20.2.1 Single-Phase Inverter

Obtaining AC from DC is based on switching the direction of current in a load. This is only possible with power electronic devices that can be controlled, i.e., those whose function can be turned on and off. These are devices such as a thyristor and an insulated gate bipolar transistor (IGBT) that we discussed in Chapter 19. The controls must also allow choosing the voltage and frequency of the generated AC signal as desired. The timing for the switching action, therefore, is quite important. In the simplest form, the output of an inverter is a square-wave AC signal, and not a sinusoidal waveform.

Figure 20.1 Arrangement for a single-phase inverter.

Figure 20.1 shows the schematics of the arrangement for a single-phase inverter. There are four switching devices to control the direction of current at each instant. These can be thyristors or gated transistors (e.g., IGBTs). For the sake of simplicity in the rest of this chapter they are only referred to as thyristors. Each of the four thyristors in the circuit allows current in only one direction and only when it is turned on through its gate. At all other times it blocks the current in either direction because of being reverse biased or turned off. These thyristors are turned on and off in an orderly manner and with an appropriate sequence and a desired frequency. The thyristors are divided into two pairs connected between the positive and negative terminals of the power supply, and the load is connected between the two pairs. The thyristors are numbered 1 to 4 for ease of reference.

Consider if with a frequency of 100 Hz sequentially thyristors T₁ and T₂ are turned on together, then turned off, immediately followed by turning on thyristors T₃ and T₄ together and then off together, with the same sequence repeated. The duration each time a pair of thyristors is on, is therefore 1/100th of a second. The result, as observed by the load, is a square-wave alternating current with 50 Hz frequency and a voltage very near the voltage of the DC power supply (three is a small voltage drop in the electronic parts). This is the way that an inverter works.

Note that the two thyristors on the same leg (e.g., T₁ and T₄ or T₂ and T₃) should never be on at the same time because they short the power supply and damage it. Also, in operation, each pair that is on must be turned off before the other pair is turned on. For the action of turning the thyristors on and off, a separate electronic circuit is necessary. This circuit is normally called driver. The driver circuit must take care of the timing and frequency of turning the thyristors on and off.

Figure 20.2 Freewheel diodes to protect the circuit from overvoltage.

If the load is purely resistive, then the current in the load will have the same pattern of the AC voltage, which is a square wave. For inductive and capacitive loads, as we discussed in Chapter 8, because of the charging effect of capacitor and the Lenz’s law in the inductor, the current and voltage are not in phase and some large voltages may develop in part of the circuit. To prevent damage because of these voltages, usually parallel with each thyristor there is a diode allowing current in the opposite direction, as shown in Figure 20.2. The role of the diode is to make a path for current to flow and prevent a voltage buildup. These diodes are called freewheel diode, or freewheeling diode. Note that these diodes are reverse biased.

Thus, they do not short the power supply and do not conduct in the normal conditions.

The inverter as stated, having zero interval between the time a thyristor is turned off and the associated thyristor in the same leg is turned on, generates a square AC waveform as shown in Figure 20.3a. However, if a short interval exists between the two actions (turning off a thyristor and turning on its pair) the generated waveform assumes a different shape, as illustrated in Figure 20.3c. This figure corresponds to the case where there exists a delay of T₂ between the instant one set of thyristors are turned off (after being in the on state for a period of T₁), and the instant the other set are turned on. Thus, for a period of T₂ all the thyristors are turned off.

As can be seen the total period for one cycle of the AC waveform is 2(T₁ + T₂). Thus, the frequency of the resulting alternating current is = $=\frac{1}{2(T_1+T_2)}$. For the square waveform of Figure 20.3a, T = 0. 2(T + T 2) 1 2

Figure 20.3 Current waveforms for an inductive load based on output voltage. (a) Voltage pattern in a square waveform. (b) Current in an inductive load with a square wave. (c) Square waveform with delay. (d) Current in an inductive load subject to voltage shown in (c).

Although the current variation in a pure resistive load is the same as the voltage, for a different type of load the current variation is different than that of the voltage, as dictated by the load. For instance, for an inductive load the current does not have a square-wave pattern, It rather looks like the waveforms in Figure 20.3b and 20.3d for the voltage waveforms in Figure 20.3a and 20.3c, respectively. As can be seen, the current in an inductive load has a form closer to a sine wave than a square wave.

By changing the time intervals T₁ and T₂ it is possible to change the current waveform to some extent, as can be seen from Figure 20.3d. It is customary to represent T₁ and T₂ in terms of angles. In such a case they do not associate with frequency anymore but can indicate the percentage of time in each cycle that a thyristor is on. For this, T₁ + T₂ = 180°, and if, for instance, T₁ = 120° and T₂ = 60°, it implies that each set of thyristors are on for 120° and stay off for the rest of a cycle (i.e., for 60° + 180° = 240°). In practice, the driver circuit (software and/or hardware) that controls the thyristors takes care of frequency by repeating the cycle after an elapsed time.

The best result, as far as having an output as close as possible to a sinusoidal waveform is concerned, is when T₁ = 133.5° and T₂ = 46.5°.

To improve the quality of the output waveform from an inverter, so that it better resembles a sinusoidal waveform, various techniques are available. One of these is pulse width modulation, which is separately discussed later (see Section 20.6). Another technique is to increase the switching steps and accordingly the necessary controls to generate stepping signals as illustrated in Figure 20.4.

Obviously, the more steps there are, the more costly the device. The steps in the configuration in Figure 20.4 have the same width (time duration). Figure 20.5 shows how a step waveform can be obtained from combining inverters with different outputs of the form in Figure 20.3c.

Practically, a waveform shown in Figure 20.5b can be made by adding the outputs of two inverters together. One way for this is indicated in Figure 20.6, where the outputs of the inverters are added together through a transformer.

The term firing is normally used for turning a thyristor on. Also, firing angle represents the delay from the start of a cycle, normally in terms of degrees, when a thyristor must be turned on. For instance, in Figure 20.5 the firing angle for the inverter to generate signal A is 22.5° and for the

Figure 20.4 A 12-step waveform.

Figure 20.5 A step waveform made up of combining simpler waveforms. (a) Two-step sine form approximation. (b) Two-step waveform made up of two square waves A and B. (c) Square wave A. (d) Square wave B.

Figure 20.6 Adding two waveforms together through a transformer. A practical way of adding two square waves.

inverter to generate signal B is 67.5°. In this sense, the two inverters in Figure 20.6 can be similar but have different firing schemes.

20.3.1 Revisiting Diode-Based Three-Phase Rectifiers

Figure 20.7 A three-phase bridge rectifier employing six diodes. (a) Schematics of a 3-phase rectifier. (b) Waveforms of the individual three phases. (c) Voltage of the DC from rectifier based on conduction of diodes.

A three-phase diode rectifier is shown in Figure 20.7a. This is the same as was depicted in Figure 16.11a, except that the diodes have been numbered. Diodes 1 and 4 are on phase A, 3 and 6 are on phase B, and 5 and 2 are on phase C. The upper diodes (1, 3, and 5) conduct when the corresponding phase is going through its positive half cycle, and the lower diodes (4, 6, and 2) conduct during the negative half cycle.

Figure 20.7b illustrates the diode numbers corresponding to each phase for each 60° interval, based on their phase and its polarity. As mentioned earlier (see Section 16.6), a diode does not conduct for the whole duration of a half cycle it is forward biased by its line voltage. At each instant, only the two diodes corresponding to the most positive voltage and the most negative voltage conduct and supply the highest possible voltage difference across the load. For example, during the interval between 0 and 30°, in each 360° cycle as represented in Figure 20.7b, diode 5 has a higher voltage than diode 1, while both A and C phases are at their positive cycle. During this period, on the basis of the voltages present at its terminals, diode 1 is reverse biased and does not conduct. The condition reverses for the period between 30° and 60°, where phase A dominates and diode 5 does not conduct.

Figure 20.7c shows the most positive and most negative voltages for two cycles of the AC line as well as the numbers for the conducting diodes at various instants during these cycles. This figure clearly shows which diodes are conducting at each interval. Take notice of the important fact that each diode conducts for 120° during its line cycle and turns off for the rest 240°.

It is customary to use a chart similar to Figure 20.8 to show the on state of each diode with respect to other diodes. This chart is more useful when switchable devices are used instead of diodes; in particular, it shows the instant when each device is turned on and starts conducting. For each diode the hatched area shows the start and end of conduction. The chart also shows which two diodes are on at each instant.

Figure 20.8 Timing chart for conducting diodes in the three-phase rectifier of Figure 20.7.

Notice the sequence the diodes start conducting, being 1, 2, 3, 4, 5, and 6. It can be seen from both Figures 20.7c and 20.8 that the functions of diodes switch after an interval of 60°, and one pair of diodes conduct for each interval. Switching action takes place automatically based on the voltage variation in the three lines, and the conducting pairs and their switching order are 1&2, 2&3, 3&4, 4&5, 5&6, and 6&1, repeated. This order also illustrates the reason why the diodes in Figure 20.7 are numbered this way.

20.3.2 Substituting Diodes by Switchable Devices for Three-Phase Rectifiers

As we have discussed, it is possible to use switchable devices such as thyristors and IGBTs instead of diodes. When switchable devices are employed, it becomes necessary to fire each device repeatedly and continuously as far as it is expected to operate. If a thyristor is not fired, then no rectification takes place, and there is no output. This firing or triggering must be based on a proper triggering scheme. For a rectifier, firing of each thyristor must take place in a timely manner and only during the half cycle when it is forward biased. In the rest of this chapter for simplicity we only use the word thyristor where it is meant to refer to a switchable device, which can be a thyristor or an IGBT.

Figure 20.9 Output of a thyristor subject to alternating current and triggered with different firing angles. (a) Thyristor circuit. (b) Firing angle = 0. (c) Firing angle = 30∘.

Figure 20.9 shows the performance of a thyristor when subject to an alternating current (two cycles are shown). If triggering occurs exactly at the zero crossing point (the instant of zero voltage) for negative to positive voltage, the thyristor starts conducting at that moment and continues to conduct as far as it is forward biased. After a current is established, the device turns off by itself as the applied AC proceeds to its negative half cycle. At the moment of zero crossing from positive voltage to negative, the voltage across the thyristor changes sign and the device becomes reverse biased. Consequently, the current drops to zero. The resulting DC output is shown by the hatched area in Figure 20.9b.

Figure 20.9, furthermore, illustrates what happens if thyristor firing does not take place exactly at the zero crossing point but with a delay. In this case the rectified signal has duration of less than half cycle of the applied AC signal. The case is depicted by the hatched area in Figure 20.9c for a 30° firing angle.

Figure 20.10 illustrates a full-wave three-phase rectifier employing six thyristors numbered T₁ to T₆, following the same numbering method as in Figure 20.7.

This rectifier has a structure and performance similar to that in Figure 20.7, except that it needs a driver to send firing signals to the gate of each thyristor to turn it on. The triggering must be sequential and based on a scheme like that shown in Figure 20.8. The thyristor driver timing must be synchronized by the time of zero crossing of one of the three phases, which is also the instant that the other two phases have the same magnitude, but opposite sign.

Figure 20.10 A three-phase bridge rectifier employing six thyristors.

Figure 20.11 Gate firing scheme for a three-phase rectifier.

Figure 20.11 shows the firing sequence of thyristors, starting from T₁ in the upper leg of phase A. After each 60° of the AC sine wave a thyristor that is already on must turn off and another one start. This is shown in Figure 20.11 for some instances. Turning off is done naturally by the voltage change across each thyristor creating a reverse bias condition.

DC voltage obtained from the above three-phase rectifier is

Thus, for 208 V, line voltage V_DC = 280 V, and for 380 V line voltage, V_DC = 513 V.

Note that this value is the maximum voltage that can be obtained. In practice, we may want a lower voltage. In such a case the output voltage can be controlled, as discussed next.

20.3.3 Voltage Control in Thyristor-Based Rectifier

One of the advantages of using switchable devices instead of diodes in a rectifier is the possibility to control the voltage of the DC output. This is done by introducing a delay in the firing of all thyristors. This delay, normally expressed in terms of an angle and measured in degrees, is defined as the firing angle (the same term we have already seen for the single-phase inverter). All thyristors must have the same delay in their triggering.

When there is a delay in triggering thyristors, those ones that should turn off were there no delays (because of receiving a voltage at their cathodes that changes their bias condition) continue to conduct. For instance, in Figure 20.12, assuming no triggering delay, T₁ starts conducting at point L and continues until point M when T₃ is triggered and caused T₁ to turn off. But, if T₃ is not triggered, then T₁ stays on. During the interval between M and N, if T₃ is triggered, it turns off T₁; otherwise, T₁ conducts until N. At point N it turns off naturally (natural commutation; see Section 19.8) and stops conducting. It starts conducting again afterward, whenever its gate receives a positive pulse, while it is forward biased. Likewise, if thyristor T₂ is not started at point P, then T₆ does not turn off and conducts until the time point Q is reached.

Note that all preceding statements are true when the load is purely resistive. In cases of inductive and capacitive loads the conditions change. A detailed study of what happens in various conditions is outside the scope of this book.

Figure 20.12 On and off conditions of a thyristor.

Figure 20.13 illustrates the case when the firing angle is not zero. The firing times and thyristor states (on or off) can be compared for three cases when the firing angle is 0, 30°, and 50°. Figure 20.13a corresponds to zero firing angle. It also shows the two thyristors that conduct for each 60° interval. For example, at the start of the period between 0 and 60°, T₁ was triggered and T₆ was already on and continued to conduct. When there is a delay in firing, then all thyristors are triggered by the same nonzero firing angle, meaning that still the same pair of thyristors as in the case of no delay conduct simultaneously. In other words, the conducting interval that each pair of thyristors share is shifted in time. In Figure 20.13b and 20.13c the delay is represented by α (alpha), which is 30° and 50°, respectively.

Figure 20.13 Comparison of DC output with (a) 0, (b) 30°, and (c) 50° firing angle and (d) input AC.

For certain firing angles at time intervals when one of the thyristors in a pair turns off by natural commutation, only one thyristor conducts, and, therefore, there is no voltage across the load and no current in it. This is especially true for resistive loads because electricity cannot be stored in them. This has happened for the 50° firing angle in Figure 20.13. It is the reason for the blank intervals in the output DC voltage, shown in Figure 20.14c. Figure 20.14 is a complement to Figure 20.13, showing the DC output voltages and their average values for the three different firing angles.

Figure 20.14 Showing examples of the DC output and its ripple. (a) 0, (b) 30°, and (c) 50° firing angle.

The result of a delay in triggering thyristors in a rectifier is the reduction of the output voltage. As can be seen from Figures 20.13 and 20.14, the average output voltage becomes smaller as the firing angle is increased. In this way the output of a thyristor-based rectifier can be controlled, as desired, by adjusting the firing angle. The output voltage is proportional to the cosine of the firing angle. If the firing angle is represented by α, then we can write

In Equation 20.2, V_DC0 represents the DC voltage from the rectifier when α = 0. Equation 20.2 also implies that if α = 90°, there is no DC output (V_DC = 0).

Figure 20.14 also depicts the DC output of the two aforementioned firing angles compared with that of zero firing angle. It is evident from the figure that the ripple frequency is the same (6 times the line frequency), but the ripple percentage gets larger as the firing angle increases.

20.4.1 180° Conduction

Thyristors must be triggered sequentially on the basis of a specific scheme so that the voltages appearing in A, B, and C have a good resemblance to a three-phase alternating current with 120° phase difference between each two phases. In the triggering scheme in Figure 20.16, each thyristor is in the on state and conducts for 180°, followed by 180° in the off state. The time for this period (i.e., the time for retriggering the same thyristor) defines the frequency of the generated alternating current. For instance, if each thyristor is fired every 20 msec, then the frequency of the output AC is 1 ÷ 0.020 = 50 Hz. In this respect, for a frequency of 60 Hz, if this firing scheme is used, then each thyristor must be fired every 1 ÷ 60 = 0.01667 sec, or 16.67 msec.

Figure 20.15 Schematic of a three-phase six-step inverter circuit.

A closer look at the firing scheme in Figure 20.16 reveals that for each interval of 60°, three out of six switches are in the on state, but they alter; one turns off and another is turned on. However, the sequence of firing thyristors is always 1, 2, 3, 4, 5, and 6. Examples of on-off alteration are shown for 120° and 180° angles. At 120°, for instance, first T₆ must be turned off and then T₃ be turned on (not the reverse). Thus, for the period between 120° and 180° the thyristors T₃, T₂, and T₁ conduct, until T₁ (the last shown in sequence T₃, T₂, T₁) is turned off and T₄ (the first shown in the next sequence) is fired. For the next period (180° to 240°) the conducting thyristors are T₄, T₃, and T₂ (as shown between the dotted lines in Figure 20.16).

Figure 20.16 Firing scheme for 180° conduction.

If the firing scheme in Figure 20.16 is employed, then the voltages on all the three lines A, B, and C continuously jump from the positive side of the power supply to the negative side (between +V_DC and zero) and vice versa. The power supply voltage is always assumed constant. Table 20.1 summarizes the connections for terminals A, B, and C. In this table V and 0 represent the DC voltage and 0, as a result of connection to the positive side or negative side of the power supply, respectively.

A three-phase load is either star connected or delta connected to them, as shown in Figure 20.17. (For simplicity, the freewheeling diodes are not shown.) Figure 20.18 depicts the states of voltages at A, B, and C for the six different intervals of a 360° cycle and the current direction in the three branches of a Y- or Δ-connected load. In columns 3, 4, and 5 a + sign denotes connection to the positive side of the DC power supply and a – sign denotes connection to the negative side. In column 6 showing the currents a 0 indicates that there is no current between the two terminals for that 60° interval.

When the load is star connected, a neutral point is practically defined for the resulting three-phase system. The phase voltage can be found from the voltage difference of the neutral point N and any of the A, B, or C terminals. Figure 20.19 can help understand the voltage at point N with respect to the power supply at each interval of a cycle. The principle of a voltage divider is used for this purpose based on a balanced three-phase resistive load connected to the inverter output.

Figure 20.17 Δ- and star-connected loads to an inverter output.

Table 20.1 Thyristors that are on and Voltages at Terminals A, B and C at 60° Intervals for 180° Conduction

Figure 20.19 also shows the currents in the three lines for a resistive load, for Y and Δ connection. These currents are proportional to the voltages for this load and have the same waveform similar to the voltage variation. For inductive and capacitive loads, current variation is affected by the load reactance and is smoothed out, as was seen for the single-phase inverter.

The line (phase-to-phase) voltages V_A − V_B, V_B − V_C, and V_C − V_A, shown as V_A–B, V_B–C, and V_C–A, respectively, are presented in Figure 20.20. As can be seen, the three alternating currents of square waveform are 120° out of phase with respect to each other, and the voltage varies between plus and minus V_DC. This figure also illustrates the line to neutral voltage (phase voltage) variation denoted by V_A–N, V_B–N, and V_C–N. On the basis of the facts depicted in Figures 20.19 and 20.20 the phase voltage fluctuates between plus and minus 2V_DC/3, and the neutral line can be connected to the negative side of the power supply.

20.4.2 120° Conduction

In the preceding firing scheme each thyristor was on for 180° period and off for 180°. It is possible to use an alternative scheme with 120°conduction instead of 180°. In this case each thyristor conducts for 120° followed by an interval of 240° silence. At each interval of 60°, thus, only two thyristors (from two different legs) conduct. When a thyristor is on for a shorter time, less energy is consumed in it. This scheme is shown in Figure 20.21.

Figure 20.18 Details of voltages and currents for 180° conduction firing scheme in Δ- and Y-connected resistive loads.

Figure 20.19 Understanding the voltage at the neutral (reference) point in an inverter with 180° firing scheme.

For this firing scheme at each 60° interval of a cycle the two points that are connected to the power supply positive and negative terminals are shown in Table 20.2. Consider a Y-connected load to the resulting three-phase electricity. The neutral point N is at equal voltages for the two connecting terminals at a time. The voltages on each of the A, B, and C terminals vary between +V_DC and zero. Thus, a point at the midrange of the power supply can be considered the neutral for the three-phase system. (If this point is available, it can be physically used as the neutral.)

Figure 20.20 Voltage relationships in the generated three-phase waveforms from a six-step inverter with 180° firing.

Figure 20.21 A 120° conduction firing scheme.

Table 20.2 Thyristors that are on (two) and Voltages at Terminals A, B and C at 60° Intervals for 120° Conduction

Figure 20.22 Waveform of a six-step inverter with 120° firing scheme.

Figure 20.22 illustrates the variation of phase voltages, based on which the line voltages V_A–B, V_B–C, and V_C–A can be found and are also shown. As depicted in Figures 20.20 and 20.22, in both 180° and 120° conduction the maximum–minimum range for the output waveforms from an inverter is two times the DC supply voltage.

As can be understood from Table 20.2 or Figure 20.21, for this triggering scheme it is not necessary to turn off a thyristor before its counterpart is turned on. This is because the two thyristors to be switched (on or off) at the end of a 60° interval do not belong to the same leg.

Figure 20.23 depicts what happens during one cycle (360°) as far as the voltage and current change in Δ-connected and star-connected loads are concerned. In columns 3, 4, and 5 of this figure, + denotes connection to the positive side of DC power supply, – represents connection to the negative side of the power supply and, NC stands for “not conducting” (see Table 20.2). This is for a pure resistive load. The corresponding currents are shown in Figure 20.24, accordingly.

20.6.1 DC Voltage Regulation by Pulse Width Modulation

The principle of pulse width modulation is shown in Figure 20.26 for a simple case of obtaining a smaller desired DC voltage from a DC power supply. In Section 20.3 (see Figure 20.14) we learned that a DC output of a rectifier may have ripples and even blank intervals. Average DC voltage then depends on the variation of the voltage level, taking into account zero values for the blank intervals.

Figure 20.26 Principle of pulse width modulation. (a) Wider pulses lead to higher average compared to (b) with narrower pulses and lower average. (c) Pulse width can be found from position of MN in triangle LPQ.

In Figure 20.26a a repeated series of pulses of constant voltage and constant width (duration), all with the same polarity, are sent across a wire. The width of pulses and the frequency of sending them are important and define the properties of their effects. Pulse width represents its duration and is denoted by a in the figure. Frequency defines the elapsed time between sending the next pulse, which is denoted by b in the figure. Thus, the pulses are sent every b second, and each pulse has a second duration. By this definition, a < b, and its maximum value can be b. If the voltage of each pulse is V volts, then the average voltage in the wire is determined by a, b, and V, i.e.,

For example, if a is half of b and V = 15 V, then the average voltage is 7.5 V. In practice, a and b can be very small, say, a few milliseconds. Suppose that b = 5 msec, a = 2 msec, and V = 15 V. Then the frequency of pulses is 200 (1/0.005) Hz, and the voltage in the line is 6 V. If a is 0, then no pulses are sent and the (average) voltage is 0.

Figure 20.26c shows the relationship in Equation 20.3. Line MN represents a and can assume any position inside the triangle. When it coincides with PQ, a = b and the available voltage is equal to the supply voltage; when it is at point L, a = 0 and the output voltage is zero. Voltage can be represented by the distance of line PQ from point L.

In practice, the width of an electric pulse and its frequency can be controlled by electronic circuits. These pulses from a low-power function generator can trigger high-power devices to turn on and off and generate high-power pulses for the required applications. For instance, the low-power pulses can be applied to the gates of thyristors and control their outputs. Any thyristor to be used for this purpose must be of the type that can be turned on and off, as necessary.

20.6.2 PWM for AC Generation

Both single-phase and three-phase inverters can use pulse width modulation technique for their operation. There are many varieties of employing pulse width modulation for the operation of inverters. The basic principle is the same as was described for DC voltage control. In an inverter, as far as the output voltage is concerned, the thyristors conduct for short periods, thus generating a series of pulses, instead of conducting for 180° or 120° duration (remember the two schemes for inverter functioning). In this way, during any given period of time, thyristors conduct for much shorter intervals but more frequently. For example, in every 20 msec (corresponding to 50 Hz) instead of switching on and then off only once, they turn on and off 40 times or more.

For AC electricity, in addition to the voltage that must be controlled as required, the waveform and frequency need also to be looked after. The average output voltage variation must be as close to a sinusoidal waveform as possible, and the voltage and frequency must be as desired. Note that all the pulses have the same voltage intensity; only their width can vary.

Figure 20.27 shows two possible scenarios for pulses in a single-phase inverter intended for a sine wave output. The difference between them is in the plus and minus polarity change for the pulse train in Figure 20.27a, whereas in Figure 20.27b all pulses have the same polarity. In other words, in Figure 20.27a the neutral point of the resulting AC waveform is defined, whereas in Figure 20.27b it is floating. Note that in both cases the number of pulses per cycle is odd and they are symmetric with respect to the zero crossing point of the intended waveform.

The frequency of the pulses and width of each pulse need to be known for pulse width modulation. Undoubtedly, frequency of sending pulses must be considerably higher than the frequency of the desired output waveform to be generated. Switching action is required for each pulse, and there are energy losses for each switching. Therefore, the number of pulses must be compromised. For a frequency of 60 Hz for the waveform the pulse train can have a frequency of several times like 1000–2000 Hz.

20.6.3 PWM-Controlled Inverter

From the above discussion it can be understood that for pulse width modulation it is essential to define the width of the pulses to be generated. To determine the pulse width at each instant of time for generating a sinusoidal waveform with a desired frequency and voltage a so-called carrier, which is normally a triangular wave, is employed. This carrier is compared to the desired waveform with the required frequency. Then based on this comparison the duration of pulses are determined.

Figure 20.27 Pulse trains for generation of AC waveforms. (a) With positive and negative power supply, (b) with only one power supply.

In practice, all this process can be done offline by a computer. The results are, then, transferred to the actual device. Figure 20.28 can help to understand the modulation process. For better clarity the triangular waveform shown has a much smaller frequency than what happens in practice.

In Figure 20.28a a triangular carrier wave, to be modulated by a reference sine wave, and the modulating wave are plotted together. This reference wave is the desired sinusoidal AC to be generated by the inverter. The points of intersection of the two waves are numbered 1 to 10. As can be observed, for the interval between 1 and 2 the carrier wave is larger (greater magnitude) than the modulating wave, but for the interval 2 to 3 the modulating function has a larger magnitude. The latter is true for the intervals between 4&5, 6&7, and 8&9. Pulses to be generates are all defined by the points of intersection of the two waves, as follows. For those intervals that the carrier is larger than the modulating wave, there is no pulse, and for the intervals that the modulating wave is larger than the carrier, there is a pulse. For each pulse, width is defined by the horizontal spacing (time interval) between the two associated intersection points. On the basis of the scenario shown in Figure 20.28a the corresponding pulses are shown in Figure 20.28b.

Figure 20.28 Modulating a triangular wave to determine the pulse width for sinusoidal wave generation. (a) Determining the width for pulses. (b) The required pulses.

The pulses will be used to trigger the thyristors in an inverter. For a single-phase inverter it is easy to see which thyristors must be turned on or off. At each instant, one thyristor from one leg and its counterpart from the other leg must be turned on at the same time (see Figure 20.1). If the scheme in Figure 20.27b is used (no polarity change for pulses), it is possible to reduce the number of switching action. For this scheme it is possible to keep one of the thyristors on for the whole half cycle and only send pulses to its counterpart. This is shown in Figure 20.29. This method is not practicable for the scheme with polarity change pulses (Figure 20.27a).

Note that to generate the pulse train in Figure 20.29 (without polarity change) the same principle is used and the pulse widths are determined based on comparison of the modulating wave and a carrier wave without polarity. The carrier wave has a larger amplitude than the reference waveform (compare Figures 20.28a and 20.29); otherwise, not sufficient intersections can be found. For thyristor numbering, see Figure 20.1.

In practice, the two waves (the carrier and the modulator) can be compared (by a comparator circuit) to determine which one has a larger magnitude. If digital systems (or a digital computer) are used, then it is possible to put the results of the comparison in memory and send the pulses using a memory look-up table. Among other alternative ways to determine the pulse widths is employing mathematical functions that result in a waveform that better resembles a sine wave. Again, the results can be read from memory-stored data.

Figure 20.29 Continuous pulses without polarity and with less switching (refer to Figure 20.1).

Note that in using pulse width modulation both the amplitude ratio and the frequency ratio of the carrier wave to the reference (desired) wave affect the number and width of the pulses. These two parameters are called frequency modulation ratio and amplitude modulation ratio and are denoted by m_f and m_a, as

To see the effect of changing amplitude modulation ratio, see Figure 20.30 and compare the pulses in the two cases shown (focus on width of the pulses). For the effect of changing frequency modulation ratio, compare Figures 20.29 and 20.30. Count the numbers of pulses and the frequency of the carrier for one cycle of the sinusoidal waveform (10 in Figure 20.29 and 9 in Figure 20.30. Also, observe the symmetry). The value of m_f is always greater than 1; it is an odd multiplier of 3 for technical reasons, and its range of values can be between 21 and 35, for instance. The value of m_a is normally less than 1. If it has a magnitude larger than 1, it represents overmodulation, which causes distortion in the resulting signal generated from the pulses.

Figure 20.30 Effect of amplitude modulation ratio.

20.6.4 Pulse Width Modulation for Three-Phase Inverter

The PWM scheme depicted in Figures 20.27 and 20.28 for a single-phase inverter can be extended to three-phase systems. Because there is a 120° phase difference between the three phases, the same pattern for one phase repeats, only with a shift in time. In other words, the same scheme for pulse widths and timing can be used for each individual phase; only they must have a timing shift corresponding to 120° with respect to each other. Figure 20.31 shows the case.

The three pulse trains shown in Figure 20.31 can be used to trigger the gates of the six thyristors in a three-phase inverter. For three-phase system it is not possible to keep any of the thyristors turned on (like the case for a single-phase inverter); thus, a lot of switching is necessary. At each instant, one thyristor of each leg is on and the other is off. At each edge of every pulse (i.e., for both rising edge and falling edge) one thyristor in a leg of the inverter is turned off and the other is turned on, as shown for phase C in Figure 20.32. The thyristor numbers correspond to the arrangement in Figure 20.15.

Figure 20.31 A, Pulse trains for three phases using PWM; B, time intersection of pulses for phase A and their counterpart pulses for B and C.

Figure 20.32 Triggering thyristors for three-phase system inverters using PWM.

For a pulse width modulation inverter to work properly the frequency of the carrier wave must be sufficiently larger than the modulating wave frequency; that is, the modulation frequency ratio needs to be greater than 20. Also, overmodulation is not desirable; thus, the amplitude modulation ratio must be smaller than 1.