Realization of Fibonacci Search Algorithm

for Single Phase Matrix Converter

Shubham Verma

Asstt Professor, Electrical Engg Dept.

Shri Ramswaroop Memorial Group of Professional Colleges

Lucknow, India

shubhamverma@live.com

Vineeta Agarwal

Professor, Electrical Engg Dept.

Motilal Nehru National Institute of Technology

Allahabad, India

vineeta@mnnit.ac.in

Abstract—Fibonacci Search Algorithm is designed to

calculate the intersection points of cosine wave and reference

wave in cosine-wave crossing method for single phase matrix

converter. The reduced number of comparisons as compared

with linear search method may be used in control circuits for

converter to provide faster response.

Keywords-Cycloconverter; Cosine Wave Crossing

Method(CWCM); Fibonacci Search; Power electronics.

I. INTRODUCTION

Naturally commutated cycloconverter (NCC) is an important power converter performing direct frequency changing [1]. In recent years there has been significant research into the design of single phase matrix converter[2]. Fully controlled frequency changers based on cycloconverter arrangement are similar in topology to those of single phase matrix converter topology [3]. It has been found that the SPMC could be used to operate as a direct AC-AC cycloconverter [4]. There have been several attempts to apply software control of cycloconverter using microprocessor-based control, DSP control or FPGA based control schemes [5]-[7]. It has been recognized that the most serious problem in all these schemes is the necessity of performing complicated real-time calculations involving a transcendental function in the control algorithm. An effective and very simple algorithm based on the cosine wave crossing pulse timing method was proposed in [8]. In this scheme, the firing angles of individual SCR are varied in such a way that the average value of the input voltage segments corresponds as closely as possible to the variation of desired sinusoidal output voltage. This can be

done by comparing a cosine signal (VCW = Vm1cos it) known as modulating wave (MW), at source frequency with an ideal sinusoidal wave, known as reference wave RW (VRW = Vm2 sin

t) at the desired output frequency. The firing instants of the SCRs are given by the intersection points when RW is equal or greater than MW. Fig.1 shows this method in which reference wave is a sinusoidal wave of output frequency, and the modulating wave is the cosine wave of the input frequency.

A conventional cosine wave crossing method (CWCM)

uses a linear search method to find the intersection points of a

cosine wave and a reference wave. The method is simple to

implement but takes more processing time. It can be reduced

by using a more efficient algorithm. In this paper, an

algorithm is proposed which is based on Fibonacci series and

requires lesser number of comparisons to calculate the

required intersections, and hence, processing time is reduced

considerably.

To find the magnitude of any kind of wave, a simple and practical method is the look-up table method, wherein the amplitude of the waveform is digitized at discrete points along the phase axis, and the digital values are stored in a linearly addressed memory. For continued generation of any periodic waveform, at most, one cycle needs to be stored. With the digitized waveform of the RW and the CW stored in a memory, the magnitude of these can be read at any time for any desired phase angle. To minimize the processing time, a half cycle of the RW and a quarter cycle of the CW are stored. The values of the RW are digitized from 0 to 180

o, and the values of the CW

are digitized from 0 to 90o at an interval of one degree for each

wave.

2

3

a1

a3

a2

VCW

= Vm 1

cos it

VRW

= Vm 2

s inot

t

t

Vi

(a )

(b )

Fig. 1. CWCM for N = 3 (a) Input voltage and trigger points (b) Intersection point of RW and CW

II. LINEAR SEARCH ALGORITHM

Linear search is a method for finding a particular value in a list or array, that consists of checking each and every one of its elements, one at a time and in sequence, until the desired value is found. Let the elements in the reference wave (RW) and the cosine wave (CW) be represented by R(l), R(2), …..., R(180) and K(1), K(2), ……, K(90), respectively. In linear search

978-1-4673-0934-9/12/$31.00 ©2012 IEEE

technique, the ith element of the CW is compared with the corresponding element of the RW, starting with i = 1. The comparison is done for each step of one degree until [K(i) - R(i/N)] becomes less than the specified tolerance. Here N is the ratio of the frequency of the CW to that of the RW. The algorithm given below shows the method:

START

Read CW, RW, and N;

Rn = 180/N (rounded to the nearest integer);

FOR j = 0 to (N - 1 ) DO

BEGIN

found = FALSE; i = 1;

WHILE found = FALSE DO

BEGIN

RP = R ( i / N ) ;

IF K(i) = RP + jRn,

THEN found = TRUE

ELSE i = i + 1;

END;

IF found = true THEN store K(Ni)

END {FOR}; END OF THE ALGORITHM

III. FIBONACCI SEARCH ALGORITHM

In order to calculate the intersection points of RW and

CW with the less number of comparisons as in linear search

algorithm, the proposed method is based on Fibonacci Search

algorithm. The Fibonacci search technique is a method of

searching a sorted array using a divide and conquer algorithm

that narrows down possible locations with the aid of Fibonacci

numbers. The Fibonacci search has an advantage over binary

search in slightly reducing the average time needed to access a

storage location.

In this method, the comparison is done in the segmented

CW which depends on the Fibonacci series. For CW, the

Fibonacci series can be written as

F = [0 1 1 2 3 5 8 13 21 34 55 89].

This covers full range of the values of the CW digitized

from 0 to 90o. Fig. 2 shows the method for both the CW and

the RW. The segments are highlighted by shaded lines.

To begin, the CW is split in three regions according to the

numbers

X1 = A+D

X2 = B -D

Here D = F [m-1], A = F[0] , B = F[m] and m is the total

number of Fibonacci series . For the case considered m is

equal to 12.

Now X1 and X2 elements of the CW is compared with the

corresponding values of RW. The difference of the two

comparisons is calculated as

H1 = [K(X1) – R(X1/N)]

H2 = [K(X2) – R(X2/N)]

The following three situations may arise:

1) H1 < H2

2) H1 = 0

3) H2 = 0 4) H1 > H2

In case 1) above, the comparison method continues by

discarding the region 1 and replacing A = X2 , while for case

4), it is continued with neglecting region 3 and replacing B

=X1. The comparison process terminates as soon as either

case 2) or case 3) is encountered. Fig. 3 shows the enlarged

view of segmented portion.

X1X2

K(X1)

K(X2)

R(X1/N)

R(0) K(90)

K(0)

H2H1

H2

R(90) R(180)

K(90)

K(90)Rn(j=1) R

n(j=2)

intersection

point RW

CW

Region 1

Region 2

Region 3R(X2/N)

Fig. 2. Comparison of CW with RW in Fibonacci search Method

X1X2R(0) K(90)

K(0)

H2H1

H2

intersection

point

R(X2/N)

R(X1/N)

K(X2)

K(X1)

Region 1

Region 2

Region 3

Fig. 3. Enlarged view of segmented portion

The following algorithm implements this method:

START

Read RW, CW,

N {ratio of input and output frequency},

m {number of total Fibonacci number =12}

FIB[series] = [0 1 1 2 3 5 8 13 21 34 55 89]

NOC=0, A = F[0] , B= F[m];

Rn = 180/N (rounded to the nearest integer);

FOR j = 0 to (N - 1 ) DO

BEGIN

found = FALSE;

WHILE found = FALSE DO

BEGIN

D = F[m-1];

X1= A + D

X2 = B - D;

H1 = [K(X1) – R(X1/N) + jRn]

H2 = [K(X2) – R(X2/N) + jRn]

CASE

H1 < H2 : A = X2;

H1 = 0 : found =TRUE; store K(X1) in array OUT[];

H2 = 0 : found =TRUE; store K(X2) in array OUT[];

H1 > H2 : B = X1;

END {CASE};

m = m -1;

NOC = NOC +1;

END {WHILE};

END {FOR}

Display the values of array OUT[]; END OF THE ALGORITHM.

IV. RESULTS

The trigger angles for the control of single phase matrix

converter are calculated by both linear search method and

Fibonacci search method for any value of N. The total

processing time for locating the intersection points for one

period of matrix converter output has also been reduced for

the proposed method, due to the less comparisons. The results

are shown in Table I, along with the number of comparisons

needed to arrive at the final intersection point of the RW and

the CW for typical values of N = 3, 5, 10, 20, 30 and 40.

The intersection points calculated are stored in a look-up

table to provide trigger pulses for Simulink model. The output

voltages for different values of N using the method are shown

in the Figures 4(a), 4(b) and 4(c) for N=3, N=5 and N=10

respectively.

TABLE I

NUMBER OF COMPARISONS FOR DIFFERENT METHODS

S.

No.

N Intersection Points

(in degree )

Number of comparisons

Linear

Search

Algorithm

Fibonacci

Search

Algorithm

1 3 68, 22, 47

137 8

2 5 75, 45, 13, 21, 68

228 20

3 10 82, 68, 49, 33, 16, 01,

21, 42, 61, 81

452 45

4 20 86, 77, 68, 60, 52, 42,

34, 26, 17, 13, 8, 13, 21, 27, 37, 47, 57, 66,

76, 86

913

77

5 30 87, 81, 76, 70, 64, 58,

52, 47,42, 34, 29, 23,

17, 13, 9, 8, 11, 15, 21,

26, 31, 37, 44, 50, 55,

63, 68, 74, 81, 87

1373 125

6 40 88, 84, 79, 75, 71, 66, 61, 57, 52, 48, 44,39,

34, 31, 26, 21, 17, 13,

13, 8,8, 8, 13, 13, 21, 21, 26, 32, 37, 42, 47,

50, 55, 60, 65, 69, 73,

78, 83, 87

1815 179

0 0.01 0.02 0.03 0.04 0.05 0.06-300

-200

-100

0

100

200

300

Time

Voltage

(a) N = 3

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-300

-200

-100

0

100

200

300

Time

Voltage

(b) N = 5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-300

-200

-100

0

100

200

300

Time

Voltage

(c) N = 10

Fig. 4. Output voltage of Matrix Converter with CWCM

V. CONCLUSIONS

In the proposed algorithm, there is, on the average, more

time taken for each comparison because of the additional time

invested by the CPU for the selection of the segments of the

CW. This, however, has been more than compensated for by

the large reduction in the number of comparisons. It may be

observed from Table I that the number of comparisons is in

the range of from one tenth to one twentieth of the

corresponding number in the conventional linear search

method. The proposed method has been found to take five to

ten times less total time. It may further be added that the worst

case of comparison in the proposed method is 12 * N as

compared with N * n of the conventional linear comparison

method. Although the algorithm has been implemented with

the cosine function of the CW and the RW, it can be expected

to work with similar CPU times for other functions as well.

This algorithm was written for the generation of switching

pulses for a single-phase matrix converter for N = 2, 3, 4, . . . ,

100.

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