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4.10: 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

    \[\begin{align}
    & \tfrac{\left(v_{2}-v_{1}\right)}{50}+\tfrac{v_{2}}{300}+\tfrac{\left(v_{2}-v_{3}\right)}{R}=0 \nonumber \\
    \Rightarrow & 6 R\left(v_{2}-v_{1}\right)+R v_{2}+300\left(v_{2}-v_{3}\right)=0 \nonumber \\
    \Rightarrow &-6 R v_{1}+(7 R+300) v_{2}-300 v_{3}=0 \label{}.
    \end{align} \nonumber \]

    Equation 7 from "Linear Algebra: Circuit Analysis" becomes

    \[\begin{align}
    \tfrac{\left(v_{3}-v_{2}\right)}{R}+\tfrac{v_{3}}{2}=0 \nonumber \\
    \Rightarrow \quad 2\left(v_{3}-v_{2}\right)+R v_{3}=0 \nonumber \\
    \Rightarrow \quad 0 v_{1}-2 v_{2}+(R+2) v_{3}=0 \label{}
    \end{align} \nonumber \]

    The matrix form of these equations is

    \[\left[\begin{array}{lll}
    1 & & 0 & 0 \\
    -6 R & 7 R & +300 & -300 \\
    0 & & -2 & R+2
    \end{array}\right]\left[\begin{array}{l}
    v_{1} \\
    v_{2} \\
    v_{3}
    \end{array}\right]=\left[\begin{array}{l}
    5 \\
    0 \\
    0
    \end{array}\right] \nonumber \]

    Write a MATLAB function file called builda to 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

    >> function A = builda(R);

    Now choose several values for \(R\). For each choice, use your function builda and solve the resulting matrix equation \(A\nu=b\) 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:

    >> det(A)

    Make a table of \(R\) and the corresponding values of \(\nu _3\):

    Screen Shot 2021-08-11 at 5.27.39 PM.png
    Figure \(\PageIndex{1}\)

    Now add a column to your table for the current through the lamp \(i=\nu _3/2\). 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.)


    This page titled 4.10: Numerical Experiment (Circuit Design) is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Louis Scharf (OpenStax CNX) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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