电机定子和转子的建模与仿真

建模与仿真
作业题目（一）
姓名：张娇
专业：电研-11
学号：2011301350117
建模与仿真演讲作业1
如下图所示直流电机电路图，电机转动惯量为J,电机扭矩方程为
ω
ωF i i K dt
d J
e a -=1
转子电流方程为：
ω
e a a a a a
i K U i R dt
di L 2-=+
定子电流方程：
e e e e e
U i R dt
di L =+
直流电机电路示意图
回答：
1. 将如上方程写为的状态空间模型形式。

2. 求解在U a = 2V ，U e = 2V 的输入条件下，系统稳定的转速和i a ,i e 大小？
3. 在2稳定条件下，若U a 和U e 从第5秒均变为5V ，试求10s 内的转子转
速的动态数值解，并给出转子电流和定子电流数值解。

解：1. 为计算和书写方便，现假定 F=1, J=1,K 1=K 2=1, L a =L e =1,R a =R e =1 • 选取状态变量为：
转速w 、转子电流i a 、定子电流i e • 输入量：转子电压U a 、定子电压U e
则系统的状态方程：
2.设U a = U e =2,系统在similink 下的仿真如下：
则仿真结果为： i a 如scope 所示：
由此图可知转子电流i a 在5s 后达到稳定值0.4。

e
e e a a e
a i u e
w i i u a
w
i
i i i w -=--=-=.
.
.
◆i e如scope1所示：
由此图可知定子电流i e在5s后达到稳定值2。

◆w如scope2所示：
由此图可知转速w在5s后达到稳定值0.8。

3.假设：U a、U e的值从第5秒突然从2跳变到5，系统在similink下仿真，仿真图如题2所示，修改参数U a、U e值如下：
则状态量相应的仿真曲线如下：
◆i a如scope所示：
由图可知转子电流i a出现震荡跳变，最后趋向于0.2。

◆i e如scope1所示：
由图可知定子电流i e随着跳变而变化最后趋向于5。

w如scope2所示：
由图可知转速w出现震荡跳变，并最终趋向于1。

求解系统变量的数值解的M文件如下所示：function xdot=lisan_shuzhi_jie(t,x)
if t<=5
xdot=[-x(1)+x(2)*x(3);
-x(1)*x(3)-x(2)+2;
-x(3)+2;]
end
if t>5
xdot=[-x(1)+x(2)*x(3);
-x(1)*x(3)-x(2)+5;
-x(3)+5;]
end
注：x(1)代表转子转速w；x(2)代表转子电流ia；x(3)代表定子电流ie；系统参数初值Ua=2V,Ue=2V,在第5秒时Ua变为5V,Ue变为5V
在MATLAB的运行窗口中输入
>> ode23(@lisan_shuzhi_jie,[0,10],[0;0;0.4]),grid on
回车后得到10s内系统状态的离散曲线如下图所示：
对应的数值解如下所示：
xdot=
2.0000
1.6000 xdot =
0.0000
2.0000
1.6000 xdot =
0.0000
1.9999
1.6000
xdot =
0.0000
1.9999
1.5999
xdot =
0.0001
1.9997
1.5998
xdot =
0.0002
1.9996
1.5997
xdot =
0.0002
1.9995
1.5996
xdot =
0.0006
1.9985
1.5988
xdot=
0.0008
1.9980
1.5984
xdot =
0.0010
1.9975
1.5980
xdot =
0.0030
1.9925
1.5940
xdot =
0.0040
1.9900
1.5920
xdot =
0.0051
1.9876
1.5900
xdot =
0.0125
1.9703
1.5763
xdot =
0.0162
1.9619
1.5696
0.0201
1.9533
1.5627 xdot =
0.0284
1.9360
1.5489 xdot =
0.0324
1.9275
1.5422 xdot =
0.0368
1.9189
1.5354 xdot =
0.0459
1.9013
1.5215 xdot =
0.0504
1.8927
1.5147 xdot =
0.0552
1.8840
1.5078 xdot =
0.0669
1.8631
1.4914 xdot =
0.0726
1.8529
1.4834 xdot =
0.0787
1.8426
1.4753 xdot =
0.0950
1.8160
1.4546
0.1028
1.8031
1.4446
xdot =
0.1113
1.7899
1.4344
xdot =
0.1340
1.7559
1.4083
xdot =
0.1446
1.7395
1.3959
xdot =
0.1566
1.7226
1.3831
xdot =
0.1887
1.6785
1.3499
xdot =
0.2031
1.6571
1.3345
xdot =
0.2200
1.6352
1.3183
xdot =
0.2669
1.5755
1.2750
xdot =
0.2866
1.5467
1.2555
xdot =
0.3111
1.5169
1.2346
0.3850
1.4275
1.1740
xdot =
0.4117
1.3844
1.1481
xdot =
0.4499
1.3394
1.1191
xdot =
0.6647
1.0858
0.9686
xdot =
0.6678
0.9616
0.9237
xdot =
0.7755
0.8371
0.8550
xdot =
0.5917
1.1671
1.0157
xdot =
0.6169
1.0834
0.9783
xdot =
0.6876
0.9965
0.9302
xdot =
0.8002
0.8156
0.8443
xdot =
0.8058
0.7308
0.8132
0.8567
0.6408
0.7732
xdot =
0.9139
0.4809
0.7112
xdot =
0.9050
0.4111
0.6877
xdot =
0.9275
0.3330
0.6587
xdot =
0.9463
0.1697
0.6031
xdot =
0.9175
0.1062
0.5824
xdot =
0.9203
0.0289
0.5564
xdot =
0.8938
-0.1365
0.5039
xdot =
0.8409
-0.1879
0.4851
xdot =
0.8218
-0.2622
0.4607
xdot =
0.7369
-0.4203
0.4090
0.6561
-0.4453
0.3919
xdot =
0.6121
-0.5101
0.3680
xdot =
0.4897
-0.6051
0.3275
xdot =
0.4147
-0.5979
0.3140
xdot =
0.3575
-0.6328
0.2953
xdot =
0.2262
-0.6654
0.2633
xdot =
0.1672
-0.6342
0.2525
xdot =
0.1095
-0.6416
0.2377
xdot =
-0.0132
-0.6211
0.2110
xdot =
-0.0517
-0.5724
0.2022
xdot =
-0.1022
-0.5577
0.1899
-0.2058
-0.4898
0.1658
xdot =
-0.2173
-0.4272
0.1583
xdot =
-0.2553
-0.3953
0.1473
xdot =
-0.3084
-0.3049
0.1287
xdot =
-0.2947
-0.2540
0.1229
xdot =
-0.3116
-0.2157
0.1144
xdot =
-0.3191
-0.1250
0.0995
xdot =
-0.2888
-0.0914
0.0949
xdot =
-0.2878
-0.0559
0.0881
xdot =
-0.3175
-0.1438
0.1026
xdot =
-0.2990
-0.1152
0.0985
-0.2982
-0.0846
0.0930 xdot =
-0.2810
-0.0255
0.0834 xdot =
-0.2579
-0.0087
0.0800 xdot =
-0.2481
0.0153
0.0756 xdot =
-0.2175
0.0557
0.0679 xdot =
-0.1954
0.0618
0.0653 xdot =
-0.1808
0.0775
0.0617 xdot =
-0.1441
0.1008
0.0553 xdot =
-0.1247
0.0985
0.0531 xdot =
-0.1085
0.1066
0.0501 xdot =
-0.0699
0.1142
0.0443
-0.0550
0.1045
0.0424
xdot =
-0.0393
0.1060
0.0398
xdot =
-0.0027
0.0977
0.0342
xdot =
0.0041
0.0812
0.0326
xdot =
0.0171
0.0771
0.0301
xdot =
0.0417
0.0547
0.0250
xdot =
0.0362
0.0378
0.0238
xdot =
0.0437
0.0306
0.0215
xdot =
0.0486
0.0054
0.0175
xdot =
0.0352
-0.0019
0.0166
xdot =
0.0368
-0.0085
0.0148
0.0288
-0.0213
0.0123
xdot =
0.0203
-0.0200
0.0117
xdot =
0.0177
-0.0237
0.0106
xdot =
0.0177
2.9763
3.0106
xdot =
0.0328
-0.0149
0.0135
xdot =
0.0297
-0.0162
0.0131
xdot =
0.0277
-0.0187
0.0125
xdot =
0.0227
-0.0226
0.0115
xdot =
0.0198
-0.0227
0.0111
xdot =
0.0175
-0.0241
0.0106
xdot =
0.0175
2.9759
3.0106
0.0252
-0.0207
0.0120
xdot =
0.0238
-0.0212
0.0117
xdot =
0.0226
-0.0220
0.0115
xdot =
0.0200
-0.0233
0.0110
xdot =
0.0187
-0.0236
0.0108
xdot =
0.0175
-0.0241
0.0106
xdot =
0.0175
2.9759
3.0106
xdot =
0.0213
-0.0227
0.0113
xdot =
0.0207
-0.0229
0.0111
xdot =
0.0200
-0.0232
0.0110
xdot =
0.0187
-0.0237
0.0108
0.0181
-0.0239
0.0107
xdot =
0.0181
2.9761
3.0107
xdot =
0.1543
-0.1247
-0.0450
xdot =
0.1543
2.8753
2.9550
xdot =
0.0193
-0.0235
0.0109
xdot =
0.0189
-0.0236
0.0108
xdot =
0.0185
-0.0238
0.0108
xdot =
0.0178
-0.0240
0.0106
xdot =
0.0178
2.9760
3.0106
xdot =
0.1521
-0.1232
-0.0442
xdot =
0.1521
2.8768
2.9558
0.1529
-0.1284
-0.0457
xdot =
0.1529
2.8716
2.9543
xdot =
0.0182
-0.0239
0.0107
xdot =
0.0182
2.9761
3.0107
xdot =
0.0846
-0.0735
-0.0167
xdot =
0.0846
2.9265
2.9833
xdot =
0.0859
-0.0757
-0.0176
xdot =
0.0859
2.9243
2.9824
xdot =
0.0183
-0.0238
0.0107
xdot =
0.0183
2.9762
3.0107
xdot =
0.0514
-0.0486
-0.0030
0.0514
2.9514
2.9970 xdot =
0.0523
-0.0497
-0.0035 xdot =
0.0523
2.9503
2.9965 xdot =
0.0184
-0.0238
0.0107 xdot =
0.0184
-0.0238
0.0107 xdot =
0.0184
2.9762
3.0107 xdot =
0.0281
-0.0312
0.0067 xdot =
0.0281
2.9688
3.0067 xdot =
0.0390
-0.0395
0.0021 xdot =
0.0390
2.9605
3.0021 xdot =
0.0444
-0.0436
-0.0002
0.0444
2.9564
2.9998
xdot =
0.0499
-0.0478
-0.0025
xdot =
0.0499
2.9522
2.9975
xdot =
0.1047
-0.0894
-0.0253
xdot =
0.1047
2.9106
2.9747
xdot =
0.1308
-0.1109
-0.0364
xdot =
0.1308
2.8891
2.9636
xdot =
0.1583
-0.1319
-0.0477
xdot =
0.1583
2.8681
2.9523
xdot =
0.4410
-0.3449
-0.1600
xdot =
0.4410
2.6551
2.8400
0.5444
-0.4702
-0.2097
xdot =
0.5444
2.5298
2.7903
xdot =
0.6840
-0.5814
-0.2639
xdot =
0.6840
2.4186
2.7361
xdot =
1.1546
-1.0067
-0.4459
xdot =
1.1546
1.9933
2.5541
xdot =
1.2414
-1.2849
-0.5187
xdot =
1.2414
1.7151
2.4813
xdot =
1.4494
-1.5068
-0.6047
xdot =
1.4494
1.4932
2.3953
xdot =
1.7375
-1.9684
-0.7458
1.7375
1.0316
2.2542
xdot =
1.7378
-2.2249
-0.8038
xdot =
1.7378
0.7751
2.1962
xdot =
1.8463
-2.4556
-0.8709
xdot =
1.8463
0.5444
2.1291
xdot =
1.9571
-2.9248
-0.9873
xdot =
1.9571
0.0752
2.0127
xdot =
1.8788
-3.1445
-1.0359
xdot =
1.8788
-0.1445
1.9641
xdot =
1.8993
-3.3672
-1.0914
xdot =
1.8993
-0.3672
1.9086
1.8414
-3.8314
-1.1974
xdot =
1.8414
-0.8314
1.8026
xdot =
1.6813
-4.0040
-1.2415
xdot =
1.6813
-1.0040
1.7585
xdot =
1.6206
-4.2088
-1.2920
xdot =
1.6206
-1.2088
1.7080
xdot =
1.3842
-4.6170
-1.3915
xdot =
1.3842
-1.6170
1.6085
xdot =
1.1514
-4.7064
-1.4326
xdot =
1.1514
-1.7064
1.5674
xdot =
1.0131
-4.8659
-1.4800
1.0131
-1.8659
1.5200
xdot =
0.6969
-5.0814
-1.5546
xdot =
0.6969
-2.0814
1.4454
xdot =
0.4964
-5.0825
-1.5864
xdot =
0.4964
-2.0825
1.4136
xdot =
0.3358
-5.1555
-1.6221
xdot =
0.3358
-2.1555
1.3779
xdot =
0.0092
-5.2246
-1.6825
xdot =
0.0092
-2.2246
1.3175
xdot =
-0.1556
-5.1684
-1.7087
xdot =
-0.1556
-2.1684
1.2913
-0.3117
-5.1763
-1.7377 xdot =
-0.3117
-2.1763
1.2623 xdot =
-0.6079
-5.1337
-1.7881 xdot =
-0.6079
-2.1337
1.2119 xdot =
-0.7327
-5.0415
-1.8102 xdot =
-0.7327
-2.0415
1.1898 xdot =
-0.8683
-5.0013
-1.8345 xdot =
-0.8683
-2.0013
1.1655 xdot =
-1.1102
-4.8803
-1.8771 xdot =
-1.1102
-1.8803
1.1229 xdot =
-1.1947
-1.7697
1.1039
-1.3011
-1.6972
1.0833
xdot =
-1.4770
-1.5308
1.0474
xdot =
-1.5249
-1.4171
1.0312
xdot =
-1.5988
-1.3277
1.0138
xdot =
-1.7051
-1.1530
0.9849
xdot =
-1.7244
-1.0520
0.9717
xdot =
-1.7667
-0.9625
0.9577
xdot =
-1.8367
-0.7501
0.9274
xdot =
-1.8279
-0.6372
0.9137
xdot =
-1.8494
-0.5323
0.8991
xdot =
-1.8749
-0.1494
0.8497
-1.7543
0.0280
0.8290
xdot =
-1.7324
0.1928
0.8055
xdot =
-1.8633
-0.3197
0.8715
xdot =
-1.8292
-0.2177
0.8590
xdot =
-1.8243
-0.1158
0.8455
xdot =
-1.7867
0.0890
0.8196
xdot =
-1.7300
0.1772
0.8078
xdot =
-1.7011
0.2722
0.7952
xdot =
-1.6165
0.4570
0.7707
xdot =
-1.5417
0.5265
0.7596
xdot =
-1.4917
0.6092
0.7477
-1.3815
0.7457
0.7272
xdot =
-1.3065
0.7904
0.7178
xdot =
-1.2473
0.8500
0.7078
xdot =
-1.1300
0.9424
0.6907
xdot =
-1.0608
0.9679
0.6828
xdot =
-1.0004
1.0071
0.6745
xdot =
-0.8935
1.0611
0.6610
xdot =
-0.8360
1.0732
0.6547
xdot =
-0.7821
1.0954
0.6481
xdot =
-0.9218
1.0468
0.6646
xdot =
-0.8803
1.0586
0.6598
-0.8406
1.0758
0.6550
xdot =
-0.7593
1.1051
0.6453
xdot =
-0.7175
1.1116
0.6407
xdot =
-0.6768
1.1237
0.6360
xdot =
-0.5756
1.1467
0.6246
xdot =
-0.5249
1.1460
0.6191
xdot =
-0.4749
1.1538
0.6135
xdot =
-0.3492
1.1629
0.5998
xdot =
-0.2891
1.1491
0.5933
xdot =
-0.2284
1.1483
0.5866
xdot =
-0.0842
1.1314
0.5709
-0.0206
1.0986
0.5637
xdot =
0.0467
1.0836
0.5561
xdot =
0.2008
1.0283
0.5386
xdot =
0.2592
0.9714
0.5307
xdot =
0.3274
0.9371
0.5222
xdot =
0.4791
0.8296
0.5025
xdot =
0.5191
0.7441
0.4938
xdot =
0.5800
0.6859
0.4843
xdot =
0.7118
0.5023
0.4605
xdot =
0.7066
0.3811
0.4504
xdot =
0.7466
0.2940
0.4391
0.7861
0.0803
0.4164 xdot =
0.7262
-0.0230
0.4068 xdot =
0.7240
-0.1129
0.3959 xdot =
0.6700
-0.2926
0.3764 xdot =
0.5847
-0.3496
0.3681 xdot =
0.5472
-0.4188
0.3587 xdot =
0.4460
-0.5265
0.3434 xdot =
0.3691
-0.5423
0.3367 xdot =
0.3172
-0.5806
0.3293 xdot =
0.1981
-0.6324
0.3160 xdot =
0.1306
-0.6204
0.3101
0.0748
-0.6338
0.3037
xdot =
-0.0434
-0.6364
0.2916
xdot =
-0.0966
-0.6036
0.2863
xdot =
-0.1486
-0.5957
0.2805
xdot =
-0.2533
-0.5565
0.2693
xdot =
-0.2885
-0.5086
0.2643
xdot =
-0.3312
-0.4833
0.2589
xdot =
-0.4137
-0.4074
0.2478
xdot =
-0.4267
-0.3482
0.2429
xdot =
-0.4560
-0.3089
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