Fundamentals of Power Electronics

The field of power electronics is concerned with the processing of electrical power using electronic devices [l–7]. The key element is the switching converter, illustrated in Fig. 1.1. In general, a switching converter contains power input and control input ports, and a power output port. The raw input power is processed as specified by the control input, yielding the conditioned output power. One of several basic functions can be performed [2]. In a dc—dc converter, the dc input voltage is converted to a dc output voltage having a larger or smaller magnitude, possibly with opposite polarity or with isolation of the input and output ground references. In an ac—dc rectifier, an ac input voltage is rectified, producing a dc output voltage. The dc output voltage and/or ac input current waveform may be controlled. The inverse process, dc-ac inversion,involves transforming a dc input voltage into an ac output voltage of controllable magnitude and frequency. Ac—ac cycloconversion involves converting an ac input voltage to a given ac output voltage of controllable magnitude and frequency.

In the previous chapter, the buck converter was introduced as a means of reducing the dc voltage, using only nondissipative switches, inductors, and capacitors. The switch produces a rectangular waveform v _s (t) as illustrated in Fig. 2.1. The voltage v _s (t) is equal to the dc input voltage V _g when the switch is in position 1, and is equal to zero when the switch is in position 2. In practice, the switch is realized using power semiconductor devices, such as transistors and diodes, which are controlled to turn on and off as required to perform the function of the ideal switch. The switching frequency f _s equal to the inverse of the switching period T _s generally lies in the range of 1 kHz to 1 MHz, depending on the switching speed of the semiconductor devices. The duty ratio D is the fraction of time that the switch spends in position 1, and is a number between zero and one. The complement of the duty ratio, D′, is defined as (1 − D).

Let us now consider the basic functions performed by a switching converter, and attempt to represent these functions by a simple equivalent circuit. The designer of a converter power stage must calculate the network voltages and currents, and specify the power components accordingly. Losses and efficiency are of prime importance. The use of equivalent circuits is a physical and intuitive approach which allows the well-known techniques of circuit analysis to be employed. As noted in the previous chapter, it is desirable to ignore the small but complicated switching ripple, and model only the important dc components of the waveforms.

We have already analyzed the operation of a number of different types of converters: buck, boost, buck—boost, Cuk, voltage-source inverter, etc. With these converters, a number of different functions can be performed: step-down of voltage, step-up, inversion of polarity, and conversion of dc to ac or vice versa.

Converter systems invariably require feedback. For example, in a typical dc-dc converter application, the output voltage v(t) must be kept constant, regardless of changes in the input voltage v _g (t) or in the effective load resistance R. This is accomplished by building a circuit that varies the converter control input [i.e., the duty cycle d(t)] in such a way that the output voltage v(t) is regulated to be equal to a desired reference value v _ref . In inverter systems, a feedback loop causes the output voltage to follow a sinusoidal reference voltage. In modern low-harmonic rectifier systems, a control system causes the converter input current to be proportional to the input voltage, such that the input port presents a resistive load to the ac source. So feedback is commonly employed.

In the design of a converter control system, it is necessary to work with frequency-dependent transfer functions and impedances, and to construct Bode diagrams. The objective of this chapter is to utilize the equivalent circuit models of the previous chapter, to construct the small-signal transfer functions and terminal impedances of switching converters. Construction of magnitude and phase Bode diagrams is reviewed, including two types of features that often appear in converter transfer functions: resonances and right half-plane zeroes. As an example, Bode diagrams of the small-signal transfer functions of the buck-boost converter are constructed, based on the model derived in Section 7.2. The transfer functions of the basic buck, boost, and buck-boost converters operating in continuous conduction mode are listed. Experimental measurement of transfer functions and impedances is discussed briefly in Section 8.4.

In all switching converters, the output voltage v (t) is a function of the input line voltage v, (t), the duty cycle d (t), and the load current i _load (t), as well as the converter circuit element values. In a dc-dc converter application, it is desired to obtain a constant output voltage v (t) = V, in spite of disturbances in v _g (t) and i _load (t), and in spite of variations in the converter circuit element values. The sources of these disturbances and variations are many, and a typical situation is illustrated in Fig. 9.1. The input voltage v _g (t) of an off-line power supply may typically contain periodic variations at the second harmonic of the ac power system frequency (100 Hz or 120 Hz), produced by a rectifier circuit. The magnitude of v _g (t) may also vary when neighboring power system loads are switched on or off. The load current i _load (t) may contain variations of significant amplitude, and a typical power supply specification is that the output voltage must remain within a specified range (for example, 5 V ± 0.1 V) when the load current takes a step change from, for example, full rated load current to 50% of the rated current, and vice versa. The values of the circuit elements are constructed to a certain tolerance, and so in high-volume manufacturing of a converter, converters are constructed whose output voltages lie in some distribution. It is desired that essentially all of this distribution fall within the specified range; however, this is not practical to achieve without the use of negative feedback. Similar considerations apply to inverter applications, except that the output voltage is ac.

So far, we have derived equivalent circuit models for dc-dc pulse-width modulation (PWM) converters operating in the continuous conduction mode. As illustrated in Fig. 10.1, the basic dc conversion property is modeled by an effective dc transformer, having a turns ratio equal to the conversion ratio M (D). This model predicts that the converter has a voltage-source output characteristic, such that the output voltage is essentially independent of the load current or load resistance R. We have also seen how to refine this model, to predict losses and efficiency, converter dynamics, and small-signal ac transfer functions. We found that the transfer functions of the buck converter contain two low-frequency poles, owing to the converter filter inductor and capacitor. The control-to-output transfer functions of the boost and buck-boost converters additionally contain a right half-plane zero. Finally, we have seen how to utilize these results in the design of converter control systems.

So far, we have discussed duty ratio control of PWM converters, in which the converter output is controlled by direct choice of the duty ratio d(t). We have therefore developed expressions and small-signal transfer functions that relate the converter waveforms and output voltage to the duty ratio.

Magnetics are an integral part of every switching converter. Often, the design of the magnetic devices cannot be isolated from the converter design. The power electronics engineer must not only model and design the converter, but must model and design the magnetics as well. Modeling and design of magnetics for switching converters is the topic of Part III of this book.

A variety of factors constrain the design of a magnetic device. The peak flux density must not saturate the core. The peak ac flux density should also be sufficiently small, such that core losses are acceptably low. The wire size should be sufficiently small, to fit the required number of turns in the core window. Subject to this constraint, the wire cross-sectional area should be as large as possible, to minimize the winding dc resistance and copper loss. But if the wire is too thick, then unacceptable copper losses occur owing to the proximity effect. An air gap is needed when the device stores significant energy. But an air gap is undesirable in transformer applications. It should be apparent that, for a given magnetic device, some of these constraints are active while others are not significant.

Let us now investigate a more general transformer design problem. It is desired to design a k-winding transformer as illustrated in Fig. 14.1. We will assume that the magnetizing inductance is not intended to store energy. Both copper loss P _cu and core loss P _fe are modeled. We will select the maximum flux density B _max to minimize the total power loss P _tot = P _fe + P _cu . In addition, it is necessary to optimally allocate the core window area among the various windings such that the total power loss is minimized.

Rectification used to be a much simpler topic. A textbook could cover the topic simply by discussing the various circuits, such as the peak-detection and inductor-input rectifiers, the phase-controlled bridge, polyphase transformer connections, and perhaps multiplier circuits. But recently, rectifiers have become much more sophisticated, and are now systems rather than mere circuits. They often include pulse-width modulated converters such as the boost converter, with control systems that regulate the ac input current waveform. So modern rectifier technology now incorporates many of the dc—dc converter fundamentals.

Conventional diode peak-detection rectifiers are inexpensive, reliable, and in widespread use. Their shortcomings are the high harmonic content of their ac line currents, and their low power factors. In this chapter, the basic operation and ac line current waveforms of several of the most common single-phase and three-phase diode rectifiers are summarized. Also introduced are phase-controlled three-phase rectifiers and inverters, and passive harmonic mitigation techniques. Several of the many references in this area are listed in the references [1–15].

To obtain low ac line current THD, the passive techniques described in the previous chapter rely on low-frequency transformers and/or reactive elements. The large size and weight of these elements are objectionable in many applications. This chapter covers active techniques that employ converters having switching frequencies much greater than the ac line frequency. The reactive elements and transformers of these converters are small, because their sizes depend on the converter switching frequency rather than the ac line frequency.

The techniques developed in earlier chapters for modeling and analysis of dc—dc converters are extended in this chapter to treat the analysis, modeling, and control of low-harmonic rectifiers. The CCM models of Chapter 3 are used in Section 18.1 to compute the average losses and efficiency of CCM PWM converters operating as rectifiers. The results yield insight that is useful in power stage design. Several converter control schemes are now in use. Peak current programming, average current control, and input voltage feedforward are all implemented in the available control ICs. Nonlinear carrier control is a relatively new scheme having a simple implementation. These schemes are discussed in Section 18.2. Modeling of the rectifier control system is covered in Section 18.3.

Part V of this text deals with a class of converters whose operation differs significantly from the PWM converters covered in Parts I to IV. Resonant power converters [1–36] contain resonant L—C networks whose voltage and current waveforms vary sinusoidally during one or more subintervals of each switching period. These sinusoidal variations are large in magnitude, and hence the small ripple approximation introduced in Chapter 2 does not apply.

Resonant switch converters are a broad class of converters in which the PWM switch network of a conventional buck, boost, or other converter is replaced with a switch cell containing resonant elements. These resonant elements are positioned such that one or more of the switching loss mechanisms is eliminated. The resulting hybrid converter combines the properties of the resonant switch network and the parent PWM converter.