1Pulse width modulation techniques
In power electronic converters, the electrical energy from one level of voltage/ current/ frequency is converted into another using semiconductor-based electronic switch. The essential characteristic of these types of circuits is that the switches are operated only in one of two states – either fully ON or fully OFF – unlike other types of electrical circuits where the control elements are operated in a linear active region.
As the power electronics industry developed, various families of power electronic converters have evolved, often linked by power level, switching devices, and topological origins [3]. Application areas of power converters improved vastly in semiconductor technology, which offer higher voltage and current ratings as well as better switching characteristics. Meanwhile, the main advantages of modern power electronic converters are high efficiency, low weight, small dimensions, fast operation, and high-power densities.
The process of switching the electronic devices in a power electronic converter from one state to another is called ‘modulation’. Each family of power converters has preferred modulation strategies associated with it that aim to optimize the circuit operation for the target criteria most appropriate for that family. Parameters such as switching frequency, distortion, losses, harmonic generation, and speed of response are typical of the issues that must be considered when developing modulation strategies for a particular family of converters. The output voltage of power inverter should be a pure sinusoidal waveform with minimum distortion. However, for practical inverters, the output voltage is a series of rectangular waveforms. The major issues for the control of the power inverters are to get suitable modulation methods to control the output rectangular waveforms to synthesize the desired waveforms. Therefore, a modulation control method is required to get a desired fundamental frequency voltage and to eliminate higher-order harmonics as much as possible.
In modern converters, pulse width modulation (PWM) is a high-speed process ranging depending on the rated power from a few kilohertz (motor control) up to several megahertz (resonant converters for power supply). Therefore, first, we discuss about the principle and different topologies regarding PWM.
The PWM technique is one of the most widely used strategies for controlling the ac output of power electronic converter. In this technique, the duty cycle of converter switches can be varied at a high frequency to achieve a target average low frequency output voltage or current. Modulation theory has been a major research area in power electronics for over three decades and continues to attract considerable attention and interest.
In principle, all modulation schemes aim to create trains of switching pulses that have the same fundamental volt-second average as a target reference waveform at any instant. The major difficulty with these trains of switched pulses is that they also contain unwanted harmonic components that should be minimized. Hence, for any PWM scheme, the primary objective can be identified, which is to calculate the converter switching ON times, which creates the desired (low-frequency) target output voltage or current. Having satisfied this primary objective, the secondary objective for a PWM strategy is to determine the most effective way of arranging the switching process to minimize unwanted harmonic distortion, switching losses, or any other specific performance criterion [7].
The dc input to the inverter is chopped by switching devices in the inverter. The amplitude and harmonic content of the ac waveform is controlled by varying the duty cycle of the switches. The fundamental voltage V1 has a maximum amplitude of 4 Vd/π for a square wave output, but by creating notches, the amplitude of V1 is reduced.
Usually, the power switches in one inverter leg are always either in ON or OFF state. Therefore, the inverter circuit can be simplified into 3 two-position switches. Either the positive or the negative dc bus voltage is applied to one of the motor phases for a short time. PWM is a method whereby the switched voltage pulses are produced for different output frequencies and voltages. A typical modulator produces an average voltage value, equal to the reference voltage within each PWM period. Considering a very short PWM period, the reference voltage is reflected by the fundamental of the switched pulse pattern. The concept of pulse width modulation is shown in Fig. 1.1.
Fig. 1.1: Principle of pulse width modulation.
There are several different PWM techniques, differing in their methods of implementation. However, in all these techniques, the aim is to generate an output voltage, which after some filtering, would result in a good-quality sinusoidal output voltage waveform of desired fundamental frequency and magnitude. However, in the case of inverters, it may not be possible to reduce the overall voltage distortion due to harmonics, but by proper switching control, the magnitudes of lower-order harmonic voltages can be reduced, often at the cost of increasing the magnitudes of higher order harmonic voltages. Such a situation is acceptable in most cases, as the harmonic voltages of higher frequencies can be satisfactorily filtered using lower sizes of filters and capacitors. Many of the loads, like motor loads, have an inherent quality to suppress high-frequency harmonic currents, and hence, an external filter may not be necessary. To judge the quality of voltage produced by a PWM inverter, a detailed harmonic analysis of the voltage waveform needs to be done.
In fact, after removing a third and multiples of third harmonics from the pole voltage waveform one obtains the corresponding load phase voltage waveform. The pole voltage waveforms of three-phase inverter are simpler to visualize and analyze, and hence, the harmonic analysis of load phase and line voltage waveforms is done via the harmonic analysis of the pole voltages. It is implicit that the load phase and line voltages will not be affected by the third and multiples of third-harmonic components that may be present in the pole voltage waveforms.
1.2Basic pulse width modulation techniques
1.2.1Single pulse width modulation
In single PWM control, the width of the pulse is varied to control the inverter output voltage, and there is only one pulse half per cycle. By comparing the rectangular reference signal with the triangular carrier wave the gating signals are generated, as shown in Fig. 1.2. The frequency of reference signal determines fundamental frequency of the output voltage.
The advantages of this technique are that the even harmonics are absent due to the symmetry of the output voltage along the x-axis and that the Nth harmonics can be eliminated from inverter output voltage if the pulse width is made equal to 2π/n. However, the disadvantages are that the output voltage introduces a great deal of harmonic content and that at a low output voltage, the distortion factors increases significantly.
1.2.2Multiple pulse width modulation
In multiple PWM, several equidistant pulses per half cycle are generated, as shown in Fig. 1.3. Using several pulses in each half cycle of the output voltage, the harmonic content can be reduced.
In this technique, the amplitudes of lower-order harmonics are reduced and the derating factor is reduced significantly. However, the fundamental component of the output voltage is less, the amplitudes of higher-order harmonics increases significantly, and switching losses are increased.
Fig. 1.2: Single pulse width modulation.
1.2.3Sinusoidal pulse width modulation
In many industrial applications, to control the inverter output voltage sinusoidal PWM (SPWM) is used. SPWM provides good performance of the drive in entire range of operation between 0 and 78% of the value that would be reached by square wave operation. If the modulation index exceeds this value, the linear relationship between the modulation index and the output voltage is not maintained, and over-modulation methods are required. The SPWM refers to the generation of PWM outputs with sine wave as the modulating signal. In this modulation method, the ON and OFF instances of the PWM signals can be determined by comparing a reference signal with a high-frequency triangular wave, as shown in Fig. 1.4. The frequency of the output voltage can be determined by the frequency of the modulation wave. The peak amplitude of the modulating wave determines the modulation index and in turn controls the RMS value of the output voltage. When the modulation index is changed, the RMS value of the output voltage also changes. This technique improves the distortion factor significantly compared to other ways of multiphase modulation. It eliminates all harmonics less than or equal to (2n − 1), where n is defined as the number of pulses per half cycle of the sine wave. The output voltage of the inverter contains harmonics. However, the harmonics are pushed to the range around the carrier frequency and its multiples.
Fig. 1.3: Multiple pulse width modulation.
Fig. 1.4: Sinusoidal pulse width modulation.
where (VA0)1 is the fundamental frequency component of the pole voltage VAO. The frequency modulation ratio (mf), which should be an odd integer, is the ratio between the PWM frequency and the fundamental frequency.
i.If mf is not an integer, subharmonics may exist at output voltage.
ii.If mf is not odd, dc component may exist and even harmonics are present at output voltage.
iii.mf should be a multiple of 3 for three-phase PWM inverter.
iv.An odd multiple of three and even harmonics are suppressed.
1.3Advanced modulation techniques
By comparing a triangular carrier wave (Vc) with a reference trapezoidal wave (Vr), the switching instance to semiconductor devices are generated as shown in Fig.1.5. This type of modulation increases the peak fundamental output voltage up to 1.05 Vd, but output voltage contains lower-order harmonics.
In staircase PWM, the modulated wave eliminates specific harmonics. To obtain the desired quality of the output voltage, the modulation frequency ratio mf and the number of steps are chosen as shown in Fig. 1.6. If the number of pulses is less than 15 per half cycle, this is optimized PWM.
Fig. 1.5: Trapezoidal modulation.
In this modulation, the signal is stepped wave. To control the magnitude of the fundamental component and to eliminate specific harmonics, this wave is divided into specific intervals, with each interval being controlled individually as shown in Fig. 1.7. When compared to that of normal PWM control, this type of control gives low distortion but higher fundamental amplitude.
1.3.4Harmonic-injected modulation
In this modulation, the signal is generated by injecting harmonics to the sine wave as shown in Fig. 1.8. The result is a flat topped wave form and it reduces the amount of over modulation. A higher fundamental amplitude and low distortion of the output voltage is provided. The amplitude of fundamental components is approximately 15% more than that of normal SPWM.
Fig. 1.6: Staircase modulation.
In this modulation, a triangular wave is allowed to oscillate within a defined window Δv above and below the reference wave Vr. From the vertices of the triangular wave Vc, the output voltage is generated as shown in Fig. 1.9. This type of modulation is also known as hysteresis modulation. If the frequency of modulating wave is changed while keeping the slope of the triangular wave constant, the number of pulses and pulse widths of modulated wave would change. The fundamental output voltage can be up to Vs and is dependent on peak amplitude Ar and frequency fr of the reference voltage. This modulation can control the ratio of voltage to frequency. Depending on the permissible harmonic content in the inverter output voltage, machine type, power level, and semiconductor switching devices employed for a particular application, the particular PWM is chosen.
1.3.6Space vector pulse width modulation
This modulation is a relatively new and popular technique in controlling motor devices. In the space vector PWM (SVPWM) method, the output voltage is approximated using the nearest three output vectors that the nodes of the triangle containing the reference vector in the space vector diagram of the inverter. When the reference vector changes from one region to another, it may induce an abrupt change in the output vector. In addition we need to calculate the switching sequences and switching time of the states at every change of the reference voltage location.
Fig. 1.7: Stepped modulation.
1.4Advantages of pulse width modulation techniques
i.Using PWM techniques, lower-order harmonics can be eliminated or minimized along with its output voltage control. The filtering requirements are also minimized.
ii.Both output voltage control and frequency control are possible in a single power stage of the inverter without any additional components.
iii.The presence of constant dc supply permits the parallel operation of several independent PWM inverters on the same rectifier power supply. The PWM inverter has a transient response that is much better than that of a quasi-square wave rectifier.
iv.The commutative ability of PWM inverters remains substantially constant compared to variable dc-link inverter, irrespective of the voltage and frequency settings.
v.The power factor of the system is good, as a diode rectifier can be employed on the line side.
vi.With constant dc supply used in PWM, we can obtain commutation even at low voltage, whereas a six-step inverter needs an auxiliary dc supply for commutating thyristors at low output voltages.
vii.The amplitude of the torque pulsations are minimized even at low speeds.
viii.A sophisticated PWM technique eliminates lower-order harmonics in the motor current, low-speed torque pulsations, and cogging effects.
Fig. 1.8: Harmonic-injected modulation.
In this chapter, the necessity of PWM in various power electronic converters is presented in detail as well as the basic and advanced PWM techniques. The advanced modulation techniques such as trapezoidal modulation, staircase modulation, stepped, harmonic-injected modulation, delta modulation, and SVPWM are specially recommended for multilevel inverters of various topologies to reduce the THD, dv/dt effect, switching frequency, and switching losses. SVPWM is more robust than THD because of its flexibility in redundant state selection in case of higher levels. Finally, the advantages of PWM techniques have been discussed to give a clear idea of PWM.
Fig. 1.9: Delta modulation.