Power Electronics: Converters and Regulators

© Springer International Publishing Switzerland 2015

Branko L. Dokić and Branko BlanušaPower Electronics10.1007/978-3-319-09402-1_1

1. Introduction

Branko L. Dokić¹   and Branko Blanuša¹  

(1)

Faculty of Electrical Engineering, University of Banja Luka, Banja Luka, Bosnia and Herzegovina

Branko L. Dokić (Corresponding author)

Email: [email protected]

Branko Blanuša

Email: [email protected]

Power electronics in a broader sense implies the part of electronics that is used in electric power. This is the equipment utilized in systems for control and regulation of electric power supplies and in systems for the regulation of electric drives. Power electronics includes various types of electric power converters, such as converters of AC to DC current, DC to AC, DC to DC, converters of different types of energy (thermal, nuclear, and light) into electric energy, etc. Since most of the equipment based on power electronics contains converters of some type, very often the concept of power electronics is understood as converter electronics.

In essence, a power electronics apparatus consists of a power part and a control part. The power component, serving for the transfer of energy from the source to the load, consists of power electronic switches, electric chokes, transformers, capacitors, fuses, and sometimes resistors. A combination of these elements is used to make different converter circuits adjusted to the mode of the primary supply and the character of the load. Energy losses within a converter should be as small as possible. Consequently, the semiconductor elements of the converter are mainly operated in the pulse (switching) mode. They may be either controllable (transistors, thyristors ) or noncontrollable (diodes). The control or information block controls the regulating (mostly switching) elements of the converter. The control, or regulation, is accomplished on the basis of the information the control block has collected from the power part of the apparatus. Mostly the information concerns the output voltage, load current or current/voltage of a critical element of the converter (e.g. transistor). The control block can functionally be a very complex electronic assembly consisting of either analogue or digital elementary assemblies.

1.1 Types of Signals

There are various types of signals (voltage/current) used in the transferring of energy from the primary source to the load and in the control of this transfer (Fig. 1.1).

Fig. 1.1

The most frequent voltage and current waveforms in power electronics circuits

Input and output voltages or currents are mainly either harmonic functions of time (Fig. 1.1a) or time-independent. The time-independent signals (Fig. 1.1b) are called direct current signals as they act in only one direction. The most frequent forms of signals inside power electronics equipment are rectangular (Fig. 1.1c). These signals are obtained at the outputs of the DC voltage supplied switching circuits as a consequence of the operation of the ON/OFF switch. A rectangular excitation of a circuit within the equipment results in responses that may be exponential (Fig. 1.1d, e), triangular (Fig. 1.1f), sawtooth (Fig. 1.1g) or harmonic functions of time. They are mostly periodic functions of time. Their values and directions are repeated after a precisely determined time interval T which is called the cycle, so that:

(1.1)

On the basis of the Fourier analysis, arbitrary periodic functions can be expanded in a series of harmonic functions with different amplitudes and frequencies. A Fourier series of any periodic function can be represented in the form of a sum of a DC component and harmonic cosine and sine functions, i.e.

(1.2)

where a 0, a n and b n are the Fourier coefficients determined by:

(1.3)

(1.4)

(1.5)

The coefficient F 0 = a 0 is the average value of a complex-periodic function, or its DC component. By using the basic trigonometric relations, the Fourier series (1.2) can be expressed in terms of cosine only or sine only, namely

(1.6)

where

(1.7)

i.e.

(1.8)

where

(1.9)

The coefficient C 1 is the amplitude of the first or the basic harmonic whose circular frequency ω = 2π/T is equal to the frequency of the complex-periodic function. The higher frequency terms (2ω, 3ω, 4ω, …) are called higher harmonics. In Fig. 1.2a symmetric rectangular signal (dash-dot line) is represented by the sum of only the first three members of the Fourier series (full line). This rectangular signal contains only odd harmonics. Its Fourier series is:

(1.10)

where F is the amplitude of the basic harmonic. With a higher number of harmonics the sum would come closer to the rectangular function, while the infinite sum would produce a complete rectangular form of the signal.

Fig. 1.2

A symmetric rectangular signal (dash-dot line) and its Fourier equivalent (full line) consisting of only the first three terms of the Fourier series

1.2 Root-Mean-Square and Average Values of Periodic Signals

The root-mean-square (RMS) value of a variable periodic current is equal to the value of a DC current which would develop the same amount of heat during the same time interval within the same resistor, i.e. which does the same amount of work. The work of the periodic current through a resistor R over a period T is determined by:

(1.11)

whereas the work of the DC current equal to the RMS value of the variable current in the same resistor over the same period T is

(1.12)

By equating these two works, i.e. W 1 = W 2, it follows that the RMS value of a periodic current is

(1.13)

Similarly, the RMS value of a periodic voltage is obtained as:

(1.14)

For example, for a harmonic voltage v(t) = V M sin(ωt) the RMS value is:

(1.15)

and the RMS value of a harmonic current of the form i(t) = I M sin(ωt) is:

(1.16)

The RMS value denotes the real influence of a harmonic current or voltage. For this reason it is mostly used without the index rms and is shortly denoted by I or V. For example, V = 220 V is the rms value of the mains voltage. Its amplitude is V M  = √2 × 220 = 310 V.

For a periodic function of a rectangular form (Fig. 1.3), determined by (1.17),

(1.17)

the rms value is

(1.18)

where D = T I /T is the duty cycle of the rectangular pulse.

Fig. 1.3

A current of a rectangular form of duty cycle 0 < D < 1, its RMS and average vales

The average value of a periodic signal within one period is defined as:

(1.19)

For a current of a rectangular form, according to Fig. 1.3, it is

(1.20)

Practically, the average value represents the area between the pulse and the time axis over a single period, divided by that period. The average value of a harmonic signal of the form f(t) = F M sin(ωt) is zero since it consists of two equal areas with opposite signs (positive and negative half-periods). In some of the circuits of power electronics, such as rectifiers, the use is made of the rectified current (Fig. 1.4), where all the parts are positive, while the original form of the wave is retained. The cycle of such a signal is T/2, and the mean value is

(1.21)

Fig. 1.4

The rectified harmonic current and its mean value

For complex-periodic currents the use is made of the form factor, as the measure of the discrepancy from the harmonic form, defined as

(1.22)

The form factor of a rectangular current according to Fig. 1.3 is k = I M √D/(I M D) = 1/√D whereas for a rectified harmonic current it is k = (I M /√2)/(2I M /π) = π/(2√2) = 1.11.

As a measure of the discrepancy of a periodic signal from the harmonic form of a current/voltage signal the use is often made of either the distortion factor

(1.23)

or of the total harmonic distortion

(1.24)

where I 1rms is the RMS value of the first harmonic and

(1.25)

is the total rms value of a complex-periodic current. In (1.25) I 0 is the DC component, and I n is the amplitude of the n-th harmonic. If the DC component is zero, the total harmonic distortion is

(1.26)

Example 1.1

Determine the effective (rms) value of

for:

(a)

(b)

When the sinusoids are of different frequencies, and the terms are orthogonal, rms value is:

(a)

First, we combine sinusoids using phasor addition:

The voltage function is then expressed as:

The rms value of voltage v is:

1.3 Power of Periodic Currents

The product of the instantaneous values of a periodic voltage across a load and the current through the load is the instantaneous power:

(1.27)

Since the instantaneous values of the voltage or current could have different signs, the instantaneous power can in general be positive or negative. The power is positive if the energy is transferred from the source to the load and negative if the energy is transferred from the load to the source. A typical example of a load involving positive and negative instantaneous power is a coil and a capacitor driven by a harmonic signal. If, for example a coil of inductance L is connected to a voltage V(t) = V M sin(ωt), the current through the coil will be shifted by −π/2 with reference to the voltage and the instantaneous power will be

The frequency of the instantaneous power will be double the voltage frequency (Fig. 1.5). The shaded area between the curve p(t) and the time axis (Fig. 1.5) represents this work. During the first and the third quarter of the cycle this work is positive, i.e., the work of the source is converted to the energy of the magnetic field of the coil. During the other two quarters of the cycle (the second and the fourth) this work is negative, meaning that the energy of the magnetic field is returned back to the source.

Fig. 1.5

Instantaneous power of a coil driven by a harmonic signal

During the intervals of negative instantaneous power, the coil behaves like a source and the source like a load. Energy is thus being exchanged between the source and the coil. Consequently, the total work of the source is zero and the average power is also zero.

The same conclusions may be drawn if a capacitor is driven by a harmonic signal. In two quarters of the cycle, the capacitor accumulates the electrostatic energy from the source and during the other two quarters this energy is returned back to the source. Consequently, here too the average power is equal to zero.

The average or active power is the one that does the work. For periodic currents it is defined by the time interval equal to one cycle:

(1.28)

It can be shown that in the case of a capacitor the average power from the source is zero. If a capacitor is driven by a rectangular signal

(1.29)

where

(1.30)

is the average current through the capacitor, then the voltage across the capacitor is

(1.31)

Since it has been assumed that the voltage across the capacitor (source voltage) was periodic, i.e. V C (t 0 + T) = V C (t 0) it follows that:

(1.32)

By comparing (1.32) and (1.30), one comes to the conclusion that the average current through the capacitor is zero, thus the average power is also zero. It is shown in the same way that the average value of the voltage across a coil driven by a periodic rectangular current is also zero.

It can thus be concluded that either a coil or a capacitor dissipate no power if driven by a periodic signal. For this reason, they are called nondissipative elements. Since minimum dissipation of power is one of the basic requirements in the design of various efficient converters, coils and capacitors are the basic elements of these circuits together with the switching circuits generating periodic voltages and currents.

Example 1.2

A coil of inductance L = 1 mH and a capacitor of capacitance 1 μF connect blocks B1 and B2 (Fig. 1.6a) and B3 and B4 (Fig. 1.6b), respectively. The current through the coil and the voltage across the capacitor are linear periodic functions determined by

(1)

(2)

Fig. 1.6

Blocks B1 and B2 connected over coil (a) and capacitor (b)

Draw the variations of the voltage across the coil and the current through the capacitor and determine their average values.

The voltage across the coils is

The current and the voltage of the coil are drawn in Fig. 1.7.

Fig. 1.7

Waveforms of the current and voltage the coil for the circuit shown in Fig. 1.6a

The areas above and below the time axis within one cycle are mutually equal but of the opposite signs.

Namely

The average value of the voltage across the coil is

The current through the capacitor is:

The voltage and the current of the capacitor are drawn in Fig. 1.8.

Fig. 1.8

Waveforms of the voltage and current the capacitor for the circuit shown in Fig. 1.6b

The areas below and above the time axis are

The average current through the capacitor is

In general, however, when the load is an impedance Z = |Z|ejφ, there will be a phase shift φ between the current and the voltage. If V = V M cos(ωt), then i = I M cos(ωt − φ) and the power active power is

(1.33)

i.e. since V M  = √2V rms and I M  = √2I rms,

(1.34)

Thus, the active power is the product of the rms values of the voltage and the current and the cosine of the angle between the load voltage and the current. The power is maximum when the load voltage and the current are in phase (φ = 0), which is the case of a purely resistive load. In a resistor the electric energy is converted to thermal energy. If φ = ±π/2, as in the case of a coil or capacitor, cos(φ) = 0, and the active power in these elements is zero.

The phasor diagram of the voltage and a current which is phase shifted by φ is shown in Fig. 1.9. Bearing in mind (1.34), the work is performed only by voltage component V cosφ which is in phase with the current, so V cosφ is called the voltage component for active power. In addition, there is a passive component V sinφ, which is orthogonal to the current vector. This component does not perform any work, i.e., it does not transform the electrical work of the source, so the corresponding power is called the reactive power and it is equal to

(1.35)

Fig. 1.9

The components of a voltage phasor for active and reactive powers (a) and power triangle (b)

The reactive power is understood as the energy alternatively exchanged between the source and the load. The vector sum of the active and reactive powers

(1.36)

is the apparent power. Its modulus is

(1.37)

Thus, the apparent power is the product of the rms values of the load voltage and the current.

The ratio of the active and apparent powers is called the power factor:

(1.38)

Therefore, the power factor of harmonic currents and voltages is cosφ. If the current or the voltage is a complex-periodic function, then (1.38) should be multiplied by the distortion factor (1.23), i.e.

(1.39)

Example 1.3

A nonsinusoidal voltage is

. This voltage is connected to the load which is a serial connection of a 10 Ω resistor and a 10 mH inductance.

(a)

Determine the power absorbed by the load, and

(b)

derive an expression for the load current.

(a)

The power absorbed by the load can be determined by the next equation:

$$ P = I_{\text{rms}}^{2} R. $$ The DC current term is: $$ I_{0} = \frac{{V_{0} }}{R} = 0.5\,\text{A} . $$

The amplitudes of the ac current terms are

The rms value of the load current is

The power absorbed by the load is

(b)

The phase angles of the ac current terms are

The load current can be expressed as

Example 1.4

The waveforms of voltage and current at a single phase load are recorded and presented in the analytical form:

Determine:

(a)

the power absorbed by the load, and

(b)

the power factor.

(a) The power absorbed by the load is determined by computing the absorbed power at each frequency

(b) The power factor is calculated by Eq. (1.38)

The rms values of the load current and voltage are:

The power factor is

1.4 Switching Elements

Switching elements are the constituent parts of the switching circuits (Fig. 1.10a). The basic switching circuit consists of a switch, a load, a power supply, and a control circuit. The control signal V cont governs the state of the switch. The ideal switch should behave as an open circuit (infinite resistance) when OFF and as a short circuit (zero resistance) when ON. The static characteristic of the switch is nonlinear (Fig. 1.10b). In the OFF state, it coincides with the abscissa and in the ON state it coincides with the ordinate. Thus, in the ON state the voltage across the ideal switch is zero and in the OFF state the current through the switch is zero. Consequently, the power of dissipation of the switch is zero in these states, p p  = V p I p  = 0. These states are called the static states. The ideal switch is instantly ON or OFF, meaning that the transition times from one state to the other are zero.

Fig. 1.10

Basic circuit (a), static characteristic of an ideal switch (b), and the current I p, voltage V p, and power dissipation p p of an ideal switch (c) for a DC power supply (V DC) and resistive load R L

No electronic switch, however, behaves ideally. A real switching element is characterized by:

finite resistance when OFF,

nonzero resistance when ON,

transition times from ON to OFF state and vice versa greater than zero, and

dissipation of power in the switch.

The static and dynamic characteristics of a real switch are shown in Fig. 1.11. In most cases, the voltage in the on state and the current in the off state are negligible. Thus, the power of dissipation of a real switch in the static state is also negligible. In the transient condition, while changing the state of the switch, both current and voltage are present (Fig. 1.11b) and the instantaneous value of dissipation may be significant.

Fig. 1.11

Characteristics of a real switch: static (a) and dynamic (b)

The transition times from one static state to the other are dependent on the frequency characteristics of the switching element, the character of the load, and the control circuit. They do not depend on the switching cycle T. Therefore, the average power dissipated by a switching element will grow with the decrease of T. Dynamic power dissipation at high frequencies may be considerable. For this reason, the maximum frequency of a switching circuit is limited not only by the turn-ON/turn-OFF times but also by the permitted power dissipation of the switch. This is particularly true for power switches and it is this type of switch that is predominantly used in power electronics.

The power semiconductor elements like diodes, bipolar or MOS transistors, thyristors , and BiMOS transistors are used as the switching elements . A common requirement for all of these elements is that the control of signals carrying considerable power has to be done by as short turn-ON/turn-OFF times as possible.

Power diodes can be classified into three groups: general purpose, very fast, and Schottky. The operating voltages and currents of general purpose diodes may range up to 3,000 V and 3,500 A, and those of very fast diodes up to 3,000 V and 1,000 A. The reverse recovery times are in the range from several hundreds nanoseconds to several microseconds. Schottky diodes have lower forward voltages and very short recovery times (below 10 ns). However, the reverse saturation current grows with the power of the diode so the characteristics are limited to 100 V and 300 A. Diodes are two terminal devices. This limits their applications as switches as the control load circuits are common.

Power bipolar transistors (PBT) are characterized by a very small collector-emitter on (saturation) resistance, from several mΩ to several tens of mΩ. Owing to this, the collector-emitter on (saturation) voltage is within the limits of 0.5–1.5 V even at very high collector currents. The maximum voltages and currents range up to 1,200 V and 400 A. The maximum frequency of the pulse DC/DC converters using PBT as switches runs up to several tens (20–30) of kilohertz. PBTs as switches are mainly used in the common-emitter connection. The control is implemented via a base circuit. If a turned-ON transistor is to reach the saturation region, in addition to the forward bias of the base-emitter junction, a sufficiently large base current is required so that the base-collector junction is also forward biased. Consequently, the control circuit requires a relatively large power.

The power MOS transistors have recently been finding an increased use in the pulse converters. They are faster than PBTs and the maximum frequency of the converters based on power MOS devices ranges up to 100–200 kHz. The rated voltages and currents are smaller than those of PBT and are within the range of 1,000 V and 50 A, respectively. The input impedance of MOS transistors is high (of the order of 10⁹ Ω), thus for their control it is sufficient to provide the corresponding gate-source voltage. Since the gate current is practically zero, there is no dissipation in the control circuit. Therefore, an MOS transistor is a voltage controlled switch compared to a PBT which is a current controlled switch.

The basic weakness of power MOSFETs is a relatively large on resistance (from several hundreds mΩ up to several Ω). This was the reason for the development of several types of BiMOS transistors which unite good properties of both bipolar transistors (small on resistance) and MOS (negligible driving current). One of these types is the insulated gate bipolar transistor (IGBT). Its input characteristics are like those of an MOS transistor and the output characteristics are like those of a PBT. The maximum voltages and currents range up to 1,200 V and 400 A and the maximum frequencies up to several tens of kHz (like PBT). The frequency characteristics of the static induction transistors (similar to JFET) have been improved. The maximum ratings of this type of transistor are 1,200 V, 300 A, and 100 kHz.

The characteristics and symbols of nonregenerative semiconductor switches (diodes, BT, and MOSFET) are shown in Table 1.1.

Table 1.1

Characteristics and symbols of nonregenerative switches

The thyristor is a representative of the regenerative switches (the switches where the change of state is supported by a positive feedback). In addition to the regenerative process, the essential difference compared to the PBT and MOSFET is in that the thyristors are turned on by feeding short pulses (several tens of milliseconds) to the gate. After switch-ON, a thyristor remains on even if the driving signal is removed from the gate. For the PBT and MOSFET devices the driving signal must be present throughout the on state. Thyristors are very powerful elements. The maximum voltages and currents range up to 10,000 V or A, respectively. Today a whole family of thyristors is commercially available. Each member of this family is specific regarding both its characteristics and its applications. The V–I characteristic, the symbol, and the equivalent circuit are shown in Table 1.2. A standard thyristor (SCR) is turned on by a positive pulse at the gate, but it cannot be turned off by a gate signal. The gate turn-off (GTO) and self-turn-off (SITH) are the self-turn-off thyristors. They are turned on by positive and turned off by negative pulses at the gate. The maximum voltages and currents of GTO thyristors are respectively 4,000 V and 3,000 A, and of SITH thyristors 1,200 V and 300 A. The maximum frequency of SITH is high and ranges up to several hundred kHz. Another thyristor type can be turned off at the gate. This is the MOS-controlled thyristor (MCT). Its maximum ratings are 1,000 V and 100 A. A triac is an AC switch. Practically, it consists of two thyristors in anti-parallel connection and its characteristic in the I and III quadrants is symmetric. Its maximum ratings are 1,200 V, 300 A, and 400 Hz.

Table 1.2

Symbols, equivalent circuits, and V–I characteristics of regenerative switches

The reverse conduction thyristor (RCT) also can conduct in both half-cycles of an AC voltage. Practically, this is a thyristor with a diode in anti-parallel connection, the diode conducting during the negative half-cycle. The maximum ratings of the RCT are 2,500 V, 1,000 A forward, and 400 A reverse current.

In addition to the triode-type thyristors, there are several types of the diode-type thyristors (two-terminal devices without a control terminal). The four-layer diode and diac belong to this group. They are mainly used as switches for triggering thyristors.

Table 1.3 gives the comparative values of the basic parameters of semiconductor power switches. The qualitative characteristics of the most frequently used switching elements are presented in Table 1.4.

Table 1.3

Characteristics of semiconductor power switches [1]

In order to obtain a better idea about the characteristics of individual elements, Fig. 1.12 illustrates their applications with respect to frequency, voltage, and current [2].

Fig. 1.12

Maximum characteristics of semiconductor power switches with respect to frequency, voltage, and current

Table 1.4

Qualitative characteristics of switching elements containing control electrode

1.5 Magnetic Elements

Pulse transformers, chokes, and resonant coils have found applications among the available magnetic elements. Transformers are used for galvanic separation and impedance matching, and chokes are used for filtering. These elements operate at frequencies above 20 kHz and their dimensions are much smaller compared to those used in linear converters. The basic equation of the mid- and high-frequency transformers can be written in the form

(1.40)

where V I is the rectangular input voltage, N I is the number of primary turns, S is the cross-section of the magnetic core, B is the maximum value of induction in the core, and f is the operating frequency. The product N I S is a measure of the volume and weight of a transformer as N I is the measure of the amount of copper used and S is the measure of the magnetic material used. For a given input voltage, the volume and weight are thus inversely proportional to the product Bf. If it is assumed that in mains transformers that B = 1.8 T, then Bf = 1.8 × 50 = 90 T/s. For pulse transformers the maximum induction is about 0.3 T. If the frequency is 30 kHz, then Bf = 9,000 T/s. This means that pulse transformers are capable of transferring considerably higher powers per unit volume and weight compared to the mains transformers.

Owing to an increased operating frequency, special materials like ferrites or highly alloyed laminated metal must be used in pulse transformers. Ferrite cores are predominantly used. Namely, it is technologically simple to fashion the required shapes of cores which facilitates the realization of optimally designed transformers. Moreover, bulk conductivity of ferrite cores is very low so that eddy current losses are practically negligible. Mainly the EC, EE, U or X cores are used. For an optimally designed transformer, it is necessary to have data about its magnetic material and the core geometry. Table 1.5 presents the data for effective lengths of magnetic force lines l e , cross-sections S e , and volumes V e of some of the standard ferrite cores.

Table 1.5

Geometric dimensions of some of the standard ferrite cores

Total losses in magnetic material consist of hysteresis, residual, and eddy current losses. In ferrite cores hysteresis losses prevail. These losses increase with frequency and maximum variation of induction ΔB per switching cycle. Catalogues specify maximum induction for bipolar symmetric driving, B ac = ΔB/2. In order to prevent shifting of the core to the saturation region, most of the time B ac < 0.3T, but at frequencies close to 1 MHz the limitation is between 30 and 50 mT. Figure 1.13 shows the losses in materials N49 and N59 (manufacturer Siemens) for B m  = 50 mT at frequencies 500 kHz and 1 MHz.

Fig. 1.13

Losses in ferrite materials N49 and N59 as functions of temperature at frequencies 500 kHz (a) and 1 MHz (b) and for B m  = 50 mT

1.5.1 Chokes

Chokes are magnetic elements made of copper wire wound around ferromagnetic cores. The job of a choke designer is to:

select the core and determine the air gap if required,

calculate the cross-section, length, and the number of turns of the copper wire, and

select the mode of winding.

The basic parameter of a choke is its inductance. If the core contains an air gap, then the inductance is approximately

(1.41)

where l is the length of magnetic force lines of each individual part of the core made of the same magnetic material and with a constant cross-section, S is the cross-section of the core, μ o = 4π × 10−7 H/m is magnetic permeability of the vacuum, μ e is the effective magnetic permeability, and N is the number of turns. The effective permeability is defined as the resulting permeability of a core consisting of materials with different permeabilities. It depends on the shape and dimensions of the core and particularly on the width of the air gap in the magnetic material.

The effective length of magnetic force lines l e is defined as

(1.42)

and the effective magnetic cross-section is

(1.43)

The effective magnetic volume is determined by

(1.44)

Now the choke inductance can be written in the form

(1.45)

It is quite difficult to determine the effective magnetic permeability of a core containing an air gap. For this reason, the manufacturers give the values of the inductance factor A L which represents the inductance of the choke consisting of the core and one turn. The inductance of the coil of a choke is

(1.46)

The inductance factor A L is determined experimentally by measuring the inductance of a coil containing only one turn and it is presented in the form of a diagram like the one in Fig. 1.14. The inductance factor A L of ferrite cores ranges from 5 to 10,000. It depends on the type of material and core dimensions. For larger cores, the inductance factor A L is larger. In addition, A L is dependent on the air gap of the core (Fig. 1.15).

Fig. 1.14

Choke inductance as function of the number of turns for different values of inductance factor A L

Fig. 1.15

Inductance factor as function of the width of the air gap of ferrite core profile E20 made of material N27 (Siemens)

1.5.2 Transformers

Transformers consist of at least two inductively coupled windings. The windings are galvanically separated, thus only the transfer of AC signals is possible. The input winding is called the primary, and the output is the secondary. Voltage induced in the secondary can be lower, or higher, or equal to the primary voltage. The ratio of the secondary and the primary voltage is determined by the ratio of the number of the secondary and the primary windings. Under the influence of the magnetic flux caused by the voltage V 1 in the primary winding, the electromotive forces E 1 and E 2 will be induced in the primary and the secondary windings, respectively

(1.47)

(1.48)

where f(Hz) is the driving frequency, N 1 and N 2 are the respective numbers of turns in the primary and in the secondary, B m (T) is the amplitude of magnetic induction in the core, and S e (cm²) is the effective cross-section of the core. If the voltage drops in the windings are neglected, then V 1 = E 1 and V 2 = E 2 and the ratio of voltage transformation is

(1.49)

Since the inductance factors of the primary and the secondary are equal, it follows that:

(1.50)

If the secondary is loaded, the current I 2 will flow. The currents of the primary I 1 and the secondary I 2 will maintain magnetic equilibrium if I 1 N 1 = I 2 N 2, giving

(1.51)

A real transformer can be replaced by the equivalent circuit in Fig. 1.16 consisting of a T equivalent circuit and an ideal transformer. Real losses within a transformer are modeled by the stray inductance L e . The coupling coefficient k depends on the degree of coupling of the magnetic fields of the windings. For ferromagnetic transformer cores, k is close to unity because almost all magnetic field lines close within the transformer core. The Ohmic resistances of the transformers used in electronics are negligible. Consequently, these transformers can be represented by a parallel connection of a coil, whose inductance is equal to the inductance of the primary winding, and an ideal transformer with a transformation ratio n = N 1 /N 2 (Fig. 1.16c).

Fig. 1.16

Real transformer (a) and its equivalent circuits (b) and (c) (L m  = kI 1, L e  = L 1 − L m )

1.6 Capacitors

Capacitors have found a very wide application in power electronics. Typical applications are:

protection circuits of power switches,

various types of filters,

resonant circuits of converters for achieving the conditions of soft commutation of switching elements,

AC circuits for power factor correction,

pulse DC/DC converters for DC component separation,

circuits for forced turning on and off of semiconductor switches (bipolar transistors and thyristors ) etc.

Globally, they can be classified in three groups:

ceramic,

film and

electrolytic.

Figure 1.17 shows the areas of capacitance and the permitted operating voltages for the three groups of capacitors.

Fig. 1.17

Areas of capacitance and permitted operating voltages

The basic function of a capacitor is to accumulate electrical energy in the form of electric charge. The electric charge Q and the accumulated energy E C are determined by

(1.52)

(1.53)

where V C is the voltage applied to the capacitor and C is its capacitance. The capacitance C is directly proportional to the surface S of the electrodes and inversely proportional to the thickness d of the dielectric layer between them, thus:

(1.54)

Relative dielectric constants ε r for different materials are shown in Table 1.6.

Table 1.6

Relative dielectric constants ε r of standard dielectric materials

A capacitor can be represented by the equivalent circuit shown in Fig. 1.18, where R S  = ESR is the equivalent series resistance and L S  = ESL the equivalent series inductance. The series resistance is the basic cause of dissipation in a capacitor and it is most often expressed by the power factor tanδ defined as

(1.55)

Fig. 1.18

Equivalent circuit of capacitor

The power factor tanδ is given in catalogue data for capacitors.

ESL represents the series inductance of the terminals of the internal structure of a capacitor. Besides being dependent on the type of capacitor, the values of the parasitic elements ESR and ESL depend on packaging and method of mounting in a circuit. The total impedance of a capacitor is determined by

(1.56)

and its modulus is shown in Fig. 1.19. The minimum of the impedance modulus is equal to the resistance R S and it is obtained at the intrinsic resonance frequency

(1.57)

Fig. 1.19

Impedance modulus of aluminum electrolytic capacitor 100 mF/63 V as a function of frequency

Below this frequency, the impedance is of the capacitive character and the influence of ESL can be neglected. For ω > ω r the influence of ESL prevails.

One electrode of electrolytic capacitors is usually made of aluminum or tantalum. A thin oxide layer serving as dielectric is formed on that electrode. The other electrode is electrolyte in either liquid or solid state. The electrode carrying the dielectric must always be at positive potential. The electrolytic capacitors should have large values of ESR and ESL. This has a negative effect on the properties of the converters where these capacitors are used in the output filters. The ripple of the output voltage is increased and the stability of the control module is decreased owing to the difficulties in the design of the compensator in the feedback loop. For this reason, it is recommendable to use a parallel connection of several capacitors of lower capacitance instead of one large capacitor. In addition, it is recommendable that a ceramic or film (polypropylene) capacitor of small ESR and ESL is connected in parallel with the electrolytic capacitor. The influence of the parasitic elements of ESR and ESL can be reduced if specially designed four-terminal capacitors are used in the output filter.

Tantalum capacitors have a high-specific capacitance (high capacitance for small dimensions and low values of ESR and ESL). ESR and ESL tantalum chip capacitors have especially low values.

The dielectric of ceramic capacitors is ceramic material. Depending on the composition of the ceramic material, two main classes of ceramic capacitors can be defined. The relative dielectric constant of class I capacitors is below 500 (ε r  < 500). The capacitance does not depend on supply voltage. These capacitors have small power losses even at high frequencies (tanδ is about 0.15 % at 1 MHz). They are used in resonant circuits, as timing elements, for filtering, etc. The relative dielectric constant of class II capacitors ranges from 1,000 to 10,000. Their capacitance is a nonlinear function of voltage and temperature. They have higher power losses (tanδ is about 3 % at 1 MHz) than the class I capacitors. Owing to high values of ε r , the capacitance is relatively high compared to dimensions.

The dielectric of film capacitors is usually a thin film of polypropylene (MKP capacitors) or of polyester (MKT capacitors). Very often, these dielectrics are combined with metallized paper, resulting in improved ability of enduring large voltage pulses. The film capacitors are mainly used in pulse circuits involving very fast voltage variations dv/dt.

1.7 Radio-Frequency Interference

In the instants of the change of state of a power switch, a very large change of voltage and current per unit time (from 10⁶ to 10⁹ A/s or V/s) is generated. This is the reason that the pulse converters generate interference, both conductive and electromagnetic. Through the input and the output contacts of a converter, conductive interference acts upon the primary source and also upon the load. This may cause erroneous operation of electronic equipment whether it is supplied from the primary source or by a converter.

A converter irradiates electromagnetic interference into the surrounding space. This may hinder the operation of the nearby electronic equipment. For this reason, the removal of radio-frequency interference is one of the key problems in the design of pulse converters. Interference cannot be entirely eliminated, but it can be reduced to within permitted limits. The limits for permitted interference are defined by various national standards and international regulations (among the best known ones is MIL-STD-461). The interference, which propagates along cables in the form of high frequency currents, is classified as symmetric (between the supply chords) and nonsymmetric, which closes through the ground. Symmetric interference at the input of a converter, as a consequence of the AC component, appears at the internal impedance of the input capacitor. Nonsymmetric interference at the input is closed, by means of the parasitic capacitance of the circuit or by inductive coupling between some of the parts, through the ground.

Symmetric interference at the output is caused by the AC component of the output current on the internal impedance of the output capacitor. It is for this reason that at the output four-terminal electrolytic capacitors with low series impedance should be used. The circuit of nonsymmetric interference at the output is closed through the load and the ground.

The AC components of the input current, generating interference on the supply lines, can be eliminated or reduced to permitted limits. This is accomplished by inserting a filter between the primary source and the stabilizer, or converter, as shown in Fig. 1.20. The filter shown in Fig. 1.20a attenuates the primary source current created by the stabilizer, which behaves like a pulse load. Considerably, more efficient are the circuits shown in Fig. 1.20b, c. In these cases, isolation of the stabilizer from the input and the output lines is accomplished. In this way the stabilizer floats in its own oscillations. This is carried out by introducing chokes in both the input and the output lines of the stabilizer. The problems caused by interference are most efficiently solved if interference is taken into consideration throughout the design and production of a converter. In doing so, the parasitic capacitance can be reduced and the lengths and the surfaces of the loops containing pulse currents can be minimized. In addition, the connection of the converter is of importance, bearing in mind the technical requirements.

Fig. 1.20

Typical filter cells for interference reduction, simple L, C filter (a) and more efficient circuits (b