Power Electronics Converters, Applications, and Design PDF

POWER ELECTRONICS Converters, Applications, and Design THIRD EDITION NED M O W Department of Electrical Engineering University of Minnesota Minneapolis, Minnesota TORE M. UNDELAND Department of Electrical Power Engineering Norwegian Uniuersity of Science and Technolom, NTNU Trondheim, Norway WILL...

POWER ELECTRONICS Converters, Applications, and Design THIRD EDITION NED M O W Department of Electrical Engineering University of Minnesota Minneapolis, Minnesota TORE M. UNDELAND Department of Electrical Power Engineering Norwegian Uniuersity of Science and Technolom, NTNU Trondheim, Norway WILLIAM P. ROBBINS Department of Electrical Engineering University of Minnesota Minneapolis, Minnesota JOHN WILEY & SONS, INC. E!XECUIIVE EDROR Bill Zobrist SENIOR EDlTORIAL ASSISTANT Jovan Yglecias MARI tc(on)r t c ( o f 0. In order to turn the switch off, a negative control signal is applied to the control terminal of the switch. During the turn-off transition period of the generic switch, the voltage build-up consists of a turn-off delay time t d ( o f f )and a voltage rise time tN. Once the voltage reaches its final value of v d (see Fig. 2-64, the diode can become forward biased and begin to conduct current. The current in the switch falls to zero with a current fall time tri as the current I , cornmutates from the switch to the diode. Large values of switch voltage and switch current occur simultaneously during the crossover interval t,(,fO, where tc(of0 = trv + tfi (2-4) The energy dissipated in the switch during this turn-off transition can be written, using Fig. 2-6c, as  2-4 DESIRED CHARACTERISTICS IN CONTKOLLABLE SWITCHES 23 where any energy dissipation during the turn-off delay interval td(off)is ignored since it is small compared to Wc(off). The instantaneous power dissipation pT(t) = v,i, plotted in Fig. 2-6c makes it clear that a large instantaneous power dissipation occurs in the switch during the turn-on and turn-off intervals. There are!, such turn-on and turn-off transitions per second. Hence the average switching power loss P, in the switch due to these transitions can be approximated from Eqs. 2-2 and 2-5 as Ps= ?'2VJo!Xtc(on)+ tc(off9 (2-6) This is an important result because it shows that the switching power loss in a semicon- ductor switch varies linearly with the switching frequency and the switching times. Therefore, if devices with short switching times are available, it is possible to operate at high switching frequencies in order to reduce filtering requirements and at the same time keep the switching power loss in the device from being excessive. The other major contribution to the power loss in the switch is the average power dissipated during the on-state Po,, which varies in proportion to the on-state voltage. From Eq. 2-3, Po, is given by ton P, = v I- On OT, (2-7) which shows that the on-stage voltage in a switch should be as small as possible. The leakage current during the off state (switch open) of controllable switches is negligibly small, and therefore the power loss during the off state can be neglected in practice. Therefore, the total average power dissipation P, in a switch equals the sum of P, and Po,. Form the considerations discussed in the preceding paragraphs, the following char- acteristics in a controllable switch are desirable: 1. Small leakage current in the off state. 2. Small on-state voltage Von to minimize on-state power losses. 3. Short turn-on and turn-off times. This will permit the device to be used at high switching frequencies. 4. Large forward- and reverse-voltage-blockingcapability. This will minimize the need for series connection of several devices, which complicates the control and protection of the switches. Moreover, most of the device types have a minimum on-state voltage regardless of their blocking voltage rating. A series connection of several such devices would lead to a higher total on-state voltage and hence higher conduction losses. In most (but not all) converter circuits, a diode is placed across the controllable switch to allow the current to flow in the reverse direction. In those circuits, controllable switches are not required to have any significant re- verse-voltage-blocking capability. 5. High on-state current rating. In high-current applications, this would minimize the need to connect several devices in parallel, thereby avoiding the problem of current sharing. 6. Positive temperature coefficient of on-state resistance. This ensures that paralleled devices will share the total current equally. 7. Small control power required to switch the device. This will simplify the control circuit design. 8. Capability to withstand rated voltage and rated current simultaneously while switching. This will eliminate the need for external protection (snubber) circuits across the device. 24 CHAPTER 2 OVERVIEW OF POWER SEMICONDUCTOR SWITCHES 9. Large dvldr and dildt ratings. This will minimize the need for external circuits otherwise needed to limit dvldt and dildt in the device so that it is not damaged. We should note that the clamped-inductive-switching circuit of Fig. 2-6a results in higher switching power loss and puts higher stresses on the switch in comparison to the resistive-switching circuit shown in Problem 2-2 (Fig. P2-2). We now will briefly consider the steady-state i-v characteristics and switching times of the commonly used semiconductor power devices that can be used as controllable switches. As mentioned previously, these devices include BJTs, MOSFETs, GTOs, and IGBTs. The details of the physical operation of these devices, their detailed switching characteristics, commonly used drive circuits, and needed snubber circuits are discussed in Chapters 19-28. 2.5 BIPOLAR JUNCTION TRANSISTORS AND MONOLITHIC DARLINGTONS The circuit symbol for an NPN BJT is shown in Fig. 2-7a, and its steady-state i-v characteristics are shown in Fig. 2-7b. As shown in the i-v characteristics, a sufficiently large base current (dependent on the collector current) results in the device being fully on. This requires that the control circuit provide a base current that is sufficiently large so that lC I, > - ~ F E where hFEis the dc current gain of the device. The on-state voltage VCE(sat)of the power transistors is usually in the 1-2-V range, so that the conduction power loss in the BJT is quite small. The idealized t v character- istics of the BJT operating as a switch are shown in Fig. 2-7c. Bipolar junction transistors are current-controlled devices, and base current must be supplied continuously to keep them in the on state. The dc current gain hFEis usually only 5-10 in high-power transistors, and so these devices are sometimes connected in a Darlington or triple Darlington configuration, as is shown in Fig. 2-8, to achieve a larger current gain. Some disadvantages accrue in this configuration including slightly higher overall VCE(sat)values and slower switching speeds. Whether in single units or made as a Darlington configuration on a single chip [a monolithic Darlington (MD)], BJTs have significant storage time during the turn-off transition. Typical switching times are in the range of a few hundred nanoseconds to a few microseconds. 'CE (61 (cl Figure 2-7 A BJT: (a) symbol, (6) i-u characteristics, (c) idealized characteristics.  2-6 METAL- OXIDE-SEMICONDUCTOR FIELD EFFECT TRANSISTORS 25 'BE - - OE - Including MDs, BJTs are available in voltage ratings up to 1400 V and current ratings of a few hundred amperes. In spite of a negative temperature coefficient of on-state resistance, modern BJTs fabricated with good quality control can be paralleled provided that care is taken in the circuit layout and that some extra current margin is provided, that is, where theoretically four transistors in parallel would suffice based on equal current sharing, five may be used to tolerate a slight current imbalance. 2-6 METAL- OXIDE- SEMICONDUCTOR FIELD EFFECT TRANSISTORS The circuit symbol of an n-channel MOSFET is shown in Fig. 2-9a. It is a voltage- controlled device, as is indicated by the C v characteristics shown in Fig. 2-9b. The device is fully on and approximates a closed switch when the gate-source voltage is below the threshold value, VGS(*).The idealized characteristics of the device operating as a switch are shown in Fig. 2-9c. Metal- oxide- semiconductor field effect transistors require the continuous applica- tion of a gate-source voltage of appropriate magnitude in order to be in the on state. No gate current flows except during the transitions from on to off or vice versa when the gate capacitance is being charged or discharged. The switching times are very short, being in D + I UGS = 7v 'DS Q - + 'GS S 0 'DS la) (b) (c ) Figure 2-9 N-channel MOSFET: (a) symbol, (b) i-u characteristics, (c) idealized characteristics. 26 CHAPTER 2 OVERVIEW OF POWER SEMICONDUCTOR SWITCHES the range of a few tens of nanoseconds to a few hundred nanoseconds depending on the device type. The on-state resistance rDSfon)of the MOSFET between the drain and source increases rapidly with the device blocking voltage rating. On a per-unit area basis, the on-state resistance as a function of blocking voltage rating BVD, can be expressed as 2.5-2.7 rDS(on) = k BVDSS (2-9) where k is a constant that depends on the device geometry. Because of this, only devices with small voltage ratings are available that have low on-state resistance and hence small conduction losses. However, because of their fast switching speed, the switching losses can be small in accordance with Eiq. 2-6. From a total power loss standpoint, 300-400-V MOSFETs compete with bipolar transistors only if the switching frequency is in excess of 30- 100 kHz. However, no definite statement can be made about the crossover frequency because it depends on the operating voltages, with low voltages favoring the MOSFET. Metal- oxide- semiconductor field effect transistors are available in voltage ratings in excess of lo00 V but with small current ratings and with up to 100 A at small voltage ratings. The maximum gate-source voltage is 2 2 0 V, although MOSFETs that can be controlled by 5-V signals are available. Because their on-state resistance has a positive temperature coefficient, MOSFETs are easily paralleled. This causes the device conducting the higher current to heat up and thus forces it to equitably share its current with the other MOSFETs in parallel. 2-7 GATE-TURN-OFF THYRISTORS The circuit symbol for the GTO is shown in Fig. 2-1% and its steady-state t v charac- teristic is shown in Fig. 2-lob. Like the thyristor, the GTO can be turned on by a short-duration gate current pulse, and once in the on-state, the GTO may stay on without any further gate current. However, unlike the thyristor, the GTO can be turned off by applying a negative gate-cathode voltage, therefore causing a sufficiently large negative gate current to flow. This negative gate current need only flow for a few microseconds (during the turn-off time), but it must have a very large magnitude, typically as large as one-third the anode current being turned off. The GTOs can block negative voltages whose magnitude depends on the details of the Turn-off 0 Off-state , V~~ "AK O1 K (al Figure 2-10 A GTO: (a)symbol, (b) i-o characteristics, (c) idealized characteristics.  2-8 INSULATED GATE BIPOLAR TRANSISTORS 27 Snubber circuit - -- to reduce I I & at AI) lI- I I I :c I ==c d t at turn turn-off ,-Off t I I G I I Gate I I drive I I circuit I D l I I t tal fbt Figure 2-11 Gate turn-off transient characteristics: (a) snubber circuit, (b) GTO turn-off characteristic. GTO design. Idealized characteristics of the device operating as a switch are shown in Fig. 2-1Oc. Even though the GTO is a controllable switch in the same category as MOSFETs and BJTs, its turn-off switching transient is different from that shown in Fig. 2-6b. This is because presently available GTOs cannot be used for inductive turn-off such as is illus- trated in Fig. 2-6 unless a snubber circuit is connected across the GTO (see Fig. 2-lla). This is a consequence of the fact that a large dvldt that accompanies inductive turn-off cannot be tolerated by present-day GTOs. Therefore a circuit to reduce dvldt at turn-off that consists of R, C, and D,as shown in Fig. 2-lla, must be used across the GTO. The resulting waveforms are shown in Fig. 2-llb, where dvldt is significantly reduced com- pared to the dvldt that would result without the turn-off snubber circuit. The details of designing a snubber circuit to shape the switching waveforms of GTOs are discussed in Chapter 27. The on-state voltage (2-3 V) of a GTO is slightly higher than those of thyristors. The GTO switching speeds are in the range of a few microseconds to 25 FS.Because of their capability to handle large voltages (up to 4.5 kV) and large currents (up to a few kilo- amperes), the GTO is used when a switch is needed for high voltages and large currents in a switching frequency range of a few hundred hertz to 10 kHz. 2-8 INSULATED GATE BIPOLAR TRANSISTORS The circuit symbol for an IGBT is shown in Fig. 2-12a and its i-v characteristics are shown in Fig. 2-12b. The IGBTs have some of the advantages of the MOSFET, the BJT, and the GTO combined. Similar to the MOSFET, the TGBT has a high impedance gate, which requires only a small amount of energy to switch the device. Like the BJT, the IGBT has a small on-state voltage even in devices with large blocking voltage ratings (for example, V,,,, is 2-3 V in a 1ooO-V device). Similar to the GTO, IGBTs can be designed to block negative voltages, as their idealized switch characteristics shown in Fig. 2-12c indicate. Insulated gate bipolar transistors have turn-on and turn-off times on the order of 1 ps and are available in module ratings as large as 1700 V and 1200 A. Voltage ratings of up to 2-3 kV are projected. 28 CHAPTER 2 OVERVIEW OF POWER SEMICONDUCTOR SWITCHES N-MCT Go----l (6) (C) Figure 2-13 An MCT: (a) circuit symbols, (b) i-u characteristic, (c) idealized characteristics.  2-10 COMPARISON OF CONTROLLABLE SWITCHES 29 2-9 MOS-CONTROLLED THYRISTORS The MOS-controlled thyristor (MCT) is a new device that has just appeared on the commercial market. Its circuit symbol is shown in Fig. 2-13a, and its i-v characteristic is shown in Fig. 2-13b. The two slightly different symbols for the MCT denote whether the device is a P-MCT or an N-MCT. The difference between the two arises from the different locations of the control terminals, a subject discussed in detail in Chapter 26. From the i-v characteristic it is apparent that the MCT has many of the properties of a GTO, including a low voltage drop in the on state at relatively high currents and a latching characteristic (the MCT remains on even if the gate drive is removed). The MCT is a voltage-controlled device like the IGBT and the MOSFET, and approximately the same energy is required to switch an MCT as for a MOSFET or an IGBT. The MCT has two principal advantages over the GTO, including much simpler drive requirements (no large negative gate current required for turn-off like the GTO) and faster switching speeds (turn-on and turn-off times of a few microseconds). The MCTs have smaller on-state voltage drops compared to IGBTs of similar ratings and are presently available in voltage ratings to 1500 V with current ratings of 50 A to a few hundred amperes. Devices with voltage ratings of 2500-3000 V have been demonstrated in prototypes and will be available soon. The current ratings of individual MCTs are sig- nificantly less than those of GTOs because individual MCTs cannot be made as large in cross-sectional area as a GTO due to their more complex structure. 2-10 COMPARISON OF CONTROLLABLE SWITCHES Only a few definite statements can be made in comparing these devices since a number of properties must be considered simultaneously and because the devices are still evolving at a rapid pace. However, the qualitative observations given in Table 2-1 can be made. It should be noted that in addition to the improvements in these devices, new devices are being investigated. The progress in semiconductor technology will undoubtedly lead to higher power ratings, faster switching speeds, and lower costs. A summary of power device capabilities is shown in Fig. 2-14. On the other hand, the forced-commutated thyristor, which was once widely used in circuits for controllable switch applications, is no longer being used in new converter designs with the possible exception of power converters in multi-MVA ratings. This is a pertinent example of how the advances in semiconductor power devices have modified converter design. Table 2-1 Relative Properties of Controllable Switches Device Power Capability Switching Speed BJT/MD Medium Medium MOSFET Low - Fast GTO High Slow IGBT Medium Medium MCT Medium Medium 30 CHAPTER 2 OVERVIEW OF POWER SEMICONDUCTOR SWITCHES 5 kV 4 kV 3 kV 2 kV - iz CUIrrent 1 kV kHz J Freque Figure 2-14 Summary of power semiconductor device capabilities. All devices except the MCT have a relatively mature technology, and only evolutionary improvements in the device capabilities are anticipated in the next few years. However. MCT technology is in a state of rapid expansion, and significant improvements in the device capabilities are possible, as indicated by the expansion arrow in the diagram. 2-11 DRIVE AND SNUBBER CIRCUITS In a given controllable power semiconductor switch, its switching speeds and on-state losses depend on how it is controlled. Therefore, for a proper converter design, it is important to design the proper drive circuit for the base of a BJT or the gate of a MOSFET, GTO, or IGBT. The future trend is to integrate a large portion of the drive circuitry along with the power switch within the device package, with the intention that the logic signals, for example, from a microprocessor, can be used to control the switch directly. These topics are discussed in Chapters 20-26. In Chapters 5- 18 where idealized switch characteristics are used in analyzing converter circuits, it is not necessary to consider these drive circuits. Snubber circuits, which were mentioned briefly in conjunction with GTOs, are used to modify the switching waveforms of controllable switches. In general, snubbers can be divided into three categories: 1. Turn-on snubbers to minimize large overcurrents through the device at turn-on. 2. Turn-off snubbers to minimize large overvoltages across the device during turn- off. 3. Stress reduction snubbers that shape the device switching waveforms such that the voltage and current associated with a device are not high simultaneously.  2-12 JUSTIFICATION FOR USING IDEALIZED DEVICE CHARACTERISTICS 31 In practice, some combination of snubbers mentioned before are used, depending on the type of device and converter topology. The snubber circuits are discussed in Chapter 27. Since ideal switches are assumed in the analysis of converters, snubber circuits are neglected in Chapters 5- 18. The future trend is to design devices that can withstand high voltage and current simultaneously during the short switching interval and thus minimize the stress reduction requirement. However, for a device with a given characteristic, an alternative to the use of snubbers is to alter the converter topology such that large voltages and currents do not occur at the same time. These converter topologies, called resonant converters, are dis- cussed in Chapter 9. 2-12 JUSTIFICATION FOR USING IDEALIZED DEVICE CHARACTERISTICS In designing a power electronic converter, it is extremely important to consider the available power semiconductor devices and their characteristics. The choice of devices depends on the application. Some of the device properties and how they influence the selection process are listed here: 1. On-state voltage or on-state resistance dictates the conduction losses in the device. 2. Switching times dictate the energy loss per transition and determine how high the operating frequency can be. 3. Voltage and current ratings determine the device power-handling capability. 4. The power required by the control circuit determines the ease of controlling the device. 5. The temperature coefficient of the device on-state resistance determines the ease of connecting them in parallel to handle large currents. 6. Device cost is a factor in its selection. In designing a converter from the system viewpoint, the voltage and current require- ments must be considered. Other important considerations include acceptable energy efficiency, the minimum switching frequency to reduce the filter and the equipment size, cost, and the like. Hence the device selection must ensure a proper match between the device capabilities and the requirements on the converter. These observations help to justify the use of idealized device characteristics in ana- lyzing converter topologies and their operation in various applications as follows: 1. Since the energy efficiency is usually desired to be high, the on-state voltage must be small compared to the operating voltages, and hence it can be ignored in analyzing converter characteristics. 2. The device switching times must be short compared to the period of the operating frequency, and thus the switchings can be assumed to be instantaneous. 3. Similarly, the other device properties can be idealized, The assumption of idealized characteristics greatly simplifies the converter analysis with no significant loss of accuracy. However, in designing the converters, not only must the device properties be considered and compared, but the converter topologies must also be carefully compared based on the properties of the available devices and the intended application. 32 CHAPTER 2 OVERVIEW OF POWER SEMICONDUCTOR SWITCHES SUMMARY Characteristics and capabilities of various power semiconductor devices are presented. A justification is provided for assuming ideal devices, unless stated explicitly, in Chapters 5- 18. The benefits of this approach are the ease of analysis and a clear explanation of the converter characteristics, unobscured by the details of device operation. PROBLEMS 2-1 The data sheets of a switching device specify the following switching times corresponding to the linearized characteristics shown in Fig. 2-6b for clamped-inductive switchings: rri = 100 ns tf, = 50 ns r,, = 100 ns tfi = 200 ns Calculate and plot the switching power loss as a function of frequency in a range of 25- 100 kHz, assuming V, = 300 V and I , = 4A in the circuit of Fig. 2-6a. 2-2 Consider the resistive-switching circuit shown in Fig. P2-2. V, = 300 V, f’ = 100 kHz and R = 75 a, so that the on-state current is the same as in Problem 2-1. Assume the switch turn-on time to be the sum of tri and tf, in Problem 2-1. Similarly, assume the turn-off time to be the sum of r,, and tfi. Figure P2-2 Assuming linear voltage- and current-switching characteristics, plot the switch voltage and current and the switching power loss as a function of time. Compare the average power loss with that in Problem 2- 1. REFERENCES 1. R. Sittig and P. Roggwiller (Eds.), Semiconductor Devices for Power Conditioning, Plenum, New York, 1982. 2. M. S. Adlsr, S. W. Westbrook, and A. J. Yerman, “Power Semiconductor Devices-An- Assessme t,” IEEE Industry Applications Society Conference Record, 1980, pp. 723-728. 3. David L. Blackburn, “Status and Trends in Power Semiconductor Devices,” EPE ’93, 5th European Conference on Power Electronics and Applications, Conference Record, 1993, Vol. 2, pp. 619-625. 4. B. Jayant Baliga, Modern Power Devices, John Wiley & Sons, Inc., New York, 1987. 5. User’s Guide to MOS Controlled Thyristors, Harris Semiconductor, 1993. CHAPTER 3 REVIEW OF BASIC ELECTRICAL AND MAGNETIC CIRCUIT CONCEPTS 3-1 INTRODUCTION The purpose of this chapter is twofold: (1) to briefly review some of the basic definitions and concepts that are essential to the study of power electronics and (2) to introduce simplifying assumptions that allow easy evaluation of power electronic circuits. 3-2 ELECTRIC CIRCUITS An attempt is made to use Institute of El ctrical and Electroni Engineers (IEEE) stan- dard letter and graphic symbols as much as possible. Moreover, the units used belong to the International System of Units (SI). The lowercase letters are used to represent instan- taneous value of quantities that may vary as a function of time. The uppercase letters are used to represent either the average or the rms values. As an example, a voltage vOi and its average value Voi are shown in Fig. 1-4b. A value that is average or rms may be stated explicitly or it may be obvious from the context. The positive direction of a current is shown explicitly by a current arrow in the circuit diagram. The voltage at any node is defined with respect to the circuit ground, for example, v, is the voltage of node a with respect to ground. The symbol v,b refers to the voltage of node a with respect to node b, where, vab = v, - vb. 3-2-1 DEFINITION OF STEADY STATE In power electronic circuits, diodes and semiconductor switches are constantly changing their on or off status. Therefore the question arises: When is such a circuit in steady state? A steady-state condition is reached when the circuit waveforms repeat with a time period T that depends on the specific nature of that circuit. 33 34 CHAPTER 3 REVIEW OF BASIC ELECTRICAL AND MAGNETIC CIRCLJlT CONCEPTS 3-2-2 AVERAGE POWER AND 11118 CURRENT Consider the circuit of Fig. 3-1, where the instantaneous power flow from subcircuit 1 to subcircuit 2 is p(t) = vi (3- 1) Both v and i may vary as a function of time. If v and i waveforms repeat with a time period T in steady state, then the average power flow can be calculated as Under the conditions stated earlier, if subcircuit 2 consists purely of a resistive load, then v = Ri and in Eq. 3-2 Pa, = R f I, T i’dt (3-3) In terms of the rms value I of the current, the average power flow can be ex- pressed as P , = R12 (3-4) A comparison of Eqs. 3-3 and 3-4 reveals that the rms value of the current is I = JF 0 (3-5) which shows the origin of the term root-mean-square. If i is a constant dc current, then Eqs. 3-4 and 3-5 are still valid with the average and the rms values being equal. 3-2-3 STEADY-STATE ac WAVEFORMS WITH SINUSOIDAL VOLTAGES AND CURRENTS Consider the ac circuit of Fig. 3-2a, with an inductive load under a steady-state operation, where v= V ~cosVwt i= ficos(wt - +> (3-6) and V and I are the rms values. The v and i waveforms are plotted as functions of ot in Fig. 3-2b. 3-2-3-1 Phasor Representation Since both v and i vary sinusoidally with time at the same frequency, they can be represented in a complex plane by means of the projection of the rotating phasors to the horizontal real axis, as shown in Fig. 3-2c. Conventionally, these phasors rotate in a Subcircuit 1 Subcircuit 2 Figure 3-1 Instantaneous power flow.  3-2 ELECTRIC CIRCUITS 35 i 4 Source Load (a) Figure 3-2 Sinusoidal steady state. counterclockwise direction with an angular frequency w, and their rms values (rather than their peak values) are used to represent their magnitudes: V = VeJO and 1 = le-J4 (3-7) Considering EQ.3-6, the phasor diagram in Fig. 3-2c corresponds to the time instant when v attains its positive-maximum value. In Eq.3-7 V and I are related by the complex load impedance Z = R + j o L = Zd4 at the operating frequency w in the following manner: where I = VIZ. 3-2-3-2 Power, Reactive Power, and Power Factor The complex power S is defined as s = VI’ = vefl. 1eJ4 = v1&4 = ~ e J 4 (3-9) Therefore, the magnitude of the complex power, which is also called the apparent power and is expressed in the units volt-amperes, is s = vz (3-10) The real average power P is P = Re[S] = VI cos 4 (3-1 1) 36 CHAPTER 3 REVIEW OF BASIC ELECTRICAL AM> MAGNETIC CIRCUIT CONCEPTS which is expressed as a product of V and the current component Zp = Z cos +, which is in phase with the voltage in the phasor diagram of Fig. 3-2c. The out-of-phase component is 1, = Z sin +. The in-phase current component iJt) and the out-of-phase current component i,(t) can be expressed as ipk + I LU l l -1 -, -11 -11 -11 -11 2 2 2 2 2 2 I I I I I I I I I I I I l-l-4 u2 u1 I F mt Figure P3-3 “i - - R (Load) 15v 3 00 ”’”-,6 ,:I I Figure P3-8 60 CHAPTER 3 REVIEW OF BASIC ELECTRICAL AND MAGNETIC ClRCUlT CONCEFTS (b) Assume that C-m so that v,(t) = V,. Calculate, ,Z and the rms value of the capacitor current i,. (c) In part (b), plot v, and iL. 3-9 The repetitive waveforms for the current into, and the voltage across a load in Fig. P3-9 are shown by linear segments. Calculate the average power P into the load. E Figure P3-9 3-10 In Fig. P3-9, it is given that the load voltage maximum (minimum) is greater (lower) than Vavg by 1%. Similarly in the current, the fluctuation around its average value is 25%. Calculate the percent error if the average power is assumed to be VavJlpg, compared to its exact value. 3-11 A transformer is wound on a toroidal core. The primary winding is supplied with a square-wave voltage with a 250-V amplitude and B frequency of 100 kHz. Assuming a uniform flux density in the core, calculate the minimum number of primary winding turns required to keep the peak flux density in the core below 0.15 Wb/m2 if the core cross-sectional area is 0.635 cm2. Plot the voltage and flux density waveforms in steady state as functions of time. 3-12 A toroidal core has distributed airgaps that make the relative permeability equal to 125. The cross-sectional area is 0.113 cm2 and the mean path length is 3.12 cm. Calculate the number of turns required to obtain an inductance of 25 p,H. 3-13 In Example 3-2, calculate the transformer voltage regulation in percent, if the input voltage is 110 V to the transformer which supplies its full-load kVA to a load at the following power factors: (a) 1.0 (b) 0.8 (lagging) Note that the transformer voltage regulation is defined as Percent regulation = 100 x vout,No-load - vout,~l-ld vout.no-10sd 3-14 Refer to Problem 3-1 1 and calculate the magnetizing inductance L,,, if the mean path length equals 3.15 cm and the relative permeability pr = 2500. REFERENCES 1. H.P. Hsu,Fourier Analysis, Simon L Schuster, New York, 1967. 2. Any introductory textbook on electrical circuits and electromagnetic fields. CHAPTER 4 COMPUTER SIMULATION OF POWER ELECTRONIC CONVERTERS AND SYSTEMS 4-1 INTRODUCTION The purpose of this chapter is to briefly describe the role of computer simulations in the analysis and design of power electronics systems. We will discuss the simulation process and some of the simulation software packages suited for this application. In power electronic systems such as shown by Fig. 1-1 and repeated as Fig. 4-1, converters for power processing consist of passive component, diodes, thyristors, and other solid-state switches. Therefore, the circuit topology changes as these switches open and close as a function of time under the guidance of the controller. Usually it is not possible, and often not desirable, to solve for the circuit states (voltages and currents) in a closed form as a function of time. However, by means of computer simulation, it is possible to model such circuits. We will use computer simulations throughout this book as optional learning aid. Computer simulations are commonly used in research to analyze the behavior of new circuits, which leads to improved understanding of the circuit. In industry, they are used to shorten the overall design process, since it is usually easier to study the influence of a parameter on the system behavior in simulation, as compared to accomplishing the same in the laboratory on a hardware breadboard [ 11. The simulations are used to calculate the circuit waveforms, the dynamic and steady- state performance of systems, and the voltage and current ratings of varims components. Usually, there will be several iterations between various steps. As the confidence in the simulation develops, it may be possible to extend simulations to include power loss Control signals Measurements Figure 4-1 Power electronics Reference system: a block diagram. 61 62 CHAFTER 4 COMPUTER SIMULATION OF POWER CONVERTERS AND SYSTEMS calculations. This allows a thermal design to ensure temperature rise within the system to acceptable levels. However, it should be noted that in power electronics, much more so than in signal- level electronics, computer simulation and a proof-of-concept hardware prototype in the laboratory are complimentary to each other. That is, computer simulation should not be looked upon as a substitute for a hardware (breadboard) prototype. 4-2 CHALLENGES IN COMPUTER SIMULATION At the outset, we need to realize that there are several factors that make simulation of power electronic systems very challenging: 1. Solid-state switches including diodes and thyristors present extreme nonlinearity during their transition from one state to the other. The simulation program ought to be able to represent this switching of states in an appropriate manner. 2. The simulation may take a long time. The time constants, or in other words the response time of various parts within the system, may differ by several orders of magnitude. For example, in a motor drive, the semiconductor switches have switching times of microseconds or less, whereas the mechanical time constant or the response time of the motor and the load may be of the order of seconds or even minutes. This requires simulation to proceed with a very small time step to have the resolution to represent the smallest time constants, for example switching, with accuracy. In the same simulation, the maximum simulation time is usually large and is dictated by the longest time constant. 3. Accurate models are not always available. This is specially true for power semi- conductor devices (even for simple power diodes) but is also the case for magnetic components such as inductors and transformers. 4. The controller in the block diagram of Fig. 4-1, which may be analog and/or digital, needs to be modeled along with the power converters. 5. Even if only the steady-state waveforms are of interest, the simulation time is usually long due to unknown values of the initial circuit states at the start of the simulation. The challenges listed above dictate that we carefully evaluate the objective of the simulation. In general, it is not desirable to simulate all aspects of the system in detail (at least not initially, but it may be done as the last step). The reason is that the simulation time may be very long and the output at the end of the simulation may be overwhelming, thus obscuring the phenomena of interest. In this respect, the best simulation is the simplest possible simulation that meets the immediate objective. In other words, we must simplify the system to meet the simulation objectives. Some of the choices are discussed in the next section. 4-3 SIMULATION PROCESS In power electronics, several types of analyses need to be carried out. For each type of analysis, there is an appropriate degree of simulation detail in which the circuit compo- nents and the controller should be represented. In the following sections, we will discuss these various types of analyses. It is important to note that at each step it may be desirable to verify simulation results by a hardware prototype in the laboratory.  4-3 SIMULATION PROCESS 63 ii I component models) 1 i, " U t Prespecified Figure 4-2 Open-loop, large- control signals signal simulation. 4-3-1 OPEN-LOOP, LARGE-SIGNAL SIMULATION In order to get a better understanding of the behavior of a new system, we often start by simulating the power processor with prespecified control signals, as shown in the block diagram of Fig. 4-2. The objective of this simulation is to obtain various voltage and current waveforms within the converters of the power processor to verify that the circuit behaves properIy, as predicted by the analytical calculations. At the end, this step pro- vides us with a choice of circuit topology and the component values. This simulation includes each switch opening and closing, and the simulation is carried out over a large number of switching cycles to reach steady state. Most often, at this stage of simulation, no benefit is gained by including very detailed models of the circuit components. Therefore, the circuit components, especially the switching devices, should be represented by their simple (idealized) models. Because the design of the controller still remains to be carried out and the dynamic behavior of the system to changes in the operating conditions is not of interest at this early stage, the controller is not represented. Therefore, it is called an open-loop simulation. 4-3-2 SMALL-SIGNAL (LINEAR) MODEL AND CONTROLLER DESIGN With a chosen circuit topology and the component values, we can develop a linear (small-signal) model (Fig. 4-3) of the power processor as a transfer function using the techniques described later in this book. The important item to note is that in such a model, the switches are represented by their averaged characteristics. Once we have a linearized model of the circuit, there are well-known methods from control theory for designing the controller to ensure stability and the dynamic response to disturbances or small changes (indicated by A in Fig. 4-3) in the input, the load, and the reference. There are specialized software packages available commercially that automate the controller design process. 4-3-3 CLOSED-LOOP, LARGE-SIGNAL SYSTEM BEHAVIOR Once the controller has been designed, the system performance must be verified by combining the controller and the circuit under a close-loop operation, in response to large disturbances such as step changes in load and inputs. The block diagram is shown in Fig. 4-4. This large-signal simulation is carried out in time domain over a long time span that ______ A Load A input Power processor 4 * (small-signal, linearized) model Control signals Figure 4-3 Small-signal (linear) model Controller A Reference and controller design. 64 CHAPTER 4 COMPUTER SIMULATION OF POWER CONVERTERS AND SYSTEMS Power processor Power input - (each switching is represented; simple switch models; saturation - Output and pertinent nonlinearities are included) Control Measurements Figure 4-4 Closed- loop, large-signal system behavior. includes many (thousands of) switching cycles. Therefore, the switching devices should be represented by their simple (idealized) models. However, saturation and other pertinent nonlinearilities and losses may be included. It is sufficient to represent the controller in such a simulation in as simple a manner as possible, rather than representing it in detail at a component level by operational amplifiers, comparators, and so on. 4-3-4 SWITCHING DETAILS In the previous simulation step, the objective was to obtain the overall system behavior that is only marginally affected by semiconductordevice nonidealities. Now the objective is to obtain overvoltages, power losses, and other component stresses due to the nonideal nature of switching devices and the stray inductances and capacitances within the power processor. This knowledge is necessary in the selection of component ratings, assessing the need for protection circuitry such as snubbers and the need to minimize stray induc- tances and capacitances. The block diagram is shown in Fig. 4-5. To obtain this information, only a few switching cycles need to be simulated with worst-case initial values of voltages and currents obtained in the previous simulations. Only a part of the circuit under investigation should be modeled in detail, rather than the entire circuit. Simulation over only a few cycles is needed to obtain the worst-case stresses because they repeat with each cycle. To this end, we need detailed and accurate models of switching devices. 4-4 MECHANICS OF SIMULATION [l] Having established the various types of analyses that need to be carried out, the next step is to determine the best tools for the job. There are two basic choices: (1) circuit-oriented simulators and (2) equation solvers. These are now discussed in a generic manner. 4-4-1 CIRCUIT-ORIENTED SIMULATORS Over the years, considerable effort has been put into developing software for circuit- oriented simulators. In these software packages, the user needs to supply the circuit Power processor I arepresented in detail: few switching cycles) I t Prespecified control signals Figure 4-5 Switching details.  4-5 SOLUTION TECHNIQUES FOR TIME-DOMAIN ANALYSIS 65 topology and the component values. The simulator internally generates the circuit equa- tions that are totally transparent to the user. Depending on the simulator, the user may have the flexibility of selecting the details of the component models. Most simulators allow controllers to be specified by means of a transfer function or by models of com- ponents such as operational amplifiers, comparators, and so on. 4-4-2 EQUATION SOLVERS An alternative to the use of circuit-oriented simulators is to describe the circuit and the controller by means of differential and algebraic equations. We must develop the equa- tions for all possible states in which the circuit may operate. There may be many such states. Then, we must describe the logic that determines the circuit state and the corre- sponding set of differential equations based on the circuit conditions. These algebraic/ differential equations can be solved by using a high-level language such as C or FOR- TRAN or by means of software packages specifically designed for this purpose that provide a choice of integration routines, graphical output, and so on. 4-4-3 COMPARISON OF CIRCUIT-ORIENTED SIMULATORS AND EQUATION SOLVERS With circuit-oriented simulators, the initial setup time is small, and it is easy to make changes in the circuit topology and control. The focus remains on the circuit rather than on the mathematics of the solution. Many built-in models for the components and the controllers (analog and digital) are usually available. It is possible to segment the overall system into smaller modules or building blocks that can be individually tested and then brought together. On the negative side, there is little control over the simulation process that can lead to long simulation times or, even worse, to numerical convergence or oscillation prob- lems, causing the simulation to halt. Steps to overcome these difficulties are usually not apparent and may require trial and error. Equation solvers, on the other hand, give total control over the simulation process, including the integration method to be used, time step of simulation, and so on. This results in a smaller simulation (execution) time. Being general-purpose tools, equation solvers can also be useful in applications other than power electronics simulation. On the negative side, a long time is usually required for the initial setup because the user must develop all possible combinations of differential and algebraic equations. Even a minor change in the circuit topology and control may require just as much effort as the initial setup. On balance, the circuit-oriented simulators are much easier and therefore are more widely used. The equation solvers tend to be used in specialized circumstances. The characteristics of some of the widely used software packages are discussed in Sections 4-6 and 4-7. 4-5 SOLUTION TECHNIQUES FOR TIME-DOMAIN ANALYSIS Both circuit-oriented simulators and equation solvers must solve differential equations as a function of time. With either approach, the user should know the concepts fundamental to the solution of these equations. In power electronics, the circuits are usually linear, but they change as a function of time due to the action of switches. A set of differential equations describes the system for each circuit state. In this section, we will discuss one numerical solution technique by 66 CHAPTER 4 COMPUTER SIMULATION OF POWER CONVERTERS AND SYSTEMS means of a simple example that allows system variables to be calculated as a function of time. 4-5-1 LINEAR DIFFERENTIAL EQUATIONS Figure 4-6a shows a simplified equivalent circuit to represent a switch-mode, regulated voltage supply of Figs. 1-3 and 1-4a in Chapter 1. In this example, change in the state of the switch in Fig. 1-3 affects the voltage voiin the circuit of Fig. 4-6a (v,; = V, when the switch is on, otherwise v,,. = 0). Thus, the same circuit topology applies in both states of the switch. The inductor resistance r, is included. The waveform of the voltage voi is shown in Fig. 4-6b, where the switch duty ratio D = t,,/T, is dictated by the controller in the actual system, based on the operating conditions. The equations are written in terms of the capacitor voltage v, and the inductor current iL, the so-called state variables, since they describe the state of the circuit. It is assumed that at time t = 0 at the beginning of the simulation, the initial inductor current iL(0)and the initial capacitor voltage v,(0) are known. By applying Kirchhoff 's current law (KCL) and Kirchhoff 's voltage law (KVL) in the circuit of Fig. 4-6a, we get the following two equations: diL rLiL + L - dt + v, = v,i (KVL) (4- 1) dv, vc i,-C---=O (KCL) (4-2) dt R [I~=I-' By dividing both sides of Eq. 4-1 by L and Eq. 4-2 by C , we can express them in the usual state variable matrix form 1 (4-3) - C CR - "d (b) Figure 4-6 Simplified equivalent circuit of a switch-mode, regulated dc power supply (same as in Fig. 1-3).  4-5 SOLUTION TECHNIQUES FOR TIME-DOMAIN ANALYSIS 67 The above equation can be written as dx(t) - Ax@) + bg(r) -- dt (4-4) where x(t) is a state variable vector and g(r) is the single input: The matrix A and the vector b are r- 1 In general, the state transition matrix A and the input vector b may be functions of time and Eq. 4-4 may be written as &(‘) - = A(t)x(r) + b(t)g(f) dt With a time step of integration At, the solution of Eq. 4-6 at time t can be expressed in terms of the solution at t - At: x(t) = X(t - At) + [A(C)x(O + b(Og(01 d5 (4-7) where 5 is a variable of integration. 4-5-2 TRAPEZOIDAL METHOD OF INTEGRATION There are many elegant numerical methods for solving the integral in Eq. 4-7. However, we will discuss only one technique, called the trapezoidal method, which is used in two of the widely used circuit-solving programs called SPICE and EMTP. This method uses an approximation of linear interpolation between the values at time t - Ar and f , assuming that x(r) is known at time t. Since x(r) is what is being calculated in Eq. 4-7, this assumption of its a priori knowledge puts this method into a category of “implicit” methods, Figure 4-7 graphically illustrates this method of integration for a single variable x(r), where A and b are scalars. As the name implies, the trapezoidal area, using the linear interpolation between the values at time r - At and t , approximates the value of the integral. Applying this method to Eq. 4-7 yields x(t) = x(t - At) + f At [A(t - At)x(t - At) + A(t)x(r)] + f At [b(r - At)& - At) Figure 4-7 Trapezoidal method t 0 t-At t of integration. 68 CHAPTER 4 (:OMPUIEH SIMIII.Al'ION OF POWER CONVERTEHS AND SYSTEMS We should note that the above equation is an algebraic equation that in this case is also linear. Therefore, rearranging terms, we get [I - i At A@)]x(t) = [I + i At A(t - At)] x(t - At) + i At [b(t - At)& - At) + b(M4l (4-9) Multiplying both sides of Eq. 4-9 by the inverse of [I - At A(t)] allows ~ ( tto ) be solved: x(t) = [I - f At A([)] - * {[I+iAtA(t -At)]x(t -At) +iAt[b(r -At)& -At) b(t)g(t)]) + (4-10) Commonly in power electronic systems, A and b change when the circuit state changes due to a switching action. However, for any circuit state over an interval during which the state is not changing, the state transition matrix A and the vector b are independent of time. Therefore, A(t - At) = A(t) = A and b(t - At) = b(t) = b. Use of this information in Eq. 4-10 results in x(t) = Mx(t - At) + N[g(t - At) + g(t)] (4-1 1) where M=[I-fArA]-'[I+~AtA] (4-12) and N = [I - At A]-' (f At) b (4-13) need to be calculated only once for any circuit state, provided the time step At chosen for the numerical solution is kept constant. An obvious question at this point is why solve the circuit state with a small time step At rather than choosing a At that takes the circuit from its previous switch state to its next switch state. In the solution of linear circuits, At must be chosen to be much smaller than the shortest time constant of interest in the circuit. However, in power electronics, Ac is often even smaller, dictated by the resolution with which the switching instants should be represented. We do not know a priori at what time the circuit will go to its next state since the values of the circuit variables themselves determine the instant of time when such a transition should take place. Another important point to note is that when the circuit state changes, the values of the state variables at the final time in the previous state are used as initial values at the beginning of the next state. 4-5-3 NONLINEAR DIFFERENTIAL EQUATIONS [l] In power electronic systems, nonlinearities are introduced by component saturation (due to component values which depend on the associated currents and voltages) and limits imposed by the controller. An example is the output capacitance of a MOSFET which is a function of the voltage across it. In such systems, the differential equations can be written as (where f is a general non-linear function) i = f(x(r),t) (4-14) The solution of the above equation can be written as (4-15)  4-6 WIDELY USED, CIRCUIT-ORIENTED SIMULATORS 69 For example, applying the Trapezoidal rule to the integral in the equation above results in At x(t) = x(t - At) +- {f(x(t),t) + f(x(t - At),t - At)}. (4-16) 2 Equation 4-16 is nonlinear, and cannot be solved directly. This is because in the right side of Eq. 4-16, f(x(t),t) depends on x ( t ). Such equations are solved by iterative procedures which converge to the solution within a reasonably small number of iterations. One of the commonly used solution techniques is the Newton-Raphson iterative procedure. 4-6 WIDELY USED, CIRCUIT-ORIENTED SIMULATORS Several general-purpose, circuit-oriented simulators are available. These include SPICE, EMTP, SABER, and KREAN, to name a few. Two of these, SPICE and EMTP, are easily available and are widely used. Both have strengths and weaknesses. SPICE was developed for simulating integrated circuits, whereas EMTP was developed for power systems modeling. Because of their widespread popularity, we will describe both briefly in the following sections. 4-6-1 SPICE The abbreviation SPICE stands for Simulation Program with Integrated Circuit Emphasis. It was developed at the University of California, Berkeley. SPICE can handle nonlinear- ities and provides an automatic control on the time step of integration. There are several commercial versions of SPICE that operate on personal computers under several popular operating systems. One commercial version of SPICE is called PSpice. In PSpice, many features are added to make it a multilevel simulator where the controllers can be represented by their behavior models, that is, by their input-output behavior, without resorting to a device-level simulation. There is an option for entering the input data by drawing the circuit schematic. In addition to its use in industry, PSpice has also become very popular in teaching undergraduate core courses in circuits and electronics. Therefore, many students are familiar with PSpice. One of the reasons for the popularity of PSpice is the availability and the capability to share its evaluation (class- room) version freely at no cost. This evaluation version is very powerful for power electronics simulations. For example, all simulations in reference 5 use only the evalu- ation version of PSpice. For these reasons, PSpice is used in this book in examples and in homework problems. To illustrate how the information about a circuit is put into a circuit-oriented program in general and PSpice in particular, a very simple example is presented. We will consider the circuit of Fig. 1-3, redrawn in Fig. 4-8a, where the control signal for the switch under an open-loop operation is the waveform shown in Fig. 4%. Note that we explicitly include the representation of the diode and the switch, whereas we could have represented this circuit by the simple equivalent circuit of Fig. 4-6, which needs to be modified if the inductor current in this circuit becomes discontinuous. This shows the power of a circuit- oriented simulators that automatically takes into account the various circuit states without the user having to specify them. In the present simulation, the diode is represented by a simple built-in model within PSpice, and the switching device is represented by a simple voltage-controlled switch. In a circuit-oriented simulator like PSpice, detailed device models can be substituted if we wish to investigate switching details. 70 CHAPTER 4 COMPUTER SIMULATION OF POWER CONVERTERS AND SYSTEMS iL -+ I fa) - Switch-control On Off On Off signal (T - 1= lops) fbl Figure 4-8 (a) Circuit for simulation. (b) Switch control waveform. As a first step, we must assign node numbers as shown in the diagram of Fig. 4-9a, where one of the nodes has to be selected as a ground (0)node. The transistor in Fig. 4-8a is modeled by a voltage-controlled switch SW in the diagram of Fig. 4-9a whose state is determined by the voltage at its control terminals. In the on state with a control voltage greater then V,, (= 1 V default value), the switch has a small on-state resistance Ron (= 1 fl default value). In the off state with a control voltage less than Voff (= 0 V default value), the switch is in its off state and is represented by a large resistance Roff (= lo6 R default value). Of course, the default values are optional and the user can specify values that are more appropriate. The listing of the input circuit file to PSpice is shown in Fig. 4-9b. The repetitive control voltage shown in Fig. 4-8b, which determines the state of the switch SW, is modeled by means of a voltage source called VCNTL within PSpice. There is a built-in model for diodes whose parameters such as the on-state resistance parameter R, and the zero-bias junction capacitance Cjo can be changed; otherwise the default values are used by the program. A sudden discontinuity in SPICE can result in the program proceeding with extremely small time steps and at worst may result in a problem of convergence, where the voltages at some node or nodes at some time step may fail to converge. If this were to happen, the simulation would stop with an error message. There are few definite rules to avoid the solution from failing to converge. Therefore, it is always better to avoid sudden discon- tinuities, such as by using an R-C “numerical snubber” across the diode in Fig. 4-9a to “soften” the discontinuity presented by the diode current suddenly going to zero. Sim- ilarly, the rise and fall times of VCNTL in Fig. 4-9a, represented by PULSE in Fig. 4-9b, are specified as 1 ns each rather than as zero. The output waveforms from the simulation are shown in Fig. 4-10. These are pro- duced by a graphical postprocessor (called Probe) within PSpice that is very easy to use. 4-6-2 EMTP SIMULATION PROGRAM Another widely used, general-purpose circuit simulation program is called EMTP (Elec- tro-Magnetic Transients Program). Unlike SPICE, which has its origin in microelectron-  4-6 WIDELY USED, CIRCUIT-ORIENTED SIMULATORS 71 iL 1 0 - 3 ,-4. + Model name = POWER-DIOOE - 2 ) , Model name = SWITCH Initial conditions: i~(0)=4A UC(0) = 5.5 v - - - (a) PSpice Example * DIODE 2 2 POWER-DIODE Rsnub 2 5 200.0 Csnub 5 2 O.2uF * sw 2 0 b 0 SWITCH VCNTL b 0 PULSE(OV,2VtOsr2nst2nst7.5ust20us) * L 2 3 5uH IC=4A rL 3 4 2m C 4 2 200uF IC=5.5V RLOAD 4 2 2.0 * VD 2 0 B.0V *.MODEL POWER-DIODE D(RS=0.02,CJO=20pF).MODEL SWITCH VSWITCH(RON=O.UI).TRAN 2Ous 5 0 0. 0 ~ s 0s 0.211s uic.PROBE.END (b) Figure 4-9 PSpice simulation of circuit in Fig. 4-8. 9.0 5.0.. 3.0 I I I I I I F 0 100 ps 200 ps 300 ps 400 ps 500 ps Time Figure 4-10 Results of PSpice simulation: it and v,. 72 CHAPI'ER 4 COMPUTER SIMULATION OF POWER CONVERTERS AND SYSTEMS ics modeling, EMTP was originally developed for the electric power industry at the Bonneville Power Administration in Portland, Oregon. ATP (Alternative Transients Pro- gram) is a version of EMTP that is also available for personal computers under MS-DOS operating system [ 7 ]. Similar to SPICE, EMTP uses a trapezoidal rule of integration, but the time step of integration is kept constant. Because of the built-in models for various power system components such as three- phase transmission lines, EMTP is a very powerful program for modeling power elec- tronics applications in power systems. Compared to SPICE, the switches in EMTP are treated quite differently. When a switch is closed, the row and column (in the network matrix) corresponding to the terminal nodes of the switch are coalesced together. There is a very powerful capability to represent analog and digital controllers that can be specified with almost the same ease as in a high-level language. The electrical network and the controller can pass values of various variables back and forth at each time step. 4-6-3 SUITABILITY OF PSpice AND EMTP For power electronics simulation, both PSpice and EMTP are very useful. PSpice is better suited for use in power electronic courses for several reasons. Its evaluation version is available at no cost (in fact, its copying and sharing is welcomed and encouraged), and it is very easy to install on either an IBM-compatible or a Macintosh computer. It is user friendly with an easy-to-learn graphical postprocessing package for plotting of results. Because of the availability of semiconductor device models, it is also well suited for applications where such detailed representations are necessary. Perhaps in the near future, the models will improve to the point where power losses can be calculated accurately for a thermal design. On the other hand, EMTP is better suited for simulating high-power electronics in power systems. It has the capability for representing controllers with the same ease as in a high-level language. The control over the time step At results in execution (run) times that are acceptable. For the reasons listed above, EMTP is very well suited for analyzing complex power electronic systems at a system level where it is adequate to represent switching devices by means of ideal switches and the controller by transfer functions and logical expressions. A large number of power electronic exercises using the evaluation (classroom) ver- sion of PSpice [ 5 ] and EMTP are available as aids in learning power electronics. They are also ideal for learning to use these software packages by examples. 4-7 EQUATION SOLVERS If we choose an equation solver, then we must write the differential and algebraic equa- tions to describe various circuit states and the logical expressions within the controller that determine the circuit state. Then, these differential/algebraicequations are simultaneously solved as a function of time. In the most basic form, we can solve these equations by programming in any one of the higher level languages such as FORTRAN, C, or Pascal. It is also possible to access libraries in any of these languages, which consist of subroutines for specific applications, such as to carry out integration or for matrix inversion. However, it is far more convenient to use a package such as MATLAB or a host of other packages where many of these convenience features are built in. Each of these packages use their own syntax and also excel in certain applications.  4-7 EQUATION SOLVERS 73 The program MATLAB can easily perform array and matrix manipulations, where, for example, y = a * b results in a value of y that equals cell-by-cell multiplication of two arrays a and b. Similarly, to invert a matrix, all one needs to specify is Y = inv(X). Powerful plotting routines are built in. MATLAB is widely used in industry. Also, such programs are used in the teaching of undergraduate courses in control systems and signal processing. Therefore, the students are usually familiar with MATLAB prior to taking power electronics courses. If not, it is possible to learn their use quickly, especially by means of examples. For these reasons, MATLAB is utilized in this book for solution of some of the examples and the homework problems. SIMULINK is a powerful graphical pre-processor or user-interface to MATLAB which allows dynamic systems to be de- scribed in an easy block-diagram form. As an example of MATLAB, the solution of the circuit in Fig. 4-8using the trape- zoidal method of integration is shown in Fig. 4-11. The circuit of Fig. 4-8reduces to the equivalent circuit shown previously in Fig. 4-6 (provided iL(t) > 0). As shown in Fig. 4-1la,the input voltage vOiis generated in MATLAB by comparing a sawtooth waveform 0 I I I +t I I k- T*++ I I (T - - )1I I - I "-fs I I I "oi % Solution of the Circuit in Fig. 4 4 using Trapezoidal flethod of Integration. clc,clq,clear I Inpui Data V d = B ; L=Se-6; C=LOOe-6; rL=Le-3; R=L.U; fs=b00e3; Vcontrol=0.75; Ts=L/fs; tmax=50*Ts; deltat=Ts/SO; % time= 0:deltat:tmax; vst= time/Ts - fix(time/Ts); voi= V d * (Vcontrol > vst); I A=[-rL/L -l/L; l/C -L/(C*R)l; b=[l/L 0 1 ' ; MN=inv(eye(Z) - deltat/2 * A); H=HN * (eye(?)+ deltat/2 * A); N=MN * deltat/Z * b ; I iL(L)=4.0; vC(L)=S.S; timelength=length(time); % for k = 2:timelength x = fl* [iL(k-I) ~C(k-b)l' + N * (Voi(k) + Voi(k-1)); iL(k) = ~ ( 1 ) ;vC(k) = ~ ( 2 ) ; end %. plot(timeriL,timervC) meta Example (b) Figure 4-11 MATLAB simulation of circuit in Fig. 4-6. 74 CHAPTEH 4 COMPm'I*;R SIMULATION OF POWER CONVERTERS AND SYSTEMS I ' I I I I i I I I * 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time x 10-4 s Figure 4-12 MATLAB simulation results. vStat the switching frequencyf, with a dc control voltage v,,,,~~~. When the control voltage is greater than vSt, vOi = V,; otherwise it is zero. The MATLAB listing is shown in Fig. 4-llb. The output waveforms are shown in Fig. 4-12. These waveforms differ slightly from the PSpice simulation results shown in Fig. 4-10 due to the on-state voltage drop across the diode in PSpice. SUMMARY Modeling and computer simulations play an important role in the analysis, design, and education of power electronic systems. Because of the challenges involved in such sim- ulations, it is important to simplify the system being simulated to be consistent with the simulation objectives. Over the years, several simulation packages have been developed. It is necessary to carefully evaluate the advantages and shortcomings of each package prior to selecting one for a given set of objectives. PROBLEMS 4-1 Generate a triangular waveform with a peak of -+ 1 V at 100 kHz using MATLAB. 4-2 Using the PSpice listing in Fig. 4-9b, obtain the switch current 'and the diode voltage waveforms. 4-3 Using the MATLAB listing in Fig. 4-llb, obtain the inductor current and the capacitor voltage waveforms similar to those in Fig. 4-12.  REFERENCES 75 4-4 In the PSpice simulation of Fig. 4-9, make RLod = 10 a.Evaluate the effects of the following changes on the simulation results: (a) Remove the R-C snubber across the diode. (b) In the pulse waveform of the control voltage VCNTL, make the rise and fall times zero. (c) Remove the R-C snubber across the diode simultaneously with the following changes: i) make the diode model to be as follows:.MODEL POWERDIODE D(IS=3e-15, RS=O.I, CJO= 1OPF) and, ii) add the following Options statement:.OPTIONS ABSTOL= lN, VNTOL= l M , RELTOL=0.015 (d) In part c, make the rise and fall times of the control voltage VCNTL zero. 4-5 To bring the results of the PSpice sirnulation of Fig. 4-9 closer to the MATLAB simulation of Fig. 4-11, insert an ideal dc voltage source of 0.7 V in series with the diode in Fig. 4-9a to compensate for the on-state voltage drop of the diode. Compare the simulation results with those from MAT- LAB in Fig. 4-12. 4-6 In the PSpice simulation of Fig. 4-9a, idealize the switch-diode combination, which allows replacing it by a pulse input voltage waveform (between 0 and 8 V) as the input vat. Compare the PSpice simulation results with the MATLAB simulation results of Fig. 4-12. 4-7 Repeat the MATLAB simulation of Fig. 4-11 by using the built-in integration routine ODE45 in MATLAB. 4-8 In the PSpice simulation of Fig. 4-8, change RLoadto be 10 which causes the inductor current i, to become discontinuous (i.e., it becomes zero for a finite interval during each switching cycle). Obtain i, and v, waveforms. 4-9 Repeat problem 4-8 (with RLoad = 10 a)using MATLAB, recognizing that iI- becomes discontin- uous at this low output power level. 4-10 Since the output capacitor C is usually large in circuits similar to that in Fig. 4-8a, the capacitor voltage changes slowly. Therefore, rewrite differential equations in terms of i, and v, by making the following assumptions: i) calculate iL ( I ) based on v,(r - At) and, ii) use the calculated value of iL(t) in the previous step to calculate v&). Simulate using MATLAB and compare results with those in Fig. 4-12. 4-1 1 In the circuit of Fig. 4-8a, ignore the input voltage source and the transistor switch. Let i,(o) = 4 A, and v,(o) = 5.5 V. Assuming the diode to be ideal, simulate this circuit in MATLAB using the trapezoidal rule of integration. Plot the inductor voltage v,. REFERENCES 1.N. Mohan, W. P. Robbins, T. M. Undeland, R. Nilssen and 0. Mo, “Simulation of Power- Electronics and Motion Control Systems- An Overview,” Proceedings of the IEEE, Vol. 82, NO. 8, Aug. 1994, pp. 1287-1302. 2. B. C. Kuo and D. C. Hanselman, MATLAB Tools for Control System Analysis and Design, Prentice Hall, Englewood (NJ), 1994. 3. L. W. Nagel, “SPICE2 A Computer Program to Simulate Semiconductor Circuits,” Memo- randum No. ERL-M520, University of California, Berkeley, 1975. 4. PSpice, MicroSim Corporation, 20 Fairbanks, Irvine, CA 92718. 5. “Power Electronics: Computer Simulation, Analysis, and Education Using Evaluation Version of PSpice,” on diskette with a manual, Minnesota Power Electronics, P.O. Box 14503, Minneapolis, MN 55414. (See page viii.) 6. W. S. Meyer and T. H. Liu, “EMTP Rule Book,” Bonneville Power Administration, Port- land. OR 97208. ‘76 CHAPTER 4 COMPUTER SIMULATION OF POWER CONVERTERS AND SYSrEMS 7. ATP version of EMTP, CanadidAmerican EMTP User Group, The Fontaine, Unit 6B, 1220 N.E., 17th Avenue, Portland, OR 97232. 8. “Computer Exercises for Power Electronics Education using EMTP,” University of Minne- sota Media Distribution, Box 734 Mayo Building, 420 Delaware Street, Minneapolis, MN 55455. 9. MATLAB, The Math Works Inc., 24 Prime Park Way, Natick, MA 01760. PART 2 GENERIC P O W R ELECTRONIC CONVERTERS CHAPTER 5 LINE-FREQUENCY DIODE RECTIFIERS : LINE-FREQUENCY ac + UNCONTROLLED dc 5-1 INTRODUCTION In most power electronic applications, the power input is in the form of a 50- or 60-Hz sine wave ac voltage provided by the electric utility, that is first converted to a dc voltage. Increasingly, the trend is to use the inexpensive rectifiers with diodes to convert the input ac into dc in an uncontrolled manner, using rectifiers with diodes, as illustrated by the block diagram of Fig. 5-1. In such diode rectifiers, the power flow can only be from the utility ac side to the dc side. A majority of the power electronics applications such as switching dc power supplies, ac motor drives, dc servo drives, and so on, use such uncontrolled rectifiers. The role of a diode rectifier in an ac motor drive was discussed by means of Fig. 1-8 in Chapter 1. In most of these applications, the rectifiers are supplied directly from the utility source without a 60-Hz transformer. The avoidance of this costly and bulky 60-Hz transformer is important in most modem power electronic systems. The dc output voltage of a rectifier should be as ripple free as possible. Therefore, a large capacitor is connected as a filter on the dc side. As will be shown in this chapter, this capacitor gets charged to a value close to the peak of the ac input voltage. As a consequence, the current through the recitifier is very large near the peak of the 60-Hz ac input voltage and it does not flow continuously;that is, it becomes zero for finite durations during each half-cycle of the line frequency. These rectifiers draw highly distorted current from the utility. Now and even more so in the future, harmonic standards and guidelines will limit the amount of current distortion allowed into the utility, and the simple diode rectifiers may not be allowed. Circuits to achieve a nearly sinusoidal current rectification at a unity power factor for many applications are discussed in Chapter 18. US Figure 5-1 Block diagram of a rectifier. 79 80 CHAPTER 5 LINE FREQUENCY DIODE RECTIFIERS Rectifiers with single-phase and three-phase inputs are discussed in this chapter. As discussed in Chapter 2 , the diodes are assumed to be ideal in the analysis of rectifiers. In a similar manner, the electromagnetic interference (EMI) filter at the ac input to the rectifiers is ignored, since it does not influence the basic operation of the rectifier. Elec- tromagnetic interference and EM1 filters are discussed in Chapter 18. 5-2 BASIC RECTIFIEH CONCEPTS Rectification of ac voltages and currents is accomplished by means of diodes. Several simple circuits are considered to illustrate the basic concepts. 5-2-1 PURE RESISTIVE LOAD Consider the circuit of Fig. 5-2a, with a sinusoidal voltage source v,. The waveforms in Fig. 5-2b show that both the load voltage vd and the current i have an average (dc) component. Because of the large ripple in vd and i, this circuit is of little practical significance. 5-2-2 INDUCTIVE IJOAD Let us consider the load to be inductive, with an inductor in series with a resistor, as shown in Fig. 5-3a. Prior to t = 0, the voltage v, is negative and the current in the circuit is zero. Subsequent to t = 0, the diode becomes forward biased and a current begins to flow. Then, the diode can be replaced by a short, as shown in the equivalent circuit of Fig. 5-3e. The current in this circuit is governed by the following differential equation: v, = Ri + L dtdi- (5- 1) where the voltage across the inductor v, = L dildt. The resulting voltages and current are shown in Figs. 5-36 and c. Until t , , v, > V , (hence vL = v, - V , is positive), the current i +udiode- - "diode \ u (b) Figure 5-2 Basic rectifier with a load resistancx:. '8, / 'diode  5-2 BASIC RECTIFIER CONCEPTS 81 (b) 0 I I I I = T I - f I I I I (cj 0 t I I I I I Area B I ! (d) 0 I I r t I Udiode I I I I I L ~ :++- +O d"+ i=O L + (e) us 'L us ?, Vd= 0 - - - - - Figure 5-3 Basic rectifier with an inductive load. builds up, and the inductor stored energy increases. Beyond t , , v, becomes negative, and the current begins to decrease. After t,, the input voltage v, becomes negative but the current is still positive and the diode must conduct because of the inductor stored energy. The instant t,, when the current goes to zero and the diode stops conducting, can be obtained as follows (also discussed in Section 3-2-5-1): The inductor equation v, = L dildt can be rearranged as 1 - vL dt = di (5-2) L Integrating both sides of the above equation between zero and t3 and recognizing that i(0) and i(r3)are both zero give 82 CHAPTER 5 LINE FREQUENCY DIODE RECTIFIERS From the above equation, we can observe that VL dt = 0 (5-4) A graphical interpretation of the above equation is as follows: Equation 5-4 can be writ- ten as I' J1:' VL dt + vL dt = 0 (5-5) which in terms of the volt-second areas A and B of Fig. 5-3c is AreaA - Area3 = 0 (5-6) Therefore, the current goes to zero at tg when area A = B in Fig. 5-3c. Beyond t,, the voltages across both R and L are zero and a reverse polarity voltage (= -v,) appears across the diode, as shown in Fig. 5-3d. These waveforms repeat with the time period T = If. The load voltage vd becomes negative during the interval from r, to t3. Therefore, in comparison to the case of purely resistive load of Fig. 5-2a, the average load voltage is less. 5-2-3 LOAD WITH AN INTERNAL dc VOLTAGE Next, we will consider the circuit of Fig. 5-4a where the load consists of an inductor L and a dc voltage Ed. The diode begins to conduct at t l when v, exceeds Ed.The current reaches its peak at t2 (when v, is again equal to Ed)and decays to zero at t3, with t3 determined by the requirement that the volt-second area A be equal to area B in the plot of v, shown in Fig. 5-4c.The voltage across the diode is shown in Fig. 5-4d. 5-3 SINGLE-PHASE DIODE BRIDGE RECTIFIERS A commonly used single-phase diode bridge rectifier is shown in Fig. 5-5. A large filter capacitor is connected on the dc side. The utility supply is modeled as a sinusoidal voltage source v, in series with its internal impedance, which in practice is primarily inductive. Therefore, it is represented by L,. To improve the line-current waveform, an inductor may be added in series on the ac side, which in effect will increase the value of L,. The objective of this chapter is to thoroughly analyze the operation of this circuit. Although the circuit appears simple, the procedure to obtain the associated voltage and current waveforms in a closed form is quite tedious. Therefore, we will simulate this circuit using PSpice and MATLAB. However, we will next analyze many simpler and hypothetical circuits in order to gain insight into the operation of the circuit in Fig. 5-5. 5 - 3 4 IDEALIZED CIRCUIT WITH Ls = 0 As a first approximation to the circuit of Fig. 5-5, we will assume L, to be zero and replace the dc side of the rectifier by a resistance R or a constant dc current source Id, as shown in Figs. 5-6a and b, respectively. It should be noted in the circuit of Fig. 5-6a that although it is very unlikely that a pure resistive load will be supplied through a diode rectifier, this circuit models power-factor-corrected rectifiers discussed in Chapter 18.  5-3 SINGLE-PHASE DIODE BRIDGE RECTIFIERS 83 vdicde i I_ Ed T (d) Figure 5-4 Basic rectifier with an internal dc voltage. cd == Ud (I w Model of the utility supply - - Figure 5-5 Single-phase diode bridge rectifier. 84 CHAPTER 5 LINE FREQUENCY DIODE RECTIFIERS id P + (a) (b) Figure 5-6 Idealized diode bridge rectifiers with L, = 0. Similarly, the representation of the load by a constant dc current in the circuit of Fig. 5-6b is an approximation to a situation where a large inductor may be connected in series at the dc output of the rectifier for filtering in Fig. 5-5. This is commonly done in phase- controlled thyristor converters, discussed in Chapter 6. The circuits in Fig. 5-6 are redrawn in Fig. 5-7, which shows that this circuit consists of two groups of diodes: the top group with diodes 1 and 3 and the bottom group with diodes 2 and 4. With L, = 0 , it is easy to see the operation of each group of diodes. The current id flows continuously through one diode of the top group and one diode of the bottom group. In the top group, the cathodes of the two diodes are at a common potential. Therefore, the diode with its anode at the highest potential will conduct id. That is, when v, is positive, diode 1 will conduct id and v, will appear as a reverse-bias voltage across diode 3. When v, goes negative, the current id shifts (commutates) instantaneously to diode 3 since L, = 0. A reverse-bias voltage appears across diode 1. In the bottom group, the anodes of the two diodes are at a common potential. Therefore, the diode with its cathode at the lowest potential will conduct id. That is, when v, is positive, diode 2 will carry id and v, will appear as a reverse-bias voltage across diode 4. When v, goes negative, the current id instantaneously commutates to diode 4 and a reverse-bias voltage appears across diode 2. The voltage and current waveforms in the circuits of Fig. 5-6 are shown in Figs. 5-8a and 6. There are several items worth noting. In both circuits, when v, is positive, diodes Y I D1 (I - P ,-. D3 f - 1 h, Gd - id D4 0. - ,-. f- D2 N In  5-3 SINGLE-PHASE DIODE BRIDGE RECTIFIERS 85 I l t O t=O (b) Figure 5-8 Waveforms in the rectifiers of (a) Fig. 5-6a and (b) Fig. 5-66. 1 and 2 conduct and vd = v, and is = id. When v, goes negative, diodes 3 and 4 conduct and, therefore, vd = -vs and is = -id. Therefore, at any time, the dc-side output voltage of the diode rectifier can be expressed as vAt) = lvsl (5-7) Similarly, the ac-side current can be expressed as id if v, > 0 is = -id if v, < 0 (5-8) 86 CHAPTER 5 LINE FREQUENCY DIODE RECTIFIERS and th