# Analysis of Peak-to-Peak Current Ripple Amplitude in Seven-Phase PWM Voltage Source Inverters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Evaluation of Peak-to-Peak Current Ripple Amplitude

#### 2.1. Load Model and Current Ripple Definitions

_{s}leads to:

_{dc}, as shown in Figure 2 (specific values of voltage levels for the 7-phase VSI are given below in Figure 5 and Figure 6).

_{g}(t) is almost sinusoidal. For these reasons, the expression of alternating voltage component can be simplified as:

#### 2.2. Multiple Space Vectors and PWM Equations

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}, x

_{6}, x

_{7}}, the three space vectors

**x**

_{1},

**x**

_{3}, and

**x**

_{5}lie in the three planes α

_{1}–β

_{1}, α

_{3}–β

_{3}and α

_{5}–β

_{5}, respectively, and are expressed as:

**α**= e(j 2π/7) and x

_{0}the zero-sequence component, always null in case of balanced systems.

_{dc}, the output space voltage vectors can be written as function of the 7 switching leg states S

_{k}= [0,1] as:

_{1}–β

_{1}, α

_{3}–β

_{3}and α

_{5}–β

_{5}are given in Figure 3.

**Figure 3.**Space vector diagrams of inverter output voltage in the planes (

**a**) α

_{1}–β

_{1}; (

**b**) α

_{3}–β

_{3}; and (

**c**) α

_{5}–β

_{5}.

**v**

_{1},

**v**

_{3}, and

**v**

_{5}in every switching period T

_{s}. In the case of symmetrical SV-PWM, the sequence is determined in T

_{s}/2 and it is repeated symmetrically in the next half of the switching period [10]. By equally sharing the application time of the zero voltage vector between the null switch configurations 0000000 and 1111111, the so-called “centered” switching pattern is realized and a nearly-optimal modulation able to minimize the RMS of current ripple is obtained, as in the case of 3-phase [28] and 5-phase inverters [15]. This SV-PWM provides the same switching pattern such as the CB-PWM when a “min/max centering” common-mode voltage is injected into the modulating signals [10,29]. As result of the SV-PWM, for each phase, the average of the inverter output voltage $\overline{v}({T}_{s})$ corresponds to the reference voltage v

^{*}.

^{*}/V

_{dc}the reference space voltage vectors become:

**Figure 4.**Space vector diagram of inverter output voltage on plane α

_{1}–β

_{1}in the range θ = [0, 90°]. Outer dashed circle is modulation limit, m

_{max}≈ 0.513. Different colored areas correspond to different equations for determining the current ripple.

_{s}/2 as [10]:

_{max}≈ 0.513, according to the generalized expression given in [30] for n phases, m

_{max}= [2cos(π/2n)]−

^{1}.

#### 2.3. Ripple Evaluation

#### 2.3.1. Evaluation in the First Sector

_{max}cosθ < 4/7. All this sub-cases are represented in Figure 5a.

_{0}, leading to:

**Figure 5.**Output voltage and current ripple in a switching period. (

**a**) For sector ①, 0 ≤ θ ≤ π/7; and (

**b**) for sector ②, π/7 ≤ θ ≤ 2π/7.

_{0}= t

_{0}/T

_{s}/2:

_{0}/2 and t

_{6}, leading to:

_{k}= t

_{k}/T

_{s}/2, the normalized current ripple becomes:

_{0}/2, t

_{6}, and t

_{5}, leading to:

_{max}cosθ <4/7 (gray area in Figure 4) is depicted in diagram ➍ of Figure 5a. In this case ${\tilde{i}}_{pp}$ can be evaluated considering the switch configurations {7F}, {7B}, {73}, and {71} with the corresponding application intervals t

_{0}/2, t

_{6}, t

_{5}, and t

_{4}, leading to:

#### 2.3.2. Evaluation in the Second Sector

_{max}cosθ < 4/7. All these sub-cases are represented in Figure 5b.

_{0}/2 and t

_{6}:

- In the first situation, yellow area (solid orange line in Figure 5b), ${\tilde{i}}_{pp}$ can be determined as in the previous sub-case by considering the switch configurations {00} and {20} with the corresponding application intervals t
_{0}/2 and t_{6}, leading to Equations (33) and (34); - In the second situation, green area (dashed orange line in Figure 5b), ${\tilde{i}}_{pp}$ can be determined by considering the switch configurations {7F} and {7B} with the corresponding application intervals t
_{0}/2 and t_{1}, leading to:$${\tilde{i}}_{pp}=\frac{2}{L}\left\{m\hspace{0.17em}{V}_{dc}cos\theta \frac{{t}_{0}}{2}+\left(m\hspace{0.17em}{V}_{dc}cos\theta -\frac{1}{7}{V}_{dc}\right)\hspace{0.17em}{t}_{1}\right\}$$

- In the first situation, yellow area (solid green line in Figure 5b), ${\tilde{i}}_{pp}$ can be determined as in the previous sub-case by considering the switch configurations {00} and {20} with the corresponding application intervals t
_{0}/2 and t_{6}, leading to Equations (33) and (34); - In the second situation, violet area (dashed green line in Figure 5b), ${\tilde{i}}_{pp}$ can be determined by considering the switch configurations {7F}, {7B}, and {79}, with the corresponding application intervals t
_{0}/2, t_{1}, and t_{2}, leading to:$${\tilde{i}}_{pp}=\frac{2}{L}\left\{m\hspace{0.17em}{V}_{dc}cos\theta \frac{{t}_{0}}{2}+\left(m{V}_{dc}cos\theta -\frac{{V}_{dc}}{7}\right){t}_{1}+\left(m{V}_{dc}cos\theta -\frac{2{V}_{dc}}{7}\right){t}_{2}\right\}$$

_{max}cosθ <4/7 (red area in Figure 4) is depicted in diagram ➍ of Figure 5b. According to this figure, ${\tilde{i}}_{pp}$ can be evaluated considering the switch configurations {7F}, {7B}, {79}, and {71} with the corresponding application intervals t

_{0}/2, t

_{1}, t

_{2}, and t

_{3}, leading to:

#### 2.3.3. Evaluation in the Third Sector

_{max}cosθ < 3/7, represented in diagrams ➊, ➋, ➌ of Figure 6a. It can be noted that there are not sub-cases, and for all the three ranges ${\tilde{i}}_{pp}$ can be evaluated considering the switch configurations {00}, {20}, and {30} with the corresponding application intervals t

_{0}, t

_{1}, and t

_{2}, leading to:

**Figure 6.**Output voltage and current ripple in a switching period. (

**a**) For sector ③, 2π/7 ≤ θ ≤ 3π/7; and (

**b**) for sector ④, 3π/7 ≤ θ ≤ π/2.

#### 2.3.4. Evaluation in the Fourth (Half) Sector

_{max}cosθ < 1/7, as depicted in Figure 6b. In this case, ${\tilde{i}}_{pp}$ can be evaluated considering the switch configurations {00}, {10}, {30}, and {38} with the corresponding application interval t

_{0}/2, t

_{6}, t

_{5}, and t

_{4}, leading to:

#### 2.4. Peak-to-Peak Current Ripple Diagrams

_{max}), corresponding to the dashed circles in Figure 4. The four ranges corresponding to the four sectors from ① to ④ are emphasized. The further sub-regions in sector ② (green-, violet-, and red-colored areas in Figure 4) can be distinguished for m = 2/7 and 3/7.Figure 7b shows the colored map of r(m, θ) for the first quadrant within the modulation limits. It can be noted that ripple amplitude is obviously zero for m = 0, since the null configurations are the only applied, increasing almost proportionally with m in the neighborhoods of m = 0. A phase angle with minimum ripple can be indentified in the range θ ≈ 30°/35°. A phase angle with maximum ripple is θ = 90°, with ripple amplitude proportional to modulation index: r(m, 90°) = 0.626 m, resulting from Equation (44). This aspect is further developed in the following sub-section.

**Figure 7.**Diagrams of the normalized peak-to-peak current ripple amplitude r(m, θ). (

**a**) As a function of the phase angle θ in the range [0, π/2], for different modulation indexes; and (

**b**) colored map in the space vector plane α–β within the modulation limits.

#### 2.5. Maximum of the Current Ripple

_{max}), also displayed in Figure 7a and further examined in simulations. It can be noted that maximum function is almost linear with the modulation index, strictly for m > 0.197. Then, on the basis of Equations (47) and (25), a simplified expression for maximum of peak-to-peak current ripple amplitude is obtained for the 7-phase inverter:

**Figure 8.**Maximum of the normalized peak-to-peak current ripple amplitude as function of modulation index.

## 3. Numerical Results

_{s}is 2.1 kHz, and the DC voltage supply V

_{dc}is 100 V. A centered symmetrical carrier-based PWM technique is considered, equivalent to the multiple space vector PWM presented in Section 2.2.

_{fund}(t), i.e.,:

_{max}), as in Section 2, to cover all the considered cases.

**Figure 9.**(

**a**) Current ripple for m = 1/7. Simulation results (blue) and evaluated peak-to-peak amplitude (red envelope) for one fundamental period, with details; (

**b**) instantaneous output current with calculated ripple envelopes (red and blue traces).

**Figure 10.**(

**a**) Current ripple for m = 2/7. Simulation results (blue) and evaluated peak-to-peak amplitude (red envelope) for one fundamental period, with details; (

**b**) instantaneous output current with calculated ripple envelopes (red and blue traces).

**Figure 11.**(

**a**) Current ripple for m = 3/7. Simulation results (blue) and evaluated peak-to-peak amplitude (red envelope) for one fundamental period, with details; (

**b**) instantaneous output current with calculated ripple envelopes (red and blue traces).

**Figure 12.**Current ripple for m= 0.513 (≈m

_{max}). (

**a**) Simulation results (blue) and evaluated peak-to-peak amplitude (red envelope) for one fundamental period, with details; (

**b**) instantaneous output current with calculated ripple envelopes (red and blue traces).

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Toliyat, H.A.; Waikar, S.P.; Lipo, T.A. Analysis and simulation of five-phase synchronous reluctance machines including third harmonic of airgap MMF. IEEE Trans. Ind. Appl.
**1998**, 34, 332–339. [Google Scholar] [CrossRef] - Xu, H.; Toliyat, H.A.; Petersen, L.J. Five-phase induction motor drives with DSP-based control system. IEEE Trans. Power Electron.
**2002**, 17, 524–533. [Google Scholar] [CrossRef] - Ryu, H.M.; Kim, J.K.; Sul, S.K. Synchronous Frame Current Control of Multi-Phase Synchronous Motor: Part I. Modeling and Current Control based on Multiple d-q Spaces Concept under Balanced Condition. In Proceedings of the 39th Annual Meeting of IEEE Industry Application Society, Seattle, WA, 3–7 October 2004; pp. 56–63.
- Parsa, L.; Toliyat, H.A. Five-phase permanent-magnet motor drives. IEEE Trans. Ind. Appl.
**2005**, 41, 30–37. [Google Scholar] [CrossRef] - Grandi, G.; Serra, G.; Tani, A. General Analysis of Multi-Phase Systems based on Space Vector Approach. In Proceedings of 12th Power Electronics and Motion Control Conference (EPE-PEMC), Portoroz, Slovenia, 30 August–1 September 2006; pp. 834–840.
- Ryu, H.M.; Kim, J.W.; Sul, S.K. Analysis of multi-phase space vector pulse width modulation based on multiple d-q spaces concept. IEEE Trans. Power Electron.
**2005**, 20, 1364–1371. [Google Scholar] [CrossRef] - Iqbal, A.; Levi, E. Space Vector Modulation Schemes for a Five-Phase Voltage Source Inverter. In Proceedings of 11th European Conference on Power Electronics and Applications (EPE), Dresden, Germany, 11–14 September 2005; pp. 1–12.
- De Silva, P.S.N.; Fletcher, J.E.; Williams, B.W. Development of Space Vector Modulation Strategies for Five Phase Voltage Source Inverters. In Proceedings of Power Electronics, Machines and Drives Conference (PEMD), Edinburgh, UK, 31 March–2 April 2004; pp. 650–655.
- Ojo, O.; Dong, G. Generalized Discontinuous Carrier-Based PWM Modulation Scheme for Multi-Phase Converter-Machine Systems. In Proceedings of 40th Annual Meeting of IEEE Industry Applications Society, Hong Kong, China, 2–6 October 2005; pp. 1374–1381.
- Grandi, G.; Serra, G.; Tani, A. Space Vector Modulation of a Seven-Phase Voltage Source Inverter. In Proceedings of the International Symposium on Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM), Taormina, Italy, 23–26 May 2006; pp. 1149–1156.
- Dujic, D.; Levi, E.; Serra, G.; Tani, A.; Zarri, L. General modulation strategy for seven-phase inverters with independent control of multiple voltage space vectors. IEEE Trans. Ind. Electron.
**2008**, 55, 1921–1932. [Google Scholar] [CrossRef] - Dujic, D.; Levi, E.; Jones, M.; Grandi, G.; Serra, G.; Tani, A. Continuous PWM Techniques for Sinusoidal Voltage Generation with Seven-Phase Voltage Source Inverters. In Proceedings of the Power Electronics Specialists Conference (IEEE-PESC), Orlando, FL, USA, 17–21 June 2007; pp. 47–52.
- Hu, J.S.; Chen, K.Y.; Shen, T.Y.; Tang, C.H. Analytical solutions of multilevel space-vector PWM for multiphase voltage source inverters. IEEE Trans Power Electron.
**2011**, 26, 1489–1502. [Google Scholar] [CrossRef] - Lopez, O.; Alvarze, J.; Gandoy, J.D.; Freijedo, F.D. Multiphase space vector PWM algorithm. IEEE Trans Ind. Electron.
**2008**, 55, 1933–1942. [Google Scholar] [CrossRef] - Casadei, D.; Mengoni, M.; Serra, G.; Tani, A.; Zarri, L. A New Carrier-Based PWM Strategy with Minimum Output Current Ripple for Five-Phase Inverters. In Proceedings of the 14th European Conference on Power Electronics and Applications (EPE), Birmingham, UK, 30 August–1 September 2011; pp. 1–10.
- Dujic, D.; Jones, M.; Levi, E. Analysis of output current ripple rms in multiphase drives using space vector approach. IEEE Trans. Power Electron.
**2009**, 24, 1926–1938. [Google Scholar] [CrossRef] - Jones, M.; Dujic, D.; Levi, E.; Prieto, J.; Barrero, F. Switching ripple characteristics of space vector PWM schemes for five-phase two-level voltage source inverters—Part 2: Current ripple. IEEE Trans. Ind. Electron.
**2011**, 58, 2799–2808. [Google Scholar] [CrossRef] - Dahono, P.A.; Supriatna, E.G. Output current-ripple analysis of five-phase PWM inverters. IEEE Trans. Ind. Appl.
**2009**, 45, 2022–2029. [Google Scholar] [CrossRef] - Dujic, D.; Jones, M.; Levi, E. Analysis of output current-ripple RMS in multiphase drives using polygon approach. IEEE Trans. Power Electron.
**2010**, 25, 1838–1849. [Google Scholar] [CrossRef] - Jiang, D.; Wang, F. Study of Analytical Current Ripple of Three-Phase PWM Converter. In Proceedings of the 27th IEEE Applied Power Electronics Conference and Exposition (APEC), Orlando, FL, USA, 5–9 February 2012; pp. 1568–1575.
- Grandi, G.; Loncarski, J.; Seebacher, R. Effects of Current Ripple on Dead-Time Distortion in Three-Phase Voltage Source Inverters. In Proceedings of the 2nd IEEE ENERGYCON Conference and Exhibition—Advances in Energy Conversion, Florence, Italy, 9–12 September 2012; pp. 207–212.
- Herran, M.A.; Fischer, J.R.; Gonzalez, S.A.; Judewicz, M.G.; Carrica, D.O. Adaptive dead-time compensation for grid-connected PWM inverters of single-stage PV systems. IEEE Trans. Power Electron.
**2013**, 28, 2816–2825. [Google Scholar] [CrossRef] - Schellekens, J.M.; Bierbooms, R.A.M.; Duarte, J.L. Dead-Time Compensation for PWM Amplifiers Using Simple Feed-Forward Techniques. In Proceedings of the 19th International Conference on Electrical Machines (ICEM), Rome, Italy, 6–8 September 2010; pp. 1–6.
- Mao, X.; Ayyanar, R.; Krishnamurthy, H.K. Optimal variable switching frequency scheme for reducing switching loss in single-phase inverters based on time-domain ripple analysis. IEEE Trans. Power Electron.
**2009**, 24, 991–1001. [Google Scholar] [CrossRef] - Ho, C.N.M.; Cheung, V.S.P.; Chung, H.S.H. Constant-frequency hysteresis current control of grid-connected VSI without bandwidth control. IEEE Trans. Power Electron.
**2009**, 24, 2484–2495. [Google Scholar] [CrossRef] - Holmes, D.G.; Davoodnezhad, R.; McGrath, B.P. An improved three-phase variable-band hysteresis current regulator. IEEE Trans. Power Electron.
**2013**, 28, 441–450. [Google Scholar] [CrossRef] - Jiang, D.; Wang, F. Variable Switching Frequency PWM for Three-Phase Converter for Loss and EMI Improvement. In Proceedings of the 27th IEEE Applied Power Electronics Conference and Exposition (APEC), Orlando, FL, USA, 5–9 February 2012; pp. 1576–1583.
- Casadei, D.; Serra, G.; Tani, A.; Zarri, L. Theoretical and experimental analysis for the RMS current ripple minimization in induction motor drives controlled by SVM technique. IEEE Trans. Ind. Electron.
**2004**, 51, 1056–1065. [Google Scholar] [CrossRef] - Iqbal, A.; Moinuddin, S. Comprehensive relationship between carrier-based PWM and space vector PWM in a five-phase VSI. IEEE Trans. Power Electron.
**2009**, 24, 2379–2390. [Google Scholar] [CrossRef] - Levi, E.; Dujic, D.; Jones, M.; Grandi, G. Analytical determination of DC-bus utilization limits in multi-phase VSI supplied AC drives. IEEE Trans. Energy Convers.
**2008**, 23, 433–443. [Google Scholar] [CrossRef] - Grandi, G.; Loncarski, J. Analysis of Dead-Time Effects in Multi-Phase Voltage Source Inverters. In Proceedings of the 6th IET Conference on Power Electronics, Machines and Drives (PEMD 2012), Bristol, UK, 27–29 March 2012.

© 2013 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Grandi, G.; Loncarski, J.
Analysis of Peak-to-Peak Current Ripple Amplitude in Seven-Phase PWM Voltage Source Inverters. *Energies* **2013**, *6*, 4429-4447.
https://doi.org/10.3390/en6094429

**AMA Style**

Grandi G, Loncarski J.
Analysis of Peak-to-Peak Current Ripple Amplitude in Seven-Phase PWM Voltage Source Inverters. *Energies*. 2013; 6(9):4429-4447.
https://doi.org/10.3390/en6094429

**Chicago/Turabian Style**

Grandi, Gabriele, and Jelena Loncarski.
2013. "Analysis of Peak-to-Peak Current Ripple Amplitude in Seven-Phase PWM Voltage Source Inverters" *Energies* 6, no. 9: 4429-4447.
https://doi.org/10.3390/en6094429