使用MATLAB快速对波形进行傅里叶分解到有限次谐波
目录
- 使用MATLAB快速对波形进行傅里叶分解到有限次谐波
- 1、解析表达式分解到有限次谐波
- 1.1、理论分析
- 1.2、全部代码
- 2、数值波形分解到有限次谐波
- 2.1、基础理论
- 2.2、对应代码
很多时候对功率放大器设计阻抗空间的分析都是从波形开始的，那么如何 从波形得到其对应的谐波分量呢？ 如何从单纯的波形得到设计所需的阻抗呢？答案就是傅里叶分解啦。
波形的分解一般分为两个情况，其一就是纯粹的解析表达式，如连续F类的波形：
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\begin{aligned} &{V_{C-F}} = {\left( {1 - \frac{2}{{\sqrt 3 }}\cos \theta } \right)^2} \cdot \left( {1 + \frac{1}{{\sqrt 3 }}\cos \theta } \right) \cdot (1 - \gamma \sin \theta )\\ &{I_{C-F}} = \frac{1}{\pi } + \frac{1}{2}\cos \theta + \frac{2}{{3\pi }}\cos (2\theta ) \end{aligned}
VC−F=(1−32cosθ)2⋅(1+31cosθ)⋅(1−γsinθ)IC−F=π1+21cosθ+3π2cos(2θ)
这种情况是相对非常直观的，对式子进行展开化简约分就能得到最终的结果，但是三角函数相乘化简也是非常的麻烦。
另一种情况就是数值表达式的波形，其解析表达式可能非常复杂难求，只能使用具有有限数值点进行求解，如考虑到无限次谐波的EF类的波形（图中是每1°一个数据，电压电流都是具有360大小的向量数据）：
对于这种情况，需要使用离散的傅里叶级数进行求解了，需要大量的计算，可以使用matlab来进行简化。
1、解析表达式分解到有限次谐波
1.1、理论分析
可以使用傅里叶级数进行分解：
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\begin{aligned} f(t)& =\frac{a_0}2+a_1cos(\omega t)+b_1sin(\omega t) \\ &+a_2cos(2\omega t)+b_2sin(2\omega t) \\ &+... \\ &=\frac{a_0}2+\sum_{n=1}^\infty\left[a_ncos(n\omega t)+b_nsin(n\omega t)\right] \end{aligned}
f(t)=2a0+a1cos(ωt)+b1sin(ωt)+a2cos(2ωt)+b2sin(2ωt)+...=2a0+n=1∑∞[ancos(nωt)+bnsin(nωt)]
使用积分计算其中的傅里叶参数：
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\begin{aligned}a_n&=\frac{2}{T}\int_{t_0}^{t_0+T}f(t)cos(n\omega t)dt\\b_n&=\frac{2}{T}\int_{t_0}^{t_0+T}f(t)sin(n\omega t)dt\end{aligned}
anbn=T2∫t0t0+Tf(t)cos(nωt)dt=T2∫t0t0+Tf(t)sin(nωt)dt
积分计算其中的傅里叶参数对应代码，周期T=2*pi：
disp('开始求取电流傅里叶分解数值');
I_A0=int(i_theta/pi,theta,int_i_low,int_i_high);
Fouier=I_A0/2;
for n=1:order
I_A(n)=(int(i_theta*cos(n*theta)/pi,theta,int_i_low,int_i_high));
I_B(n)=(int(i_theta*sin(n*theta)/pi,theta,int_i_low,int_i_high));
Fouier=Fouier+I_A(n)*cos(n*theta)+I_B(n)*sin(n*theta);
end
disp(Fouier)
disp('开始求取电压傅里叶分解数值');
V_A0=int(v_theta/pi,theta,int_v_low,int_v_high);
Fouier=V_A0/2;
for n=1:order
V_A(n)=(int(v_theta*cos(n*theta)/pi,theta,int_v_low,int_v_high));
V_B(n)=(int(v_theta*sin(n*theta)/pi,theta,int_v_low,int_v_high));
Fouier=Fouier+V_A(n)*cos(n*theta)+V_B(n)*sin(n*theta);
end
disp(Fouier)
使用电流电压的傅里叶参数可以计算最终波形对应的阻抗：
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{Z_{n,f}} =- \frac{{{a_{V,n}} + j{b_{V,n}}}}{{{a_{I,n}} + j{b_{I,n}}}}
Zn,f=−aI,n+jbI,naV,n+jbV,n
阻抗计算对应代码（阻抗参考是Ropt，因此要乘以0.5，具体参考Ropt定义）：
% 阻抗参考是Ropt，因此要乘以0.5
disp('最佳阻抗');
for n=1:order
disp(simplify(-0.5*(V_A(n)+1j*V_B(n))/(I_A(n)+1j*I_B(n))));
end
可以看到，对于上面的连续F类，得到的结果为：
和理论结果一致：
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\begin{cases}Z_{\mathrm{l}f,C-F}=\left(\frac{2}{\sqrt{3}}+j\gamma\right)R_{opt}\\Z_{2f,C-F}=-j\frac{7\sqrt{3}\pi}{24}R_{opt}\\\\Z_{3f,C-F}=\infty\end{cases}
⎩
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⎧Zlf,C−F=(32+jγ)RoptZ2f,C−F=−j2473πRoptZ3f,C−F=∞
1.2、全部代码
clear all
close all
clc
syms theta gamma
v_theta=(1-2/(sqrt(3))*cos(theta))^2*(1+1/(sqrt(3))*cos(theta))*(1-gamma*sin(theta));
i_theta=1/pi+1/2*cos(theta)+2/(3*pi)*cos(2*theta);
order=3;
int_i_low=0;
int_i_high=2*pi;
int_v_low=0;
int_v_high=2*pi;
disp('开始求取电流傅里叶分解数值');
I_A0=int(i_theta/pi,theta,int_i_low,int_i_high);
Fouier=I_A0/2;
for n=1:order
I_A(n)=(int(i_theta*cos(n*theta)/pi,theta,int_i_low,int_i_high));
I_B(n)=(int(i_theta*sin(n*theta)/pi,theta,int_i_low,int_i_high));
Fouier=Fouier+I_A(n)*cos(n*theta)+I_B(n)*sin(n*theta);
end
disp(Fouier)
disp('开始求取电压傅里叶分解数值');
V_A0=int(v_theta/pi,theta,int_v_low,int_v_high);
Fouier=V_A0/2;
for n=1:order
V_A(n)=(int(v_theta*cos(n*theta)/pi,theta,int_v_low,int_v_high));
V_B(n)=(int(v_theta*sin(n*theta)/pi,theta,int_v_low,int_v_high));
Fouier=Fouier+V_A(n)*cos(n*theta)+V_B(n)*sin(n*theta);
end
disp(Fouier)
% 阻抗参考Ropt，因此要乘以0.5
disp('最佳阻抗');
for n=1:order
disp(simplify(-0.5*(V_A(n)+1j*V_B(n))/(I_A(n)+1j*I_B(n))));
end
2、数值波形分解到有限次谐波
2.1、基础理论
我们现在只知道离散个点的波形点，如上面的EF类的波形，也就是0-360度对应的电压电流纵坐标：
v_theta=[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.129750090188117 0.255289136488565 0.376625563240092 0.493771567461190 0.606743078675062 0.715559714306197 0.820244730715221 0.920824969944700 1.01733080225395 1.10979606452633 1.19825799463809 1.28275716188279 1.36333739355047 1.44004569776601 1.51293218269540 1.58204997223385 1.64745511829383 1.70920650981569 1.76736577862760 1.82199720228559 1.87316760402839 1.92094624998539 1.96540474377936 2.00661691866925 2.04465872738108 2.07960812977794 2.11154497852306 2.14055090289194 2.16670919089223 2.19010466985182 2.21082358563754 2.22895348066865 2.24458307089029 2.25780212187380 2.26870132421120 2.27737216837235 2.28390681919350 2.28839799016632 2.29093881769660 2.29162273550150 2.29054334931401 2.28779431206257 2.28346919969278 2.27766138779744 2.27046392921947 2.26196943279093 2.25226994336973 2.24145682333331 2.22962063568691 2.21685102894105 2.20323662391102 2.18886490258756 2.17382209922559 2.15819309379440 2.14206130792919 2.12550860352063 2.10861518407504 2.09145949897413 2.07411815075892 2.05666580555838 2.03917510677888 2.02171659216579 2.00435861434412 1.98716726494012 1.97020630238054 1.95353708346157 1.93721849877388 1.92130691206479 1.90585610361357 1.89091721768970 1.87653871415869 1.86276632429423 1.84964301084941 1.83720893243417 1.82550141223995 1.81455491114661 1.80440100524056 1.79506836776717 1.78658275553402 1.77896699977591 1.77224100148585 1.76642173121067 1.76152323330329 1.75755663461776 1.75453015762705 1.75244913793754 1.75131604616785 1.75113051415408 1.75188936543721 1.75358664998267 1.75621368307635 1.75975908833542 1.76420884476677 1.76954633780040 1.77575241421946 1.78280544090333 1.79068136729513 1.79935379149957 1.80879402991220 1.81897119027659 1.82985224806066 1.84140212603937 1.85358377696624 1.86635826921214 1.87968487524548 1.89352116282429 1.90782308876684 1.92254509516368 1.93764020789113 1.95306013728256 1.96875538081114 1.98467532763490 2.00076836485232 2.01698198531430 2.03326289683643 2.04955713265309 2.06581016295376 2.08196700733989 2.09797234803965 2.11377064371706 2.12930624371044 2.14452350253532 2.15936689448609 2.17378112817058 2.18771126081174 2.20110281215137 2.21390187779056 2.22605524180278 2.23751048845669 2.24821611288614 2.25812163054717 2.26717768530275 2.27533615597819 2.28255026123205 2.28877466258980 2.29396556548978 2.29808081819375 2.30108000841734 2.30292455753854 2.30357781224585 2.30300513349109 2.30117398261556 2.29805400452179 2.29361710776788 2.28783754146454 2.28069196886001 2.27215953750263 2.26222194587468 2.25086350639677 2.23807120470647 2.22383475512025 2.20814665219245 2.19100221829117 2.17239964711551 2.15234004308466 2.13082745653473 2.10786891466503 2.08347444818129 2.05765711358891 2.03043301109573 2.00182129808950 1.97184419816107 1.94052700565101 1.90789808570285 1.87398886981256 1.83883384687003 1.80247054969468 1.76493953707278 1.72628437131108 1.68655159132716 1.64579068130262 1.60405403493198 1.56139691530578 1.51787741047259 1.47355638473006 1.42849742570208 1.38276678726365 1.33643332838164 1.28956844794501 1.24224601566347 1.19454229911915 1.14653588706108 1.09830760903752 1.04994045146629 1.00151947024801 0.953131700032240 0.904866060251106 0.856813258039101 0.809065688162828 0.761717330088043 0.714863642315586 0.668601454121624 0.623028854841491 0.578245080839374 0.534350400309946 0.491445996060881 0.449633846428079 0.409016604478074 0.369697475654887 0.331780094030513 0.295368397320433 0.260566500827536 0.227478570479249 0.196208695124241 0.166860758256219 0.139538309333334 0.114344434862345 0.0913816294174346 0.0707516667636549 0.0525554712551816 0.0368929896782647 0.0238630637086112 0.0135633031520352 0.00608996013661750 0.00153780442336330];
i_theta=[2.71877140023093e-15 0.0105802447184730 0.0214593627914783 0.0326318770606530 0.0440919543587784 0.0558334102470919 0.0678497141808513 0.0801339950975745 0.0926790474218736 0.105477337480283 0.118521010318992 0.131801896916889 0.145311521785851 0.159041110949733 0.172981600293053 0.187123644269917 0.201457624963267 0.215973661484159 0.230661619700282 0.245511122282607 0.260511559058602 0.275652097660105 0.290921694453584 0.306309105740139 0.321802899212305 0.337391465654379 0.353063030872676 0.368805667841881 0.384607309053348 0.400455759050972 0.416338707140022 0.432243740254109 0.448158355965254 0.464069975621858 0.479965957599197 0.495833610646935 0.511660207318022 0.527432997463224 0.543139221775473 0.558766125368123 0.574300971371190 0.589731054529576 0.605043714787323 0.620226350841903 0.635266433652606 0.650151519887139 0.664869265290589 0.679407437961026 0.693753931516094 0.707896778135093 0.721824161461166 0.735524429348392 0.748986106438761 0.762197906554188 0.775148744888967 0.787827749988275 0.800224275498601 0.812327911676234 0.824128496640252 0.835616127356703 0.846781170341063 0.857614272066312 0.868106369064369 0.878248697708976 0.888032803668483 0.897450551017397 0.906494130995960 0.915156070407433 0.923429239643184 0.931306860326153 0.938782512563674 0.945850141801153 0.952504065268514 0.958738978011872 0.964549958503340 0.969932473822392 0.974882384402731 0.979395948339108 0.983469825249066 0.987101079685117 0.990287184093406 0.993026021315432 0.995315886629956 0.997155489332780 0.998543953852620 0.999480820401845 0.999966045161431 1 0.999583471727408 0.998717660883866 0.997404180066151 0.995645051793016 0.993442705912439 0.990799976553921 0.987720098629568 0.984206703888232 0.980263816527522 0.975895848369025 0.971107593602574 0.965904223105957 0.960291278346913 0.954274664874798 0.947860645409769 0.941055832537829 0.933867181020519 0.926301979728526 0.918367843208913 0.910072702896094 0.901424797977131 0.892432665922306 0.883105132692350 0.873451302634046 0.863480548076357 0.853202498639505 0.842627030269838 0.831764254013592 0.820624504542979 0.809218328448336 0.797556472310320 0.785649870566400 0.773509633186142 0.761147033169997 0.748573493886507 0.735800576263023 0.722839965845228 0.709703459740842 0.696402953463102 0.682950427689634 0.669357934952504 0.655637586275242 0.641801537772738 0.627861977229913 0.613831110675097 0.599721148964041 0.585544294390456 0.571312727338955 0.557038592996174 0.542733988135819 0.528410947993227 0.514081433244961 0.499757317108799 0.485450372579319 0.471172259814115 0.456934513685486 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;]
此时需要使用离散傅里叶级数进行傅里叶分量的计算，理论公式如：
可以使用傅里叶级数进行分解：
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\begin{aligned} f(t)& =\frac{a_0}2+a_1cos(\omega t)+b_1sin(\omega t) \\ &+a_2cos(2\omega t)+b_2sin(2\omega t) \\ &+... \\ &=\frac{a_0}2+\sum_{n=1}^\infty\left[a_ncos(n\omega t)+b_nsin(n\omega t)\right] \end{aligned}
f(t)=2a0+a1cos(ωt)+b1sin(ωt)+a2cos(2ωt)+b2sin(2ωt)+...=2a0+n=1∑∞[ancos(nωt)+bnsin(nωt)]
使用求和计算其中的傅里叶参数：
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\begin{array}{l} {a_n} = \frac{2}{N}\sum {f(\theta )cos(n\theta )d\theta } \\ {b_n} = \frac{2}{N}\sum {f(\theta )\sin (n\theta )d\theta } \end{array}
an=N2∑f(θ)cos(nθ)dθbn=N2∑f(θ)sin(nθ)dθ
对应代码：
disp('开始求取电流傅里叶分解数值');
I_A0=sum(i_theta)/(length(theta_num))*2;
Fouier=I_A0/2;
for n=1:order
I_A(n)=2*sum(i_theta.*cos(n*theta_num))/(length(theta_num));
I_B(n)=2*sum(i_theta.*sin(n*theta_num))/(length(theta_num));
Fouier=Fouier+I_A(n)*cos(n*theta)+I_B(n)*sin(n*theta);
end
disp(vpa(Fouier))
使用上面的EF类带入，得到结果为（2次谐波短路，三次谐波开路，画Smith圆图匹配时由于匹配方向问题需要对阻抗取共轭）：
结果和理论一致。
2.2、对应代码
clear all
close all
clc
syms theta
theta_num=(0:pi/180:2*pi-pi/180);
% 连续F类，相乘变量，自由因子
% gamma=0;
% v_theta=((1-2/(sqrt(3)).*cos(theta_num)).^2).*(1+1/(sqrt(3)).*cos(theta_num)).*(1-gamma.*sin(theta_num));
% i_theta=1/pi+1/2*cos(theta_num)+2/(3*pi)*cos(2*theta_num);
% EF类波形
v_theta=[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.129750090188117 0.255289136488565 0.376625563240092 0.493771567461190 0.606743078675062 0.715559714306197 0.820244730715221 0.920824969944700 1.01733080225395 1.10979606452633 1.19825799463809 1.28275716188279 1.36333739355047 1.44004569776601 1.51293218269540 1.58204997223385 1.64745511829383 1.70920650981569 1.76736577862760 1.82199720228559 1.87316760402839 1.92094624998539 1.96540474377936 2.00661691866925 2.04465872738108 2.07960812977794 2.11154497852306 2.14055090289194 2.16670919089223 2.19010466985182 2.21082358563754 2.22895348066865 2.24458307089029 2.25780212187380 2.26870132421120 2.27737216837235 2.28390681919350 2.28839799016632 2.29093881769660 2.29162273550150 2.29054334931401 2.28779431206257 2.28346919969278 2.27766138779744 2.27046392921947 2.26196943279093 2.25226994336973 2.24145682333331 2.22962063568691 2.21685102894105 2.20323662391102 2.18886490258756 2.17382209922559 2.15819309379440 2.14206130792919 2.12550860352063 2.10861518407504 2.09145949897413 2.07411815075892 2.05666580555838 2.03917510677888 2.02171659216579 2.00435861434412 1.98716726494012 1.97020630238054 1.95353708346157 1.93721849877388 1.92130691206479 1.90585610361357 1.89091721768970 1.87653871415869 1.86276632429423 1.84964301084941 1.83720893243417 1.82550141223995 1.81455491114661 1.80440100524056 1.79506836776717 1.78658275553402 1.77896699977591 1.77224100148585 1.76642173121067 1.76152323330329 1.75755663461776 1.75453015762705 1.75244913793754 1.75131604616785 1.75113051415408 1.75188936543721 1.75358664998267 1.75621368307635 1.75975908833542 1.76420884476677 1.76954633780040 1.77575241421946 1.78280544090333 1.79068136729513 1.79935379149957 1.80879402991220 1.81897119027659 1.82985224806066 1.84140212603937 1.85358377696624 1.86635826921214 1.87968487524548 1.89352116282429 1.90782308876684 1.92254509516368 1.93764020789113 1.95306013728256 1.96875538081114 1.98467532763490 2.00076836485232 2.01698198531430 2.03326289683643 2.04955713265309 2.06581016295376 2.08196700733989 2.09797234803965 2.11377064371706 2.12930624371044 2.14452350253532 2.15936689448609 2.17378112817058 2.18771126081174 2.20110281215137 2.21390187779056 2.22605524180278 2.23751048845669 2.24821611288614 2.25812163054717 2.26717768530275 2.27533615597819 2.28255026123205 2.28877466258980 2.29396556548978 2.29808081819375 2.30108000841734 2.30292455753854 2.30357781224585 2.30300513349109 2.30117398261556 2.29805400452179 2.29361710776788 2.28783754146454 2.28069196886001 2.27215953750263 2.26222194587468 2.25086350639677 2.23807120470647 2.22383475512025 2.20814665219245 2.19100221829117 2.17239964711551 2.15234004308466 2.13082745653473 2.10786891466503 2.08347444818129 2.05765711358891 2.03043301109573 2.00182129808950 1.97184419816107 1.94052700565101 1.90789808570285 1.87398886981256 1.83883384687003 1.80247054969468 1.76493953707278 1.72628437131108 1.68655159132716 1.64579068130262 1.60405403493198 1.56139691530578 1.51787741047259 1.47355638473006 1.42849742570208 1.38276678726365 1.33643332838164 1.28956844794501 1.24224601566347 1.19454229911915 1.14653588706108 1.09830760903752 1.04994045146629 1.00151947024801 0.953131700032240 0.904866060251106 0.856813258039101 0.809065688162828 0.761717330088043 0.714863642315586 0.668601454121624 0.623028854841491 0.578245080839374 0.534350400309946 0.491445996060881 0.449633846428079 0.409016604478074 0.369697475654887 0.331780094030513 0.295368397320433 0.260566500827536 0.227478570479249 0.196208695124241 0.166860758256219 0.139538309333334 0.114344434862345 0.0913816294174346 0.0707516667636549 0.0525554712551816 0.0368929896782647 0.0238630637086112 0.0135633031520352 0.00608996013661750 0.00153780442336330];
i_theta=[2.71877140023093e-15 0.0105802447184730 0.0214593627914783 0.0326318770606530 0.0440919543587784 0.0558334102470919 0.0678497141808513 0.0801339950975745 0.0926790474218736 0.105477337480283 0.118521010318992 0.131801896916889 0.145311521785851 0.159041110949733 0.172981600293053 0.187123644269917 0.201457624963267 0.215973661484159 0.230661619700282 0.245511122282607 0.260511559058602 0.275652097660105 0.290921694453584 0.306309105740139 0.321802899212305 0.337391465654379 0.353063030872676 0.368805667841881 0.384607309053348 0.400455759050972 0.416338707140022 0.432243740254109 0.448158355965254 0.464069975621858 0.479965957599197 0.495833610646935 0.511660207318022 0.527432997463224 0.543139221775473 0.558766125368123 0.574300971371190 0.589731054529576 0.605043714787323 0.620226350841903 0.635266433652606 0.650151519887139 0.664869265290589 0.679407437961026 0.693753931516094 0.707896778135093 0.721824161461166 0.735524429348392 0.748986106438761 0.762197906554188 0.775148744888967 0.787827749988275 0.800224275498601 0.812327911676234 0.824128496640252 0.835616127356703 0.846781170341063 0.857614272066312 0.868106369064369 0.878248697708976 0.888032803668483 0.897450551017397 0.906494130995960 0.915156070407433 0.923429239643184 0.931306860326153 0.938782512563674 0.945850141801153 0.952504065268514 0.958738978011872 0.964549958503340 0.969932473822392 0.974882384402731 0.979395948339108 0.983469825249066 0.987101079685117 0.990287184093406 0.993026021315432 0.995315886629956 0.997155489332780 0.998543953852620 0.999480820401845 0.999966045161431 1 0.999583471727408 0.998717660883866 0.997404180066151 0.995645051793016 0.993442705912439 0.990799976553921 0.987720098629568 0.984206703888232 0.980263816527522 0.975895848369025 0.971107593602574 0.965904223105957 0.960291278346913 0.954274664874798 0.947860645409769 0.941055832537829 0.933867181020519 0.926301979728526 0.918367843208913 0.910072702896094 0.901424797977131 0.892432665922306 0.883105132692350 0.873451302634046 0.863480548076357 0.853202498639505 0.842627030269838 0.831764254013592 0.820624504542979 0.809218328448336 0.797556472310320 0.785649870566400 0.773509633186142 0.761147033169997 0.748573493886507 0.735800576263023 0.722839965845228 0.709703459740842 0.696402953463102 0.682950427689634 0.669357934952504 0.655637586275242 0.641801537772738 0.627861977229913 0.613831110675097 0.599721148964041 0.585544294390456 0.571312727338955 0.557038592996174 0.542733988135819 0.528410947993227 0.514081433244961 0.499757317108799 0.485450372579319 0.471172259814115 0.456934513685486 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;]
% 分析到的阶数
order=3;
disp('开始求取电流傅里叶分解数值');
I_A0=sum(i_theta)/(length(theta_num))*2;
Fouier=I_A0/2;
for n=1:order
I_A(n)=2*sum(i_theta.*cos(n*theta_num))/(length(theta_num));
I_B(n)=2*sum(i_theta.*sin(n*theta_num))/(length(theta_num));
Fouier=Fouier+I_A(n)*cos(n*theta)+I_B(n)*sin(n*theta);
end
disp(vpa(Fouier))
disp('开始求取电压傅里叶分解数值');
I_V0=sum(v_theta)/(length(theta_num))*2;
Fouier=I_V0/2;
for n=1:order
V_A(n)=2*sum(v_theta.*cos(n*theta_num))/(length(theta_num));
V_B(n)=2*sum(v_theta.*sin(n*theta_num))/(length(theta_num));
Fouier=Fouier+I_A(n)*cos(n*theta)+I_B(n)*sin(n*theta);
end
disp(vpa(Fouier))
disp('最佳阻抗');
for n=1:order
tmp=(-0.5*(V_A(n)+1j*V_B(n))./(I_A(n)+1j*I_B(n)));
disp(['n=',num2str(n),'时阻抗：',num2str(tmp)]);
end
% 注意！！！画Smith圆图匹配时由于匹配方向问题需要对阻抗取共轭
