Implications of Kirchhoff’s Voltage

1.5.2 Implications of Kirchhoff’s Voltage

and Current Laws

A major challenge of switch circuits is their capacity to Input

“violate” circuit laws. First, consider the simple circuits shown source

Load

in Fig. 1.13. We might try the circuit shown in Fig. 1.13a for 2,1

ac–dc conversion, but there is a problem. According to Kirchhoff ’s Voltage Law (KVL), the “sum of voltage drops around a closed loop is zero.” However, with the switch closed, the sum of voltages around the loop is not zero. In reality,

FIGURE 1.11 H-bridge configuration of a 2 × 2 switch matrix. this is not a valid result. Instead, a very large current will flow

1 Introduction 9

(a)

(b)

V ac Switch

V dc I 1

must remain

Switch

open

must remain

open

FIGURE 1.13 Hypothetical power converters: (a) possible ac–dc converter (b) possible dc–dc converter. (From [2], c Inc.; used by permission.)

and cause a large I ·R drop in the wires. KVL will be satisfied

(a)

(b)

by the wire voltage drop, but a fire or, better yet, fuse action, might result. There is, however, nothing that would prevent an operator from trying to close the switch. KVL, then, implies a crucial restriction: a switch matrix must not attempt to inter- connect unequal voltage sources directly. Notice that a wire, or dead short, can be thought of as a voltage source with V = 0, so KVL is a generalization of avoiding shorts across an individual voltage source.

FIGURE 1.14 A similar constraint holds for Kirchhoff ’s Current Law Short-term KVL and KCL problems in energy storage cir- (KCL) that states that “currents into a node must sum to zero.” cuits: (a) an inductor cannot sustain dc voltage indefinitely; (b) a capacitor

cannot sustain dc current indefinitely.

When current sources are present in a converter, we must avoid any attempts to violate KCL. In Fig. 1.13b, if the current sources are different and if the switch is opened, the sum of the cur- rents into the node will not be zero. In a real circuit, high because the long-term effect is similar to shorting the source. In voltages will build up and cause an arc to create another cur- Fig. 1.14b, the current source will cause the capacitor voltage to rent path. This situation has real potential for damage, and a ramp toward infinity. This causes a “KCL problem”; eventually, fuse will not help. As a result, KCL implies the restriction that an arc will be formed to create an additional current path, just

a switch matrix must not attempt to interconnect unequal cur- as if the current source had been opened. Of course, these con- rent sources directly. An open circuit can be thought of as a nections are not problematic if they are only temporary. How- current source with I = 0, so KCL applies to the problem of ever, it should be evident that an inductor will not support opening an individual current source.

dc voltage, and a capacitor will not support dc current. On In contrast to conventional circuits, in which KVL and KCL average, over an extended time interval, the voltage across an are automatically satisfied, switches do not “know” KVL or inductor must be zero, and the current into a capacitor must KCL. If a designer forgets to check, and accidentally shorts two

be zero.

voltages or breaks a current source connection, some problem or damage will result. KVL and KCL place necessary constraints on the operation of a switch matrix. In the case of voltage

1.5.3 Resolving the Hardware Problem –

sources, switches must not act to create short-circuit paths

Semiconductor Devices

among unlike sources. In the case of KCL, switches must act

A switch is either on or off. When on, an ideal switch will carry to provide a path for currents. These constraints drastically any current in any direction. When off, it will never carry cur-

reduce the number of valid switch-operating conditions in a rent, no matter what voltage is applied. It is entirely lossless and switch matrix, thereby leading to manageable operating design changes from its on-state to its off-state instantaneously. A real problems.

switch can only approximate an ideal switch. The following are When energy storage is included, there are interesting impli- the aspects of real switches that differ from the ideal:

cations of the circuit law restrictions. Figure 1.14 shows two “circuit law problems.” In Fig. 1.14a, the voltage source will

• limits on the amount and direction of on-state current; cause the inductor current to ramp up indefinitely, since

a nonzero on-state voltage drop (such as a diode forward

V = L di/dt. We might consider this to be a “KVL problem”

voltage);

10 P. T. Krein • some levels of leakage current when the device is sup-

to high-power controllable rectifiers, they have been posed to be off;

applied to all power-conversion applications. • limitations on the voltage that can be applied when off;

Some of the features of the most common power semicon- • operating speed. The duration of transition between the ductors are listed in Table 1.1. The table shows a wide variety on-states and off-states is important. of speeds and rating levels. As a rule, faster speeds apply to

The degree to which the properties of an ideal switch must be lower ratings. For each device type, cost tends to increase both met by a real switch depends on the application. For example, for faster devices and for devices with higher power-handling

a diode can easily be used to conduct dc current; the fact that capacity. it conducts only in one direction is often an advantage, not a

Conducting direction and blocking behavior are fundamen- weakness.

tally tied to the device type, and these basic characteristics Many different types of semiconductors have been applied in constrain the choice of device for a given conversion function. power electronics. In general, these fall into three groups:

Consider again a diode. It carries current in only one direc- tion and always blocks current in the other direction. Ideally,

1. Diodes, which are used in rectifiers, dc–dc converters, the diode exhibits no forward voltage drop or off-state leak- and in supporting roles.

age current. Although an ideal diode lacks the many features Transistors, which in general are suitable for control of of an ideal switch, it is an important switching device. Other single-polarity circuits. Several types of transistors are real devices operate with polarity limits on current and voltage applied to power converters. The IGBT type is unique and have corresponding ideal counterparts. It is convenient to to power electronics and has good characteristics for define a special type of switch to represent this behavior: the applications such as inverters.

restricted switch.

Thyristors, which are multijunction semiconductor devices with latching behavior. In general, thyristors

A restricted switch is an ideal switch with can be switched with short pulses and then maintain

D EFINITION

the addition of restrictions on the direction of current their state until current is removed. They act only as

flow and voltage polarity. The ideal diode is one example switches. The characteristics are especially well suited

of a restricted switch.

TABLE 1.1 Examples of semiconductor devices used in power electronics Device type

Characteristics of power devices Diode

Current ratings from under 1 A to more than 5000 A. Voltage ratings from 10 V to 10 kV or more. The fastest power devices switch in less than

10 ns, whereas the slowest require 100 µs or more. The function of a diode applies in rectifiers and dc–dc circuits.

BJT (Bipolar junction transistor) Conducts collector current (in one direction) when sufficient base current is applied. The function applies to dc–dc circuits. Power BJTs have mostly been supplanted by FETs and IGBTs.

FET (Field effect transistor) Conducts drain current when sufficient gate voltage is applied. Power FETs (nearly always enhancement-mode MOSFETs) have a parallel connected reverse diode by virtue of their construction. Ratings from about 0.5 A to about 150 A and 20 V up to 1200 V. Switching times are fast, from 20 ns or less up to 200 ns. The function applies to dc–dc conversion, where the FET is in wide use, and to inverters.

IGBT (Insulated gate bipolar transistor) A special type of transistor that has the function of a BJT with its base driven by an FET. Faster than a BJT of similar ratings, and easy to use. Ratings from 10 A to more than 600 A, with voltages of 600 to 2500 V. The IGBT is popular in inverters from about 1 to 200 kW or more. It is found almost exclusively in power electronics applications.

SCR (Silicon-controlled rectifier) A thyristor that conducts like a diode after a gate pulse is applied. Turns off only when current becomes zero. Prevents current flow until a pulse appears. Ratings from 10 A up to more than 5000 A, and from 200 V up to 6 kV. Switching requires 1 to 200 µs. Widely used for controlled rectifiers. The SCR is found almost exclusively in power electronics applications, and is the most common member of the thyristor family.

GTO (Gate turn-off thyristor) An SCR that can be turned off by sending a negative pulse to its gate terminal. Can substitute for transistors in applications above 200 kW or more. The ratings approach those of SCRs, and the speeds are similar as well.

TRIAC A semiconductor constructed to resemble two SCRs connected in reverse parallel. Ratings from 2 to 50 A and 200 to 800 V. Used in lamp dimmers, home appliances, and hand tools. Not as rugged as many other device types, but very convenient for many ac applications.

IGCT (Integrated gate commutated thyristor) A combination device that includes a high-power thyristor and external electronics to control it. This device is a member of a larger family of combination devices, in which multiple semiconductor chips packaged together perform a single power function. The IGCT provides a high-performance GTO function for power levels above 1 MW or more.

1 Introduction 11 TABLE 1.2 The types of restricted switches

Action

Restricted switch symbol Device symbol Carries current in one direction, blocks in the

other (forward-conducting reverse-blocking)

Carries or blocks current in one direction

BJT

(forward-conducting forward-blocking)

Carries in one direction or blocks in both

GTO

directions (forward-conducting bidirectional-blocking)

Carries in both directions, but blocks only in

FET

one direction (bidirectional-carrying forward-blocking)

Fully bidirectional

Ideal switch

The diode always permits current flow in one direction, while and dc voltage. Regardless of the physical arrangement of the blocking flow in the other direction. It therefore represents a matrix, we would expect bidirectional-conducting forward- forward-conducting reverse-blocking restricted switch and oper- blocking switches to be useful for this conversion. This is ates in one quadrant on a graph of device current versus.

a correct result: modern inverters operating from dc volt- voltage. This function is automatic – the two diode terminals age sources are built with FETs or with IGBTs packaged with provide all the necessary information for switch action. Other reverse-parallel diodes. As new power devices are introduced restricted switches require a third gate terminal to determine to the market, it is straightforward to determine what types their state. Consider the polarity possibilities given in Table 1.2. of converters will use them. Additional functions such as bidirectional-conducting reverse- blocking can be obtained by reverse connection of one of the five types in the table.

1.5.4 Resolving the Software Problem –

The quadrant operation shown in the table indicates

Switching Functions

polarities. For example, the current in a diode will be posi- tive when on, and the voltage will be negative when off. This The physical m × n switch matrix can be associated with a means diode operation is restricted to the single quadrant com- mathematical m × n switch state matrix. Each element of this prising the upper vertical (current) axis and the left horizontal matrix, called a switching function, shows whether the corre- (voltage) axis. Other combinations appear in the table. Sym- sponding physical device is on or off. bols for restricted switches can be built up by interpreting the

A switching function, q(t), has a value of diode’s triangle as the current-carrying direction and the bar

D EFINITION

as the blocking direction. Five types of symbols can be drawn

1 when the corresponding physical switch is on and 0 as shown in Table 1.2. These symbols are used infrequently,

when it is off. Switching functions are discrete-valued but are useful for showing the polarity behavior of switching

functions of time, and control of switching devices can devices. A circuit drawn with restricted switches represents an

be represented with them.

idealized power converter. Figure 1.15 shows a typical switching function. It is periodic, Restricted switch concepts guide the selection of devices. with period T, representing the most likely repetitive switch For example, consider an inverter intended to deliver ac load action in a power converter. For convenience, it is drawn on current from a dc voltage source. A switch matrix built to

a relative time scale that begins at 0 and draws out the square perform this function must be able to manipulate ac current wave period by period. The actual timing is arbitrary, so the

12 P. T. Krein Absolute time reference

Fig. 1.9, the loop and node equations change depending on 1

which switch is acting at a given moment. The two possible circuit configurations each have distinct equations. Switching functions allow them to be combined. By assigning switch-

ing functions q 1 (t) and q 2 (t) to the left- and right-switching

0 devices, respectively, we obtain

0 t 0 DT T T + DT 2T

Relative time period T di L

A generic switching function with period T, duty ratio D,

dt

and time reference t 0 .

dv C v C

=0 , left switch on (1.8)

dt

center of the first pulse is defined as a specified time t 0 in the

di L

figure. In many converters, the switching function is generated

as an actual control voltage signal that might drive the gate

of either a MOSFET or some other semiconductor switching dv C v C

device.

=i L , right switch on (1.9)

dt

The timing of switch action is the only alternative for control of a power converter. Because switch action can be represented Because the switches alternate, and the switching functions with a discrete-valued switching function, timing can be repre- must be 0 or 1, these sets of equations can be combined to give sented within the switching function framework. On the basis of Fig. 1.15, a generic switching function can be characterized

di L

dv C v C

V v , C i (1.10) completely with three parameters:

in −L

2 C 2 dt L =q dt + R =q

1. The duty ratio, D, is the fraction of time during which The combined expressions are simpler and easier to analyze the switch is on. For control purposes, the on-time than the original equations. interval or pulse width can be adjusted to achieve

For control purposes, the average of equations such as (1.10)

a desired result. We can term this adjustment pro- often proceeds with the replacement of switching functions q cess pulse-width modulation (PWM), perhaps the most with duty ratios d. The discrete time action of a switching func- important process for implementing control in power tion thus will be represented by an average duty ratio param- converters.

eter. Switching functions, the advantages gained by averaging,

2. The frequency f switch = 1/T (with radian frequency ω = and control approaches such as PWM are discussed at length 2π f switch ) is most often constant, although not in all in several chapters in this handbook. applications. For control purposes, frequency can be adjusted. This strategy is sometimes used in low-power dc–dc converters to manage wide load ranges. In other

1.5.5 Resolving the Interface Problem – Lossless

converters, frequency control is unusual because the

Filter Design

operating frequency is often dictated by the appli- Lossless filters for power electronic applications are sometimes cation.

called smoothing filters [9]. In applications in which dc outputs

The time delay t 0 or phase ϕ 0 = ωt 0 . Rectifiers often use

are of interest, such filters are commonly implemented as sim- phase control to provide a range of adjustment. Phase- ple low-pass LC structures. The analysis is facilitated because in shifted bridge circuits are common for high-power most cases the residual output waveform, termed ripple, has a dc–dc conversion. A few specialized ac–ac converter predictable shape. Filter design for rectifiers or dc–dc convert- applications use phase modulation. ers is a question of choosing storage elements large enough to

With just three parameters to vary, there are relatively few keep ripple low, but not so large that the whole circuit becomes possible ways to control any power electronic circuit. dc–dc unwieldy or expensive. converters and inverters usually rely on duty ratio adjustment

Filter design is more challenging when ac outputs are (PWM) to alter their behavior. Phase control is common in desired. In some cases, this is again an issue of low-pass filter controlled rectifier applications.

design. However, in many applications, low-pass filters are not Switching functions are powerful tools for the general rep- adequate to meet low noise requirements. In these situations, resentation of converter action [7]. The most widely used active filters can be used. In power electronics, the term active control approaches derive from averages of switching functions filter refers to lossless switching converters that actively inject [2, 8]. Their utility comes from their application in writing cir- or remove energy moment-by-moment to compensate for dis- cuit equations. For example, in the boost converter shown in tortion. The circuits (discussed in Chapter 41 – Active Filters

1 Introduction 13 in this handbook) are not related to the linear active filter

1.7 Summary

op-amp circuits used in analog signal processing. In ac applica- tions, there is a continuing opportunity for innovation in filter Power electronics is the study of electronic circuits for the con- design.

trol and conversion of electrical energy. The technology is a critical part of our energy infrastructure and is central for a wide range of uses of electricity. For power electronics design, we consider only those circuits and devices that, in princi-

1.6 Sample Applications

ple, introduce no loss and achieve near-perfect reliability. The two key characteristics of high efficiency and high reliability

Although power electronics is becoming universal for elec- are implemented with switching circuits, supplemented with tronic systems, a few emerging applications have generated energy storage. Switching circuits can be organized as switch wide interest. Some are discussed briefly here to introduce the matrices. This facilitates their analysis and design. breadth of activity in the field.

In a power electronic system, the three primary challenges

A hybrid electric vehicle typically has two major power elec- are the hardware problem of implementing a switch matrix, tronic systems and dozens or even hundreds of smaller systems the software problem of deciding how to operate that matrix, [10]. The two large units are the inverter system, which controls and the interface problem of removing unwanted distortion the electric drive motor, and a rectifier system, which manages and providing the user with the desired clean power source. battery charging. Smaller systems include motor controllers The hardware is implemented with a few special types of power in electric power steering units, lighting electronics for high- semiconductors. These include several types of transistors, intensity headlamps, controllers for the wide range of small especially MOSFETs and IGBTs, and several types of thyristors, motors that actuate everything from windshield wipers to DVD especially SCRs and GTOs. The software problem can be rep- players, and power supplies for the host of microcontrollers resented in terms of switching functions. The frequency, duty embedded in a modern car. As plug-in hybrid and electric cars ratio, and phase of switching functions are available for opera- continue to develop, the efficiency and sophistication of battery tional purposes. The interface problem is addressed by means chargers and drive motor controllers will increase.

of lossless filter circuits. Most often, these are lossless LC passive In a hybrid car, a typical inverter rating is about 50 kW. Fully filters to smooth out ripple or reduce harmonics. Active filter electric automobiles have inverter ratings to about 200 kW. circuits also have been applied to make dynamic corrections in Hybrid and electric drive power levels increase from there to power conversion waveforms. include multimegawatt inverters that power high-speed trains.

Improvements in devices and advances in control concepts At the other power extreme, designers have been working on have led to steady improvements in power electronic circuits small power electronic systems that extract energy from various and systems. This is driving tremendous expansion of applica- ambient sources for local purposes [11]. These energy harvest- tions. Personal computers, for example, would be unwieldy and ing applications will be an emerging growth area. An example inefficient without power electronic dc supplies. Mobile devices application is a corrosion sensor, built into a highway bridge, and laptop computers would be impractical. High-efficiency that gathers energy from vibration. The vibration energy is lighting, motor controls, and a wide range of industrial con- converted and stored, then used for intermittent communica- trols depend on power electronics. Strong growth is occurring tion with a central monitoring computer. Typical power levels in automotive applications, in lighting, in dc power supplies for can be less than 0.001 W.

portable devices, in high-end converters for advanced micro- Solar, wind, and other alternative energy resources are tightly processors, and in alternative and renewable energy. During the linked to power electronics. A typical solar panel produces next generation, we will reach a time when almost all electrical about 30-V dc, with tolerances from 20 to 40 V or more. This energy is processed through power electronics somewhere in randomly changing dc source must be converted to clean, the path from generation to end use. tightly regulated ac voltage that synchronizes to the electric- ity grid and delivers energy. Solar inverter efficiency, reliability, and cost are major factors in successful energy innovation.