Analytical Calculation of Harmonics and Harmonic Losses in Five-Phase Carrier-Based PWM Voltage Source Inverters

The main purpose of this paper is to analyze a five-phase Voltage Source Inverter (VSI) that is operated with the Pulse Width Modulation (PWM) switching technique. Double Fourier integral (DFI) analysis has been used to extract the harmonics of the line-to-line voltages. Moreover, the harmonic current ripple has been calculated for an effective inductive five-phase load with a regular pentagon connection. Correspondingly, a new closed-form solution for calculating the harmonic losses in five-phase VSIs has been derived. In addition, a new equation for the weighted total harmonic distortion (WTHD) index has been extracted for five-phase VSIs. To validate the suggested analytical solutions (i.e., harmonic losses and WTHD equations), the results are compared with the DFI calculation method and the method of applying the fast Fourier transform (FFT) to a simulated waveform. The results show that the proposed analytical method has high accuracy and requires less mathematical effort, especially at high-frequency ratio values. Finally, the study includes simulation results and the implementation of an experimental setup.


I. INTRODUCTION
Over the past few decades, multi-phase motor drives (those with more than three phases) have been the subject of many studies owing to their specific advantages [1]- [9]. Multi-phase motor drives benefit from higher reliability, more power density, higher efficiency, lower torque ripple, and lower current ratings in the inverter arms [10], [11]. Furthermore, the utilization of multi-phase machines offers the capability of controlling more motors simultaneously with a single inverter [12]- [14]. Recently, with the development of semiconductor technology and digital processors, the industry has been trending toward multi-phase drives in medium-and high-power applications such as aircrafts, ships, traction applications, and electric vehicles.
The associate editor coordinating the review of this manuscript and approving it for publication was Qiuye Sun .
In the literature, most studies in the area of multi-phase systems have investigated multi-phase motors [15], [16] and their controlling schemes [3]- [14].
In the field of modulation techniques of multi-phase VSIs, a variety of studies have been reported in recent decades [4]- [6], [17]- [21]. However, some areas still require further investigation.
The most well-known modulation techniques for multiphase VSIs are the carrier-based PWM, the space vector modulation (SVM), and the selective harmonic elimination (SHE) techniques. The PWM method in multi-phase VSIs is an extension of the famous PWM strategy for three-phase VSIs. The PWM technique is less complicated than other switching methods and is independent of the phase numbers. These advantages become more significant in applications with a large number of phases, where other methods are more complex [21], [22].
Another phenomenon that has been witnesses in recent decades is the increased utilization of nonlinear loads such as power electronic converters which produce harmonics [23]. Harmonics have several disadvantages including equipment failure and malfunction, increasing core losses in machines and transformers, overheating and noise in machines and transformers, and vibration in machines. Moreover, harmonics might result in unexpected resonances and power system instability. Therefore, harmonic stability and resonance analysis have been widely researched. For stabilization studies, the line impedance cooperative stability region identification method has been proposed for grid-tied inverters under weak grids [24]. Another important issue caused by harmonics is the increased switching losses in power converters, which reduces their energy efficiency [25].
Due to the issues discussed above, the precise calculation of the harmonic components and determination of the harmonic losses are two significant issues in multiphase VSIs [25]- [27]. These issues become even more important in PWM-controlled VSIs used in high-power applications. However, the harmonic evaluation of a PWM waveform and determining the harmonic components is a complex issue [27]. A simple and general method is to apply the FFT to a computer simulated signal. Although this method requires fewer mathematical manipulations, the precision of the results is highly affected by the simulation's numerical round-off errors and detrimental effects arising from spectral leakage and picket fence effect phenomena. Moreover, this method requires high computing capacity. In contrast, analytical methods guarantee high precision extraction of harmonic components of the voltage signals in a PWM-controlled VSI [26], [27]. DFI analysis is a wellknown analytical solution for extracting the harmonics of PWM signals [25], [27].
Nevertheless, the harmonic losses of the load, which originate from the load current harmonic components, are an important part of the load losses. Therefore, extracting a closed-form equation to calculate the harmonic losses is of great interest in high-power industrial applications. In [25], the harmonic components of a three-phase VSI with carrier-based PWM have been extracted based on DFI analysis, and accordingly, the harmonic losses have been calculated.
Therefore, the main goal of this paper is to provide a more accurate description of a five-phase VSI operated with the PWM switching technique. As a result, this work concentrates on the analytical calculation of harmonic components and harmonic losses in five-phase inverters. This investigation has been carried out using simulations and experiments.
The paper is organized as follows. Section II discusses the DFI analysis of the five-phase PWM inverter. A new closed-form equation for calculating the harmonic losses of a five-phase VSI is presented in Section III. Then in Section IV, the WTHD factor is analytically computed for the five-phase VSI and compared with that of the three-phase VSI. In Sections V and VI, the simulation and experimental results of the five-phase VSI are presented, respectively. The main contributions of this paper are that it: • Presents the harmonic solution for the line-to-line voltages of a five-phase VSI.
• Provides a new closed-form equation for calculating the harmonic losses in a five-phase VSI with an effective inductive load.
• Develops a new closed-form equation for calculating the WTHD of a five-phase VSI. Fig. 1 shows a single-phase VSI that is controlled by comparing a sinusoidal reference and a triangular carrier. The harmonic solution of the phase voltage v an (t) can be extracted using DFI analysis as follows [25]:

II. DFI ANALYSIS OF A FIVE-PHASE PWM INVERTER
v an (t) where ω o and ω c are the fundamental angular frequency and carrier angular frequency, respectively. Note that m f = ω c /ω o is the frequency modulation index and M is the amplitude modulation index. θ o and θ c are the initial phases of the fundamental and carrier waveform, respectively. m and n correspond to the carrier harmonics and the sideband harmonics, respectively. In addition, J n is the n-th order Bessel Function of the first kind. Fig. 2 represents the configuration of a five-phase PWM inverter. To generate the switching pulses of the inverter, the PWM technique with five sinusoidal modulating references and a triangular carrier has been applied, as shown in Fig 3-(a).
In Fig. 3-(b)-(c), the inverter phase voltages (v an and v cn ) and the line-to-line output voltage (v an ) are depicted (p.u. of V dc ), respectively.
The vector diagram of the five-phase VSI is shown in Fig. 4-(a). It is clear that in a five-phase VSI, the maximum line-to-line voltages that can be generated are v ac (t), v bd (t), v ce (t), v da (t), and v eb (t). It can be proved that − → v ac = 2 sin(2π/5) − → v az ≈ 1.902 − → v az . Consequently, to achieve maximum line-to-line voltages in a five-phase VSI, the load should be connected like a regular pentagon with an appropriate configuration, as shown in Fig. 4-(b).
In the following paragraphs, the analytical solution of VSI phase voltages will be developed. Then, using the Fourier series of phase voltages, the harmonic spectrum of the lineto-line voltages will be derived.
By considering θ c = 0 in (1) and applying a phase shift of 2π /5 to the reference signal, the phase voltages   v an (t) . . . , v en (t) can be expressed as follows: v an (t) = V dc + MV dc cos(ω o t) cos(mω c t + nω o t) (2) As mentioned above, with respect to the vector diagram shown in Fig. 4-(a), the line-to-line voltages (v ac (t), v bd (t), v ce (t), v da (t), and v eb (t)) are considered and calculated in this work. By considering (2)-(4) and using trigonometric relations, the line-to-line voltages will be computed. Details of these calculations are given in the Appendix. The maximum line-to-line voltages can be expressed as: . . .
The maximum line-to-line voltages for three-phase and seven-phase VSIs can be calculated in the same way. The results are represented in Table 1 of the Appendix.
In Fig. 5, the dominant harmonic components of the lineto-line voltage v ac are given as a function of M in the fivephase VSI.  It is clear that the harmonic amplitudes in the line-to-line voltage spectrums achieved from the analytical and simulation methods are relatively close. However, the simulation method takes more time, and the precision of its results is highly affected by numerical round-off errors and detrimental effects arising from spectral leakage and the picket fence effect.

III. LOSSES CALCULATION
To determine a closed-form formula for calculating the harmonic losses in five-phase VSIs, correct and logical assumptions must first be made. Therefore, some indispensable simulation results, which help evaluate the waveform variations more precisely and make reasonable assumptions, are initially provided.
In Fig. 8, the line-to-line voltage (v ac ), its fundamental component, and the corresponding harmonic current ripple, i ac , are represented for m f = 21, M = 0.8 with an effective inductive load. It can be seen that the load current has a high-frequency ripple that is twice the switching frequency (2f c ).
In Fig. 9-(a)-(b), the inverter phase voltages (v an and v cn ) and the triangular carrier are represented in an arbitrary switching period, T = 1/f c . In Fig. 9-(c), the inverter line-to-line voltage (v ac ), the harmonic current ripple ( i ac ), and the average load EMF (e ac ) are shown in an arbitrary switching period. It is evident that the average load EMF, e ac , can be assumed to be constant during the switching period T . Moreover, for an effective inductive load, the current ripple, i ac , has linear variations.
In Fig. 10, the current ripple ( i ac ) and the five-phase line-to-line voltages (v ac (t), v bd (t), . . . , v eb (t)) are depicted in an arbitrary switching period T . It is clear from the figures that every line-to-line voltage waveform has two active pulses in each switching period T , and that each of the pulses occurs in half of the cycle of the switching period T . However, as shown in Fig. 10, the active pulses are not necessarily located in the middle of the half cycles T /2.
Based on the above discussion, in Fig. 11, an expanded sketch of Fig. 8 over an arbitrary switching interval T is represented using the following assumptions: 1) The average load EMF, e ac , is assumed to be constant during the switching period T = 1/f c .
2) The ohmic loss of the fundamental current component, which does not affect the harmonic losses, is not considered in the upcoming discussion.
3) It is supposed that the five-phase load is effective inductive, which leads to linear variations of the current ripple, i ac .
4) The active pulses in the output line-to-line voltages are not located in the middle of the half cycles T /2. According to Fig. 11, the current ripple i ac , which is the difference between the actual current and the fundamental component, can be calculated for all the periods T 1 , T 2 , and T 3 as follows: For 0 ≤ t ≤ T 1 : where L σ is the load inductance and u 1 − u 2 = e ac /V dc , with u 1 = e an /V dc and u 2 = e dn /V dc . Furthermore, it can be proved that: VOLUME 10, 2022 The average squared current ripple i 2 ac (t) can be defined as follows: Substituting (8)-(10) into (14) and simplifying gives: On the other hand, from the modulation strategy of the fivephase inverter, it can be shown that: Substituting (16)- (17) into (15) and manipulating the equation, results in: −8M 3 sin 3 2π 5 cos 3 (θ + π 10 ) −2M 4 sin 2π 5 cos(θ + π 10 ) The harmonic current ripple can be defined as: Accordingly, the harmonic losses can be written as: where R e is the load resistance. It should be noted that the above integration is only valid for e ac ≥ 0, which leads to: e ac ≈ (v ac ) 1 = 2 sin 5MV dc cos(ω 0 t + π 10 ) 37334 VOLUME 10, 2022  The valid integration interval is thus given by: Using (22), the average squared harmonic current ripple can be written as: Substituting (18) into (23) and simplifying gives:   Consequently, by substituting (24) into (20), the harmonic loss expression can be calculated as: where f (M ) = 2M 2 sin 2 2π 5 − 32 3π M 3 sin 3 2π 5 + 3 2 M 4 sin 2 2π 5 . Note that for a three-phase VSI, f (M ) can be written as [25]: Fig. 12 represents f (M ) as a function of M for the five-and three-phase VSI. It is clear that for higher M values, the harmonic loss in the five-phase VSI is lower than the threephase VSI. The harmonic loss for three-phase and sevenphase VSIs can be expressed in the same way. The results are given in Table 1 in the Appendix.

IV. WTHD CALCULATION
For a voltage signal, the WTHD0 and WTHD can be defined as: where V n is the magnitude of the n-th harmonic in the lineto-line voltage and V 1 is the fundamental component magnitude of the line-to-line voltage at M = 1 and is equal to 2 sin(2π/5)V dc ≈ 1.902V dc .
The magnitude of the current harmonics can be determined from the line-to-line voltage harmonics as follows: Accordingly, the Root-Mean-Square (RMS) value of the harmonic current can be written as: Substituting (29) into (30) gives: Or alternatively: Substituting (32) into (27) leads to:  Further substituting (24) into (33) and simplifying, considering T = 1/f c and ω o = 2πf o , gives: where f c /f o is the frequency modulation index m f . In Fig. 13, the WTHD is plotted according to M for different values of m f (m f = 9, 11,13,15,17,19,21). As expected, the WTHD decreases as the m f value increases.
Furthermore, the WTHD in five-phase and three-phase VSIs with respect to M is represented for m f = 15 in Fig. 14. It is clear that the WTHD of the five-phase VSI is smaller than that of its three-phase counterpart, and this difference increases at higher M values.

V. SIMULATION RESULTS
To confirm the accuracy of the derived equations, several tests have been performed under various operating conditions. The simulation of the complete five-phase VSI has been performed using PSCAD/EMTDC software. The five-phase VSI supplies a resistive-inductive load with a regular pentagon connection, according to Fig. 4-(b).
The simulation parameters are compatible with the experimental setup discussed in the next section. The load has a resistance of 10 and an inductance of 20 mH in each phase. The dc-link voltage is equal to 40 V. Some of the simulation results are presented below.
In Fig. 15, the line-to-line voltage v ac , the line current i a , and the phase current i as of the five-phase PWM inverter are illustrated for M = 0.8 and m f = 9. It is evident that the ratio of the line current magnitude to the phase current magnitude is equal to 2sin(2π/5) ≈ 1.902.
In Fig. 16, the harmonic losses of the five-phase VSI have been extracted and represented using the analytical method and the DFI analysis method. The harmonic losses are depicted along with the amplitude modulation index M for different values of m f (m f = 9, 11, 13, 15, 17, 19, 21).
In the analytical method, the harmonic losses are calculated using the closed-form equation (25) derived from the harmonic current ripple. In the DFI analysis approach, the voltage harmonic components are first extracted using (5)- (7). Then, to compute the current harmonic components, each voltage component is divided by its corresponding impedance value. Subsequently, the RMS values of the current harmonic components can be extracted. Finally, the harmonic losses are calculated from the summation of the ohmic losses of all harmonic components.
The results from the analytical method are clearly in accordance with those of the DFI-based analysis method, which proves the precision of the former. However, at lower m f values, the accuracy of the analytical method will decrease because its required assumptions are not fully met.
To achieve a more complete evaluation, the normalized value of the harmonic losses with respect to the current loss of the fundamental component should be considered. Fig. 17 demonstrates the normalized harmonic losses of the five-phase VSI according to M for different values of m f (m f = 9, 11, 13, 15, 17, 19, 21) for both the analytical and the double Fourier integral based analysis methods.
In Fig. 18, the harmonic losses and normalized harmonic losses of the five-phase VSI (with m f = 9) have been represented and compared using three different methods:

• Analytical method (closed-form equation)
• DFI analysis method • The method of applying the FFT to a simulated waveform In the method of applying the FFT to a simulated waveform, the first step is to generate the harmonic current waveform of the VSI using PSCAD/EMTDC software. Then, the harmonic components of the simulated current waveform are extracted by applying the FFT with the frequency resolution of 5 Hz. Subsequently, the harmonic losses are calculated from the summation of the ohmic losses of all harmonic components. This process should be repeated for different m f and M values. Therefore, the outcome is a set of distinct values and is thus discontinuous.
In order to make a more valuable comparison, the error and the calculation times of each of the three methods are provided in Tables 2-4 in the Appendix. For example, for m f = 9 and a fixed set of M values, the average error of the harmonic loss calculation in the analytical method is 2.85%. Therefore, as compared to the simulation method, which has an error of 6.42%, the analytical method provides greater accuracy in calculating the harmonic loss. Moreover, using the analytical method for the same set of M values, the average error of the harmonic loss calculation for VOLUME 10, 2022  m f = 21 is 1.09%, which demonstrates that the accuracy increases in higher frequency ratio values. Unlike the DFI analysis method and the simulation and FFT method, the proposed closed-form equation for harmonic losses requires no simulation or program writing, and it has considerably fewer mathematical operations. Fig. 19 shows the WTHD0 results of the five-phase VSI in terms of the amplitude modulation index M for different values of m f (m f = 9, 11, 13, 15, 17, 19, 21) using both the analytical method and the DFI analysis method. It is evident that the results from the analytical method are in accordance with the double Fourier integral analysis method. This proves the precision of the analytical method, especially at high m f values. In Fig. 20, the harmonic losses, normalized harmonic losses, and current WTHD of the seven-phase, five-phase, and three-phase VSIs are represented with respect to M for m f = 9.
As can be seen in the figures, by increasing the phase numbers, the harmonic losses and WTHD decrease, and the difference becomes more significant in high amplitude modulation index values.

VI. EXPERIMENTAL RESULTS
To verify the simulation results, a five-phase inverter setup has been built utilizing PM50CTJ060 600V/50A Mitsubishi IPMs. The five-phase VSI is supplied through a dc power supply. A Gate driver board has been designed to fire the IGBT switches. The 74HCT244N Buffer IC has been utilized for amplifying the current values, and TLP521 Photo-couplers have been used for isolating the power circuit and the DSP board. In addition, a protection circuit has been added to  the Gate driver board; it prevents simultaneous conduction of up and down switches by controlling the enable pin of the Buffer IC using 74ALS08 NAND and 74HC4078 OR ICs. The PWM switching technique is programmed with the Code Composed Studio (CCS). It is implemented utilizing a TMS320F2812 Digital Signal Processor (DSP) board and connected to a PC through an XDS100 Emulator to compile and load the programs. An Instek GDS-820 digital oscilloscope has been utilized to monitor the results. Furthermore, the current and voltage waveforms are measured using LA55/P and LV25P/SP2 LEM sensors, respectively. Moreover, the waveforms are sampled utilizing an Advantech portable data acquisition module, USB-4711A, with a sampling rate of 10 kHz.
Photographs of the implemented five-phase VSI experimental setup and the five-phase resistive-inductive load with a regular pentagon connection are shown in Figs  The experimental results are compatible with the simulation results, which validates the accuracy of the simulation results. The negligible differences between the two are due to the diodes' forward voltage drops and the small imbalance in the five-phase loads. Fig. 24 shows the v ac line-to-line voltage frequency spectrum with respect to the fundamental in the five-phase VSI using the analytical method, simulation method, and experimental results for M = 0.8 with m f = 7. The simulation parameters, such as snubbers and the turn-on delay time The results show negligible differences in the magnitude of the harmonics in the frequency spectrums. However, the harmonic orders of the spectrum in the simulation and experimental results correlate closely with that of the analytical method.

VII. CONCLUSION
Due to the importance of power quality evaluation and the calculation of harmonic losses of multi-phase VSIs in high-power applications, this paper focuses on the analysis of harmonics and harmonic losses in five-phase VSIs with PWM switching schemes. In this paper, using the DFI analysis, the harmonic content of the line-to-line voltages in the five-phase PWM inverter has been extracted. Accordingly, new closed-form solutions for calculating harmonic losses and the WTHD index have been derived for the five-phase VSI with a regular pentagon load connection.
The results show that the harmonic loss in the five-phase VSI is lower than that of the three-phase VSI for high amplitude modulation index values. Also, the WTHD of the five-phase VSI is smaller than that of its three-phase counterparts, and this difference increases at higher amplitude modulation index values. Moreover, the extracted closed-form solutions require less calculation effort and are sufficiently accurate, especially at high-frequency ratio values. For m f = 9 and a fixed set of M values, the average error of harmonic loss calculation in the analytical method is 2.85% whereas the simulation method has an error rate of 6.42%; the analytical method thus calculates the harmonic loss more accurately. In addition, for the same set of M values, the average error of harmonic loss calculation in the analytical method for m f = 21 is 1.09%, which indicates that the accuracy increases in higher frequency ratio values. Unlike the DFI analysis method and the simulation and FFT method, the proposed closed-form equation for harmonic loss requires neither simulation nor program writing; it also involves considerably fewer mathematical operations.

APPENDIX
The phase voltages of the five-phase VSI can be written as:  The following trigonometry formulas are considered: