistor (PUT) Reverse-blocking diode thyristor (RBDT) Reverse-blocking tetrode thyristor (SCSsilicon-controlled switch) Reverse-blocking triode thyristor (SCR) Reverse-conducting diode thyristor (RCDT) Reverse-conducting triode thyristor (RCTT) Silicon bilateral switch (SBS) Silicon unilateral switch (SUS) |
The power module will continue to have a significant impact on power electronics and motion control. The advantages of using power modules are uniformity, relatively low cost, ease of assembly, low maintenance costs and most of all - simple diagnostic. Ultimately, we can expect to see smart power devices and integrated circuits employed in complex power modules that minimize component count and increase reliability. However, as the intelligence levels of the integrated circuits and power devices become higher, thermal and packaging problems become more acute. A number of module manufacturers are working on innovative packages and thermal interface problems. Undoubtedly, these new electrical and mechanical smart power devices will appear in the near future.
The SIT stands for Static Induction Transistor or Static Induction Thyristor. SIT was first invented by Dr. Nishizawa (Japan) in 1950s who also invented the PIN diode. This new SIT technology may challenge SCRs, GTOs, MOSFETs, IGBTs, magnetrons and most other power semiconductors (Ohmi 1979). SITs have extremely low forward voltage drop, very fast swithing speeds and greater radiation tolerance than most semiconductors. They operate at thousands of volts, hundreds of amperes, frequency up to 10 GHz and levels up to one megawatt. It is believed that this new technology will bring revolution in power electronics industry.
| Atlantic Semiconductors/Diodes Power Tech Edal Industries RCA EE Tech Sarkes Tarzian Electronic Devices Semtech Corporation Fairchild Semiconductor Solitron Devices General Electric Solid State Devices General Instrument Teccor Electronics General Semiconductor Industries Texas Instruments Hewlett-Packard TRW Hughes Tungsol Intersil Unitrode Corporation Lansdale Varo, Inc. Motorola Semiconductor Products Westcode Power Physics Westinghouse Electronic Power Semiconductors, Inc. |
3 APPLICATIONS
Power semiconductors have a wide spectrum of applications. Table 3.1 is a list of applications by product type. This is not intended to be all inclusive but rather to illustrate the spectrum of uses of power semiconductors in commercially available products. Table 3.2 is a listing of typical products by function. New applications are still being found in every fast growing technologies. Special attentions will be focused on the applications in the fields of motor drives and power converting systems in this course. The functional block diagram of a typical power converter is shown in Fig. 3.1.
Their functions are described in the followings:
- The power circuits, the output of which may be a variable direct or alternating voltage or current source or may be an alternating source of variable voltage and frequency.
- The digital circuits, which in response to the signals from the controlling system switch the power switching devices of the power circuits on and off at appropriate instants.
- The controlled system, which may simply be a rotating machine and driven load with appropriate feedback output, or may be something considerably more complicated.
- The controlling system, which in response to the command and feedback signals issues the appropriate control signals to the digital circuits.
| AEROSPACE AND DEFENSE Oil-well drilling Aircraft power supplies Paper mill Laser power supplies Printing press Radar/sonar power supplies Pump and compressor Solid-state relays, contactors, and Still mill circuit breakers Synthetic fiber Sonobuoy flashers INDUSTRIAL (continued) Space power supplies Power Supplies VLF transmitters Aluminum reduction CONSUMER Battery charger Audio amplifiers Computer Electric door openers Electrochemical Heat controls Electroplating Electric blanket Electrostatic precipitator Electric dryer Induction heating Food-warmer tray Laboratory Furnace Mining Oven Particle accelerator Range surface unit Welding High-frequency lighting Static relays and circuit breakers Light dimmers Ultrasonic generators Light flashers POWER SYSTEMS Motor controls Gas turbine starting Air conditioning Generator exciters Blender HVDC Electric fan Nuclear-reactor control-rod drives Food mixer Solar power supplies Garage-door opener Synchronous machine starting Hand power-tool Uninterruptible power supplies Model train VAR compensation Movie projector Wind generator converters Sewing machine TRANSPORTATION Slot car Electronic ignition RF amplifiers Linear induction motor control Security systems Motor drives TV deflection Electric vehicle INDUSTRIAL Elevator Mercury-arc lamp ballasts Fork-lift truck Motor drives Locomotive Cement kiln Mass transit Conveyor People movers Crane and hoist Traffic-signal control Machine tool Voltage regulator Mining |
| STATIC SWITCHING Solid-state relays, contactors, and circuit breakers Logic systems Circuit protectors---crowbars, limit activated interrupters AC PHASE CONTROL Light dimmers Motor speed controls Voltage regulators VAR regulators PHASE-CONTROLLED RECTIFIER/INVERTER dc motor drives Regulated dc power supplies HVDC Wind generator converters CYCLOCONVERTER Aircraft VSCF systems Variable-frequency ac motor drives FREQUENCY MULTIPLIER Induction-heating supplies High-frequency lighting TRANSISTOR LINEAR AMPLIFIER dc-dc buck, boost, and buck-boost converters High-performance regulated power supplies THYRISTOR CHOPPER Electric transportation propulsion control Generator exciters High-performance, high-power regulated supplies DC-AC INVERTER Aircraft and space power supplies Uninterruptible power supplies |
4 FUTURE DEVELOPMENT ON POWER ELECTRONICS
Lower voltages, higher current and better regulation are needed for the coming generation of VLSI. Highly regulated voltages on the order of three volts, or less, currents in tens of hundreds of amperes and load dynamics of 100 A/sec will be required. Narain G. Hingorani, vice president of the Electrical System Division for EPRI (Electric Power Research Institute, Palo Alto, CA, USA), says power electronics should be a national priority. He points out (1987) that over the next 20 years the development of semiconductor devices for high-voltage, high current applications will bring about a major transformation in industrial system. He says this "second electronics revolution" has always begun and is being used to make AC/DC converters for high-voltage DC (HVDC) transmission, static VAR compensators, UPS to protect sensitive equipment and drives for adjustable speed motors (Miller 1987).High frequency resonant converters will be found wide applications in the switching power supplies for personal computers in the near future. Built-in UPS for the personal computer will possible in the future.
Microprocessors will play a more important role in the power converting system in the future. Dedicated microcontroller and power control IC will be developed. Smaller size, lower cost, higher power handling capability and more reliable power converting system will be on the market.
5 UNIQUE ASPECTS OF POWER ELECTRONICS
5.1 Switching Operation
When semiconductors are used in high-power applications, they are generally operated as switches. There are only a very few exceptions to this, such as linear amplifiers and series or shunt transistor-regulated linear power supplies. When the semiconductor is used as a switch, it is possible to control large amounts of power to a load with a relatively low power dissipation in the switching device. For example, in an ideal switch with zero voltage drop when "on," zero leakage current when "off," and zero switching time, there is no power dissipation in the switching device regardless of the on-state voltage drop across the switching devices limits the on-state current because of device dissipation limits. All practical power semiconductor devices also have a limited off-state voltage rating. When high-frequency switching is employed, there also may be significant switching loss due to the large instantaneous device power dissipation because of the finite times of practical power semiconductors. Practical semiconductors always have some losses. In the following section power losses in a switching power transistor are considered. This can be treated as a reference for the analysis of power losses of other power semiconductor devices.
Power Losses in a Switching Power Transistor
During the "on" period of power transistor-switch "conduction losses" will appear. These energy losses are independent of the switching frequency and are equal to about the on-state voltage drop VCE(sat) multiplied by the average output current. During the "off" period "turn-off losses" will appear. This energy losses are usually very small and are equal to about the off-state voltage drop Vcc multiplied by the turn-off leakage current. Energy losses during the on-off and off-on transition names as "switching losses" will produce power losses proportional to the switching frequency. Power losses of a switching device in power converter are the summation of conduction losses, turn-off losses, and switching losses. Excessive power losses which in turn means excessive junction temperature and may incur the thermal run-away phenomenon this degrades the performance and reliability of the power switching device. The problem resides in how to reduce the conduction losses and switching losses of the power transistor in turn means to reduce the junction temperature to increase the safe operating range and reliability.
Fig. 5.1 illustrates the switching action of a simple switching transistor circuit. When the transistor is off or open, the collector-emitter voltage is assumed to be E volts. Actually, the off-state transistor voltage would be slightly less than the dc supply voltage because of the load voltage drop due to the leakage current. The collect current flowing when the transistor is on is assumed to be I amperes. In addition, the instantaneous collect-emitter voltage v(t) and collect current i(t) are both assumed to change in a linear fashion during the switching interval. Although this is not exactly the case in practical situation, the linear variations are a reasonably good approximation, and they most clearly illustrate the important switching characteristics.
The instantaneous power dissipated during the switching interval can be expressed as
(5.1)
In the expressions for v(t) and i(t), the beginning of the switching interval is assumed to be t=0. Also, the saturated voltage drop and collect leakage current are both assumed to be negligible.
The average power dissipated during a switching interval is important since it determines the maximum number of switchings possible in a given time interval. The average dissipation during the interval TSW is given by
(5.2)
Combining with 5.1,
(5.3)
The total average dissipation in a switching element is obtained by adding the on-state, off-state, and switching losses. For example, with a switching period of T, assuming linear switching with a switching times of TSW for both turn-on and turn-off and an on-time and off-time of TON and TOFF respectively.
Total device average dissipation = PT
(5.4)
where VCE(SAT) and Ileakage are assumed small enough such that the switching loss is approximately given by 5.3.
5.2 Commutation
The IEEE Standard Dictionary of Electrical and Electronic Terms contains the following definitions of commutation:
The transfer of unidirectional current between rectifier circuit elements or thyristor converter circuit elements that conduction in succession.
Thyristor commutation is the process to turn off a thyristor. The circuit that is used to realize commutation process is called the commutation circuit. The duration for which the commutation circuit is able to apply reverse bias across the thyristor is called the "circuit turn-off time" which should be greater than the "thyristor turn-off time". In general, complete commutation may involve a number of events. The most important of these are:
(1) The reduction of forward current to zero in one power semiconductor switching elements.
(2) The delay of reapplication of forward voltage to this element until it has regained its forward-blocking capability.
(3) The provision of an alternative path for the load-current flow until the next element to conduct is turned on.
(4) The build-up of forward current in the next element which is to conduct.
These events may occur concurrently or in sequence. Reliable operation of most power electronic equipment is critical dependent upon how well the commutation process is handled.
When power transistors, power MOSFET, or GTO devices are used as the switching elements, the commutation process is simplified. The transistor is a current-driven switching device, its current flow stops when the base drive is removed or driven negative. Also, the transistor will almost immediately revert to the blocking or off-state when this occurs. While as the MOSFET is a voltage-driven switching device, its current flow controlled by a positive-biased gate-to-drain voltage, its current flow stops when the positive biased voltage is removed. A GTO regains its blocking capability in the order of ten microseconds after the required negative gate current is applied. However, it is still very important to provide an alternative path for load current flow during transistor or GTO switching when the load is inductive.
The "commutation circuits" or "triggering circuits" will depend on the power switching devices used, connected load, operation condition, required performance and reliability.
For reliable operation the switching devices should be right chosen and the commutation circuits should be carefully designed.
5.3 Power Semiconductor Devices Characteristics
The characteristics of semiconductor devices are mainly electrical and thermal. When semiconductors are used for power applications, the device characteristics of great importance are the on-state voltage drop, the on-state current handling ability, the off-state voltage blocking ability, the switching speed, the rate of recovery of blocking capability, the power dissipation limits, and the power gain. The detailed characteristics and ratings of each power semiconductor device are very involved and need a separated paragraph to discuss. In the followings, however, brief description of power semiconductor characteristics are introduced. These are mainly in terms of thyristor; it is to be assumed that it is equally valid, where applicable, to diodes, transistors, GTOs, and MOSFETs.
Effects of Temperature
Temperature affects semiconductor devices in two ways: firstly it affects their electrical characteristics, and secondly it may have a direct effect on the materials used or inadvertently included in their construction.
The dissipation of losses in a semiconductor device leads to a rise in its internal temperature which is a function of the amount of the dissipation and the thermal resistance of the cell. The latter is normally defined (except for very small wire-ended devices) as between the virtual junction and the mounting base or stud, so that the temperature rise considered from the base to the junction is
(5.5)
where Rj-b is the thermal resistance and W is the power dissipation, the base assumes a temperature above that of its surroundings by an amount depending on the total thermal resistance external to the cell. If Rb-a is the thermal resistance from the base to the ambient fluid or heat sink, the total temperature rise is
(5.6)
The above is a slight oversimplification in that a small proportion of the heat produced is lost directly from the housing of the cell, or along the flexible top connection in the case of a single-ended cell.
Voltage Ratings
Absolute voltage limits are set to the permissible operating voltage by the breakover voltage in the forward direction, and by the sharp increase in reverse current in the avalanche region. The "breakover voltage" is defined as the voltage on the voltage-current characteristic for which the differential resistance is zero and where the voltage applied to the thyristor increases a maximum value. In practice a maximum instantaneous reverse voltage rating is usually determined by reference to a particular value of leakage current, based on the known spread of characteristics for a particular type of cell; this constitutes a peak transient or non-repetitive rating, related to a specified duration: e.g. the amplitude of a 10 msec half-sine-wave. Lower values of voltage, arrived at by applying an empirical ratio or testing to a lower current limits, may be specified as repetitive instantaneous or dc ratings, limiting the average or continuous reverse loss to a reasonable, low level.
A repetitive voltage rating dose not signify an appropriate nominal operating voltage for a semiconductor device, which can only be arrived at (unless by trial and error) through a knowledge of possible transient voltages, repetitive or nonrepetitive, superimposed on the normal supply.
The effects of an increasing junction temperature on the reverse characteristic of a silicon diode or thyristor are to increase the leakage current (ideally according to an exponential law but in practice not necessarily, owing to leakage at the edge of the element) and to increase the avalanche voltage. The effect is thus illustrated in Fig. 5.2. Fig. 5.2 also shows the effects on the forward blocking characteristic of a thyristor, which are, again, to increase the leakage current, but, in consequence, to reduce the breakover voltage.
Relative to measurements at normal room temperature (
for the purposes of standardization), the reverse voltage rating as determined by measurement at a particular current may be higher at the maximum rated junction temperature, because of the increased avalanche voltage, or lower, because of the increased leakage current, depending on the shape of the characteristics; these alternative possibilities are illustrated in Fig. 5.3. At low temperatures, the avalanche effect is normally predominant, and the voltage rating therefore reduced. The overall voltage ratings of a particular thyristor, which have to conform to a standardized relationship, are based on the least favorable of all the forward and reverse tests, at room temperature and at the maximum junction temperature, allowing small safety margins for possible errors in measurements.
Forward Current Ratings
There are three main considerations which may set limits to the permissible forward currents namely
(1) The requirement that the rated junction temperature should not be exceeded as a result of losses.
(2) The safe current rating of the external connections, particularly flexible conductors and crimps.
(3) The temperature gradients within the cell, which should not be such as to cause excessive stress through differential expansion.
Since the junction temperature depends on the internal temperature rise and the base temperature, the limiting current determined by (1) is a function of base temperature. On the other hand, (3) is largely independent of base temperature, and (2) is commonly defined without reference to temperature. A graph of current rating against base temperature is therefore normally in the form of a region constant current adjoining a region in which the current falls with increasing base temperature, as illustrated in Fig. 5.4.
If the reverse loss is so small as to be considered negligible, as is usually the case in a sizable silicon device, the current rating falls to zero at a base temperature equal to the rated continuous junction temperature, as in Fig. 5.4(a); otherwise a margin of temperature is left below that level corresponding to the effect of the reverse loss alone, as in Fig. 5.4(b).
Gate Firing Characteristics and Gate Ratings
The direct effect of applying a positive gate current to a thyristor can be considered to be a reduction in the breakover voltage, typically as illustrated by the graph of Fig. 5.5. In a practical application, the minimum effective firing current is that which will reduce the breakover voltage below the voltage applied to the cell, and to cover the majority of applications it is defined with respect to a suitable low anode-to-cathode voltage-typically five volts. The figure quoted is the maximum limit of what is normally a fairly wide production spread, at a particular temperature, together with the corresponding maximum gate firing voltage. In addition, it is normal to quote the maximum gate voltage that is guaranteed not to fire any thyristor at its quoted minimum breakover voltage, typically 0.25V.
di/dt Effects
The rate of rise of current, commonly referred to as 'di /dt', for which a thyristor can safely be rated varies greatly according to its operating conditions, being reduced by factors which increase the energy dissipated and increased by those which improve the thyristor's switching characteristics; Thus the rating is reduced for
(a) an increasing forward blocking voltage immediately before switching;
(b) an increasing peak forward current;
(c) an increasing designed voltage rating, which generally implies a thicker silicon element and higher forward voltage drop; and is increased by
(d) an increased gate overdrive i.e., the ratio between the gate current and the firing current limit for the cell (since the latter varies with temperature, the minimum operating junction temperature is also a factor)
(e) a reduced gate current rise.
dv/dt Effects
A thyristor may be triggered as a result of a high rate of rise of forward voltage, even though no gate current is applied. The effect may be thought of as due to capacitive current within the element performing the same function as a gate current, albeit not necessarily in the region of the gate. The rate of rise, commonly 'dv/dt', at which a thyristor may break over is basically a characteristic, but, as in the case of breakover due to excessive forward voltage, it may effectively constitute a rating, in that the cell may be damaged as a result of the irregular mode of switching.
As with thyristor gate current, the effect of dv/dt may be regarded as a reduction of the breakover voltage, the effect becoming greater with increasing junction temperature. Thus, dv/dt rating are normally given for the most restrictive case of maximum junction temperature, and some qualification as to the applied voltage, for example, a certain proportion of the quoted breakover voltage.
Forward Recovery and Reverse Recovery
These terms are usually related to the recovery capabilities of switching diodes. The existence of capacitance effects in both the forward (ON) and reverse (OFF) conditions of the junction diode indicates that when switching between these states there are going to be transient and that they are certain to be different in the two directions OFF to ON and ON to OFF. For switching diodes in high frequency power electronic system their forward and reverse recovery time are usually in the range of several to tens of nano-seconds. These properties will be described in detail the chapter 3.
Secondary Breakdown
This is a Potentially destructive phenomenon that occurs in all bipolar transistors. When the transistor is switched off, the emitter-base junction is reverse-biased and a lateral or transverse electric field is produced in the base region which causes current crowding under the center of the emitter. This constricts the current flow to a small portion of the base region, producing localized hot spots in the transistor pellet. If the energy in these hot spots is sufficient, the excessive localized heating can destroy the transistor. Forward-biased second breakdown also can occur during turn-on of the transistor. However, the localized current densities are greater during reverse-biased second breakdown. Thus, the energy capability of the transistor is much lower when it is reverse biased.
The second-breakdown limiting effect is most severe during switching transients when relatively high voltages and high current occur simultaneously for finite time intervals Power transistors with narrow base regions, which are required to achieve high-frequency switching characteristics, are more severely limited by second breakdown.
FBSOA and RBSOA
The safe operating area (SOA) is defined as the region of v - i characteristics (Fig. 5.6a) of the transistor within which the Vce /Ic loci must fall. During turn-on the switching loci of Vce /Ic must be within the forward bias SOA (FBSOA) as shown in Fig. 5.6(b). While during turn-off the switching loci of Vce /Ic must be within the reverse bias SOA (RBSOA) as shown in Fig.5.6(c). In practice, to allow for component tolerances, potential product aging and to achieve a low reject rate at final test and high reliability, the equipment designed should keep well away from the limit lines on the data-sheets. The switching loci of a power transistor is greatly dependent upon the connected load, base driver, and the snubber circuits.
and (c) turn-off SOA (FBSOA).
6. FUNDAMENTAL PRINCIPLES OF POWER ELECTRONICS
In the text book of William E. Newell, he has listed "ten cornerstones of power electronics". These ten rules are fundamental tools for every power electronic engineer and should be clearly understood. In the followings, these ten rules are introduced with some given examples.
Ten Cornerstones of Power Electronics
1. Kirchoff's Voltage Law
For any lumped electric circuit, for any of its loops, and at any time, the algebraic sum of the branch voltages around the loop is zero.
Notes:
There are two kinds of circuits: lumped circuits and distributed circuits. For two terminal lumped elements, the current through the element and the voltage across it are well defined quantities. For lumped elements with more than two terminals, the current entering any terminal and the voltage across any pair of terminals are well defined at all times. In practice, any interconnection of lumped elements such that the dimensions of the circuit are small compared with the wavelength associated with the highest frequency of interest will be called a lumped circuit.
Remark:
1. KVL imposes a linear constraint between branch voltages of a loop.
2. KVL is independent of the circuit elements.
2. Kirchoff's Current Law
For any lumped electric circuit, for any of its nodes, and at any times, the algebric sum of all branch currents leaving the node is zero.
Remarks:
1. KCL imposes a linear constraint on the branch currents of a node.
2. KCL is independent of the nature of the circuit elements.
3. KCL expresses the conservation of charge at every node.
3. Ohm's Law V = iR
A linear time-invariant resistor, by definition, has a V-I characteristic (see Fig. 6.1a) that does not vary with time and is also a straight line through the origin. Therefore, the relation between its instantaneous voltage v(t) and current i(t)is expresses by Ohm's law as follows:
(6.1)
R and G are constants independent of i, v, and t. R is called the resistance (unit is Ohm) and G called conductance.
where C is a constant (independent of v, q, and t) which measures the slope of the characteristic and is called capacitance (unit is Farad) and S is called the elastance. Equation 6.2 can be further written as:
(6.3)
(6.4)
It is interesting to note that the value of v at time t, v(t), depends on its initial value v(0) and all the values of the current between time 0 and time t; this fact is often alluded to by saying that "capacitor have memory", whereas the resistor is a memoryless element.
5. 
The voltage across the inductor (measure with reference direction indicated in Fig. 6.1c) is given by Faraday's induction law as
(6.5)
where v is in volts and £pis in webers. An inductor is called time-invariant if its characteristic does not change with time; an inductor is called linear if at all times its characteristic is a straight line through the origin of the i - plane. By definition the characteristic of the linear time-invariant inductor has an equation of the form
(6.6)
where L is a constant (independent of , i, and t) and is called the inductance (unit is Henrry). The equations relating the terminal voltage and current be further derived as:
(6.7)
(6.8)
It is interesting to note that the value of i at any time t, i(t), depends on its inertial value i(0) and all the values of the current between time 0 and time t; this fact is often alluded to by saying that "inductor have memory", whereas the resistor is a memoryless element.
6. Average Value and Root-Mean-Square
(6.9)
(6.10)
A period sinusoidal waveform with amplitude of A, its half period average value is 
and RMS value is 
.
7. Instantaneous Power and Average Power
(6.11)
(6.12)
8. Average voltage across an inductor over a full cycle in steady sate is zero.
Consider an inductor L, with a voltage across it of 
and with a current through it of 
:
(6.13)
or
(6.14)
By definition, in steady state the current at the end of a period, 
, must be equal to the current at the beginning of the period, 
. Thus,
(6.15)
If Eq. (6.15) is multiplied by the constant, L /T, then the left side of this equation is the average voltage across the inductor. Since the right side remains equal zero, the principle "the average voltage across an inductor over a full cycle in steady state is zero" is shown.
9. Average current through a capacitor over a full cycle in steady state ia zero.
Consider a capacitor C with a voltage across it of 
and a current through it of 
.
(6.16)
or
(6.17)
In steady state, the voltage across the capacitor at the end of a period, 
, must be equal to the voltage at the beginning of the period, 
. Therefore,
(6.18)
If 6.18 is multiplied by the constant, C/T, then the left side of this equation is the average current. Since the right side remains equal zero, the principle "the average current through a capacitor over a full cycle in steady state is zero" is shown.
EXAMPLE 1
Consider the switching circuit in Fig. 6.2 with E = 200V and R = 10.Assume that the periodically operated switch is closed for 10s and opened for 40s, and that the inductor is large enough such that its current is always greater than zero during both the on- and off-intervals of the switch. Used the steady-state principles for the inductor and capacitor to calculate the average current through the resistor R and through the inductor L in steady state.
Solution:
When the switch is closed, the diode is reverse-biased and the voltage 
is equal to E. The inductor current 
is assumed to be greater than zero for all time during each switching period in steady state. When the switch is opened, the inductor current cannot change abruptly, so that it immediately flows through the diode, which is referred to as a coating or freewheeling diode. The diode current decays during the freewheeling interval with a time function determined by the particular E, R, L, C, switching period T, and the switching duty ratio d. The duty ratio of the switching is defined as
(6.19)
The average value of the voltage 
is
(6.20)
In steady state, the average voltage across the inductor must be zero. Thus, the average value of 
must all appear across the resistor. Therefore, the average current through R is
(6.21)
Since no average current can flow through the capacitor in steady state, the average value of the inductor current is equal to the average current through R.
(6.22)
The inductance and capacitance values would have to be known in order to determine the specific time functions for the circuit voltages and currents.
It should be noted that 
and 
are not composed of simple exponential functions of time, since they are given by the solutions of the second-order differential equations representing the circuit during the on- and off-intervals.
EXAMPLE 2
Consider the circuit of Fig. 6.2 again with E = 200V, R = 10, 
, and d = 0.2. However, now assume that capacitor is so large that there is negligible ripple in the load voltage 
and assume an inductance L = 400H. Determine the steady-state waveforms for 
, 
, and 
.
Solution:
Since the supply voltage E and the duty ratio d are the same as in the previous example, 
. With the assumption of a large enough capacitor such that there is negligible ripple in the load voltage 
, this voltage is essentially a constant, as shown in Fig. 6.3. Now the slope of the current through the inductor is determined from 
as follows:
(6.23)
(6.24)
(6.25)
Since the two triangular portions of the current 
have the same height, the average value of 
is the same during the on-interval and during the off-interval. The average value of the inductor current must also equal the average of the current through the resistor R, as there can be no average current through the capacitor in steady state. Figure 6.3(b) shows the steady-state current waveforms.
A useful check on the results for this example is to calculate the power delivered from the dc supply, which must equal the power delivered to the load resistor.
(6.26)
(6.27)
The average value of the dc supply current is obtained easily from the waveform for 
in Fig. 6.3(b). It is very important to note that in this case the voltage across R is essentially a constant. Thus
(6.28)
In a more general case, with ripple in the voltage 
, it would be necessary to calculate the RMS value of 
to determine the power delivered to the resistor.
10. Harmonic Analysis
In most power electronic circuits the voltage and current waveforms are distorted and nonsinusoidal. Thus harmonic analysis is a very useful analytical tool for every power electronic engineer. A brief review of the essential elements of harmonic analysis is presented here. A method to calculate the Fourier coefficients of a generalized piecewise linear waveforms are also presented in this section.
All periodic waveforms arising in physical system can be represented by a Fourier series of the form
(6.29)
where 
is the fundamental frequency with period T, n takes on all positive integer values, and the An's and Bn's are the amplitudes of the cosine and sine terms in the Fourier series.
The Fourier representation is based on the very useful property of sinusoidal functionsevery arbitrary f(wt) for a physical system can be represented by the summation of a set of sine and cosine waves.
In general, an infinite number of sine and cosine waves are required, each having unique amplitudes, An and Bn. It is worthy of note that f(wt) is restricted to functions arising in physical systems. The mathematician is always capable of defining a function which does not permit the application of even quite generally accepted mathematical operations. Fortunately, functions arising in physical systems are singlevalued and have only a finite number of discontinuities in any finite interval, which together are sufficient conditions for convergence of the Fourier series.
The equations which are used to determine the coefficients of the terms in the Fourier series are as follows:
(6.30)
(6.31)
(6.32)
where n = 1, 2, 3, (all positive integers).
Equation (6.30) is the average value or dc component of the waveform f(t). These equations can also be written in terms where the variable of integration is the angle t, as follows:
(6.33)
(6.34)
(6.35)
generally, these latter equations are more convenient to apply.
It is quite useful in the determination of the coefficient for the Fourier series to use symmetry properties of the distorted waveforms. The following types of symmetry occur often in practical waveforms:
1. Odd-function symmetry; f(wt) = -f(-wt). Waveforms with this symmetry contain only sine terms.
2. Even-function symmetry; f(wt) = f(-wt). Waveforms with this symmetry contain only cosine terms.
3. Half-wave symmetry; |f(wt) = -f(wt + p)|. Waveforms with this symmetry contain only odd harmonics.
Example 3
Determine the harmonics presented in the square wave 
shown in Fig. 6.4(a).
Solution:
With the wt = 0 axis chosen as indicated in Fig. 6.4(a), this square wave has both even-function symmetry, f(wt) = -f(-wt), and half-wave symmetry, f(wt) = -f(wt + p), is said to have an even quad symmetric property, and its Fourier series will contain odd harmonic of cosine terms only. From (6.34),
(6.36)
It is quite clear that (6.36) becomes zero for even integer values of n, since the arguments of both terms becomes an even multiple of . When n is an odd integer,
Thus, the Fourier series representation for the square wave of Fig. 6.4(a) is
(6.37)
with its harmonic spectrum shown in Fig. 6.4(b).
The generalized piecewise linear periodic waveform as shown in Fig. 6.6(a) with M+1 subintervals can be represented as:
(6.38)
is defined as the k-th interval unit sample function. The k-th interval of the piecewise linear waveform is shown in Fig. 6.6(b) and can be used as the basic switching unit for PWM waveform spectra analysis.
Fourier Coefficients Calculation Methods
In PWM inverter applications, output waveforms are usually either even or odd functions. The desired output is its fundamental component, higher order harmonics are supposed to be minimized by the modulation strategy, and no dc component should exist. From Fourier series analysis, it can be shown that a periodic function f(t) has half-cycle symmetry and in addition, is either an even or odd function, then f(t) is said to have even or odd quarter-cycle symmetry, and consists of odd harmonics of cosine or sine terms only. For three phase applications all harmonic multiples of three are suppressed. The Fourier coefficients are functions of the switching instants and levels of the PWM waveforms. It is therefore desired to numerically compute the Fourier coefficients exactly and efficiently. The output waveforms of a three-phase PWM inverter are identical with one another but displaced by 
. These waveforms can be represented by Fourier series as shown:
(6.39)
(6.40)
(6.41)
If the inverter output waveforms are symmetrical about the time axis, only the interval from 
need be considered. These symmetrical waveforms can be divided into subintervals and then used straight line approximations; for pulse-like waveforms as the PWM modulation signals, this will be the exact representation. The switching function 
of the PWM modulator can be defined as:
(6.42)
where 
is in electrical radians. The harmonic analysis for the PWM output waveform consists of determining the Fourier coefficients of Eq. (6.42) which can be expressed as:
(6.43)
(6.44)
(6.45)
S() in the k-th interval can be expressed as:
(6.46)
Let's define 
, then S() can be expresses as
(6.47)
The coefficients of Eq. (6.43)-(6.45) can then be expressed as:
(6.48)
(6.49)
(6.50)
where Xk , Yk and Zk are determined from the modulation methods. For PWM modulation signals, 
will be zero and the computation required will be much reduced. For the calculation of the nth-order sine and cosine terms, the following recursive formulas can be used to speed up the computation.
(6.51)
(6.52)
It should be noted that harmonics computed from Eq.(6.48)-(6.50) are exact solutions and can be of any order as long as the computation time and accuracy of the computer are concerned. This computation method can be used for the calculation of Fourier coefficients of pulsewidth modulated waveforms with great ease. A computer program of this computation algorithm written by the C-language is listed in Appendix D.
Power Electronics Homework 1.
Problem: For the given single-phase half-wave rectifier with R-L load as shown in Fig. 1, the source voltage 
, and L=10mH, with initial condition 
, the switching SW closes at t = 0, calculate the following:
(a) The extinction angle = ?
(b) The average value of the load current 
.
(c) The rms value of the load current 
.
(d) The rms magnitudes of the fundamental, second, third, and fourth harmonics in 
.
(e) The power factor 
= ?
(f) The output voltage 
and current 
ripple factor ?
(g) Dolt the computer calculation or simulation results of the following signals in full cycles 
(h) Plot the relationship between the inductive load factor 
and the extinction angle .
(i) Plot the normalized average current In and normalized rms current versus .
(j) Plot the normalized output average voltage
versus .
