FOURIER ANALYSIS OF A SINGLE-PHASE FULL BRIDGE RECTIFIER USING MATLAB

Transcription

1 -774 FOURIER ANALYSIS OF A SINGLE-PHASE FULL BRIDGE RECTIFIER USING MATLAB Bruno Osorno California Sate University Nortridge 8 Nordoff St Nortridge CA bruno@ecs.csun.edu Pone: (88) Abstract Te use of trigonometric Fourier series is applied wen repetitive waveforms are found. Tis situation occurs at te output of full bridge single-pase rectifiers. As it will be seen, te application of a very powerful matematical tecnique as given te rigt solution of a typical electrical engineering circuit. Te waveforms analyzed are in teir steady-state mode wit a repetitive period T tat depends on te pysical electric circuit. []. We will indicate tat te use of te fundamental signal of a distorted waveform becomes te most important piece of information. Tis is because in power electronics te calculation of Power factor, Displacement power factor and Total Harmonic Distortion make use of te fundamental value. Teory and simulation Non-sinusoidal waveforms, f(t), tat ave angular frequencies ω can be obtained as: f ( t) = F + = f ( t) = a + = { a cos( ω t) + b sin( ωt)} Were: Te average value is: F = a Te average value of a periodic function is called te DC value of tat function. Te MATLAB statement for te average value of a waveform v is: vdc=mean(v) Proceedings of te American Society for Engineering Education Annual Conference & Exposition Copyrigt, American Society for Engineering Education Page 7.3.

2 -774 Now we will generate a waveform in order to obtain its dc value. Te MATLAB code is given below. % Waverfom period divided in 5 points Ts=;dt=Ts/5 t=:dt:ts/4-dt; t=ts/4:dt:ts/-dt; t3=ts/:dt:ts; t=[t,t,t3]; %Amplitudes for te waveform v=[*ones(size(t)),-5*ones(size(t)),zeros(size(t3))]; %Plotting te waveform plot(t,v) %Axis limits axis([,ts,-,]) %Setting te grid on grid %Setting te labels xlabel('time in seconds') %Calculation of te average DC value Vdc=mean(v) Vdc = time in seconds Figure. Waveform created wit MATLAB..476 Te coefficient a becomes: Proceedings of te American Society for Engineering Education Annual Conference & Exposition Copyrigt, American Society for Engineering Education Page 7.3.

3 -774 a = π Te coefficient b becomes: b = π f ( t)cos( ω t) d( ωt) =,..., 3 π f ( t)sin( ωt) d( ωt) It is a well known fact tat [] te average value of f(t) is: π F = a = f ( t) d( ωt) = π T Were te radian frequency is: T f ( t) =,... 4 ω = ω 6 T Tis average was obtained using MATLAB for te waveform generated. Te value was.476 Volts. Tis can be easily proved using equation 5. Now, moving on to te frequency domain, we can obtain te RMS (root mean square) of f(t) as F. Tis notation usually is described as te pasor form of f(t). Ten: F = F 7 e j θ Anoter pasor form, using a and b coefficients, is given as follows: dt 5 a + F = b 8 Were te angle θ is obtained as follows: b tan(θ ) = 9 a Finally te RMS value of f(t) is F, ten we ave te following equation: F = ( + F ) = F Te MATLAB RMS of a waveform v can be obtained wit te following statement: Vrms=sqrt(mean(v^)) Te RMS value of te waveform v is obtained wit te following program. Te only addition to te previous code (Vdc) is te statement V rms =sqrt(mean(v^)). Proceedings of te American Society for Engineering Education Annual Conference & Exposition Copyrigt, American Society for Engineering Education Page 7.3.3

4 -774 Ts=;dt=Ts/5 t=:dt:ts/4-dt; t=ts/4:dt:ts/-dt; t3=ts/:dt:ts; t=[t,t,t3]; %Amplitudes for te waveform v=[*ones(size(t)),-5*ones(size(t)),zeros(size(t3))]; %Plotting te waveform plot(t,v) %Axis limits axis([,ts,-,]) %Setting te grid on grid %Setting te labels xlabel('time in seconds') %Calculation of te average DC value Vdc=mean(v) %Calculation of te RMS value. Vrms=sqrt(mean(v.^)) Vrms =» Te MATLAB armonic value of a waveform v is obtained as follows:» Ts=;dt=Ts/5 t=:dt:ts/4-dt; t=ts/4:dt:ts/-dt; t3=ts/:dt:ts; t=[t,t,t3]; %Amplitudes for te waveform v=[*ones(size(t)),-5*ones(size(t)),zeros(size(t3))]; %Plotting te waveform plot(t,v) %Axis limits axis([,ts,-,]) %Setting te grid on grid %Setting te labels xlabel('time in seconds') % %Calculation of te spectrum Proceedings of te American Society for Engineering Education Annual Conference & Exposition Copyrigt, American Society for Engineering Education Page 7.3.4

5 -774 % [f,y,p]=armonic(v,6); title('spectrum of "v" waveform') xlabel('armonic number') ylabel('amplitude') p*8/pi ans =» Spectrum of "v" waveform 5 4 amplitude armonic number Figure. Spectrum of figure using MATLAB. To obtain te spectrum we created a function tat is stored in te work file of MATLAB. Tis function is listed below: %Creation of a function called armonic % function[f,y,p]=armonic(x,n) N=size(x,); f=::n-; y=fft(x); y=*(abs(y)/n); y()=y()/n; p=angle(y())+pi/; stem(f,y, r ) axis([,n-,,.*max(y)]) % Proceedings of te American Society for Engineering Education Annual Conference & Exposition Copyrigt, American Society for Engineering Education Page 7.3.5

6 -774 Fourier analysis of voltage and current As stated earlier, te trigonometric Fourier analysis of a repetitive waveform can be obtained using equations troug. If we let vs( t) = vs sin( ωt) be te input voltage for a typical circuit (see figure ) te input current can be obtained as: i s( t) = is ( t) + i ( ) s t Were: i s ( ) is te fundamental component (at line frequency f ). t i s (t) is te component at te armonic frequency (f ). D3 DN448 D3 DN448 VOFF = VAMPL = 6 FREQ = 6 V R D33 DN448 D3 DN448 Figure3. PSPICE circuit of a single-pase rectifier * source FOURIER V_V N379 N3754 +SIN 6 6 R_R N75 D_D3 N379 N75 DN448 D_D3 N3754 N75 DN448 D_D3 N3754 DN448 D_D33 N379 DN448 Ten we ave []: f = f Proceedings of te American Society for Engineering Education Annual Conference & Exposition Copyrigt, American Society for Engineering Education Page 7.3.6

7 -774 Ten equation can be expanded as follows: i ( t) = I sin( ωt θ ) + I sin( ωwt θ ) 3 s s s Typical distortion occurs at te current level, voltages remain, for te most part, undistorted. Figure 4 sows a very typical voltage-current waveform were te distortion of currents is apparent and te process of te Fourier trigonometric analysis can be seen. v s i s i dis t q i s Figure 4. Typical voltage-current output and its Fourier components Te RMS current is: T I s = ( ( is ( t) + ( )) ( ) is t = I s + I s 4 T Te IEEE [] standard recommends te maximum allowed armonic content in power electronics depending on te type of circuit. Tis value is calculated using equation 5. THD stands for total armonic distortion and it is usually given in %. PSPICE determines automatically tis value. In MATLAB we ave to calculate it. In te next section we sow a PSPICE output for a simple single-pase rectifier. I s I s % THD = 5 I s Full bridge single-pase rectifier Using PSPICE we obtained te output of a single-pase full bridge rectifier. Figure, sows te circuit and figure 5 sows te output. Te trigonometric Fourier analysis is performed wit PSPICE. Proceedings of te American Society for Engineering Education Annual Conference & Exposition Copyrigt, American Society for Engineering Education Page 7.3.7

8 s ms ms 3ms 4ms -I(R) V(D3:) Time 4.A.A DC component nd Harmonic 4t Harmonic A Hz 5Hz Hz 5Hz Hz 5Hz 3Hz -I(R) Frequency Figure5. PSPICE output of te single-pase rectifier sown in figure 3 Upper grap indicates te voltage and current output. Lower grap indicates te Harmonic analysis. Conclusions Power electronics and Power quality are one of te major fields in electrical engineering tat require te understanding of trigonometric Fourier series and its applications. It is of no surprise tat tis tecnique works very well in obtaining te necessary information from input/output voltage/current signals. By doing so, we go a step furter and determine te %THD tat gives us an indication of te goodness of our electrical design. Again, by using one of te most powerful matematical tecniques we arrive to a simple, yet, very important solution of a problem tat in te past was tedious and cumbersome in its solution. Recently, wit te powerful use of Personal Computers and, peraps te most widely used, software simulator and matematical packages te application and solution of trigonometric Fourier series as become a lot simpler. Consequently te design of electrical circuits in general is getting better and better. BIBLIOGRAPHY: [] Moan, Undeland, Robbins, Power Electronics Converters Applications and Design, Second edition, Wiley 995. [] Jai P. Agrawal, Power Electronics Systems teory and Design, Prentice-Hall,. [3] Pilip T. Krein, Elements of Power Electronics, Oxford University Press, 998. [4] Muammad H. Rasid, Power Electronics Circuits, Devices, and Applications. Second Edition. Prentice-Hall,993. Proceedings of te American Society for Engineering Education Annual Conference & Exposition Copyrigt, American Society for Engineering Education Page 7.3.8

9 -774 [5] W. Seperd, L.N. Hulley, D.T.W. Liang, Power Electronics and Motor Control, Second Edition. Cambridge University Press [6] Josep Vitayatil, Power Electronics Principles and Applications, McGraw-Hill, 995. [7] Duane Hanselman, Bruce Littlefield, Mastering Matlab 5, Prentice-Hall, 998. [8] Stepen J. Capman, Matlab Programming for Engineers, Brooks/Cole Tomson Learning,. [9] J.N. Ross, Te Essence of Power Electronics, Prentice-Hall, 997. [] IEEE Standard 59. [] Gordon W. Roberts, Adel S. Sedra, Spice, Second Edition, Oxford University Press, 999. BRUNO OSORNO, is a professor of electrical and computer engineering at California State University Nortridge. He is te lead faculty member in te Power Systems and Power Electronics program. Professor Osorno as written over tecnical papers. His current interest in researc is Fuzzy Logic applications in power electronics and electric motor drives. Proceedings of te American Society for Engineering Education Annual Conference & Exposition Copyrigt, American Society for Engineering Education Page 7.3.9