Engineering and Scientific Computations Using MATLAB phần 10 docx

Chapter 6: SIMULINK 196 Solution. Using the differential equations, one must build the SIMULINK block diagram. From the differential equations given above, we obtain the following set of equations to be used in the SIMULINK mdl model: = w, . dB, dt To guarantee the balanced operating condition, the following phase voltages should be applied to the induction motor windings: uus(t) = Jzu, cos(uyt), Ubs(t) = Jzu, COS(W/t - +), u, (t) = Jzu, COS(U/f + 4.). The SIMULINK block diagram to simulate three-phase squirrel-cage induction motors is developed and illustrated in Figure 6.29. The Derivative blocks are used. Chapter 6: SIMULINK 197 Figure 6.29. SIMULINK block diagram to simulate squirrel-cage (c6 __ 2 4.mdl) induction motors Chapter 6: SIMULINK 198 350 ~ I I 0 L-L 0 005 01 015 02 025 03 035 04 Tme (seconds) Figure 6.30. Transient dynamic for the angular velocity w,(t) Chapter 6: SIWLINK 150- 100.1; 54- 0- -50- -lw -150 I99 I , , 1 I - Stator current ias. [A] 200, 150- 100 50- 0- -50 -100 -1%. I .' , i j - - + -200 ! I 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 lime (seconds) SIaIa current its, [A] 150 1W 50 0 i'$ I \.: b -50 -100 -150 1 -200L L ' 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 lime (seconds) Stator current ics. [A] 2w( I:, -2WlY , I_ I 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Time (seconds) Rotor current bar. [A] 200 -2w -J 8- 0 005 01 015 02 025 03 035 04 lime ICrmnnrlSi Rotor current ibr, [A] - -~ 1- Rotor current icr. [A] I -200 1 0 005 01 015 02 025 03 035 04 lime (seconds) Figure 6.3 1. Transient dynamic of the currents in the stator and rotor windings 0 Chapter 6: SIMULINK 200 Example 6,2.5. Simulation of permanent-magnet synchronous motors In SIMULINK simulate three-phase permanent-magnet synchronous motors described by five nonlinear differential equations [4]: - - UCS 7 Lm 2 L:, - L,,L, - t;5, 'bs + 2 L,, - Em Lrn + 'as + 2 L:, - L,,E, - L'5, 2 L:, LJ, - c - vm Ern - w, cosq - w,co 8 n - vm Ern - w, COs(0, + 5.) 2 Lf, - L, L, - L, 4 ) 2L2,,-LssL,-L, Yrn(2Lss - Ern) 2 L:s - L,, L, - L, - - 7 'cs7 2 L, - z, Lrn 7 'as + - 7 'bs + 2 - 2 L', - L, L, - L, rs Ern 2 Lfs - L, L, - L, + Lrn 4s - 2 Lf, - L, z, - L, 2 L, - L, L, - L, 2 Lf, - L, - 7 'as - 7 'bs - - ~Lrn At 2 L;, - L,?, Em - L, - - 2 L,, - Ern 2 Lt, - L,, L, - L, + - 1 'as + - 7 'bs + - 7 UCS7 Lrn 2 L;, - L, L, - L, Lrn 2 L', - L,, L, - L, wr * d'r - dt The following phase voltages are applied to guarantee the balance motor operation: ua,(t) = Jzu, cos~, , Uhs(t) = Jzu, cos(~, - 5n) and ucs(t> = Jzu, co 4, B + -n :) . The motor parameters are: uM = 40 V, r, = 1 ohm, L, = 0.002 H, L,, = 0.0002 H, Em = 0.0012 H, ty, = 0.08V-sec/rad7 tym = 0.08 N-&A, B, = 0.000008 N-m-sec/rad, and J= 0.00004 kg-m2. Solution. As the differential equations are known, one can develop the SIMULINK block diagram (mdl model) to simulate permanent-magnet synchronous motors. The resulting SIMULINK block diagram is illustrated in Figure 6.32. Chapter 6: SIMULINK 201 Figure 6.32. SIMULINK block diagram to simulate permanent-magnet synchronous motors (c6 -_ 2 5.mdl) The transient dynamics are studied as the motor accelerates and the rated voltage is u, (t) = A40 cos 0, , ubS (t) = ,h40 cos(0, - f X) and U, (t) = A40 COS(~, + f x). The motor parameters are downloaded using the following statement typed in the supplied to the stator windings. In particular, Command Window: The motor accelerates from stall, and the load torque 0.5 N-m is applied at 0 sec. synchronous motor. Figure 6.33 illustrates the evolution of four states for the three-phase permanent-magnet Chapter 6: SIMULINK 202 Figure 6.33. Transient dynamics of the permanent-magnet synchronous motor variables These state variables can be plotted using plot. In particular, the following m-file can be used to plot the transient data: Chapter 6: SIMULINK 203 Example 6.2.6. Simulation of permanent-magnet DC motors using the state-space model Simulate permanent-magnet DC motors in SIMULMK using the state-space form. The linear differential equations to model permanent-magnet DC motors are (see Examples 5.3.6 and 6.2.1) dwr -‘a . Bm 1 -I, dt J J J The motor parameters to be used in numerical simulations are: r, = 1 ohm, La = 0.02 H, k, = 0.3 V-sechad, J = 0.0001 kg-m2, and B,,, = 0.000005 N-m-sechad. The applied voltage is [::o]=[::]=[lb]- u, = 25rect(t) V. The initial conditions to be used are Solution. Using the differential equations which model permanent-magnet DC motors we obtain the model in the state-space form as Denoting the state and control variables to be x1 = ia, x2 = w, and u = u, , we find In general, dx we have - dt For our example, x = L:ll,A=l 2 landB= LJ JJ The output equation is y = w, . 1 -La]. - 0 Hence, we have the following state-space model for permanent-magnet DC motors: y=w,=Cx+Du=[O 1 +[Oba=[O 1 +[Oh, C=[O l]andD=[O] E:I E3 Chapter 6: SIMULINK 204 Using the State-Space block, the simulation can be performed. To attain flexibility, symbolic notations are used. The State-Space block is illustrated in Figure 6.34. Figure 6.34. State-Space block with parameters of permanent-magnet DC motors The developed SIMULINK mdl model is documented in Figure 6.35. Figure 6.35. SIMULINK block diagram to simulate the motor dynamics (c 6 __ 2 6 . mdl) The simulations are performed assigning the motor parameters and initial conditions. In particular, the motor parameters ( r, = 1 ohm, L, = 0.02 H, k, = 0.3 V-sec/rad, J = 0.0001 kg- m2, and B, = 0.000005 N-m-sechad), applied voltage u, = 25rect(t) V, and the initial conditions [ 1 1 0IT are downloaded. We input the following in the Command Window: Running the simulation and using the following plotting statement >> plot(x(:,l),x(:,2)); xlabel('Time (seconds)'); title('Angu1ar velocity wr, [rad/sec] ') the dynamics of the motor angular velocity result (Figures 6.36). Chapter 6: SIMULINK 205 400 300 200 100 0 -1 00 Angular velocity wr, [radlsec] 7 r - T I -1 1 I -400 I' - L I 0 0.2 0.4 0.6 0 8 1 12 14 1.6 Time (seconds) Figure 6.36. Angular velocity dynamics if u, = 40rect(t) V 1.8 2 0 Again it should be emphasized that different illustrative educational examples in aerospace and automotive applications are readily available. These examples with the corresponding SIMULINK block diagrams can be easily accessed and used to master the MATLAB environment. [...]... Lyshevski, S E., Control Systems Theory with Engineering Applications, Birkhauser, Boston, MA, 2002 Engineering and ScientEfic Computations Using MATLAB@ .Sergey E Lyshevski Copyright 02003 John Wiley & Sons, Inc ISBN: 0-471-46200-4 Appendix: MATLAB Functions, Operators, Characters, Commands, and Solvers 207 APPENDIX MATLAB Functions, Operators, Characters, Commands, and Solvers ['I 0 > >= -r= & I , , XOT... function handle functions func2str str2func Ab s Eva1 Real 1 Strings I MATLABdata type that is a handle to a function Return information about a function handle Constructs a function name string from a function handle Constructs a function handle from a function name string Absolute value and complex magnitude Interpret strings containing MATLAB expressions Real part of complex number MATLAB string handling... volume data set I Return coordinate and color limits for volume (scalar and vector) Table A 10. 4 Domain Generation gr iddata Data gridding and surface fitting me s h g rid Generation of X and Y arrays for 3D plots 213 Appendix: MATLAB Functions, Operators, Characters, Commands, and Solvers 2 14 Table A 10. 5 SDecialized Plotting area I Area plot box I Axis box for 2D and 3D dots comet Comet plot compass... cell array Create an array of all ones Appendix: MATLAB Functions, Operators, Characters, Commands, and Solvers rand randn zeros : (colon) Uniformly distributed random numbers and arrays Normally distributed random numbers and arrays Create an array of all zeros Regularly spaced vector ans computer ePs i Recent answer Identify the computer on which MATLAB running is Floating-point relative accuracy... SYmmlq Symmetric LQ method Appendix: MATLAB Functions, Operators, Characters, Commands, and Solvers expm funm logm sqrtm Matrix exponential Evaluate general matrix function Matrix logarithm Matrix square root qrdelete qrinsert Delete column from QR factorization Insert column in QR factorization spdiags sPeYe sprand sprandn sp rand sym Extract and create sparse band and diagonal matrices Sparse identity... multidimensional functions and interpolation ndims Number of array dimensions permute Rearranee the dimensions of a multidimensional arrav reshape Reshape array s h i f tdim Shift dimensions squeeze Remove singleton dimensions sub2 i n d Single index from subscripts Table A 10 MATLAB Functions 21 1 Appendix: MATLAB Funclions, Operators, Characters, Commands, and Solvers 2 12 1 Table A 10. 2 Plotting and Data Visualization... cursor Zoom in and out on a 2D plot Table A 10. 19 Region of Interest Dragrect I Drag XOR rectangles with mouse Drawnow I Complete any pending drawing Rbbox I Rubberband box Table A l l Polynomial and Interpolation Functions Table A 1 1.2 Data Internolation convhull I Convex hull convhu1In I Multidimensional convex hull I Appendix: MATLAB Functions, Operators, Characters, Commands, and Solvers 2 18... Appendix: MA TLAB Functions, Operators, Characters, Commands, and solvers ylabel zlabel Y-axis labels for 2D and 3D plots Z-axis labels for 3D plots Table A 10. 2.4 Surface, Mesh, and Contour Plots Table A .10. 3 Volume Visualization I Plot velocitv vectors as cones in 3D vector field contourslice I Draw contours in volume slice plane curl I Compute the curl and angular velocity of a vector field divergence... String to Function Handle Conversion I Constructs a function name string from a function handle str2iunc I Constructs a function handle from a function name string I func~str Deblank Findstr Lower Strip trailing blanks from the end of a string Find one string within another Convert string to lowercase 1 Appendix: MATLAB Functions, Operators, Characters, Commands, and Solvers 2 10 strvcat symvar texlabel... Variant of HSV lines Line color colormap prism Colormap of prism colors spring Shades of magenta and yellow colormap 215 Appendix: MATLAB Functions, Operators, Characters, Commands, and Solvers 2 16 summer winter getappdata isappda ta rmappda ta setappdata I Shades of green and yellow colormap I Shades of blue and green colonnaD Get value of application data True if application data exists Remove application . Table A. 10. MATLAB Functions Appendix: MATLAB Funclions, Operators, Characters, Commands, and Solvers 2 12 barh hist histc hold loglog pie DlOt Table A. 10. 2. Plotting and Data. Constructs a function handle from a function name string Ab s Eva1 Real 1 Strings I MATLAB string handling Absolute value and complex magnitude Interpret strings containing MATLAB expressions. function handle functions func2str st r2 func MATLAB data type that is a handle to a function Return information about a function handle Constructs a function name string from a function handle