Chapter 4.7 LU Decomposition More Examples Electrical Engineering Example Three-phase loads e common in AC systems. When the system is balanced the analysis can be simplified to a single equivalent rcuit model. However, when it is unbalanced the only practical solution involves the solution of simultaneous line equations. n one model the following equations need to be solved..746.456..8..8 2.456.746.8..8....8.7787.525..8 6..8..525.7787.8..9..8..8.88.64 6..8..8..64.88.9 Find the values of,,,,, and using LU decomposition. Solution u u2 u u4 u5 u6 2 u22 u2 u24 u25 u26 2 u u4 u5 u6 [ A] [ L][ U ] 4 42 4 u44 u45 u46 5 52 5 54 u55 u56 6 62 6 64 65 u66 The [ U ] matrix is the same as the one found at the end of the forwd elimination steps of the naïve Gauss elimination method. Forwd Elimination of Unknowns Since there e six equations, there will be five steps of forwd elimination of unknowns. First step Divide Row by.746 and multiply it by.456, that is, multiply Row by.456.746.656. Row.656 ( ) [.456.278.656.48429.656.48429] [ 72.64] Subtract the result from Row 2 to get 4.7.
4.7.2 Chapter 4.7.746..8..8.456.94.8..8...9464.7787.525..8.8.484.525.7787.8...9464..8.88.64.8.484.8..64.88 2 6..9 6..9 Divide Row by.746 and multiply it by., that is, multiply Row by..746.45. Row (.45)..656.45.724.45.724.686 Subtract the result from Row to get.746.456..8..8 2.94.9464.484.9464.484.9464.77857.526.9866.78928 6.69.8..525.7787.8..9..8..8.88.64 6..8..8..64.88.9 Divide Row by.746 and multiply it by.8, that is, multiply Row by.8.746.724. Row (.724) 5 5.8.48429.724 8.579.724 8.579.2869 Subtract the result from Row 4 to get.746.456..8..8 2.94.9464.484.9464.484.9464.77857.526.9866.78928 6.69.484.529.77879.78928.86 5.9..8..8.88.64 6..8..8..64.88.9 Divide Row by.746 and multiply it by., that is, multiply Row by..746.45. Row (.45)..656.45.724.45.724 Subtract the result from Row 5 to get [ ] [.686]
LU Decomposition-More Examples: Electrical Engineering 4.7..746.8.456.94.9464.484.9464...9464.77857.529.9866.8.8.484.526.77879.78928...9464.9866.78928.8787.64.8.484.78928.86.689.88 2 6.69 5.9 6.69.9 Divide Row by.746 and multiply it by.8, that is, multiply Row by.8.746.724. Row (.724) 5 5.8.48429.724 8.579.724 8.579.2869 Subtract the result from Row 6 to get.746.456..8..8 2.94.9464.484.9464.484.9464.77857.529.9866.78928 6.69.484.529.77879.78928.86 5.9.9464.9866.78928.8787.689 6.69.484.78928.86.689.889 2.6 Second step Divide Row 2 by.94 and multiply it by.9464, that is, multiply Row 2 by.9464.94.994. Row 2 (.994) 6 5 6 5.9464.764 2.84.764 2.84.87 Subtract the result from Row to get.746.456..8..8 2.94.9464.484.9464.484.77857.526.98697.78644 6.747.484.529.77879.78928.86 5.9.9464.9866.78928.8787.689 6.69.484.78928.86.689.889 2.6 Divide Row 2 by.94 and multiply it by.484, that is, multiply Row 2 by.484.94.456. Row 2 (.456) 5 5.484 2.84.262 2.84.262.577 Subtract the result from Row 4 to get
4.7.4 Chapter 4.7.746.456.94.9464.484..9464.77857.526.9866.78928.8.484.526.77857.78928.86..9464.98697.78644.8787.689.8.484.78644.98697.689.889 2 6.747 4. 6.69 2.6 Divide Row 2 by.94 and multiply it by.9464, that is, multiply Row 2 by.9464.94.994. Row 2 (.994) 6 5 6 5.9464.764 2.84.764 2.84.87 Subtract the result from Row 5 to get.746.456..8..8 2.94.9464.484.9464.484.77857.526.98697.78644 6.747.526.77857.78644.98697 4..98697.78644.8787.686 6.747.484.78928.86.689.889 2.6 Divide Row 2 by.94 and multiply it by.484, that is, multiply Row 2 by.484.94.456. Row 2 (.456) 5 5 [.484 2.84.262 2.84.262] [.577] Subtract the result from Row 6 to get.746.456..8..8 2.94.9464.484.9464.484.77857.526.98697.78644 6.747.526.77857.78644.98697 4..98697.78644.8787.686 6.747.78644.98697.686.8787.67 Third step Divide Row by.77857 and multiply it by.526, that is, multiply Row by.526.77857.6686. Row (.6686) [.526.4779.65965.5256] [ 4.269] Subtract the result from Row 4 to get
LU Decomposition-More Examples: Electrical Engineering 4.7.5.746.456.94..9464.77857.98697.78644.8.484.526.264.78644.98697..9464.98697.2679.8787.686.8.484.78644.526.686.8787 2 6.747 62.86 6.747.67 Divide Row by.77857 and multiply it by.98697, that is, multiply Row by.98697.77857.2677. Row (.2677) 5 [.98697.65965.25 9.9695 ] [.78275] Subtract the result from Row 5 to get.746.456..8..8 2.94.9464.484.9464.484.77857.526.98697.78644 6.747.264.2679.526 62.86.2679.8774.676 6.965.78644.98697.686.8787.67 Divide Row by.77857 and multiply it by.78644, that is, multiply Row by.78644.77857.. Row (.) 5 5 [.78644.5256 9.9695 7.949 ] [.6272] Subtract the result from Row 6 to get.746.456..8..8 2.94.9464.484.9464.484.77857.526.98697.78644 6.747.264.2679.526 62.86.2679.8774.676 6.965.526.676.8795 4.29 Fourth step Divide Row 4 by.264 and multiply it by.2679, that is, multiply Row 4 by.2679.264.257. Row 4 (.257) 6 5.2679.427.727.776
4.7.6 Chapter 4.7 Subtract the result from Row 5 to get.746.456..8.94.9464.484.77857.526.264.526..9464.98697.2679.8775.676.8.484.78644.526.675.8795 2 6.747 62.86 6.5 4.29 Divide Row 4 by.264 and multiply it by.526, that is, multiply Row 4 by.526.264.429. Row 4 (.429) 5 [.526.727.696] [.678] Subtract the result from Row 6 to get.746.456..8..8 2.94.9464.484.9464.484.77857.526.98697.78644 6.747.264.2679.526 62.86.8775.675 6.5.675.8775 4.97 Fifth step Divide Row 5 by.8775 and multiply it by.675, that is, multiply Row 5 by.675.8775.74745. Row 5 (.7474) [.675.4527] [ 45.62] Subtract the result from Row 6 to get.746.456..8..8 2.94.9464.484.9464.484.77857.526.98697.78644 6.747.264.2679.526 62.86.8775.675 6.5.259 5.76 The coeffient matrix after the completion of the forwd elimination steps is the [ ] U matrix.
LU Decomposition-More Examples: Electrical Engineering 4.7.7 [ U ].746 L. Now find [ ].456.94..9464.77857.8.484.526.264 2 2 [ L ] 4 42 4 5 52 5 54 6 62 6 64 65 From Step of the forwd elimination process.456 2.656.746..45.746.8 4.724.746. 5.45.746.8 6.724.746 From Step 2 of the forwd elimination process.9464 2.994.94.484 42.456.94.9464 52.994.94.484 62.456.94 From Step of the forwd elimination process.526 4.6686.77857.98697 5.2677.77857..9464.98697.2679.8775.8.484.78644.526.675.259
4.7.8 Chapter 4.7.78644 6..77857 From Step 4 of the forwd elimination process.2679 54.257.264.526 64.429.264 From Step 5 of the forwd elimination process.675 65.74745.8775 Hence.656.45.994 [ L ].724.456.6686.45.994.2677.257.724.456..429 L Z C Now that [ L ] and [ U ] e known, solve [ ].656.45.994.724.456.6686.45.994.2677.724.456. This provides the six equations z 2.656z + z2..45z +.994 z + z ( ) 6..257.429.74745.74745 z z z z z z 2.724z +.456z2 +.6686z + z4.9.45z +.994 z2 +.2677z +.257 z4 + z5.724z +.456z2 +.z +.429z4 +.74745z5 + z6 ( ) ( ) 6. Forwd substitution stting from the first equation gives z 2 Substituting the value of z into the second equation, z2. 656z ( ).656 2 72.64 2 4 5 6 2. 6..9 6..9.9
LU Decomposition-More Examples: Electrical Engineering 4.7.9 Substituting the values of z and z 2 into the third equation, z 6..45z (. 994) z 2 6..45( 2) (.994)( ) 6.747 Substituting the values of z, z 2, and z into the fourth equation, z4.9.724z.456z2. 6686z.9.724( 2).456( ).6686( 6.747) 62.86 Substituting the values of z, z 2, z, and z 4 into the fifth equation, z 5 6..45z.994 z 2.2677z. 257 z 6..45( 2) (.994)( ).2677( 6.747) (.257)( 62.86) 6.5 Substituting the values of z, z 2, z, z 4, and z 5 into the sixth equation, z6.9.724z.456z2.z.429z4. 74745z5.9.724( 2).456( ).( 6.747).429( 62.86).74745( 6.5) 5.76 Hence z 2 z2 z 6.747 [ Z ] z4 62.86 z 6.5 5 z6 5.76 U Z. Now solve [ ] ( ) ( ) 4.746.456..8..8 2.94.9464.484.9464.484.77857.526.98697.78644 6.747.264.2679.526 62.86.8775.675 6.5.259 5.76 This provides the six equations for [ ]..746 + (.456) +. + (.8) +. + (.8) 2.94.9464 +.484 +.9464 +.484 72.64 +
4.7. Chapter 4.7 (.526) +.98697 + (.78644) 6. 747.77857 +.264 +.2679 +.526 62.86.8775 + (.675) 6. 5.259 5.76 From the sixth equation.259 5.76 5.76.259 9.74 Substituting the value of into the fifth equation,.8775 + (.675) 6. 5 6.5 (.675).8775.94 Substituting the values of and into the fourth equation,.264 +.2679 +.526 62.86 62.86.2679.526.264 57.42 Substituting the values of,, and into the third equation,.77857 + (.526) +.98697 + (.78644) 6. 747 6.747 (.526).98697 (.78644).77857 6.66 Substituting the values of,,, and into the second equation,.94 +.9464 +.484 +.9464 +.484 72.64.9464.484.9464.484.94 7.97 Substituting the values of,,,, and into the first equation,.746 +.456 +. +.8 +. +.8 2 (.456). (.8). (.8) 9. The solution vector is ( ) ( ) ( ) 2.746
LU Decomposition-More Examples: Electrical Engineering 4.7. 9. 7.97 6.66 57.42.94 9.74 SMULTANEOUS LNEAR EQUATONS Topic LU Decomposition More Examples Summy Examples of LU decomposition Major Electrical Engineering Authors Aut Kaw Date August 8, 29 Web Site http://numericalmethods.eng.usf.edu