交流异步电机的Modelica模型
Modelica标准库里的异步电机模型过于复杂，为了便于学习，我用最基本的异步电机方程写了一个Modelica模型，公式参照陈伯时的《电力拖动自动控制系统--运动控制系统》第3版的190页到195页的内容，实际的电机模型参数参照了Novotny和Lipo的《Vector Control and Dynamics of AC Drives》第78页的一个例子参数并稍作修改。这个模型没有使用dq坐标系。
本模型中使用的电机主要参数为：
- 额定电压（相电压）：220 V
- 额定频率：50 Hz
- 极对数：2
- 转动惯量：0.1 kg.m^2
- 定子电阻：0.531 Ohm
- 转子电阻：0.408 Ohm
- 定子漏感：2.52 mH
- 转子漏感：2.52 mH
- 互感：8.47 mH
上述参数可根据实际电机的参数进行修改，负载转矩和负载惯量可根据实际仿真情况加以修改。
Modelica模型如下。
model SACIM "A Simple AC Induction Motor Model"
type Voltage=Real(unit="V");
type Current=Real(unit="A");
type Resistance=Real(unit="Ohm");
type Inductance=Real(unit="H");
type Speed=Real(unit="r/min");
type Torque=Real(unit="N.m");
type Inertia=Real(unit="kg.m^2");
type Frequency=Real(unit="Hz");
type Flux=Real(unit="Wb");
type Angle=Real(unit="rad");
type AngularVelocity=Real(unit="rad/s");
constant Real Pi = 3.1415926;
Current i_A"A Phase Current of Stator";
Current i_B"B Phase Current of Stator";
Current i_C"C Phase Current of Stator";
Voltage u_A"A Phase Voltage of Stator";
Voltage u_B"B Phase Voltage of Stator";
Voltage u_C"C Phase Voltage of Stator";
Current i_a"A Phase Current of Rotor";
Current i_b"B Phase Current of Rotor";
Current i_c"C Phase Current of Rotor";
Frequency f_s"Frequency of Stator";
Torque Tm"Torque of the Motor";
Speed n"Speed of the Motor";
Flux Psi_A"A Phase Flux-Linkage of Stator";
Flux Psi_B"B Phase Flux-Linkage of Stator";
Flux Psi_C"C Phase Flux-Linkage of Stator";
Flux Psi_a"a Phase Flux-Linkage of Rotor";
Flux Psi_b"b Phase Flux-Linkage of Rotor";
Flux Psi_c"c Phase Flux-Linkage of Rotor";
Angle phi"Electrical Angle of Rotor";
Angle phi_m"Mechnical Angle of Rotor";
AngularVelocity w"Angular Velocity of Rotor";
Torque Tl"Load Torque";
parameter Resistance Rs = 0.531"Stator Resistance";
parameter Resistance Rr = 0.408"Rotor Resistance";
parameter Inductance Ls = 0.00252"Stator Leakage Inductance";
parameter Inductance Lr = 0.00252"Rotor Leakage Inductance";
parameter Inductance Lm = 0.00847"Mutual Inductance";
parameter Frequency f_N = 50"Rated Frequency of Stator";
parameter Voltage u_N = 220"Rated Phase Voltage of Stator";
parameter Real p =2"number of pole pairs";
parameter Inertia Jm = 0.1"Motor Inertia";
parameter Inertia Jl = 0.1"Load Inertia";
initial equation
Psi_A = 0;
Psi_B = 0;
Psi_C = 0;
Psi_a = 0;
Psi_b = 0;
Psi_c = 0;
phi = 0;
w = 0;
equation
u_A = Rs * i_A + 1000 * der(Psi_A);
u_B = Rs * i_B + 1000 * der(Psi_B);
u_C = Rs * i_C + 1000 * der(Psi_C);
0 = Rr * i_a + 1000 * der(Psi_a);
0 = Rr * i_b + 1000 * der(Psi_b);
0 = Rr * i_c + 1000 * der(Psi_c);
Psi_A = (Lm+Ls)*i_A + (-0.5*Lm)*i_B + (-0.5*Lm)*i_C + (Lm*cos(phi))*i_a + (Lm*cos(phi+2*Pi/3))*i_b + (Lm*cos(phi-2*Pi/3))*i_c;
Psi_B = (-0.5*Lm)*i_A + (Lm+Ls)*i_B + (-0.5*Lm)*i_C + (Lm*cos(phi-2*Pi/3))*i_a + (Lm*cos(phi))*i_b + (Lm*cos(phi+2*Pi/3))*i_c;
Psi_C = (-0.5*Lm)*i_A + (-0.5*Lm)*i_B + (Lm+Ls)*i_C + (Lm*cos(phi+2*Pi/3))*i_a + (Lm*cos(phi-2*Pi/3))*i_b + (Lm*cos(phi))*i_c;
Psi_a = (Lm*cos(phi))*i_A + (Lm*cos(phi-2*Pi/3))*i_B + (Lm*cos(phi+2*Pi/3))*i_C + (Lm+Lr)*i_a + (-0.5*Lm)*i_b + (-0.5*Lm)*i_c;
Psi_b = (Lm*cos(phi+2*Pi/3))*i_A + (Lm*cos(phi))*i_B + (Lm*cos(phi-2*Pi/3))*i_C + (-0.5*Lm)*i_a + (Lm+Lr)*i_b + (-0.5*Lm)*i_c;
Psi_c = (Lm*cos(phi-2*Pi/3))*i_A + (Lm*cos(phi+2*Pi/3))*i_B + (Lm*cos(phi))*i_C + (-0.5*Lm)*i_a + (-0.5*Lm)*i_b + (Lm+Lr)*i_c;
Tm =-p*Lm*((i_A*i_a+i_B*i_b+i_C*i_c)*sin(phi)+(i_A*i_b+i_B*i_c+i_C*i_a)*sin(phi+2*Pi/3)+(i_A*i_c+i_B*i_a+i_C*i_b)*sin(phi-2*Pi/3));
w = 1000 * der(phi_m);
phi_m = phi/p;
n= w*60/(2*Pi);
Tm-Tl = (Jm+Jl) * 1000 * der(w);
if time <= 100 then
u_A = 0;
u_B = 0;
u_C = 0;
f_s = 0;
Tl = 0;
else
f_s = f_N;
u_A = u_N * 1.414 * sin(2*Pi*f_s*time/1000);
u_B = u_N * 1.414 * sin(2*Pi*f_s*time/1000-2*Pi/3);
u_C = u_N * 1.414 * sin(2*Pi*f_s*time/1000-4*Pi/3);
Tl = 10;
end if;
end SACIM;
在模型中，我们定义了电机的定子电压方程（由于仿真软件的时间单位是毫秒，所以所有求导操作前都乘以1000，下同）：
u_A = Rs * i_A + 1000 * der(Psi_A);
u_B = Rs * i_B + 1000 * der(Psi_B);
u_C = Rs * i_C + 1000 * der(Psi_C);
转子电压方程（针对鼠笼转子，如果采用绕线式转子，可以把外接的电阻和电抗参数引入方程）：
0 = Rr * i_a + 1000 * der(Psi_a);
0 = Rr * i_b + 1000 * der(Psi_b);
0 = Rr * i_c + 1000 * der(Psi_c);
定子磁链方程：
Psi_A = (Lm+Ls)*i_A + (-0.5*Lm)*i_B + (-0.5*Lm)*i_C + (Lm*cos(phi))*i_a + (Lm*cos(phi+2*Pi/3))*i_b + (Lm*cos(phi-2*Pi/3))*i_c;
Psi_B = (-0.5*Lm)*i_A + (Lm+Ls)*i_B + (-0.5*Lm)*i_C + (Lm*cos(phi-2*Pi/3))*i_a + (Lm*cos(phi))*i_b + (Lm*cos(phi+2*Pi/3))*i_c;
Psi_C = (-0.5*Lm)*i_A + (-0.5*Lm)*i_B + (Lm+Ls)*i_C + (Lm*cos(phi+2*Pi/3))*i_a + (Lm*cos(phi-2*Pi/3))*i_b + (Lm*cos(phi))*i_c;
转子磁链方程：
Psi_a = (Lm*cos(phi))*i_A + (Lm*cos(phi-2*Pi/3))*i_B + (Lm*cos(phi+2*Pi/3))*i_C + (Lm+Lr)*i_a + (-0.5*Lm)*i_b + (-0.5*Lm)*i_c;
Psi_b = (Lm*cos(phi+2*Pi/3))*i_A + (Lm*cos(phi))*i_B + (Lm*cos(phi-2*Pi/3))*i_C + (-0.5*Lm)*i_a + (Lm+Lr)*i_b + (-0.5*Lm)*i_c;
Psi_c = (Lm*cos(phi-2*Pi/3))*i_A + (Lm*cos(phi+2*Pi/3))*i_B + (Lm*cos(phi))*i_C + (-0.5*Lm)*i_a + (-0.5*Lm)*i_b + (Lm+Lr)*i_c;
电磁转矩方程：
Tm =-p*Lm*((i_A*i_a+i_B*i_b+i_C*i_c)*sin(phi)+(i_A*i_b+i_B*i_c+i_C*i_a)*sin(phi+2*Pi/3)+(i_A*i_c+i_B*i_a+i_C*i_b)*sin(phi-2*Pi/3));
转子的角速度和机械角位移存在导数关系：
w = 1000 * der(phi_m);
下面两个公式实现转子电角度与机械角度，角速度和电机转速之间的单位转换。
phi_m = phi/p;
n= w*60/(2*Pi);
电机的实际速度由电机的电磁转矩、负载转矩以及电机和负载的共同负载惯量决定(采用最简化的刚体动力学模型，可逐渐扩展):
Tm-Tl = (Jm+Jl) * 1000 * der(w);
在此基础上，我们就可以通过设定外部条件的变化来仿真电机的运行，如改变定子电压和频率，定子串电阻，转子串电阻，改变负载大小等。
这里只给出了最简单的额定电压直接启动的例子
if time <= 100 then
u_A = 0;
u_B = 0;
u_C = 0;
f_s = 0;
Tl = 0;
else
f_s = f_N;
u_A = u_N * 1.414 * sin(2*Pi*f_s*time/1000);
u_B = u_N * 1.414 * sin(2*Pi*f_s*time/1000-2*Pi/3);
u_C = u_N * 1.414 * sin(2*Pi*f_s*time/1000-4*Pi/3);
Tl = 10;
end if;
运行如下命令可进行模型的仿真：
simulate(SACIM,startTime=0,stopTime=2000)
