As discussed in the introduction to Chapter 1, the role of power electronics is to facilitate power flow, often in a controlled manner, between two systems shown in Figure 5.1: one of them a “source” and the other a “load.” Typically, power is provided by a single-phase or a three-phase utility source, for example, in adjustable-speed motor drives. (Of course, there are exceptions, for example, in wind turbines, where the wind-turbine generator is the source of power to the utility grid that acts like a “load.”)
Such power-electronic interfaces often consist of a voltage-link structure, discussed in Section 1.5.1, where the input from the AC source is first rectified into a DC voltage across a large capacitor. If reversing power flow is not an objective, it is possible to rectify the AC input, single-phase or three-phase, by means of diode rectifiers discussed in this chapter. The knowledge of such systems is essential for learning about thyristor converters, discussed in Chapter 13, which are used in important applications such as high-voltage DC transmission (HVDC) systems.
In diode rectifiers, unless corrective action is taken as described in the next chapter, power is drawn by means of highly distorted currents, which have a deleterious effect on the power quality of the utility source. This issue and the basic principles of diode-rectifier operation are examined in this chapter.
To quantify distortion in the current drawn by power electronic systems, it is necessary to define certain indices. As a base case, consider the linear load shown in Figure 5.2a, which is supplied by a sinusoidal source in steady state. The voltage and current phasors are shown in Figure 5.2b, where is the angle by which the current lags the voltage. Using RMS values for the voltage and current magnitudes, the average power supplied by the source is
The power factor at which power is drawn is defined as the ratio of the real average power P to the product of the RMS voltage and the RMS current:
where is the apparent power. For a given voltage, from Equation (5.2), the RMS current drawn is
This shows that the power factor PF and the current are inversely proportional. This current flows through the utility distribution lines, transformers, and so on, causing losses in their resistances. This is the reason why utilities prefer unity power factor loads that draw power at the minimum value of the RMS current.
The sinusoidal current drawn by the linear load in Figure 5.2 has zero distortion. However, power electronic systems with diode rectifiers as the front-end draw currents with a distorted waveform such as that shown by in Figure 5.3a. The utility voltage is assumed sinusoidal. The following analysis is general, applying to the utility supply that is either single-phase or three-phase, in which case the analysis is on a per-phase basis.
The current waveform in Figure 5.3a repeats with a time period . By Fourier analysis of this repetitive waveform, we can compute its fundamental frequency () component , shown dotted in Figure 5.3a. The distortion component in the input current is the difference between and the fundamental-frequency component :
where using Equation (5.4) is plotted in Figure 5.3b. This distortion component consists of components at frequencies that are the multiples of the fundamental frequency.
To obtain the RMS value of in Figure 5.3a, we will apply the basic definition of RMS:
Using Equation (5.4),
In a repetitive waveform, the integral of the products of the two harmonic components (including the fundamental) at unequal frequencies, over the repetition time-period, equals zero:
Therefore, substituting Equation (5.6) into Equation (5.5) and making use of Equation (5.7) that implies that the integral of the third term on the right side of Equation (5.6) equals zero (assuming the average component to be zero),
or
where the RMS values of the fundamental-frequency component and the distortion component are as follows:
and
Based on the RMS values of the fundamental and the distortion components in the input current , a distortion index called the total harmonic distortion (THD) is defined in percentage as follows:
Using Equation (5.9) into Equation (5.12),
The RMS value of the distortion component can be obtained based on the harmonic components (except the fundamental) as follows using Equation (5.7):
where is the RMS value of the harmonic component “h.”
By Fourier analysis, any distorted (non-sinusoidal) waveform that is repetitive with a fundamental frequency , for example, in Figure 5.3a, can be expressed as a sum of sinusoidal components at the fundamental frequency and its multiples (harmonic frequencies):
where the average value is DC,
The sinusoidal waveforms in Equation (5.15) at the fundamental frequency () and the harmonic components at frequencies times can be expressed as the sum of their cosine and sine components,
The cosine and the sine components above, given by Equations (5.17) and (5.18), can be combined and written as a phasor in terms of its RMS value,
where the RMS magnitude in terms of the peak values and equals
and the phase can be expressed as
It can be shown that the RMS value of the distorted function can be expressed in terms of its average and the sinusoidal components as
In Fourier analysis, by appropriate selection of the time origin, it is often possible to make the sine or the cosine components in Equation (5.15) to be zero, thus considerably simplifying the analysis, as illustrated by a simple example below.
A current of a square waveform is shown in Figure 5.4a. Calculate and plot its fundamental frequency component and its distortion component. What is the %THD associated with this waveform?
Solution From Fourier analysis, by choosing the time origin as shown in Figure 5.4a, in Figure 5.4a can be expressed as
The fundamental frequency component and the distortion component are plotted in Figures 5.4b and 5.4c.
From Figure 5.4a, it is obvious that the RMS value of the square waveform is equal to . In the Fourier expression of Equation (5.23), the RMS value of the fundamental-frequency component is
Therefore, the distortion component can be calculated from Equation (5.9) as
Therefore, using the definition of THD,
Next, we will consider the power factor at which power is drawn by a load with a distorted current waveform such as that shown in Figure 5.3a. As before, it is reasonable to assume that the utility-supplied line-frequency voltage is sinusoidal, with an RMS value of and a frequency . Based on Equation (5.7), which states that the product of the cross-frequency terms has a zero average, the average power P drawn by the load in Figure 5.3a is due only to the fundamental-frequency component of the current:
Therefore, in contrast to Equation (5.1) for a linear load, in a load that draws distorted current, similar to Equation (5.1),
where is the angle by which the fundamental-frequency current component lags behind the voltage, as shown in Figure 5.3a.
At this point, another term called the displacement power factor (DPF) needs to be introduced, where,
Therefore, using the DPF in Equation (5.25),
In the presence of distortion in the current, the meaning and therefore the definition of the power factor, at which the real average power P is drawn, remains the same as in Equation (5.2), that is, the ratio of the real power to the product of the RMS voltage and the RMS current:
Substituting Equation (5.27) for P into Equation (5.28),
In linear loads that draw sinusoidal currents, the current-ratio in Equation (5.29) is unity, hence . Equation (5.29) shows the following: a high distortion in the current waveform leads to a low power factor, even if the DPF is high. Using Equation (5.13), the ratio in Equation (5.29) can be expressed in terms of the total harmonic distortion as
Therefore, in Equation (5.29),
The effect of THD on the power factor is shown in Figure 5.5 by plotting versus THD. It shows that even if the displacement power factor is unity, a total harmonic distortion of 100% (which is possible in power electronic systems unless corrective measures are taken) can reduce the power factor to approximately 0.7 (or , to three decimal places, to be exact) which is unacceptably low.
There are several deleterious effects of high distortion in the current waveform and the poor power factor that results due to it. These are as follows:
In order to prevent degradation in power quality, recommended guidelines (in the form of the IEEE-519) have been suggested by the IEEE (Institute of Electrical and Electronics Engineers). These guidelines place the responsibilities of maintaining power quality on the consumers and the utilities as follows: (1) on the power consumers, such as the users of power electronic systems, to limit the distortion in the current drawn, and (2) on the utilities to ensure that the voltage supply is sinusoidal with less than a specified amount of distortion.
The limits on current distortion placed by the IEEE-519 are shown in Table 5.1, where the limits on harmonic currents, as a ratio of the fundamental component, are specified for various harmonic frequencies. Also, the limits on the THD are specified. These limits are selected to prevent distortion in the voltage waveform of the utility supply. Therefore, the limits on distortion in Table 5.1 depend on the “stiffness” of the utility supply, which is shown in Figure 5.6a by a voltage source in series with an internal impedance . An ideal voltage supply has zero internal impedance. In contrast, the voltage supply at the end of a long distribution line, for example, will have a large internal impedance.
TABLE 5.1 Harmonic current distortion ()
Odd Harmonic Order h (in %) | Total Harmonic Distortion (%) | |||||
Isc/I1 | h < 11 | 11 ≤ h < 17 | 17 ≤ h < 23 | 23 ≤ h <35 | 35 ≤ h | |
<20 | 4.0 | 2.0 | 1.5 | 0.6 | 0.3 | 5.0 |
20–50 | 7.0 | 3.5 | 2.5 | 1.0 | 0.5 | 8.0 |
50–100 | 10.0 | 4.5 | 4.0 | 1.5 | 0.7 | 12.0 |
100–1000 | 12.0 | 5.5 | 5.0 | 2.0 | 1.0 | 15.0 |
>1000 | 15.0 | 7.0 | 6.0 | 2.5 | 1.4 | 20.0 |
To define the “stiffness” of the supply, the short-circuit current is calculated by hypothetically placing a short circuit at the supply terminals, as shown in Figure 5.6b. The stiffness of the supply must be calculated in relation to the load current. Therefore, the stiffness is defined by a ratio called the short-circuit ratio (SCR):
where is the fundamental-frequency component of the load current. Table 5.1 shows that a smaller short-circuit ratio corresponds to lower limits on the allowed distortion in the current drawn. For the short-circuit ratio of less than 20, the total harmonic distortion in the current must be less than 5%. Power electronic systems that meet this limit would also meet the limits of more stiff supplies.
It should be noted that the IEEE-519 does not propose harmonic guidelines for individual pieces of equipment but rather for the aggregate of loads (such as in an industrial plant) seen from the service entrance, which is also the point of common coupling (PCC) with other customers. However, the IEEE-519 is frequently interpreted as the harmonic guidelines for specifying individual pieces of equipment such as motor drives. There are other harmonic standards, such as the IEC-1000, which apply to individual pieces of equipment.
Interaction between the utility supply and power electronic systems depends on the “front-ends” (within the power-processing units), which convert line-frequency AC into DC. These front-ends can be broadly classified as follows:
All of these front-ends can be designed to interface with single-phase or three-phase utility systems. In the following discussion, a brief description of the diode interface shown in Figure 5.7a is provided, supplemented by an analysis of results obtained through computer simulations. Interfaces using switch-mode converters in Figure 5.7b and thyristor converters in Figure 5.7c are discussed later in this book.
Most power electronic systems use diode-bridge rectifiers, such as the one shown in Figure 5.7a, even though they draw currents with highly distorted waveforms and the power through them can flow only in one direction. In switch-mode DC power supplies, these diode-bridge rectifiers are supplemented by a power-factor-correction circuit to meet current harmonic limits, as discussed in the next chapter.
Diode rectifiers rectify line-frequency AC into DC across the DC-bus capacitor without any control over the DC-bus voltage. For analyzing the interaction between the utility and the power electronic systems, the switch-mode converter and the load can be represented by an equivalent resistance across the DC-bus capacitor. In our theoretical discussion, it is adequate to assume the diodes are ideal.
In the following subsections, we will consider single-phase as well as three-phase diode rectifiers operating in steady state, where waveforms repeat from one line-frequency cycle to the next.
At power levels below a few kW, for example, in residential applications, power electronic systems are supplied by a single-phase utility source. A commonly used full-bridge rectifier circuit is shown in Figure 5.8a, in which is the sum of the inductance internal to the utility supply and an external inductance, which may be intentionally added in series. Losses on the AC side can be represented by the series resistance .
To understand the circuit operation, the rectifier circuit can be drawn as in Figure 5.8b, where the AC-side inductance and resistance have been ignored. The circuit consists of a top group and a bottom group of diodes. If the DC-side current is to flow, one diode from each group must conduct to facilitate the flow of this current. In the top group, both diodes have their cathodes connected together. Therefore, the diode connected to the most positive voltage will conduct; the other will be reverse biased. In the bottom group, both diodes have their anodes connected together. Therefore, the diode connected to the most negative voltage will conduct; the other will be reverse biased.
As an example, resistance is connected across the terminals on the DC-side, as shown in Figure 5.9a. The circuit waveforms are shown Figure 5.9b. During the positive half-cycle of the input voltage , diodes 1 and 2 conduct the DC-side current , equal to , and the DC-side voltage is . During the negative half-cycle of the input voltage , diodes 3 and 4 conduct the DC-side current , equal to , and the DC-side voltage is
The average value of the voltage across the DC-side of the converter can be obtained by averaging the waveform in Figure 5.9b over only one half-cycle (by symmetry):
and
As another example, consider the load on the DC-side to have a large inductance, as shown in Figure 5.10a, such that the DC-side current is essentially DC. Assuming that to be purely DC, the waveforms are as shown in Figure 5.10b.
Note that the waveform of in Figure 5.10b is identical to that in Figure 5.9b for a resistive load. Therefore, the average value of the voltage across the DC-side of the converter in Figure 5.10a is the same as in Equation (5.33), that is . Similarly, the average value of the DC-side current is related to by , as .
Figures 5.9b and 5.10b show that the DC-side voltage has a large ripple. To eliminate this, in order to achieve a voltage waveform that is fairly DC, a large filter capacitor is connected on the DC-side, as shown in Figure 5.8a. As shown in Figure 5.11, at the beginning of the positive half-cycle of the input voltage , the capacitor is already charged to a DC voltage . So long as exceeds the input voltage magnitude, all diodes remain reverse biased, and the input current is zero. Power to the equivalent resistance is supplied by the energy stored in the capacitor up to .
Beyond , the input current increases, flowing through diodes and . Beyond , the input voltage becomes smaller than the capacitor voltage, and the input current begins to decline, falling to zero at . Beyond , until one half-cycle later than , the input current remains zero, and the power to is supplied by the energy stored in the capacitor. At during the negative half-cycle of the input voltage, the input current flows through diodes and . The rectifier DC-side current continues to flow in the same direction as during the positive half-cycle; however, the input current , as shown in Figure 5.11. Figure 5.12 shows waveforms obtained by LTspice simulations for two values of the AC-side inductance, with current THD of 86% and 62%, respectively (higher inductance reduces THD, as discussed in the next section).
The fact that flows in the same direction during both the positive and the negative half-cycles represents the rectification process. In the circuit of Figure 5.8a in steady state, all waveforms repeat from one cycle to the next. Therefore, the average value of the capacitor current over a line-frequency cycle must be zero so that the DC-bus voltage is in steady state. As a consequence, the average current through the equivalent load-resistance equals the average of the rectifier DC-side current; that is, .
As Figures 5.11 and 5.12 show, power is drawn from the utility supply by means of a pulse of current every half-cycle. The larger the “base” of this pulse during which the current flows, the lower its peak value and the total harmonic distortion (THD). This pulse widening can be accomplished by increasing the AC-side inductance . Another parameter under the designer’s control is the value of the DC-bus capacitor . At its minimum, it should be able to carry the ripple current in and in (which in practice is the input DC-side current, with a pulsating waveform, of a switch-mode converter) and keep the peak-to-peak ripple in the DC-bus voltage to some acceptable value, for example, less than 5% of the DC-bus average value. Assuming that these constraints are met, the lower the value of , the lower the THD in current and the higher the ripple in the DC-bus voltage.
In practice, it is almost impossible to meet the harmonic limits specified by IEEE-519 by using the above techniques. Rather, the power-factor-correction circuits described in the next chapter are needed to meet the harmonic specifications.
The simulation of a single-phase diode-bridge rectifier is demonstrated by means of an example:
Example 5.2
In the single-phase diode-bridge rectifier shown in Figure 5.8a, , , , and . The supply voltage is RMS at Simulate this rectifier using LTspice.
Solution The simulation file used in this example is available on the accompanying website. The LTspice model is shown in Figure 5.13, and the steady-state waveforms from the simulation of this model are shown in Figure 5.14.
It is preferable to use a three-phase utility source, except at a fractional kilowatt, if such a supply is available. A commonly used full-bridge rectifier circuit is shown in Figure 5.15a.
To understand the circuit operation, the rectifier circuit can be drawn as in Figure 5.15b. The circuit consists of a top group and a bottom group of diodes. Initially, is ignored, and the DC-side current is assumed to flow continuously. At least one diode from each group must conduct to facilitate the flow of . In the top group, all diodes have their cathodes connected together. Therefore, the diode connected to the most positive voltage will conduct; the other two will be reverse biased. In the bottom group, all diodes have their anodes connected together. Therefore, the diode connected to the most negative voltage will conduct; the other two will be reverse biased.
Ignoring and assuming that the DC-side current is a pure DC, the waveforms are as shown in Figure 5.16. In Figure 5.16a, the waveforms (identified by the dark portions of the curves) show that each diode, based on the principle described above, conducts during . The diodes are numbered so that they begin conducting sequentially: 1, 2, 3, and so on. The waveforms for the voltages and , with respect to the source-neutral, consist of -segments of the phase voltages, as shown in Figure 5.16a. The waveform of the DC-side voltage is shown in Figure 5.16b. It consists of -segments of the line-line voltages supplied by the utility. Line currents on the AC-side are as shown in Figure 5.16c. For example, phase-a current flows for during each half-cycle of the phase-a input voltage; it flows through diode during the positive half-cycle of and through diode during the negative half-cycle.
The average value of the DC-side voltage can be obtained by considering only a -segment in the 6-pulse (per line-frequency cycle) waveform shown in Figure 5.16b. Let us consider the instant of the peak in the -segment to be the time-origin, with as the peak line-line voltage. The average value can be obtained by calculating the integral from to (the area shown by the hatched area in Figure 5.16b) and then dividing by the interval :
This average value is plotted as a straight line in Figure 5.16b.
In the three-phase rectifier of Figure 5.15a with the DC-bus capacitor filter, the input current waveforms obtained by computer simulations are shown in Figure 5.17.
Figure 5.17a shows that the input current waveform within each half-cycle consists of two distinct pulses when is small. For example, in the waveform during the positive half-cycle, the first pulse corresponds to the flow of DC-side current through the diode pair and then through the diode pair . At larger values of , within each half-cycle, the input current between the two pulses does not go to zero, as shown in Figure 5.17b.
The effects of and on the waveforms can be determined by a parametric analysis, similar to the case of single-phase rectifiers. The THD in the current waveform of Figure 5.17b is much smaller than in Figure 5.17a (23% versus 82%). The AC-side inductance is required to provide a line-frequency reactance that is typically greater than 2% of the base impedance , which is defined as follows:
where is the three-phase power rating of the power electronic system, and is the RMS value of the phase voltage. Therefore, typically, the minimum AC-side inductance should be such that
The simulation of a three-phase diode-bridge rectifier is demonstrated by means of an example:
Example 5.3
In the three-phase diode-bridge rectifier shown in Figure 5.15a, , , and . The supply voltage is line-line RMS at . Simulate this rectifier using LTspice.
Solution The simulation file used in this example is available on the accompanying website. The LTspice model is shown in Figure 5.18, and the steady-state waveforms from the simulation of this model are shown in Figure 5.19.
Examination of single-phase and three-phase rectifier waveforms shows the differences in their characteristics. Three-phase rectification results in six identical “pulses” per cycle in the rectified DC-side voltage, whereas single-phase rectification results in two such pulses. Therefore, three-phase rectifiers are superior in terms of minimizing distortion in line currents and ripple across the DC-bus voltage. Consequently, as stated earlier, three-phase rectifiers should be used if a three-phase supply is available. However, three-phase rectifiers, just like single-phase rectifiers, are also unable to meet the harmonic limits specified by IEEE-519 unless corrective actions such as those described in Chapter 12 are taken.
In power electronic systems with rectifier front-ends, it may be necessary to take steps to avoid a large inrush of current at the instant the system is connected to the utility source. In such power electronic systems, the DC-bus capacitor is very large and initially has no voltage across it. Therefore, at the instant the switch in Figure 5.20a is closed to connect the power electronic system to the utility source, a large current flows through the diode-bridge rectifier, charging the DC-bus capacitor.
This transient current inrush is highly undesirable; fortunately, several means of avoiding it are available. These include using a front-end that consists of thyristors, discussed in Chapter 14, or using a series semiconductor switch as shown in Figure 5.16b. At the instant of starting, the resistance across the switch lets the DC-bus capacitor to get charged without a large inrush current, and subsequently, the semiconductor switch is turned on to bypass the resistance. There are other techniques that can also be employed.
In stop-and-go applications such as elevators, it is cost effective to feed the energy recovered by regenerative braking of the motor drive back into the utility supply. Converter arrangements for such applications are considered in Chapter 12.