1. PHY 202 (Blum)1 Use’s Excel Array Formulas to Solve Simultaneous Equations

2. Simultaneous Linear Equation PHY 202 (Blum)2

3. Present Example The example we will consider here is from resistor circuits and goes by the name Kirchhoff’s circuit laws. The variables are currents in different “loops” of the circuit. The coefficients (numbers multiplying the variables) are resistances. The constants are voltages. PHY 202 (Blum)3

4. Example with numbers 5 = 3.2 J A – 1.0 J B - 2.2 J C 0 = -1.0 J A + 5.5 J B – 3.0 J C 0 = -2.2 J A – 3.0 J B + 6.9 J C In the above we have three current variables: J A, J B and J C. PHY 202 (Blum)4

5. 5 Loop equations as matrix equation 5 = 3.2J A – 1 J B - 2.2J C 0 = -1J A + 5.5 J B – 3J C 0 = -2.2J A – 3 J B + 6.9J C

6. 6PHY 202 (Blum) Enter matrix in Excel, highlight a region the same size as the matrix.

7. 7PHY 202 (Blum) In the formula bar, enter =MINVERSE(range) where range is the set of cells corresponding to the matrix (e.g. B1:D3). Then hit Crtl+Shift+Enter

8. 8PHY 202 (Blum) Result of matrix inversion

9. 9PHY 202 (Blum) Prepare the “voltage vector”, then highlight a range the same size as the vector and enter =MMULT(range1,range2) where range1 is the inverse matrix and range2 is the voltage vector. Then Ctrl-Shift-Enter. Voltage vector

10. 10PHY 202 (Blum) Results of Matrix Multiplication

11. The answer The current vector is the answer J A =3.152 mA J B =1.470 mA J C =1.644 mA PHY 202 (Blum)11

12. Use the matrix approach in Excel to find the solution for 10 = 7 J A – 3 J B - 4 J C 0 = -3 J A + 6 J B – 1 J C 0 = -4 J A – 1 J B + 10J C PHY 202 (Blum)12

13. Atomic Isotope Problem Atomic Masses Isotopes: The element indium exists naturally as two isotopes. 113 In has a mass of 112.9043 amu, and 115 In has a mass of 114.9041 amu. The average atomic mass of indium is 114.82 amu. Calculate the percent relative abundance of the two isotopes of indium. http://eppe.tripod.com/stoictry.htm PHY 202 (Blum)13

14. Use the matrix approach in Excel to find the solution for 112.9043 x + 114.9041 y = 114.82 x + y = 1 where x is the fraction of the first isotope and x is the fraction of the second isotope. PHY 202 (Blum)14