Solving Scale Linear Systems (Example system) Lecture 13 MA/CS 471 Fall 2003
First – Brief Reintroduction to Linear Systems • First we will use an example physical system to construct a set of 5 couple linear equations in 5 unknowns. • We will seek a solution using Matlab • Later we will consider generalizations to larger systems (with correspondingly more unknowns to find).
Circuit Problem 5W 1W 4W 6W 3W 7W + - 30V 1W 2W Problem: Find the current running through each closed loop
Circuit Problem 5W 1W 4W DC Battery Resistor (resistance in ohms) Resistance free wire 6W 3W 7W + - 30V 1W 2W Notation
Circuit Problem 5W 1W 4W 6W 3W 7W 30V 1W 2W Find the current (in amperes) traveling in the shown closed loops
Kirchoff’s Second Law • Kirchoff's 2nd Law states that for any closed loop path around a circuit the sum of the voltage gains and voltage drops equals zero. In the circuit shown, there is a voltage gain for each electron traveling through the voltage source and a voltage drop across the resistor.
Loop 1 Balance Consider LOOP 1 5W 1W 4W 6W I2 3W 7W I1 30V I3 1W 2W The gain is 30V. The loop 1 loss (by Ohm’s law) is: The gain due to current from loop 2 is: The gain due to current from loop 3 is: Kirchoff’s 2nd law states gain=loss, =>
All Loop Balances 5 5W 1W 4W 6W 3W 4 2 7W 30V 1 1W 2W 3
Rearranging Linear System Arranging unknown Loop currents on lefthand side and known voltage sources on right hand side: Divide through by Ohms:
Final System Simplifying the coefficients: Matrix form:
Final Form • Negating both sides: • This is the enemy. • We will create systems with a large number of degrees of freedom later on.
Solution (by Matlab) 1.08A 5W Solution: 1W 4W 6W 1.97A 2.74A 3W 8.19A 7W 5.46A 30V 1W 2W
Homework/Lab work Q1) • Create a non-trivial circuit with 15 sub loops. Use a range of resistor values between 1 and 10. • Using a sparse matrix (see MA375/Lecture 8 intro), solve for loop currents with Matlab • Draw a diagram indicating current along each segment of circuit (to two significant figures). • Verify Kerchoff’s first law (look it up) by checking the sum of currents at three of the wire intersections. • Count the number of non-zeros of your 15x15 matrix and report the amount of fill (i.e. number of non-zeros/225) • Include print out of matlab window used for matrix solution. Q2) Review: a) LU factorization b) condition number of a matrix