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Math: Matrix Equations / Solutions

Many methods of solving circuits in ECE result in matrix equations.  You will see this first in Kirchoff's method (KVL/KCL), for instance.  You can always just 'do the algebra', but eventually as the circuits get more complicated, it is nice to use matrix math to solve for your unknowns.

We have talked about having as many (N) equations as we do (N) unknowns.  Solving a matrix is just solving this set of simultaneous equations in order to obtain the N unknown values.  You should have already seen this in math. Your book,Ulaby Textbook Appendix B, has review information on Matrix Math.

Here are a few links to help you review:

1.  How to convert a circuit question into a matrix equation (KVL/KCL): 

Download L(2-2) -- Kirchhoffs laws KCL KVL - Slide 17.pdf

2. Solve the Matrix Equation:

a. Method 1 -- By Hand Substitution ('do the algebra').  Please be sure you are able to do this algebra by hand, always useful.

http://www.purplemath.com/modules/systlin4.htm Links to an external site.

http://www.youtube.com/watch?v=cwHR_B9zK7k Links to an external site.

Here is an example. Links to an external site.

b. Method 2 -- By Hand Gaussian Elimination ('do the algebra' another way)

Gauss Elim Links to an external site.

http://en.wikipedia.org/wiki/Gaussian_elimination Links to an external site.

c. Method 3 -- Use Your Calculator (this is best for ECE1240, especially on the exam)

Here is an example with the TI89.  For other calculators, see your instruction manual.

Download simultaneous_equations_ti89.pdf

d. Method 4 -- Matlab (very good method, commonly used by engineers in practice)

Download Matrices_in_Matlab.pdf

Here is the m-file, so you can run this example (open it in Matlab): KVL_KCL.m Download KVL_KCL.m

 

More Advanced:  Ok!  If you really enjoy computer programming, There is more ... Gaussian Elimination and other matrix math come with some interesting numerical challenges.  Check out the sections on pivoting and scaling in the Numerical EM class Links to an external site.. 

And, if you want a really sweet code, check out the iteration solution (Successive Over Relaxation, SOR):

http://www.ece.utah.edu/~cfurse/ece6340/   Links to an external site.

http://en.wikipedia.org/wiki/Successive_over-relaxation Links to an external site.