Overview
This lecture explains how to formulate and solve a system of equations using Kirchhoff’s Laws, set up the corresponding coefficient matrix, use a calculator for matrix math, and interpret your results.
Writing and Arranging the Equations
- Combine like terms in each loop/branch equation to get three equations in three unknowns (i1, i2, i3).
- Arrange each equation so all variables (currents) are on the left, constants on the right.
- Example equations:
- -i1 - i2 + i3 = 0
- -6i1 + 3i2 = -24
- -3i2 - 6i3 = -12
Creating the Coefficient Matrix
- Write the coefficients of i1, i2, i3, and the constants into a 3x4 matrix.
- Example setup:
- Row 1: -1, -1, 1, 0
- Row 2: -6, 3, 0, -24
- Row 3: 0, -3, -6, -12
Solving Using Calculator and rref
- Input the matrix into your calculator as described.
- Use the rref (reduced row-echelon form) function to solve for current values.
- The solution gives i1, i2, i3 as the last column of the final matrix.
Interpreting Results
- The sign of each current indicates its actual direction relative to your diagram.
- Report currents with magnitudes and specify "same" or "opposite" direction as drawn in your circuit diagram.
Grading and Troubleshooting
- Most points are awarded for correct equation setup matching your diagram.
- Different students may have different equations due to loop and current direction choices.
- If you get an all-zero row in your matrix, replace one equation with a different loop or branch equation.
Key Terms & Definitions
- Kirchhoff’s Laws — Rules for analyzing current and voltage in circuits (Junction Rule and Loop Rule).
- Coefficient Matrix — An array of equation coefficients and constants used for solving systems with matrices.
- rref — Reduced row-echelon form; calculator function to solve linear systems.
Action Items / Next Steps
- Draw a clear, labeled circuit diagram indicating current directions.
- Write and arrange your three equations as explained.
- Construct and submit your coefficient matrix along with your final currents and directions.
- If results are unexpected (e.g., a row of zeros), revise your set of equations.