Comparison based Selective Harmonic Elimination ... - IEEE Xplore

Comparison based Selective Harmonic Elimination using a Modified Reference Signal Sri Nikhil Gupta Gourisetti, Hirak Patangia, and Sraddhanjoli Bhadra Donaghey College of Engineering and Information Technology University of Arkansas, Little Rock Little Rock, USA [email protected]

By focusing on reference signal modification, the sensitivity of PWM generator system has been found to be much improved compared to modified carrier approach. The motivation for the work originated from our original desire to develop low cost switching modulators which require lower switching frequency for RF applications [6].

Abstract— A novel method has been proposed where selective harmonic elimination is achieved by comparing a modified reference signal against a high frequency carrier. A triangular or sinusoidal carrier can be used in its implementation. Unlike other selective harmonic elimination techniques, no on-the-fly signal processing is necessary. The tunability feature is similar to that of sine-triangle PWM generator and easy for implementation. The method allows maintaining a desired fundamental strength while eliminating unwanted harmonics, thereby improving SNR and lowering THD. Simulation and laboratory studies are provided to support the proposed approach.

The proposed method relies on finding the switching transition points that eliminate the selected harmonics with a specific fundamental strength [2], and use curve fitting to derive the reference signal for sinusoidal or triangular carrier. The following equation for curve fitting has been used since it satisfies the symmetry requirements necessary to produce the desired PWM.

Index Terms—SPWM, PWM Generators, inverter, sine-sine PWM, SHE, Harmonic Elimination

I. INTRODUCTION

f (θ) = A1 sin (k1θ) + A2 sin (k2θ) + A3 sin (k3θ) +…

Total Harmonic Distortion (THD) is an important criteria considered for practical implementation of a PWM generator because richer the harmonic content higher the THD. Selective Harmonic Elimination (SHE) is an efficient method of eliminating harmonics at a lower switching frequency. The disadvantage is that a digital signal processor is required [1] to store the transition points which are recalled in specific applications. The sine-triangle SPWM has the edge in that a comparison method can be used and thus implementation is simple. Recently, a modified carrier technique has been presented [2, 3, 4] that combines SHE with the comparison method of SPWM. Here a sinusoidal carrier is employed. This method has enabled harmonic elimination as high as 21st harmonic while the SPWM switching frequency is that of SHE. However, it has been found that the harmonic cancellation is rather sensitive to component tolerances which may not be acceptable in certain applications. Recently, some results have been published where a modified reference is used to improve the THD with no specific attention to harmonic elimination [5]. The basic idea is to strengthen the fundamental to reduce the THD. In many applications, a harmonic-free baseband floor is necessary to reduce complexity of harmonic eliminating circuitry. This paper focuses on modifying the sinusoidal reference signal in a way that unwanted harmonics are eliminated and fundamental is maintained at a desired strength.

where k is an odd integer, and k1< k2< k3…… II. HARMONIC ELIMINATION FOR VARIOUS VALUES OF CARRIER TO REFERENCE RATIOS A. Case-I: Frequency ratio of 5 The first set of side bands of an SPWM signal exists at (k-2)fo and (k+2)fo where kfo is the carrier frequency with fo being the reference frequency [8]. Eliminating the (k-2) component for a ratio of 5 will suppress third harmonic giving a baseband until the carrier at 5fo. It has been shown in [3] that the PWM transition points in that case are: 0.1081msec, 0.1497msec, 0.3503msec, and 0.3919msec if the reference time period is 1msec. Reflecting these time points on the sine signal with unity amplitude, we can calculate the amplitude points that are necessary for the reference signal to pass through. Fig.1 depicts a 5 kHz sine signal with the above time points, its FFT magnitude and dB plot, and phase plot.

The work was partially supported by NSF (Grant #0942327) Fig.1 5kHz sine signal, FFT and Phase plot

978-1-4673-4569-9/13/$31.00 ©2013 IEEE

(1)

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fundamental strenght at 0.75. Without the constraints in A1 and k1, (2a-2d) are solved directly in Matlab to obtain: A1=-1.08, A2=-0.43, k1=1, k2=7. Hence the reference signal equation can be written as:

The coordinates (θ and f) of the points for the modified refernce to pass through are: (0.6786, -0.2505), (0.9408, 0.9999), (2.201, -0.9999), (2.4630, -0.2505). Inserting these values in (1), we get four equations as shown below. (2a) (2b) (2c) (2d)

f (θ 1 ) = A1 sin( 0 .6786 k 1 ) + A 2 sin( 0 . 6786 k 2 ) = − 0 . 2505 f (θ 2 ) = A1 sin( 0 . 9408 k 1 ) + A 2 sin( 0 . 9408 k 2 ) = − 0 . 9999 f (θ 3 ) = A 1 sin( 2 . 201 k 1 ) + A 2 sin( 2 . 201 k 2 ) = − 0 . 9999 f (θ 4 ) = A1 sin( 2 . 4630 k 1 ) + A 2 sin( 2 . 4630 k 2 ) = − 0 . 2505

f (t ) = −1.08 sin(ω0t ) − 0.43 sin(7ω0t )

(5)

When (5) is used as a reference and compared against a 5 kHz sine signal, the resulting PWM and the FFT are shown in Fig.3.

Due to the half wave symmetry produced by the signal function, only four equations are sufficient. Instead of solving (2a-2d) directly, we make the following assumptions to insure that the dominating component is the fundamental frequency at 1 kHz. With A1 = 1 and k1 = 1, (2a-2d) are modified as:

f (θ 1 ) = −0.2505 = 0.6277 + A2 sin( 0.6786 k 2 ) f (θ 2 ) = −0 .9999 = 0.8080 + A2 sin( 0.9408 k 2 ) f (θ 3 ) = − 0 .9999 = 0 . 8080 + A 2 sin( 2 . 201 k 2 ) f (θ 4 ) = − 0 .2505 = 0 .6277 + A 2 sin( 2 .4630 k 2 )

(3a)

(3b) (3c) (3d) Fig.3 Ref signal as in (5), carrier, PWM-its FFT & phase plot

Solving (3a-3d) using MATLAB the variables are calculated as: A2= -0.34 and k2=7. Since A2 is much smaller than A1 the net polarity of the modified reference signal is dominated by the first component of the equation. It can be seen from (2a2d) that the y-coordinates of the signal are negative, inorder to ensure that the modified reference signal meets the carrier signal at those points, the net polarity of the reference signal should be negative. Hence the A1 polarity is changed to negative. The reference signal can be written as:

f (t ) = − sin(ω 0t ) − 0.34 sin( 7ω 0 t )

The magnitude, dB values and phase angles in radians are shown below in Table-II. TABLE II.

MAGNITUDE, PHASE & DECIBEL VALUES FOR FIG.3

Harmonic Fundamental 3rd 5th

Magnitude 0.7891 0.0069 0.7419

dB -2.0570 -43.1632 -2.5934

Phase (rad) 1.5708 -1.5708 1.5708

It can be observed from Fig.3 that the 3rd harmonic≈0 and is almost 41 dB below the fundamental maintaining the fundamental strenght at 0.79. One advantage of (5) over (4) is the fundamental strength but there is a loss in the degree of suppression. But in a more generalized perspective, both (4) and (5) are valid and each satisfies the required conditions.

(4)

Using (4) as a reference and the sinusoidal carrier at 5 kHz, the following simulation results are obtained.

This method was also tested for a 5kHz triangular carrier signal.The amplitudes were recalculated for this waveform and the resulting equations were solved analytically taking advantage of quarter wave symmetry. The resulting reference signal is given in (6), and the FFT and PWM results are shown in Fig.4.

f (t ) = −1.07 sin(ω0 t ) − 0.53 sin( 7ω 0 t ) Fig.2 Ref signal as in (4), carrier, PWM-its FFT & phase plot

The magnitude, dB values and phase angles in radians are tabulated below in Table-I. TABLE I.



Magnitude 0.7440 0.0003 0.8132

dB -2.5686 -70.1468 -1.7961

Phase (rad) 1.5708 -1.5708 1.5708

It can be observed from Fig.2 that the 3rd harmonic≈0 and is almost 68 dB below the fundamental maintaining the

Fig.4 Ref signal as in (6), carrier, PWM-its FFT & phase plot

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(6)

The magnitude, dB values and phase angles in radians are tabulated below in Table-III. TABLE III.

from Table-IV. It can be seen that the fundamental strength is higher than in previous cases and all harmonics between fundamental and kth harmonic are eliminated.



Magnitude 0.8304 0.0042 0.6816

dB -1.6147 -47.5633 -3.3293

Phase (rad) 1.5756 2.842 1.5767

It can be observed from Fig.4 that the 3rd harmonic≈0 and is almost 46 dB below the fundamental maintaining the fundamental strength at 0.83. This is higher than those given by (4) and (5). B. Case-II: k=7 Using the same approach as above and using the transition points from [4] of first quarter cycle i.e., 0.0767msec, 0.1221msec, 0.2343msec, the y-corordinates (respective amplitude points) are determined as: -0.2298, -0.7913, -0.7709 respectively. The calculated x and y coordinates are used to write 6 transecendental equations:

Fig.5 Ref signal as in (8), carrier, PWM-its FFT & phase plot TABLE IV.


Harmonic Fundamental 3rd 5th 7th

f (θ1 ) = A1 sin(0.482k1 ) + A2 sin(0.482k 2 ) + A3 sin(0.482k 3 ) = −0.2298(7a) f (θ 2 ) = A1 sin(0.767k1 ) + A2 sin(0.767k 2 ) + A3 sin(0.767k3 ) = −0.7913(7b)

Magnitude 1.0043 0.0051 0.0077 0.2367

dB 0.0371 -45.8258 -42.2506 -12.5166

Phase (rad) -1.5748 2.4691 -2.1157 1.5877

Unlike the signal in Fig.2-4 where the reference signal looks as if a small high frequency ripple is added to a low frequency sine signal, Fig.5 looks rather different. Now it looks as if the entire signal is being modified yet maintaining the overall signal frequency at 1kHz. The peak amplitude of reference signal in Fig.5 is higher (peak amplitude≈1.8) than in Fig.2-4 (peak amplitude≈1.3) which is mainly due to the second and third component in (8). It is clear from Table-IV that the 3rd harmonic is about 46 dB below the fundamental and 5th harmonic is about 42 dB below the fundamental which is similar to the results demonstrated for k=5 case.

f (θ 3 ) = A1 sin(1.472k1 ) + A2 sin(1.472k 2 ) + A3 sin(1.472k 3 ) = −0.7709(7c) f (θ 3 ) = A1 sin(1.669k1 ) + A2 sin(1.669k 2 ) + A3 sin(1.669k 3 ) = −0.7709(7d) f (θ 4 ) = A1 sin(2.375k1 ) + A2 sin(2.375k 2 ) + A3 sin(2.375k 3 ) = −0.7913(7e) f (θ 5 ) = A1 sin(2.660k1 ) + A2 sin(2.660k 2 ) + A3 sin(2.660k 3 ) = −0.2298 (7f)

In (2a-2d) each transcendental equation was in 2 component 4 variable form whereas in (7a-7f), each transcendental equation is in 3 component 6 variable form. This assumption was based on number of crossing points in a quarter cycle. For k=5, there are two crossing points in a quarter cycle and for k=7 there are three crossing points in a quarter cycle. This was just an assumption as the case of component dependency on transition points doesn’t necessarily have to be proportional. There is a possibility to achieve a 2 component or 3 component universal equation that can eliminate all harmonics up to kth harmonic and this is currently under investigation.

It was noted earlier that for k=7, two sets of equations are found. The second set of values acquired are used to formulate the following reference signal equation:

f (t ) = 0.62sin(ω0t ) + 0.82sin(7ω0t ) − 0.5sin(11ω0t) (9) Comparing (8) with (9), there are minor changes in the vales of coefficients but k3 is changed by a factor of 2 i.e., now k3 is k+4. Even though numerically the changes look minor but they have an impact on the PWM signal generated. The plots related to (9) are shown in Fig.6 and the respective harmonic values are given in Table-V. It is clear from Fig.6 that the conditions to eliminate 3rd and 5th harmonic are satisfied. This means that for a particular set of conditions, there could be more than one solutions. This means for a system of transcendental equations for a given k value, there could be many local maximas and not just one global maxima.

Equations (7a-7f) are solved to obtain the variables A1-A3 and k1-k3. Interestingly, when these equations are solved two different solution sets i.e., two different modified reference signal are obtained that serves the same purpose of eliminating all the harmonics in between fundamental and kth harmonic. First solution set is used to formulate the following reference signal:

f (t) = 0.63sin(ω0t ) + 0.87sin(7ω0t ) − 0.45sin(9ω0t ) (8) In (8) k2 and k3 doesn’t completely look random. The second component k2 is nothing but k and 3rd component k3 is simply k+2. (8) is used as a reference signal to plot against a 7 kHz triangular carrier signal and is depicted in Fig.5. The magnitude, dB, phase values till kth harmonic can be observed

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Fig.6 Ref signal as in (9), carrier, PWM-its FFT & phase plot TABLE V.

Fig.7 Ref signal as in (10), carrier, PWM-its FFT & phase plot


Harmonic Fundamental 3rd 5th 7th

Magnitude 1.0365 0.0041 0.0057 0.0755

dB 0.3111 -47.7392 -44.8089 -22.4365

TABLE VI.


Harmonic Fundamental 3rd 5th 7th 9th

Phase (rad) -1.5747 2.9177 2.3404 1.6238

Equation (9) signal has a better suppression of harmonics than (8) signal. Not only 3rd and 5th harmonics are eliminated but also 7th harmonic is suppressed to a large extent with an improved fundamental strength. Even the net reference signal amplitude of 1.66 is a bit less than that of the reference signal in (8). Looking at phase angles of the harmonics, except the fundamental, all the other angles are affected by a noticeable amount. This implies that phase angle of the harmonics could play a critical role in harmonic elimination.

Magnitude 1.0485 0.0052 0.0160 0.0100 0.2620

dB 0.4116 -45.7519 -35.9349 -39.9746 -11.6354

Phase (rad) -1.5746 2.4585 1.8240 -1.9810 1.5861

It is clear that the 3rd, 5th, and 7th are suppressed to the levels of 46 dB, 36 dB, 40 dB respectively below the fundamental. The reference signal in Fig.7 looks very similar to the reference signal in Fig.5 so this could mean that by scaling an existing signal equation to desired frequency, unwanted harmonics can be eliminated without fiding the transition points and solving transcendental equations. III. EXPERIMENTAL DATA

C. Case-III: k=9 For k=9, the equations obtained for eliminating 3rd and 5th harmonic for k=7 case are used and scaled such that 3rd, 5th and 7th harmonics are eliminated. Hence in this case, the equation is not obtained by solving a set of transcendental equations, instead by a simple approximation and scaling of the frequency components. As an example, (8) is used and scaled to give (10). It should be mentioned that this is not a unique solution. There are many other local maxima solutions satisfying the condition set to eliminate everything between the fundamental and the 9th harmonic. The reference equation for this particular case is:

A Monte-Carlo sensitivity analysis test is performed on the acquired results. A total of 53 solution sets were acquired out of which 16 simulations were performed and the THD window was found to be 0.6% (best possible case) to 6.67% (worst possible case). This shows that the method is not overly sensitive and the THD is very good. The data points of the signals generated are acquired from MatLab and are uploaded into an arbitrary waveform generator to observe its practical accuracy. All the derived equations are simulated and respective signals are viewed in a mixed-signal oscilloscope.

f (t ) = 0.6 sin(ω0t ) + 0.81sin(9ω0t ) − 0.45sin(11ω0t ) (10) Here in (10), k2 and k3 follow a similar pattern as in (8) i.e., k2=k and k3=k+2. This information can be used to conclude that one local maxima set can be obtained by assuming that the second component frequency can depend upon the carrier signal frequency and the third component can depend up on k+2 value. Fig.7 and Table-VI represent plots using (10) and the harmonic values respectively.

Fig.8 Modified reference as in (4), carrier, PWM and FFT

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The experimental results match with MatLab simulations. FFT signal in Fig.11-13 are scaled down to avoid the loss of peak point vision since the fundamental in these cases is greater than one. The FFT in Fig.8-10 shows virtually no trace of 3rd harmonic and the baseband floor is clean between 0 to 5f0; Fig.11-12 shows no trace of 3rd, 5th harmonics and the baseband floor is clean between 0 to 7f0; Fig.13 shows no trace of 3rd, 5th & 7th harmonics with a clear baseband floor between 0 to 9f0. No spurious sidebands was generated in any cases. IV. CONCLUSIONS The paper presents a novel method to eliminate unwanted harmonics using both sine-triangle and sine-sine PWM models. The elimination technique is based on real-time comparison between carrier and reference signals. This provides tuning flexibility since the modified reference and a fast moving carrier signal can be implemented from a master clock at the reference frequency. Unlike the sine-triangle PWM that uses a high switching frequency to push out harmonics to higher frequencies of the spectrum, this method optimizes switching frequency and provides pulse widths suitable for practical implementation. The THD sensitivity of the method is superior to that of the modified carrier presented in [3, 4]. The modified reference signals presented are not unique and other solutions exist. Currently, work is underway to seek a solution for f(θ) that would be applicable to a wide range of k. Relationship of Ais to control the fundamental amplitude is also under investigation.



ACKNOWLEDGMENT The authors wish to acknowledge the support of NSF under grant number 0942327. REFERENCES [1] P. T. Krein, B. M. Nee, and J. R. Wells, “Harmonic elimination switching through modulation,” in proc. IEEE Workshop Comput. Power Electron., pp. 123–126, 2004. [2] Hirak Patangia, Sri Nikhil Gupta Gourisetti, “A Harmonically Superior Modulator with Wide Baseband and Real-Time Tunability,” IEEE International Symposium on Electronic Design, India, pp.18-23, 2011. Fig.12 Modified reference as in (9), carrier, PWM and FFT

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[3] Hirak Patangia, Sri Nikhil Gupta Gourisetti, “Real time harmonic elimination using a modified carrier,” CONIELECOMP, Mexico, pp. 273-277, 2012. [4] Hirak Patangia, Sri Nikhil Gupta Gourisetti, “A novel strategy for selective harmonic elimination based on a sine-sine PWM model,” MWSCAS, U.S.A, pp. 310-313, 2012 [5] R. N. Ray, D. Chatterjee and S. K. Goswamie, “A modified reference approach for harmonic elimination in pulse-width modulation inverter suitable for distribution generators”, Electric Power Components and Systems, pp. 815-827, 2008.

[6] Amir Ghaffari, Eric A. M. Klumperink, Bram Nauta, “A differential 4-path highly linear widely tunable on-chip bandpass filter,” IEEE Radio Frequency Integrated Circuits Symposium, pp. 299–302, 2010. [7] D. G. Holmes and T. A. Lipo, “Pulse Width Modulation for Power Converters Principles and Practice, Hoboken, NI: IEEE Press, 2003.

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