5.9 Linear algebra: numerical experiment (circuit design)

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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 $R$ . Equation 2 is unchanged, but Equation 6 becomes

$\frac{\left({v}_{2}-{v}_{1}\right)}{50}+\frac{{v}_{2}}{300}+\frac{\left({v}_{2}-{v}_{3}\right)}{R}=0⇒6R\left({v}_{2}-{v}_{1}\right)+R{v}_{2}+300\left({v}_{2}-{v}_{3}\right)=0⇒-6R{v}_{1}+\left(7R+300\right){v}_{2}-300{v}_{3}=0.$
$\frac{\left({v}_{3}-{v}_{2}\right)}{R}+\frac{{v}_{3}}{2}=0⇒2\left({v}_{3}-{v}_{2}\right)+R{v}_{3}=0⇒0{v}_{1}-2{v}_{2}+\left(R+2\right){v}_{3}=0.$

The matrix form of these equations is

$100-6R7R+300-3000-2R+2{v}_{1}{v}_{2}{v}_{3}=500.$

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

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

Make a table of $R$ and the corresponding values of ${v}_{3}$ :

Now add a column to your table for the current through the lamp $i={v}_{3}/2$ . Add rows to your table until you have found a value of $R$ for which the lamp will glow. ( $i$ needs to be between 0.05 and 0.075 ampere.)

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