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Linear Algebra: Numerical Experiment (Circuit Design)

Module by: Louis Scharf. E-mail the author

Note:

This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.

Our analysis in Example 1 from "Linear Algebra: Circuit Analysis" and Problem 1 from "Linear Algebra: Circuit Analysis" indicates that not enough current will flow through the lamp to make it glow. We now wish to change the resistance of the 100 ohm resistor to a new value so that the lamp will glow. We replace 100 in the equations with an unknown resistance R R. Equation 2 is unchanged, but Equation 6 becomes

( v 2 - v 1 ) 50 + v 2 300 + ( v 2 - v 3 ) R = 0 ⇒ 6 R ( v 2 - v 1 ) + R v 2 + 300 ( v 2 - v 3 ) = 0 ⇒ - 6 R v 1 + ( 7 R + 300 ) v 2 - 300 v 3 = 0 . ( v 2 - v 1 ) 50 + v 2 300 + ( v 2 - v 3 ) R = 0 ⇒ 6 R ( v 2 - v 1 ) + R v 2 + 300 ( v 2 - v 3 ) = 0 ⇒ - 6 R v 1 + ( 7 R + 300 ) v 2 - 300 v 3 = 0 .
(1)

Equation 7 from "Linear Algebra: Circuit Analysis" becomes

( v 3 - v 2 ) R + v 3 2 = 0 ⇒ 2 ( v 3 - v 2 ) + R v 3 = 0 ⇒ 0 v 1 - 2 v 2 + ( R + 2 ) v 3 = 0 . ( v 3 - v 2 ) R + v 3 2 = 0 ⇒ 2 ( v 3 - v 2 ) + R v 3 = 0 ⇒ 0 v 1 - 2 v 2 + ( R + 2 ) v 3 = 0 .
(2)

The matrix form of these equations is

1 0 0 - 6 R 7 R + 300 - 300 0 - 2 R + 2 v 1 v 2 v 3 = 5 0 0 . 1 0 0 - 6 R 7 R + 300 - 300 0 - 2 R + 2 v 1 v 2 v 3 = 5 0 0 .
(3)

Write a MATLAB function file called builda to accept R R as an input and return the matrix A A in Equation 3 as an output. The first line of your function file should be

function A = builda(R);

Now choose several values for R R. For each choice, use your function buildabuilda and solve the resulting matrix equation Av=bAv=b for the voltages. Each time you choose a different R R to build a different matrix A A, check the determinant of A to be sure the equations have a unique solution:

>> det(A)

Make a table of R R and the corresponding values of v 3 v 3 :

Figure 1
Figure one is an empty table, with the leftmost column titled R, the middle column titled v_3, and a rightmost column left unlabeled.

Now add a column to your table for the current through the lamp i=v3/2i=v3/2. Add rows to your table until you have found a value of R R for which the lamp will glow. ( i i needs to be between 0.05 and 0.075 ampere.)

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