同步电机三相短路电流和转矩计算

1、同步电机三相短路电流和电磁转矩计算 编写 佘名寰本文是按照陈珩教授所著的同步电机运行基本理论与计算机算法一书介绍的算法和例题计算同步电机的三相短路电流。计算程序用MATLAB语言编写，计算结果与书中结果基本一致。本文可供电力系统电气技术人员和大专院校电力专业学生参考。1 计算方法1.1初始数据计算由短路前的机端电压u0, 定子绕组电流io, 和功率因数角0 求得短路前的功率角0=tan-1u0sin0+xqi0u0cos0+ri0-0从而得u0, i0 的正、交轴分量ud0=u0sin0uq0=u0cos0id0=i0sin(0+0)iq0=i0cos(0+0)短路前的空载电势是Eq0=uq0

2、+riq0+xdid0励磁电流为if0= Eq0/xaf式中 xd 为同步电机正轴同步电抗xq 同步电机交轴同步电抗 xaf 定子绕组与劢磁绕组间的互感电抗 r 定子绕组电阻1.2电流变化量的状态空间方程式同步电机突然短路时各绕组电流的变化量 id iq if iD iQ 的计算可运用以派克分量表示的状态空间方程式uduquf00=-xdxafxaD-xqxaQ-xafxfxfD-xaDxfDxD-xaQxQidiqifiDiQ+-rxq-xaQ-xd-rxafxaDrfrDrQidiqifiDiQ方程中各下标变量的含义为d-纵轴，q-横轴，f-励磁绕组，D-纵轴阻尼绕组，Q-横轴阻尼绕组，a

3、-定子绕组上式可简化为udq0=Xdq0(3)Idq0+Zdq0(3)Idq0化作电流变化量的常系数一阶微分方程组形式Idq0=-Xdq0（3）-1Zdq0(3)Idq0+Xdq0（3）-1udq0在三相短路时若励磁电压不可调，则udq0=-ud0 -uq0 0 0 0 t由于电流不能突变，t=0瞬间电流变化量的初值idq0 0= 0 0 0 0 0 t将电压变化量和电流变化量的初值代入微分方程，用数值计算的龙格-库塔法即可求出t=0+h 时刻的各电流变化量，反复计算则可求得各个时刻的idq0 ，叠加短路前绕组电流idq0 0= id0 iq0 if0 00t可得短路时电流全量idq0= id

4、 iq if iD iQ t用派克逆变换可得定子三相电流，以a相为例 ia=idcost+0-iqsint+0 0 短路t=0时转子位置角2.同步电机三相短路电流计算例题与程序电机参数r=0.005, rf=0.000656，rD=0.00151, rQ=0.00159xd=1,0, xq=0.60, xf=1.03, xD=0.95, xQ=0.70xaf=0.85, xaD=0.85, xfD=0.85, xaQ=0.45原始运行条件为额定负载 U0=1, i0=1, 0=0.5548 (单位为弧度，相对于cos=0.8) 短路时的转子位置角 0=3.1416 三相短路计算程序：CMSHO

5、RT3.M% part 1ra=0.005;rf=.000656;rzd=.00151;rzq=.00159;xd=1.0; xq=.60;xf=1.03;xzd=.95;xzq=.70;xaf=.85;xazd=.85;xfzd=.85;xazq=.450; u0=1.0;i0=1.0;phas=.5548;cita0=3.1416;p=31.4160;h=.5236; x1=-1.0,0.0,0.85,0.85,0.0;0.0,-0.60,0.0,0.0,0.45;-0.85,0.0,1.03,0.85,0.0; -0.85,0.0,0.85,0.95,0.0;0.0,-0.45,0.0,

6、0.0,0.70; z1=-0.005,0.6,0.0,0.0,-0.45;-1.0,-0.005,0.85,0.85,0.0;0.0,0.0,0.000656,0.0,0.0; 0.0,0.0,0.0,0.00151,0.0;0.0,0.0,0.0,0.0,0.00159; g0=(u0*sin(phas)+xq*i0)/(u0*cos(phas)+ra*i0); g0=atan(g0)-phas; ud0=u0*sin(g0); uq0=u0*cos(g0); di0=i0*sin(g0+phas); qi0=i0*cos(g0+phas); eq0=uq0+ra*qi0+xd*di0;

7、fi0=eq0/xaf; du=-ud0,-uq0,0.0,0.0,0.0; x2=inv(x1); z2=-x2*z1; i1=x2*du' y=0.0,0.0,0.0,0.0,0.0;% part 2 t=0.0 ; for i=1:5 b(i)=y(i); end dy=z2*(y).'+i1; tt(1)=t; di(1)=y(1)+di0; qi(1)=y(2)+qi0; fi(1)=y(3)+fi0; zdi(1)=y(4); zqi(1)=y(5); ai(1)=di(1)*cos(t+cita0)-qi(1)*sin(t+cita0); tm(1)=qi(1)*

8、(-xd*di(1)+xaf*fi(1)+xazd*zdi(1)-di(1)*(-xq*qi(1)+ xazq*zqi(1); % while (tt(n)<p)% k=0; for n=2:10 a(1)=h/2; a(2)=a(1); a(3)=h; a(4)=h; for k=1:3 for i=1:5 c(i)=b(i)+a(k)*dy(i); y(i)=y(i)+a(k+1)*dy(i)/3.0; end dy=z2*(c).'+i1; end for i=1:5 y(i)=y(i)+h*dy(i)/6.0; end t=t+h; for i=1:5 b(i)=y(i)

9、; end dy=z2*(y).'+i1; % k=k+1;% part 3 tt(n)=t; di(n)=y(1)+di0; qi(n)=y(2)+qi0; fi(n)=y(3)+fi0; zdi(n)=y(4); zqi(n)=y(5); ai(n)=di(n)*cos(t+cita0)-qi(n)*sin(t+cita0); tm(n)=qi(n)*(-xd*di(n)+xaf*fi(n)+xazd*zdi(n)-di(n)*(-xq*qi(n)+ xazq*zqi(n); endplot(tt,ai); xlabel('t(rad)');ylabel('

10、ia'); title('3-phase fault of synchronous machine'); grid3程序说明：X1为状态空间方程的 Xdq0 矩阵Z1 为状态空间方程的 Zdq0 矩阵X2=Xdq0（3）-1I1=Xdq0（3）-1udq0程序 % part 1 第一部分输入初始数据，建立状态空间方程；程序 % part 2 第二部分用定步长四阶龙格-库塔法解一阶微分方程组，计算 tn+1=tn+h 时刻各绕组电流的变化量。时间步长 h用标么值，单位为弧度,h 取1/6 定步长四阶龙格-库塔法解一阶微分方程组的基本公式：y1=f1(t,y1,y2,ym)

11、, y1(t0)=y10y2=f2(t,y1,y2,ym), y2(t0)=y20.Ym=fm(t,y1,y2,ym), ym(t0)=ym0微分方程组由 tj 积分一步到 tj+1=tj+h Yi,j+1=yij+(k1i+2k2i+2k3i+k4i)h/6, i=1,2,mk1i=fi(tj,y1j,y2j,ymj), i=1,2,mk2i=fi(tj+h/2,y1j+k11h/2,ymj+k1mh/2), i=1,2,mk3i=fi(tj+h/2,y1j+k21h/2,ymj+k2mh/2), i=1,2,mk4i=fi(tj+h,y1j+k31h,ymj+k3mh), i=1,2,m程

12、序 % part 3 第三部分输出计算结果，tm 为转矩Tm =iqd+idqd=-xdid+xafif+xaDiDq=-xqiq+xaQiQ本程序计算时间周期为t=010X1/6,绘画取204.程序计算结果：T=0-10X1/6,取10个点纵轴短路电流didi = Columns 1 through 7 Columns 8 through 10 8.066892631590545 5.980344929918005 3.734356462313833A相短路电流aiai = Columns 1 through 7 Columns 8 through 10横轴短路电流qiqi = Columns 1 through 7 Columns 8 through 10励磁绕组电流fifi = Columns 1 through 7 Columns 8 through 10纵轴阻尼绕组电流zdizdi = Columns 1 through 7 Columns 8 through 10横轴阻力尼绕组电流zqizqi = Columns 1 through 7 Columns 8 through 10电磁转矩tmtm = Columns 1 through 7 C