Chapter 11
Applications of Silicon Carbide Devices in Power Systems
11.1 Introduction to Power Electronic Systems
The topic of power electronic systems is both broad and deep, and we will only present an overview in this chapter. Our objective is to consider those systems in which the substitution of silicon carbide devices may produce significant advantages in performance, efficiency, reliability, and/or overall system cost. The discussion will be limited to the basic circuit topology and device requirements, and will not explore second-order effects that are also important for a practical design. For these, the reader is referred to one of the specialty texts on power systems [1], and thence to the literature.
A block diagram of a generic power processing system is shown in Figure 11.1. This system provides an interface between two ports, typically a source of electric power and a load to which electric power is delivered. In the general case, the power processor may consist of three elements: an electronic converter connected to port 1, an electronic converter connected to port 2, and an energy storage element between the two converters. The converters may include one or more power semiconductor devices, along with passive components such as resistors, inductors, and capacitors. The energy storage element between the converters is typically either an inductor or a capacitor. In most cases the power processor is designed to be unidirectional, with power flowing from the source port to the load port, but in some cases the power flow can be bidirectional, as in motor drives for electric vehicles where regenerative braking is used to return kinetic energy to a storage device.
Figure 11.1 Schematic of a generic power processor.
Electronic converters may be classified based on whether their inputs and outputs are DC or AC. The four possible input–output combinations are listed in Table 11.1. Converters may also be classified according to the switching mode upon which their operation is based. The four possible switching modes are
- Uncommutated (e.g., diode rectifiers)
- Line-frequency commutated (e.g., thyristor rectifiers and inverters)
- Switch-mode (e.g., pulse-width modulated waveform generators)
- Resonant (switching occurs at a zero crossing of the voltage or current waveform).
Table 11.1 Classification of electronic power converters.
| Input | Output | Designation | Possible switching modes | |||
| Uncommutated (e.g., diodes) | Line-frequency commutated | Switch mode | Resonant | |||
| AC | DC | Rectifier | X | X | — | X |
| DC | AC | Inverter | — | X | X | X |
| AC | AC | AC converter | — | X | X | |
| DC | DC | DC converter | — | — | X | — |
Converters may employ any of the semiconductor devices discussed in Chapters 7–10, with the type of device depending on the application and the circuit topology employed. Often, the designer has the choice of several possible devices. For example, when a switching device is required, the designer might choose a JFET (junction field-effect transistor), a MOSFET (metal-oxide-semiconductor field effect transistor), a BJT (bipolar junction transistor), or an IGBT (insulated-gate bipolar transistor), depending on the requirements of the application.
This chapter is organized as follows. In Section 11.2 we introduce three basic converter circuits: (i) line-frequency-commutated rectifiers and inverters, (ii) switch-mode DC converters and power supplies, and (iii) switch-mode inverters. In Section 11.3 we discuss motor drives for DC motors, induction motors, synchronous motors, and hybrid and electric vehicles. Section 11.4 covers the applications of SiC power devices in renewable energy, and Section 11.5 deals with switch-mode power supplies. Finally, in Section 11.6 we summarize the present state-of-the-art of SiC power devices, as compared to the silicon devices with which they compete.
11.2 Basic Power Converter Circuits
11.2.1 Line-Frequency Phase-Controlled Rectifiers and Inverters
Line-frequency phase-controlled converters are used to transfer power between a line-frequency AC environment and a controlled DC environment. Thyristor-based line-frequency converters are used primarily in high-power three-phase applications, especially in cases where bidirectional power flow is desired. Examples include high-voltage DC transmission systems and high-power AC and DC motor drives where regenerative braking is employed. In line-frequency converters, turn-off of the thyristors occurs at zero crossings of the thyristor current, which are naturally synchronized with the terminal voltage of the AC port.
A basic thyristor converter driving a resistive load is illustrated in Figure 11.2, along with its operational waveforms. The thyristor is triggered at an arbitrary phase angle 
by a short gate pulse. Once triggered, the thyristor remains in its forward-conducting mode until the cathode voltage changes sign at 
, whereupon it enters its reverse-blocking mode. When the cathode voltage again becomes positive at 
, the thyristor enters its forward-blocking mode until the next gate trigger pulse at 
. In this analysis the forward voltage drop of the thyristor is neglected, and the load voltage 
is equal to the source voltage during the period when the thyristor is conducting. The current waveform is a truncated half-sinusoid, and the average power delivered to the load can be varied from zero to 
by adjusting the phase angle 
of the triggering pulse.
Figure 11.2 Schematic of a simple thyristor-based line-frequency phase-controlled converter driving a resistive load.
Figure 11.3 shows a thyristor driving an inductive load from a sinusoidal source. Prior to triggering of the thyristor, the current is zero. Once the thyristor is triggered, a current begins to flow and the inductor voltage depends on the current according to
Figure 11.3 A line-frequency phase-controlled converter driving an inductive load.
In Figure 11.3, the inductor voltage is shown graphically as the difference between the source voltage 
and the resistor voltage 
when the thyristor is on. As long as 
is positive, 
is positive and the current increases. When the resistor voltage 
equals the source voltage 
, the inductor voltage 
changes sign and the current begins to decrease. When the current reaches zero, the thyristor enters its reverse-blocking mode and the current remains zero until the source voltage becomes positive again and the next trigger pulse arrives. When the source voltage is negative and the current is positive, reactive power stored in the inductor is being returned to the source. Stored power is also delivered to the resistive load during this period, since the resistor voltage and current are both positive.
Figure 11.4 shows a thyristor driving a load consisting of an inductor in series with a voltage source. This type of circuit is representative of a DC motor, where the voltage source 
represents the back-emf induced in the stator windings by the rotating magnetic field of the rotor. The current is initially zero and the thyristor voltage is 
. As 
increases during the first half-cycle, 
eventually becomes positive and the thyristor enters its forward-blocking mode. The current remains zero until the thyristor is triggered, at which point the load is effectively connected to the source 
and the current begins to increase. The inductor voltage is given by
and is shown in Figure 11.4. When 
becomes negative, 
is negative and the current decreases. When the current reaches zero, the thyristor enters its reverse-blocking mode. With zero current, all the source voltage develops across the thyristor, and 
. The current remains zero until the thyristor is triggered after the start of the next AC cycle.
Figure 11.4 A line-frequency phase-controlled converter driving an inductive load and an opposing voltage source.
The converter of Figure 11.4 only delivers power to the load during the first half-cycle. A converter capable of delivering power during both half-cycles is shown in Figure 11.5. We assume that the load inductance is large, and may be represented by an equivalent DC current source 
. The operation can be understood as follows. During the half-cycle preceding 
, thyristors 3 and 4 are conducting and the load voltage 
is equal to 
, since T3 and T4 cross-connect the load to the source. When the source voltage goes positive at 
, thyristors 1 and 2 enter their forward-blocking modes and the load voltage 
goes negative through the conduction of T3 and T4 (the ideal current source develops whatever voltage is necessary to maintain a constant current 
). T1 and T2 are triggered at 
. With T1 and T2 conducting, the load voltage 
switches rapidly to 
, and T3 and T4 enter their reverse-blocking modes.
Figure 11.5 A line-frequency phase-controlled converter that delivers power during both half cycles to an inductive load, represented by a DC current source.
The line-frequency phase-controlled converters of Figures 11.2–11.5 can operate in two quadrants of the 
plane as either rectifiers or inverters, as illustrated in Figure 11.6. During the portion of the ac cycle where 
and 
are positive, power is delivered from port 1 to port 2, and the converter is operating as a rectifier. For the portion of the cycle where 
is negative and 
is positive, power is delivered from port 2 to port 1, and the converter is operating as an inverter. The portions of the AC cycle over which rectification and inversion occur are determined by the triggering angle 
.
Figure 11.6 The 
plane. Operation in the first quadrant means that port 2 is absorbing power from port 1, and represents rectification. Operation in the fourth quadrant means that port 2 is delivering power to port 1, and corresponds to the inverter mode of operation. Bidirectional converters may operate in either quadrant, depending on the triggering angle.
Consider the converter of Figure 11.5. For triggering angles 
the converter operates in the rectifier mode, while for triggering angles 
it operates in the inverter mode. In the rectifier mode, the converter can be used as a battery charger or a DC motor drive. In this case, the generalized load, Figure 11.7a, is replaced by a voltage source 
representing the battery or the back-emf in the stator windings of the motor, as shown in Figure 11.7b. The same converter can be used in the inverter mode to transfer power from an energy source such as a solar array to the AC power grid. In this case the load is replaced by a DC source of the opposite polarity, as shown in Figure 11.7c. This configuration can also be used as a motor drive for high-power AC synchronous motors.
Figure 11.7 The generalized load of Figure 11.5, shown in (a), can be replaced by a positive DC source (b) to represent a battery or a DC motor, or by a negative DC source (c) to represent a photovoltaic power source.
11.2.2 Switch-Mode DC–DC Converters
DC–DC converters are used to transfer power between two DC environments. Typical applications include switch-mode DC power supplies and DC motor drives. Consider the generic power processor of Figure 11.1 where the source at port 1 is the AC line and the load at port 2 requires regulated DC power. In this case, a switch-mode DC converter would be used as converter 2 and a rectifier used as converter 1. In this section we will focus on converter 2, which converts unregulated DC to regulated DC. The circuit topologies to be considered are (i) step-down (buck) converters, (ii) step-up (boost) converters, (iii) step-down/step-up (buck/boost) converters, and (iv) full-bridge DC converters.
Figure 11.8 shows a schematic of a step-down or buck converter. In this figure we use a generic circuit symbol for a transistor switch, keeping in mind that the actual switching device might be a JFET, MOSFET, BJT, or IGBT, depending on the application. As the name implies, the buck converter delivers regulated DC power to the load at a lower voltage than the unregulated DC power at the source. The regulation is accomplished by adjusting the duty factor of a periodic rectangular waveform applied to the control electrode of the transistor. When the transistor is on, current flows from the source to the load through a low-pass filter formed by the inductor and capacitor. When the transistor is off, the current through the inductor cannot change instantaneously and the current path is through the inductor, the load, and the diode. If the period of the switching waveform is short compared to the time constant of the filter, the load voltage 
and load current 
may be regarded as DC quantities. In this case, the load voltage is simply the source voltage multiplied by the duty factor: 
, where 
.
Figure 11.8 A step-down or buck converter. A generic switch symbol is used for the transistor, which may be either a JFET, MOSFET, BJT, or IGBT.
Figure 11.9 shows a step-up or boost converter. With the transistor on, current flows through the inductor, storing reactive energy in the inductor. When the transistor turns off, the inductor current cannot change instantaneously, and flows through the diode to the capacitor and load resistor. With the transistor on, the diode prevents the capacitor from discharging through the transistor, and the capacitor instead supplies current to the load. In this way the inductor and capacitor function as a low-pass filter, keeping the current through the load constant, provided the period of the switching waveform is short compared to the RC and LC time constants of the circuit.
Figure 11.9 A step-up or boost converter.
In steady state, the integral of the inductor voltage over one period must be zero. To see this, recall that
Cross-multiplying and integrating over one period yields
since in steady state 
. When the transistor is on, 
, and when the transistor is off, 
(neglecting voltage drops across the transistor and diode). Thus Equation 11.4 can be written
from which we obtain
Since 
, we are guaranteed that 
, hence the name “boost converter.” Note that 
can become arbitrarily large as 
approaches unity.
It is sometimes necessary to provide an output voltage that may be either larger or smaller than the input voltage. This can be done using the step-down/step-up or buck/boost converter shown in Figure 11.10. This converter is obtained from the circuit of Figure 11.8 by interchanging the inductor and diode. When the transistor is on, current flows through the inductor, storing reactive energy. When the transistor turns off, the inductor current cannot change instantaneously, and flows through the capacitor and load resistor and back through the diode. With the transistor on, the diode prevents source current from flowing to the load, and the load current is supplied by the capacitor. Note that the polarity of the output voltage is opposite to the two previous converters. Since the integral of the inductor voltage over one period must be zero, we can write
where we have again neglected voltage drops across the transistor and diode. Solving for 
gives
If 
and the circuit functions as a buck converter, while if 
and the circuit functions as a boost converter.
Figure 11.10 A step-down/step-up or buck/boost converter.
The final DC converter to be considered is the full-bridge converter of Figure 11.11. This same basic circuit configuration appears often in power electronics, and is used in switch-mode inverters to be discussed in the next section. Here we consider only the conversion of unregulated DC to regulated DC. In this implementation, one of the transistors in the 
pair is on and the other off at all times, and similarly for the transistors in the 
pair. Switching of transistors in the 
pair occurs simultaneously with switching of transistors in the 
pair. In this way, the load is continually connected to the source, either directly connected through 
and 
, or cross-connected through 
and 
. The diodes are used to clamp excursions in the load voltage that exceed the source voltage, either positive or negative.
Figure 11.11 A full-bridge DC–DC converter.
In DC converter applications, the load contains one or more energy storage elements, such as the inductance of a DC motor winding illustrated in Figure 11.11. The transistors in the converter are switched at a frequency whose period is short compared to the RL time constant of the load. If the load is a DC motor, this ensures that the load current and the emf 
induced in the motor winding are DC quantities.
The magnitude and polarity of 
are determined by the duty factor 
of the switching waveforms. The time-average load voltage can be written
Equation 11.9 shows that the average load voltage varies linearly with the duty factor, increasing from 
when 
, to 
when 
. Thus, both polarities of output voltage can be obtained simply by varying the duty factor, independent of the direction of the current. Consider the DC motor load shown in Figure 11.11. If the induced emf 
exceeds the source voltage, as happens during regenerative braking, the current is in the opposite direction to that indicated in the figure, and power flows from the load back to the source. Thus the full-bridge converter is capable of operation in all four quadrants of the 
plane, as illustrated in Figure 11.12.
Figure 11.12 Operation of the full-bridge DC–DC converter in the 
plane. In the first or third quadrants power is transferred from port 1 to port 2, while in the second or fourth quadrants power is transferred from port 2 to port 1.
As an aside, we note that four-quadrant operation is available in this converter even if the transistors have a preferential direction for current flow. For example, BJTs and IGBTs have much higher gain in the forward direction than in the reverse direction. In the full-bridge converter, BJTs or IGBTs are connected so that their preferential current direction is opposite to that of their parallel diode. This way the diode carries most of the current when the current flow is opposite to the preferential direction of the transistor.
11.2.3 Switch-Mode Inverters
Switch-mode inverters convert unregulated DC into sinusoidal AC of variable amplitude and frequency. Typical applications are in AC motor drives and uninterruptible AC power supplies. If the application calls for conversion from line-frequency AC, the generic power processor of Figure 11.1 will consist of a rectifier as converter 1 and a switch-mode inverter as converter 2. The switch-mode inverter uses pulse-width-modulated (PWM) switching to synthesize a sine wave output.
Figure 11.13 shows a single-phase half-bridge switch-mode inverter. The DC source 
is bridged by two equal capacitors, with each capacitor charging to a voltage of 
. We assume the capacitors are large enough that their voltages remain essentially constant during one cycle. Transistors 
and 
are switched with opposite polarity signals so that at any given time one transistor in the pair is on and the other off. When 
is on, the A terminal of the load is connected to 
and the B terminal to the capacitor midpoint at 
. When 
is on, the A terminal of the load is connected to 
and the B terminal to the capacitor midpoint at 
. Thus the load voltage 
switches between 
and 
, as shown in Figure 11.14. The 
output waveform is pulse-width modulated at frequency 
to synthesize a sine wave of frequency 
at the output.
Figure 11.13 A single-phase half-bridge switch-mode inverter. The switching waveform is modulated to synthesize a sine wave across the load, as shown in Figure 11.14.
Figure 11.14 Waveforms of the half-bridge switch-mode inverter (a) and its frequency spectrum (b). The fundamental Fourier component of the output waveform is the dashed sinusoid at frequency 
.
The harmonic content of the output waveform can be obtained by Fourier analysis and contains components at the fundamental frequency 
, and at integral multiples of 
with the (much higher) switching frequency 
, along with sidebands as shown in the figure. It is desirable that 
so that these harmonics are well above the response capability of the load being driven. When this is the case, the load responds as if driven by the fundamental Fourier component.
The frequency 
of the switching waveform should satisfy the following criteria:
, where 
is the frequency of the synthesized sine wave. Higher 
values push the harmonics to higher frequencies and make the load respond as if driven by a pure sinusoid at the fundamental frequency 
.- For
, 
should be an odd integer multiple of 
. This eliminates the even harmonics of 
from the Fourier analysis (as shown in Figure 11.14) so that only odd harmonics are present, that is, 
Although the even harmonics are removed, their sidebands remain, but they have lower amplitude and are less disruptive. - For
the harmonics are small, and 
does not have to be an integer multiple of 
, that is, the switching waveform and the output waveform may be asynchronous. This makes it possible to vary the frequency of the output waveform without changing the switching waveform. (An exception occurs when driving an AC motor, since even small sub-harmonics can produce undesirably large stator currents.) 
should lie outside the audible frequency range. In most cases 
is chosen to be either below 6 kHz or above 20 kHz. Higher 
produces a higher quality sine wave, but leads to proportionally higher switching losses in the transistors of the converter.
For low-frequency applications 
, 
may be in the range 9–15, whereas for high-frequency applications 
, 
may be larger than 100. The switching frequency is an important parameter in selecting the optimum device for the application, especially when high blocking voltages are required, since switching loss in bipolar devices such as BJTs, IGBTs, and thyristors is often the dominant loss, and is proportional to 
.
Figure 11.15 shows a single-phase full-bridge switch-mode inverter. This is the same circuit as the full-bridge DC converter of Figure 11.11, with the only operational difference being the PWM waveforms applied to the transistors. Like the converter of Figure 11.11, the full-bridge inverter of Figure 11.15 is capable of operation in all four quadrants, permitting bidirectional power flow. The full-bridge inverter can also be obtained from the half-bridge inverter of Figure 11.13 by connecting the B terminal of the load to the positive and negative terminals of the source through a second pair of switching transistors 
and 
.
Figure 11.15 A single-phase full-bridge switch-mode inverter. The control waveforms are designed to synthesize a sinusoidal output, shown dashed in Figure 11.16.
There are two possible PWM schemes for the full-bridge inverter. In bipolar modulation, the waveforms applied to transistors 
and 
in the full-bridge inverter are the same as for the half-bridge inverter, while the waveforms applied to transistors 
and 
are the inverse. The control waveform thus consist of two sub-periods. During the first sub-period, transistors 
and 
are on while transistors 
and 
are off. This connects the load directly across the source so that 
. During the second sub-period, transistors 
and 
are off while transistors 
and 
are on. This cross-connects the load to the source so that 
. As a result, the output voltage 
alternates between 
and 
(hence the name “bipolar”), and the fraction of time allotted to each state is modulated to synthesize a sine wave output similar to that of Figure 11.14, with one important difference: Because of the symmetrical transistor configuration, the output voltage swing of the full-bridge inverter, shown in Figure 11.16, is twice as large 
as the half-bridge inverter 
. Thus the full-bridge inverter can deliver the same output power using half the current. This is an important advantage, since it reduces the need for paralleling devices in high-power applications.
Figure 11.16 Waveforms (a) and output spectrum (b) of the full-bridge switch-mode inverter when operated using bipolar PWM.
The second PWM scheme for the full-bridge inverter is called unipolar modulation. This scheme has the same effect as doubling the switching frequency, in terms of the harmonics present in the output waveform, without actually changing the frequency 
at which the transistors are switched. Stated another way, with unipolar modulation one can achieve the same harmonic content at half the switching frequency of bipolar modulation. This is a significant advantage, since transistor switching loss is proportional to switching frequency. Unipolar modulation requires separately-timed control signals to transistors 
and 
and transistors 
and 
, rather than inverse-polarity signals with the same timing, as in bipolar modulation. The resulting output waveform, shown in Figure 11.17, steps between 
and zero during the first half-cycle and between 
and zero during the second half-cycle, hence the name “unipolar”. In unipolar modulation, the ratio 
should be an even integer, as compared to an odd integer in bipolar modulation. With this choice, all odd harmonics are eliminated from the output spectrum along with their sidebands, as seen in Figure 11.17. What remains are the sidebands of the even harmonics at 
, 
, and so on. Note that the principal harmonics at 
, 
, and so on are also suppressed, and only their sidebands remain.
Figure 11.17 Waveforms (a) and output spectrum (b) of the full-bridge switch-mode inverter when operated using unipolar PWM.
Many applications, such as uninterruptible AC power supplies and AC motor drives, require three-phase AC outputs. Figure 11.18 shows a three-phase switch-mode inverter driving a three-phase AC motor. The three-phase inverter can be envisioned as three single-phase half-bridge inverter sections of the type shown in Figure 11.13, driven by waveforms of the type shown in Figure 11.14, where the control waveforms for the a, b, and c phases are displaced 120° with respect to each other. Unlike the single-phase half-bridge inverter of Figure 11.13, the line-to-neutral voltages of the three-phase inverter 
, and 
switch between 
and zero rather than from 
to 
. Because of the 120° relationship between phases, the line-to-line voltages 
, and 
swing between 
and 
. Since one of the two switches in each leg is always on at any instant in time, the output voltage is independent of the magnitude and direction of the output current. Thus, this inverter is capable of operating in all four quadrants, permitting bidirectional power flow.
Figure 11.18 A three-phase switch-mode inverter driving a three-phase AC motor.
In three-phase inverters, the only harmonics of concern are those of the line-to-line voltages. The frequency spectra of the line-to-neutral voltages are the same as the half-bridge inverter shown in Figure 11.14. However, when the line-to-neutral signals are combined algebraically to obtain the line-to-line voltages, their 120° phase shift results in cancellations that eliminate some of the harmonics. This is particularly true if the ratio 
is an odd integer multiple of three (i.e., 3, 9, 15…), since this removes all principal harmonics of 
from the spectrum, leaving only their sidebands. An example of a line-to-line waveform and the associated frequency spectrum are shown in Figure 11.19.
Figure 11.19 Line-to-line waveforms (a) and output spectrum (b) of the three-phase switch-mode inverter of Figure 11.18.
The switching waveform of the three-phase switch-mode inverter should satisfy the following criteria:
- For
should be an odd integer multiple of 3 (i.e., 3, 9, 15, or 21). This is required to cancel all direct harmonics of 
from the output spectrum. - For
the harmonics are small, and 
does not have to be an integer multiple of 
, that is, the switching waveform and the output waveform may be asynchronous. This makes it possible to vary the frequency of the output waveform without changing the switching waveform. (An exception occurs when driving an AC motor, since even small sub-harmonics can produce undesirably large stator currents.)
11.3 Power Electronics for Motor Drives
11.3.1 Introduction to Electric Motors and Motor Drives
Electric motors can be classified into three primary types: DC motors, induction (or asynchronous) motors, and synchronous motors. The drive requirements for the three types are different, and will be considered below. Electric motor applications range from low power (a few watts) to very high power (megawatts), and from high precision, such as servo drives for robotics, to less critical applications, such as adjustable speed drives for pumps and fans. The applications may call for single-quadrant operation (motoring), two-quadrant operation (motoring plus regenerative braking), or four-quadrant operation (reversible motoring and braking). All of these factors play a role in the design and performance specifications of the electronic motor drive.
In general, the current rating of the motor drive is dictated by the torque required of the motor in the particular application, since electromechanical torque is proportional to current. The voltage rating of the motor drive is determined by the rotational speed and controllability requirements, based on the following considerations. In both DC and AC motors, rotation produces a back-emf in the motor windings, and the equivalent circuit presented by the motor to the drive circuit can be represented as a voltage source (the back-emf) in series with the winding inductance, as illustrated in Figure 11.20. The rate of change in current (and therefore torque) is given by
where 
is the output voltage of the drive, 
is the back-emf of the motor, and 
is the inductance of the motor windings. The back-emf, in turn, is proportional to the rotational speed of the rotor. To achieve a short response time to speed and position commands, the output voltage of the drive 
must exceed the back-emf 
by a sufficient margin. Thus the voltage rating of the motor drive is determined by the speed of the motor (through the back-emf) and the rate at which motor torque needs to be changed. We now consider drive circuits for the three primary types of motors.
Figure 11.20 Generic equivalent circuit of a motor and its associated drive circuit.
11.3.2 DC Motor Drives
DC motors are typically used for speed and position control in applications where low initial cost and good performance characteristics are desired. In DC motors, the stator establishes a stationary magnetic field using either permanent magnets or stator field windings. When the field is provided by windings, the stator current controls the field flux 
. If magnetic saturation is neglected, the field flux is proportional to the field current,
where 
is the field constant of the motor. The rotor carries the armature windings that supply variable power to the motor and the load. The armature windings are connected to a segmented copper commutator that rotates with the shaft and is contacted by stationary carbon brushes mounted on the stator.
In a DC motor, the electromechanical torque is produced by an interaction between the stator's field flux and the rotor's armature flux. The rotor flux is proportional to the armature current, and the electromechanical torque can be written
where 
is the torque constant of the motor. In addition, a back-emf is induced in the armature windings by their rotation through the stator field. This back-emf is proportional to the field flux and the angular velocity,
where 
is the voltage constant of the motor and 
is the angular velocity of the rotor. Setting the electrical power delivered to the motor (
) equal to the mechanical power delivered by the motor to the load (
), we find that 
, with 
in units of [N m / A Wb] and 
in units of [V s / Wb].
The right side of Figure 11.11 shows the equivalent circuit of the armature windings in a DC motor, where 
is the winding resistance, 
the winding self-inductance, and 
the back-emf given by Equation 11.13. In the normal mode of operation, 
and 
are positive, and the motor produces positive torque, Equation 11.12, and a positive rotational velocity, Equation 11.13, delivering mechanical power 
to the load. However, it is often desirable to use the motor for regenerative breaking. To do this, the terminal voltage 
is reduced below the induced emf 
so that the direction of current is reversed, that is, 
becomes negative. This has the effect of reversing the torque, Equation 11.12, thereby slowing the rotation of the motor and the load. In addition, the mechanical power delivered to the load 
and the electrical power drawn from the source 
both become negative, which represents net power taken from the kinetic energy of the load and returned to the source (note that 
remains positive, since the rotational velocity 
has not changed sign). Eventually the back-emf is reduced to zero when the motor comes to a stop. If the terminal voltage 
is made negative, the torque is negative, and the motor rotates in the opposite direction, producing a negative back-emf. Thus it is possible to reverse the direction of a DC motor simply by reversing the voltage and current polarities applied to the armature, and the DC motor is capable of operating in all four quadrants of Figure 11.12.
The selection of a power converter to drive a DC motor depends on whether single-quadrant, two-quadrant, or four-quadrant operation is desired. If the rotation is unidirectional and braking is not required, single-quadrant operation can be provided by the simple buck converter of Figure 11.21. If rotation is unidirectional but braking is required, two-quadrant operation can be obtained using the converter of Figure 11.22. In this circuit, 
and 
are switched so that only one is on at any time. When 
is on and 
is off, 
and 
are positive (motoring). When 
is on and 
is off, the back-emf of the motor causes 
to reverse direction (braking).
Figure 11.21 A DC motor with single-quadrant operation driven by a simple buck converter similar to that of Figure 11.8.
Figure 11.22 A DC motor with unidirectional rotation and regenerative braking, driven by a simple two-quadrant converter.
Applications that require reversible-speed operation at moderate power, along with regenerative braking, call for a four-quadrant converter such as the full-bridge DC converter of Figure 11.11. An approach for high-power, fully reversible applications is to connect two line-frequency phase-controlled inverters of the type shown in Figure 11.5 in anti-parallel to achieve four-quadrant operation, as shown in Figure 11.23. For forward rotation, converter 1 operates in the rectifier mode for motoring, while converter 2 operates in the inverter mode for braking. For reverse rotation, converter 2 operates in the rectifier mode for motoring and converter 1 operates in the inverter mode for braking.
Figure 11.23 A DC motor with reversible rotation and regenerative braking, driven by two anti-parallel line-frequency phase-controlled converters of the type shown in Figure 11.5.
11.3.3 Induction Motor Drives
Induction motors or “asynchronous motors” are AC motors that supply power to the rotor by electromagnetic induction rather than by brushes or slip rings. Induction motors are preferred in applications where low cost and rugged construction are desired. They operate at nearly constant rotational speed determined by the angular frequency of the AC drive signal.
Most induction motors are driven by a three-phase AC power source. The stator of an induction motor usually contains multiple poles of three-phase windings, as illustrated in Figure 11.24. The wiring diagram for the stator windings is shown at the top. Phase 1 produces four poles, two “N” and two “S”. In the motor diagram, stator current is flowing away from the reader in the “
” segments and toward the reader in the “
” segments, producing magnetic field lines that penetrate the rotor. The rotor itself has no external electrical connections, and can be one of three types. Squirrel-cage rotors have a series of conducting bars around the periphery, oriented parallel to the rotor axis and shorted at each end by conducting rings, thereby forming a cage-like structure. Slip-ring rotors have windings connected to slip rings that replace the bars of a squirrel-cage design. Solid-core rotors are made from magnetically soft steel.
Figure 11.24 Schematic diagram of a four-pole three-phase squirrel-cage induction motor.
In operation, the stator windings create a rotating magnetic field that rotates at the synchronous speed 
given by
where 
is the angular frequency of the driving voltage and 
is the number of poles. The field lines from the stator penetrate the rotor, inducing currents in the bars or windings of the rotor. These currents, in turn, create a rotor field that rotates at the synchronous speed with respect to the stator. However, the rotor itself does not rotate at the synchronous speed, because if it did there would be no relative motion between the rotor and the rotating stator field, and hence no induced currents in the rotor. Instead, the rotor rotates in the same direction as the stator field, but at a speed 
that is slightly less than 
. This means the rotor is “slipping” with respect to the stator field at a relative speed, called the “slip speed” 
, given by
It is customary to refer to the “slip” of the motor, where the slip 
is the normalized slip speed defined by
The rotor field is synchronous with the stator field, but rotates at speed 
with respect to the rotor, since the rotor is slipping by that amount with respect to the stator field.
The electrical response of the motor can be represented by the per-phase equivalent circuit of Figure 11.25, where 
is the rms line-to-line voltage of the three-phase motor drive and 
is the rms phase current. Here 
is the back-emf induced in the stator windings by the rotor field. The stator current 
consists of two components: 
is the portion of the stator current that establishes the air-gap flux, and 
represents the interaction with the rotor field that produces the electromechanical torque. 
and 
are the resistance and self-inductance of the stator windings, and 
is the magnetizing inductance of the stator windings. 
and 
are the resistance and inductance of the rotor. 
is the effective resistance of the rotor as reflected back into the stator circuit, where 
is the slip between the rotor and the stator fields. For small values of slip corresponding to normal operation, the rotor current 
is small compared to the magnetizing current 
, and 
and 
can be neglected, making 
.
Figure 11.25 Per-phase equivalent circuit of an induction motor.
The electromagnetic torque is produced by the interaction between the rotating stator field flux 
and the flux of the rotor field. Since the rotor field is proportional to the rotor current, we can write
where 
is the torque constant of the motor. Equation 11.17 for the induction motor is analogous to Equation 11.12 for the DC motor. In the induction motor, the induced rotor current 
is proportional to the air-gap flux 
and the slip frequency 
, so Equation 11.17 can also be written
For zero slip (synchronous rotation), the effective resistance 
in the rotor leg of the equivalent circuit of Figure 11.25 is infinite and no rotor current flows, meaning no torque is delivered to the load, as indicated by Equations 11.17 and 11.18. Torque is only produced when the rotor speed 
is less than the synchronous speed 
, and Equations 11.18 and 11.16 tells us the torque is proportional to this difference.
Under normal operating conditions, the voltage drops across the stator winding resistance 
and self-inductance 
are small compared to the back-emf 
. The back-emf, in turn, is proportional to the stator field and the angular frequency. Thus, we can write
where 
is the voltage constant of the motor. Equation 11.19 for the induction motor is analogous to Equation 11.13 for the DC motor.
The torque-speed relation given by Equation 11.18 is only valid for small values of slip, which is the normal operating regime of the motor. The full torque-speed curve is illustrated in Figure 11.26. The rated torque and rated speed are normally specified on the nameplate of the motor, and correspond to the upper limit of torque (and lower limit of speed) for which the linear relationship in Equation 11.18 applies. The steady-state speed is determined by the intersection of the torque-speed curves of the load and the motor, as shown in Figure 11.27 for different values of synchronous frequency 
. The synchronous frequency is controlled by the power electronic converter driving the motor. For the torque provided by the motor to equal the rated torque at any frequency, Equation 11.18 requires that the air-gap flux remain constant as frequency is varied. Equation 11.19 then shows that the source voltage 
provided by the drive electronics must scale directly with frequency 
.
Figure 11.26 Torque-speed curve of an induction motor. The normal operating region is the linear regime at the far right, where the slip is small and the air-gap flux is constant.
Figure 11.27 Torque-speed diagram of an induction motor and its load. The steady-state operating point is the intersection of the load torque and motor torque curves, and can be modulated by varying the frequency 
of the driving waveform.
Figure 11.27 also allows us to visualize how an induction motor accelerates from a dead stop. At zero rotation 
the motor torque exceeds the load torque by a wide margin, resulting in angular acceleration of the motor and load. As the rotor speed increases, the motor and load torques eventually balance, resulting in a steady-state rotation at a speed determined by the synchronous frequency of the drive electronics.
Induction motors are also capable of electromagnetic braking, which occurs when the rotor speed 
exceeds the rotational speed of the stator field 
. In this mode the induction motor acts as a generator, transferring power from the rotating shaft back to the source. This can be understood by extending the torque-speed curve of Figure 11.26 to rotational speeds greater than 
, as shown in Figure 11.28. For 
the torque is negative, which decelerates the rotor and transfers inertial energy back into the power source. Consider a specific example. Suppose the motor is initially operating with positive torque at a steady-state speed 
. If the drive frequency is reduced to 
, the rotor speed (assuming it has not yet changed) is given by 
. The rotor speed now exceeds the synchronous speed of the stator field, placing the motor in the negative-torque regime, and the motor decelerates until the torque again becomes positive. The motor can be decelerated smoothly to a full stop if the stator frequency is reduced gradually.
Figure 11.28 Torque-speed diagram of an induction motor for rotor speeds above and below the synchronous speed. For 
the rotor currents, torque, and slip are negative, and the motor is in the generator mode, transferring inertial energy back to the source.
Based on the above considerations, power electronic converters for induction motor drives must produce three-phase AC with variable frequency and magnitude. The frequency is adjusted to control the speed of the motor, and the magnitude is adjusted to insure constant air-gap flux as the speed is varied, in accordance with Equation 11.19. If electromagnetic braking is desired, the converter must be capable of at least two-quadrant operation.
Referring to the generic power processor of Figure 11.1, the induction motor drive may utilize either a diode rectifier or a four-quadrant switch-mode converter as converter 1, depending on whether regenerative braking is required. The three-phase switch-mode inverter of Figure 11.18 can be used as converter 2 to provide variable-frequency, variable-magnitude AC power to the motor. This inverter is capable of four-quadrant operation, allowing for electromagnetic braking. During braking, inertial energy is absorbed from the load and transferred back to the power electronics through converter 2. This energy must either be dissipated within the power processor or returned to the AC source. If dissipative braking is used, a resistor is switched in parallel with the filter capacitor in the power processor during braking, as illustrated in Figure 11.29a. If regenerative braking is employed, a four-quadrant switch-mode converter must be used as converter 1, allowing power to be returned to the AC line, as shown in Figure 11.29b.
Figure 11.29 Induction motor drives capable of dissipative braking (a) and regenerative braking (b).
11.3.4 Synchronous Motor Drives
Synchronous motors are AC motors that employ a permanent-magnet rotor or a DC-wound rotor along with a three-phase AC stator, where the rotor rotates at the synchronous speed of the stator field. Synchronous motors are used as servo drives in robotic applications and as adjustable-speed drives in high-power applications. Low-power applications usually employ permanent-magnet synchronous motors, also called “brushless DC” motors, while high-power applications use wound-rotor synchronous motors, also called “salient-pole” motors. Figure 11.30 illustrates these two motor types.
Figure 11.30 Schematic diagram of synchronous motors using a permanent magnet rotor (a) and a wound rotor (b). The rotor establishes a DC magnetic field that rotates at the synchronous speed of the stator field.
In both brushless and wound-rotor motors, the rotor field flux 
is stationary with respect to the rotor, and rotates at the synchronous speed 
with respect to the stator and motor housing. The angular frequency of the stator currents is 
, where 
is the number of poles on the rotor (two are shown in Figure 11.30). The rotor field induces a voltage in the stator windings that is proportional to the rotor flux and the synchronous speed. For stator phase “a”, the rms value of the induced voltage is given by
where 
is the equivalent number of turns in the stator phase windings.
The three-phase stator windings produce a stator field 
that rotates at the synchronous speed 
with an amplitude proportional to the fundamental frequency component of the stator current. Like the rotor field, the rotating stator field induces a voltage in the stator windings. For phase “a”, the induced voltage due to all three stator windings can be written
where 
is the armature inductance (3/2 the self-inductance of phase “a”), 
is the stator current in phase “a,” and 
is a phasor having a torque angle 
with respect to 
. The fluxes of the stator and rotor add in phasor fashion to produce the air gap flux 
. Similarly, the induced voltages 
and 
add in phasor fashion to produce the net air-gap voltage 
in the stator windings. These relationships can be understood using the per-phase stator equivalent circuit shown in Figure 11.31, where 
and 
are the resistance and self-inductance of the stator phase windings and 
and 
are induced emfs due to the rotor and stator fields, respectively. The per-phase stator terminal voltage is
Figure 11.31 Per-phase equivalent circuit of a synchronous motor.
The electromagnetic torque is proportional to the phasor product of the rotor and stator fluxes, and can be written
where 
is the torque constant of the motor and 
is the torque angle between the rotor and stator flux phasors.
Synchronous motors are often operated with sinusoidal induced-voltage waveforms, particularly in servo applications. In this case the torque angle 
is maintained at 90°, and for a fixed field flux 
, the electromagnetic torque is proportional to the stator current, as given by Equation 11.23. The stator current waveforms are sinusoidal with a 120° phase relationship, and are synchronized to the position of the rotor, which is monitored by a rotational position sensor.
Synchronous motors may also be designed with magnetic structures that produce trapezoidal induced-voltage waveforms. In this case the drive waveforms are rectangular, with constant amplitude 
for 120° of rotation, zero amplitude for the next 60° of rotation, constant amplitude 
for the next 120° of rotation, then zero amplitude for another 60° of rotation. As before, the three drive waveforms have a 120° phase relationship with each other, resulting in a constant torque with minimum ripple.
For either sinusoidal or trapezoidal operation, the three-phase current-regulated voltage-source inverter of Figure 11.18 may be used to drive the stator windings, and the waveforms 
, and 
are either sinusoidal or rectangular, depending on the type of motor. The complete drive circuit includes position feedback from a sensor monitoring the rotor position, along with electronics to generate the control signals to the inverter. Since the timing of the stator currents is synchronized to the instantaneous rotor position, the torque angle is always maintained at its optimal value of 90°, and the torque is determined by the magnitude of the stator drive currents according to Equation 11.23.
In very high power applications 
, thyristor-based load-commutated inverters are used. Figure 11.32 shows a thyristor-based current-source inverter (CSI) driven by a line-voltage-commutated thyristor-based rectifier. The phase waveforms 
, and 
are rectangular, as described above for the trapezoidal motor operation, and although the induced-voltage waveforms in each phase are trapezoidal, the line-to-line voltages are sinusoidal. The line-voltage-commutated rectifier has the per-phase equivalent circuit of Figure 11.5, and the load-commutated inverter has the same per-phase circuit, but is operated in the inverter mode, as in Figure 11.7c.
Figure 11.32 Thyristor-based motor drive used for high-horsepower synchronous motors. The drive employs a line-commutated rectifier as converter 1 and a load-commutated current-source inverter as converter 2. The per-phase equivalent circuit of both the rectifier and inverter is given in Figure 11.5.
11.3.5 Motor Drives for Hybrid and Electric Vehicles
With the growing emphasis on energy independence and environmental sustainability, electric vehicles and hybrid electric vehicles are assuming increased importance. This represents a large potential market for SiC power devices.
In a pure electric vehicle (EV), the wheels are driven by an AC electric motor powered by a rechargeable storage battery. In a hybrid electric vehicle (HEV), an internal combustion engine is combined with an electric drive system. Figure 11.33 presents block diagrams of a pure internal combustion drive, a pure electric drive, and two forms of hybrid drives: the parallel hybrid and the series hybrid. The parallel hybrid drive obtains motive power from either the internal combustion engine or from an electric motor powered by a battery. The series hybrid drive obtains motive power from an electric motor energized either by the battery or by a generator driven by the internal combustion engine. Both series and parallel drives are capable of regenerative braking, where kinetic energy of motion is returned to the battery.
Figure 11.33 Block diagrams of a conventional internal combustion drive (a), a pure electric drive (b), a parallel hybrid drive (c), and a series hybrid drive (d).
Figure 11.34 shows a high-capacity series drive (a) typical of diesel-electric locomotives or military tanks, where separate traction motors are placed on each wheel. Figure 11.34 also shows a combined hybrid drive (b) used in automotive applications. The combined hybrid allows energy to flow along four different pathways: (i) from the internal combustion engine through the transmission to the wheels, (ii) from the internal combustion engine through a generator and power converter to the battery, (iii) from the battery through a power converter to the electric motor and thence to the wheels, and (iv) from the wheels to the battery by regenerative braking, with the electric motor operating in the generator mode and the bi-directional converter operating in the rectifier mode.
Figure 11.34 Block diagrams of a high-capacity series hybrid drive (a) and a combined series-parallel hybrid drive (b).
The drive motor in electric vehicles is typically either an AC induction motor (or “asynchronous motor”) or an AC synchronous motor with a permanent-magnet rotor (also called a “brushless DC” motor). Neither motor requires electrical connection to the rotor. In the induction motor, the rotor field is established by currents induced in the rotor by the rotating field of the stator. In the permanent-magnet synchronous motor, the rotor field is established by a permanent magnet. Both motors require three-phase AC drive to the stator windings to establish a rotating magnetic field in the air gap. This drive is supplied by the three-phase converter located between the battery and the motor in Figures 11.33 and 11.34. The frequency of the AC drive is adjusted to control the speed of the motor, and the magnitude is adjusted to control the torque produced by the motor, as discussed below.
The control circuits for the converter will be different depending on whether the inverter is driving an induction motor or a permanent-magnet synchronous motor. In an induction motor, the rotor slips with respect to the stator field, and the torque is given by Equation 11.18. For constant slip 
, the torque is proportional to the square of the air-gap flux, which is established by the stator drive current. In a permanent-magnet synchronous motor, the rotor turns at synchronous speed established by the stator and the torque is given by Equation 11.23. If the synchronous motor is operated with sinusoidal waveforms, the torque angle 
is maintained at 90° by position sensors, and the torque is proportional to the stator drive current.
Electric motors in EVs and HEVs operate at AC drive voltages in the 600 V range, while the battery voltage is typically 200–300 V. The power converter between the battery and electric motor in Figures 11.33 and 11.34 employs the bidirectional three-phase switch-mode inverter shown in Figure 11.18. A 600 V output voltage requires switching transistors with blocking voltages around 1200 V. In the past, these converters have been implemented with silicon IGBTs. However, as will be discussed in Section 11.6, SiC unipolar power switching devices (MOSFETs and JFETs) have performance parameters superior to silicon MOSFETs and IGBTs at 600 V and above, and are expected to replace silicon components in many EV and HEV applications.
11.4 Power Electronics for Renewable Energy
Solar and wind-energy farms provide renewable energy with minimal pollution, and are rapidly growing components of worldwide energy production. Both these energy sources require power electronics to convert DC power (solar) or asynchronous AC power (wind) to AC power synchronized to the electric utility grid.
11.4.1 Inverters for Photovoltaic Power Sources
Solar cells are large-area pn diodes specially designed to admit light into the depletion region, where photogenerated carriers are separated by the built-in electric field. Electrons and holes flow to their majority-carrier sides of the junction, giving rise to a reverse photocurrent 
. The 
characteristics are given by the Shockley diode equation, Equation 7.26, with an additive term that represents the reverse photocurrent,
The photocurrent 
is proportional to the photon flux 
, and the proportionality constant 
is only a weak function of junction voltage. The 
characteristics are illustrated schematically in Figure 11.35. The photocurrent subtracts from the ideal diode current, causing the 
characteristics to enter the fourth quadrant, where power is delivered by the cell to the external circuit. Silicon solar cells operate with an open-circuit voltage in the range 0.5–0.65 V, depending on photon flux and temperature. The power delivered by the cell is given by 
, and the maximum power point is obtained when the 
product is a maximum, illustrated by the points in Figure 11.35. Commercial solar arrays are maintained at the optimum power point by a perturb-and-adjust technique, where slight adjustments are made to the operating point every few seconds to automatically seek the maximum power point.
Figure 11.35 Current–voltage characteristics of a solar cell under different illumination intensities, expressed as the ratio of photocurrent 
to diode saturation current 
.
Since the solar array produces low-voltage DC, power electronics is needed to convert this to high-voltage sinusoidal AC that is synchronized to the utility grid at near-unity power factor. For power levels below a few kilowatts, such as produced by single-home solar arrays, the connection is typically made to a single-phase AC line. The conversion occurs in four stages: (i) low-voltage DC from the solar array is converted to high-frequency low-voltage AC by a switch-mode inverter, (ii) this is passed through a step-up transformer to produce high-voltage high-frequency AC, (iii) which is rectified and filtered, yielding high-voltage DC, (iv) which is converted to high-voltage line-frequency AC, synchronized to the AC utility grid. Figure 11.36 shows a solar converter consisting of a switch-mode inverter, a high-frequency step-up transformer, a diode rectifier, and a thyristor-based line-commutated inverter. The thyristor-based inverter is the same converter shown in Figure 11.5, now operating in the inverter mode. For single-phase utility connections, the converter output voltage is typically in the range 208–240 V.
Figure 11.36 A single-phase solar converter consisting of a switch-mode inverter, a high-frequency step-up transformer, a diode rectifier and filter, and a thyristor-based line-commutated inverter.
At power levels above a few kilowatts, such as produced by commercial solar arrays, the interface to the utility grid is a three-phase connection. Figure 11.37 shows a solar converter for this application. This circuit is similar to the single-phase converter of Figure 11.36 except the output inverter has been implemented as a three-phase switch-mode inverter similar to that shown in Figure 11.18. For three-phase connections, the converter output is typically 480 V, and is stepped to 12 kV or higher by a line-frequency transformer connected to the converter output (not shown).
Figure 11.37 A three-phase solar converter. This converter is similar to the single-phase converter of Figure 11.36, but employs a three-phase switch-mode inverter at the output.
11.4.2 Converters for Wind Turbine Power Sources
A growing fraction of renewable energy worldwide is supplied by wind farms. Each wind turbine drives a three-phase AC generator with a permanent-magnet rotor. The power produced is proportional to the cube of wind velocity, and the optimum shaft speed varies with wind conditions, making it impractical to generate AC at a constant frequency. Therefore, power electronics is used to interconnect this variable-frequency power to the utility grid.
Figure 11.38 shows a three-phase AC converter suitable for moderate-power wind turbines or small hydroelectric power sources. In this converter, variable-frequency three-phase AC is converted to DC by a diode rectifier, filtered, and converted to three-phase line-frequency AC by a switch-mode inverter similar to that of Figure 11.18. The inverter is controlled to ensure that its output is synchronized to the utility grid at near-unity power factor.
Figure 11.38 A three-phase wind turbine converter for use at moderate power levels.
The permanent-magnet AC generators in large commercial wind turbines have voltage outputs of 3–5 kV and power ratings up to 10 MVA. The converter interface to the utility grid is also around 3–5 kV. At these power levels, thyristor-based converters are typically used. Figure 11.39 shows a high-power converter that incorporates a three-phase source-commutated thyristor rectifier followed by a three-phase line-commutated thyristor inverter. Both the rectifier and inverter are three-phase versions of the phase-commutated converter shown in Figure 11.5. Commercial units currently use silicon gate-turn-off thyristors (GTOs) or integrated gate-commutated thyristors (IGCTs).
Figure 11.39 A three-phase wind-energy converter suitable for high-power turbine systems.
11.5 Power Electronics for Switch-Mode Power Supplies
Regulated power supplies deliver stable DC power to a load. Over a range of output loading, the power supply acts as an ideal voltage source, providing a controlled voltage that is independent of the load current. In most cases the output must be electrically isolated from the input, that is, the output must be floating with respect to the ground reference of the input supply. In the past, most regulated power supplies used analog circuits, but the availability of advanced semiconductor devices has made it possible to implement switch-mode power supplies that are smaller, lighter, and more efficient than analog supplies.
Figure 11.40 shows a generic block diagram of a regulated switch-mode power supply. Line-frequency AC is rectified and filtered to produce unregulated DC, which is inverted to high-frequency AC, passed through a high-frequency isolation transformer, rectified, and filtered to produce regulated DC at the output.
Figure 11.40 Block diagram of a switch-mode power supply.
In this section we will focus on the DC–DC conversion portion of the power supply, and consider five topologies: (i) flyback converters, (ii) forward converters, (iii) push-pull converters, (iv) half-bridge converters, and (v) full-bridge converters. The first two converters take the isolation transformer only into the first quadrant of its B–H loop, because the magnetizing current in the primary winding is always positive. Depending on the topology of the converter, this can lead to residual magnetization of the core that must be eliminated by a demagnetizing winding. The last three converters produce bidirectional magnetizing currents in the primary winding, taking the transformer alternately into the first and third quadrants of its B–H loop, and residual magnetization is not an issue.
We begin with the flyback converter, shown in Figure 11.41. This topology is derived from the buck-boost converter of Section 11.2.2, shown in Figure 11.10. In the flyback converter, the simple inductor has been replaced by a two-winding inductor that functions as an isolation transformer, shown in Figure 11.41 along with its magnetizing inductance 
. The operation of the circuit can be understood as follows. When the transistor is on, current flows through the primary winding of the transformer, and magnetic energy is stored in the core of the transformer. During this time, the diode is reverse biased and the capacitor supplies current to the load 
. The flux 
in the transformer can be found using Ampere's law,
where 
is the number of turns in the primary winding. From Faraday's law we obtain the voltage across the primary winding,
Figure 11.41 A flyback converter derived from the buck-boost converter of Figure 11.10.
Using Equations 11.25 and 11.26 we can obtain an expression for the build-up of flux as a function of time,
where 
is the flux at the start of the switching cycle. During the time when the transistor is on, 
and Equation 11.27 can be written
Equation 11.28 shows that the flux increases linearly with time while the transistor is conducting. The maximum flux is attained at the point when the transistor switches off,
where 
is the length of time the transistor is on. When the transistor is off, the voltage across the primary is determined by the turns ratio of the transformer 
times the secondary voltage 
. In analogy with Equation 11.28, since the primary voltage is now negative, the flux decreases as
In steady state, the flux at the end of the switching period 
is equal to the flux at the start of the switching period 
. Inserting Equation 11.29 into Equation 11.30, we can write
Using the relation in the last equality to solve for 
yields
where 
is the duty factor, 
. Equation 11.32 is analogous to Equation 11.8 for the non-isolated buck-boost converter. In the present circuit, the output voltage is related to the input voltage by the turns ratio of the transformer and the duty cycle of the switching waveform.
Using Equation 11.32, the maximum voltage across the transistor in the off state can be written
Equation 11.33 specifies the voltage rating of the switching transistor in the flyback converter.
The transformer in the flyback converter serves a dual purpose. It functions as an energy storage device, like the inductor in the buck-boost converter, and it also provides the desired input–output isolation. When the transistor is on, current flows in the primary while the diode blocks reverse current in the secondary. Energy is stored in the magnetic field while the transistor is conducting, and that energy is delivered to the load when the transistor is off. The longer the transistor is on, the greater the energy that is transferred to the load. At a 50% duty factor, the output voltage is equal to 
, and the output voltage can be controlled to be above or below 
by adjusting the duty factor.
The second converter topology is the forward converter, shown in Figure 11.42. This circuit is derived from the buck converter of Figure 11.8 by inserting a transformer in series with the transistor. When the transistor is on, current flows through the secondary of the transformer, through the series diode and the inductor to the load. The inductor voltage during this period is
Figure 11.42 A forward converter derived from the buck converter of Figure 11.8.
When the transistor is off, the inductor current flows through the shunt diode, and the voltage across the inductor is
Since
we can write
Using Equations 11.34 and 11.35 in Equation 11.37 yields
from which we obtain
Because of the current flow directions in the transformer, the forward converter tends to build up residual magnetization in the transformer core. This magnetization can be removed by including a third demagnetizing winding on the transformer core, wound in the opposite direction to the primary and connected across 
through a diode (not shown). Alternatively, the core can be demagnetized by connecting a zener diode across the transistor.
The remaining three converters establish bi-directional current flow in the primary winding of the transformer, taking it into both the first and third quadrant of its B–H loop. A two-transistor push-pull converter is shown in Figure 11.43. Transistors T1 and T2 are turned on alternately, each for a time 
, with a short blanking interval 
during which both are off. Hence 
. We define the duty factor 
and note that with this definition, 
. During the first half-cycle, T1 is on and current flows through the inductor through the upper diode, as shown in the figure. The inductor voltage during this period is given by Equation 11.34. During the blanking interval 
when both transistors are off, the inductor current splits through the two secondary windings and the inductor voltage is given by
Figure 11.43 A push-pull converter, derived from the buck converter discussed earlier. This converter requires two transistors and a center-tapped transformer.
Setting the integral of the inductor voltage over one half-cycle to zero and using Equations 11.34 and 11.40 allows us to solve for 
, yielding
The half-bridge converter is shown in Figure 11.44. The switching waveforms are the same as the push-pull converter just described. Transistors T1 and T2 are turned on alternately, each for a time 
, with a blanking interval 
during which both are off. Again, 
and the duty factor 
. During the first half-cycle, T1 is on and current flows through the inductor through the upper diode, as shown in the figure. The inductor voltage during the period when T1 is on is given by
Figure 11.44 A half-bridge converter. This converter has the same output circuit as the push-pull converter of Figure 11.43.
During the blanking interval both transistors are off, and the inductor voltage is given by Equation 11.40. Setting the integral of the inductor voltage over one half-cycle to zero and using Equations 11.40 and 11.42 allows us to solve for 
,
The full-bridge converter is shown in Figure 11.45. Transistors T1 and T2 are switched together, and transistors T3 and T4 are switched together, using the same switching waveforms as the two previous converters. The analysis of the full-bridge converter proceeds in the same way as the previous converters, and the output voltage is again given by Equation 11.41 with the duty factor 
. The full-bridge converter has the advantage that the current flowing through the transistors is half that of the half-bridge converter, allowing the use of smaller transistors. However, the transistors in both converters must support the full supply voltage 
.
Figure 11.45 A full-bridge converter. This converter is similar to the half-bridge converter of Figure 11.44, and employs four transistors, each with half the current rating of the half-bridge converter.
11.6 Performance Comparison of SiC and Silicon Power Devices
In the previous sections, we reviewed the fundamentals of power electronic systems, discussed basic converter circuit topologies, and considered several power systems that can benefit from SiC devices. Having established the general device requirements for various systems, it is natural to ask how the performance of today's SiC power devices compares to the existing technology. In this section we will present actual performance data for several types of SiC power devices and compare their performance to existing silicon devices and emerging GaN devices (where available). We caution that the discussion in this section represents only a snapshot in time. Silicon technology is mature and approaching its fundamental limits dictated by material parameters. SiC is in its adolescence and experiencing rapid growth. GaN is in its infancy, with great promise for the future. Because of the continuing evolution of all three technologies, the comparisons presented in this section are fleeting, and the reader should consult the literature for the latest information.
The most straightforward device comparison deals only with on-state performance, and neglects switching losses. This approach is acceptable when comparing unipolar devices such as Schottky diodes, JFETs, and MOSFETs, but a more sophisticated approach is needed to evaluate bipolar devices such as pin diodes, BJTs, IGBTs, and thyristors.
As discussed in Section 7.1, the unipolar device figure of merit (FOM) is 
. This FOM has a theoretical maximum dictated by the mobility, dielectric constant, and breakdown field of the material, and is given by Equation 7.13, repeated here for a punch-through design:
On a logarithmic plot of 
versus 
, the theoretical maximum unipolar FOMs are diagonal lines with a slope close to 2 (a precise calculation deviates slightly from 2 due to the dependence of mobility and critical field on doping). Figure 11.46 shows the unipolar limits for silicon, SiC, and GaN devices (note that these limits must be modified in the case of superjunction devices, which are vertical RESURF structures whose FOM has a slope of 1 instead of 2). Highest performance corresponds to the lower right corner of the figure, and all real devices have on-resistances above their respective limit lines. The right-hand axis shows maximum on-state current density at a power dissipation of 
, calculated using Equation 7.5. Owing to their higher critical field, the theoretical performance limits for SiC and GaN are significantly higher than for silicon, which accounts for the great interest in these technologies.
Figure 11.46 A standard device performance plot relating specific on-resistance to blocking voltage. The theoretical maximum unipolar FOMs are shown for silicon, SiC, and GaN.
The first SiC devices to enter commercial production were Schottky barrier diodes (SBDs) and their cousins, JBS (junction-barrier Schottky) diodes. These are used as clamping diodes across transistor switches in all the half-bridge and full-bridge converter circuits discussed in this chapter. Inserting SiC SBDs and JBS diodes instead of silicon pin diodes reduces the switching loss in these circuits due to the faster turn-off of unipolar diodes, as discussed in Chapter 7. This can lead to significant increase in efficiency in switch-mode power supplies used, for example, in file servers in large data centers. (It is not practical to use silicon SBDs for these applications because of their large reverse leakage currents, arising from the lower barrier heights of Schottky metals on silicon.)
Figure 11.47 shows the FOMs for a number of SiC SBDs reported to date [2–7]. Since Schottky diodes have an offset voltage in their 
characteristics, the maximum current cannot be calculated directly from Equation 7.5, which assumes no offset voltage. The power dissipation must include both the offset voltage and the voltage across the differential on-resistance and can be written
To calculate the on-state current for a given power dissipation, Equation 11.44 is solved for 
, yielding
Figure 11.47 On-resistance (diamonds) and maximum current (crosses) for several SiC Schottky barrier and JBS diodes.
The maximum on-state current is obtained from Equation 11.45 by setting the on-state power to some limiting value, say 
, and the offset voltage to the appropriate value for the device, which for SiC SBDs is around 1 V. In the plot of Figure 11.47, the gray diamonds are the on-resistances of the SBDs reading to the left-hand axis, and the black crosses are the maximum on-state currents of the same SBDs reading to the right-hand axis.
Figure 11.48 shows representative performance metrics for silicon LDMOSFETs, superjunction MOSFETs, and IGBTs [8]. IGBTs are bipolar devices and are not subject to the unipolar device theoretical limits, since their drift regions are conductivity modulated. Moreover, IGBTs have an offset voltage of about one forward-biased diode drop in their 
characteristics, so Equation 11.45 is used to calculate their maximum current, assuming an offset voltage of 0.8 V (for silicon). In the plot of Figure 11.48, the light gray crosses are the on-resistances of silicon IGBTs reading to the left-hand axis, and the black crosses are the maximum on-state currents of the same IGBTs reading to the right-hand axis. The MOSFET points may be read to both the left- and right-hand axes, since they have no offset voltage.
Figure 11.48 State-of-the-art performance of silicon lateral DMOSFETs, superjunction MOSFETs, and IGBTs ([8] reproduced with permission from IEEE).
Figure 11.49 presents the reported on-state performance for a number of SiC power DMOSFETs and UMOSFETs [8]. For blocking voltages in the range from 600 V to 2 kV, both types of SiC MOSFETs have maximum currents much higher than silicon devices. For example, at a blocking voltage of 1 kV, SiC UMOSFETs provide current densities as much as 
than silicon IGBTs at the same power dissipation. We also note that the best SiC UMOSFETs carry about 50% higher on-current than SiC DMOSFETs, as indicated by the SoA lines in the figure. Above 2 kV the on-state performance of SiC DMOSFETs approaches that of silicon IGBTs, but extrapolation of the UMOS SoA line to higher blocking voltages suggests that SiC UMOSFETs may retain their advantage over silicon IGBTs to 5 kV or higher.
Figure 11.49 State-of-the-art performance of SiC DMOSFETs and UMOSFETs, with silicon MOSFET and IGBT regimes indicated ([8] reproduced with permission from IEEE).
It is important to emphasize that any comparison of MOSFETs and IGBTs must take into account the switching losses of both devices. The switching loss in IGBTs is substantial, due to the large stored minority charge that must be removed before the device can fully turn off. This loss mechanism is absent in unipolar devices such as MOSFETs and JFETs. Switching loss is proportional to switching frequency, as shown by Equation 7.15, and switching loss can become the dominant power dissipation mechanism at high frequencies. This can force a reduction in maximum current to keep the total power dissipation below 
. The reader is referred to Section 10.3 for a more complete discussion.
Figure 11.50 shows performance metrics for SiC JFETs, SiC BJTs, and GaN HEMTs (high-electron-mobility transistors) [8–13]. SiC JFETs have achieved blocking voltages up to 10 kV with on-state currents similar to SiC MOSFETs [10, 11]. Because they do not rely on a gate oxide, JFETs avoid issues related to oxide reliability, and are particularly suited for high-temperature operation. SiC BJTs have been demonstrated with blocking voltages as high as 21 kV and performance very close to the SiC unipolar limit [9]. Like JFETs, BJTs do not depend on a gate oxide and are therefore capable of operation at temperatures well above 200 °C. A drawback of BJTs is their base drive requirement, which adds a power dissipation term that has been neglected in this figure. However, BJTs with common-emitter current gains as high as 257 have recently been reported [14], easing the demands on base drive circuits.
Figure 11.50 State-of-the-art performance of SiC JFETs, BJTs, and GaN heterojunction field-effect transitors (HFETs), with SiC MOSFETs in the background ([8] reproduced with permission from IEEE).
GaN HEMTs have demonstrated impressive performance in the lower blocking voltage regime. GaN technology is in an earlier stage of development than SiC, but has the potential for significant improvement in the future. The reader is referred to the current literature for details.
Figure 11.51 shows on-state performance for selected SiC IGBTs and thyristors reported to date [15–19]. The on-resistance axis has been removed from this figure, since the plotted points indicate only the maximum current density. Like silicon IGBTs, SiC IGBTs and thyristors have offset voltages, and their maximum on-state current is calculated using Equation 11.45 with an offset voltage of 2.8 V. From the figure we see that SiC IGBTs and thyristors have performance metrics similar to the best SiC BJTs. All three are bipolar devices, capable of operation beyond the SiC unipolar limit. It is expected that as material quality improves and more applications arise at blocking voltages above 10 kV, development of high-voltage SiC BJTs, IGBTs, and thyristors will accelerate.
Figure 11.51 On-state performance of SiC IGBTs and thyristors, with SiC MOSFETs, JFETs, and BJTs in the background ([8] reproduced with permission from IEEE).
References
- [1] Mohan, N., Undeland, T.M. and Robbins, W.P. (2003) Power Electronics, 3rd edn, John Wiley & Sons, Inc., Hoboken, NJ.
- [2] Raghunathan, R., Alok, D. and Baliga, B.J. (1995) High-voltage 4H-SiC Schottky barrier diodes. IEEE Electron. Device Lett., 16 (6), 226–227.
- [3] Rupp, R., Treu, M., Mauder, A. et al. (1999) Performance and reliability issues of SiC-Schottky diodes. Mater. Sci. Forum, 338–342, 1167–1170.
- [4] Singh, R., Cooper, J.A., Melloch, M.R. et al. (2002) SiC power Schottky and PiN diodes. IEEE Trans. Electron. Devices, 49 (4), 665–672.
- [5] Zhao, J.H., Alexandrov, P. and Li, X. (2003) Demonstration of the first 10-kV 4H-SiC Schottky barrier diodes. IEEE Electron. Device Lett., 24 (6), 402–404.
- [6] Nakamura, T., Miyanagi, T., Kamata, I. et al. (2005) A 4.15 kV 9.07 mΩ-cm2 4H-SiC Schottky-barrier diode using Mo contact annealed at high temperature. IEEE Electron. Device Lett., 26 (2), 99–101.
- [7] Hull, B.A., Sumakeris, J.J., O'Loughlin, M.J. et al. (2008) Performance and stability of large-area 4H-SiC 10-kV junction barrier Schottky rectifiers. IEEE Trans. Electron. Devices, 55 (8), 1864–1870.
- [8] Ohashi, H. (2012) Power devices then and now, strategy of Japan, in 24th International Symposium on Power Semiconductor Devices and ICs (ISPSD 2012), IEEE, pp. 9–12.
- [9] Miyake, H., Okuda, T., Niwa, H. et al. (2012) 21-kV SiC BJTs with space-modulated junction termination extension. IEEE Electron. Device Lett., 33 (11), 1598–1600.
- [10] Veliadis, V., Stewart, E.J., Hearne, H. et al. (2010) A 9-kV normally-on vertical-channel SiC JFET for unipolar operation. IEEE Electron. Device Lett., 31 (5), 470–472.
- [11] Zhao, J.H., Alexandrov, P., Zhang, J. and Li, X. (2004) Fabrication and characterization of 11-kV normally off 4H-SiC trenched-and-implanted vertical junction FET. IEEE Electron. Device Lett., 25 (7), 474–476.
- [12] Zhao, J.H., Kiyoski, T., Alexandrov, P. et al. (2003) 1710-V 2.77 mΩ-cm2 4H-SiC trenched and implanted vertical junction field-effect transistors. IEEE Electron. Device Lett., 24 (2), 81–83.
- [13] Zhang, Y., Sheng, K., Su, M. et al. (2007) 1000-V 9.1 mΩ-cm2 normally off 4H-SiC lateral RESURF JFET for power integrated circuit applications. IEEE Electron. Device Lett., 28 (5), 404–407.
- [14] Miyake, H., Kimoto, T. and Suda, J. (2011) 4H-SiC BJTs with record current gains of 257 on (0001) and 335 on (000-1). IEEE Electron. Device Lett., 32 (7), 841–843.
- [15] Ryu, S.-H., Cheng, L., Dhar, S. et al. (2012) Development of 15 kV 4H-SiC IGBTs. Mater. Sci. Forum, 717–720, 1135–1138.
- [16] Sui, Y., Wang, X. and Cooper, J.A. (2007) High voltage self-aligned p-channel DMOS IGBTs in 4H-SiC. IEEE Electron. Device Lett., 28 (8), 728–730.
- [17] Sugawara, Y., Takayama, D., Asano, K. et al. (2004) 12.7 kV ultra high voltage SiC commutated gate turn-off thyristor: SICGT. International Symposium on Power Semiconductor Devices and ICs, Kitakyushu, Japan, pp. 365–368.
- [18] Zhang, Q.J., Agarwal, A., Capell, C. et al. (2012) 12 kV, 1 cm2 SiC GTO thyristors with negative bevel termination. Mater. Sci. Forum, 717-720, 1151–1154.
- [19] Cheng, L., Agarwal, A.K., Capell, C. et al. (2013) 15 kV, large-area (1 cm2), 4H-SiC p-type gate turn-off thyristors. Mater. Sci. Forum, 740-742, 978–981.