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Solving systems of linear equations: resistor networks

Developed by Eric Ayars - Published August 1, 2016

These exercises are designed to get you started on using computers to solve systems of linear equations. The systems of equations we'll be using come from resistor networks typical of what you might see in a second-semester physics course, but the same techniques can be used for any system of linear equations regardless of their source. We can of course solve these systems of equations using algebra, but for larger systems of equations it's a lot easier to use a computer.
Subject Area Electricity & Magnetism First Year and Beyond the First Year Python, Mathematica, and Easy Java Simulations Students who complete these exercises will * Be able to convert from a system of equations to a matrix equation. * Be able to use a computational package to solve the matrix equation. 30 min

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).

## Exercise 1: Introductory Problem Consider the network of resistors and batteries shown in the first figure below. ![two-loop circuit](images/linearEquations/2_loop.png) There are three unknowns in that circuit, $I_1$, $I_2$, and $I_3$. We can solve for these unknowns by building a system of equations using Kirchhoff's Laws: 1. The sum of voltages around any loop is zero. 2. The sum of currents at any junction is zero. Applying the voltage law to the left-hand loop, we get $$V_1 - I_1R_1 - I_3R_3 = 0$$ From the right-hand loop, we get $$V_2 + I_2R_2 - I_3R_3 = 0$$ We need one more equation, for which we can use the junction at top center and the current law: $$I_1 - I_2 - I_3 = 0$$ We now have the requisite three independent equations, which we can solve using methods learned in high-school algebra. There's another way of solving this, using matrix methods. First, rewrite the equations just a bit. \begin{align} R_1 I_1 + 0 I_3 + R_3 I_3 &=& V_1 \\\ 0 I_1 - R_2 I_2 + R_3 I_3 &=& V_2 \\\ I_1 - I_2 - I_3 &=& 0 \end{align} And now we can see that this can be written as a matrix equation! $$\begin{bmatrix} R_1 & 0 & R_3 \\\ 0 & -R_2 & R_3 \\\ 1 & -1 & -1 \end{bmatrix} \begin{bmatrix} I_1 \\\ I_2 \\\ I_3 \end{bmatrix} = \begin{bmatrix} V_1 \\\ V_2 \\\ 0 \end{bmatrix}$$ This matrix equation, $$\mathbf{M}\mathbf{x} = \mathbf{b}$$ has solution $$\mathbf{x} = \mathbf{M}^{-1}\mathbf{b}$$ where $\mathbf{M}^{-1}$ is the _inverse_ of $\mathbf{M}$. Most computational packages have built-in capability for inverting matrices. ### Assignment 1. By carrying out the matrix multiplication explicitly, show that the matrix equation above reduces to the system of equations from which it is derived. 2. Use a matrix-solving package to find the currents. 3. Check your answers by substituting the currents into the equations and verifying that they are solutions. ## Exercise 2: More complicated problem The resistor network below is perhaps a bit more challenging. ![Five-loop circuit](images/linearEquations/5_loop.png) ### Assignment 1. Write a system of equations that can be used to solve for the currents in the circuit above. 2. Re-write the system of equations from part 1 of this assignment as a matrix equation. 3. Use a matrix-solving package to find the currents. 4. Check your answers: do the values of currents you found solve the equations with which you started? 5. You may notice something interesting if you compare your solution here to your solution for exercise 1. (Assuming the same values of $R$ and $V$ were used in both exercises.) Explain why this happened. Unless your instructor provides you with other values, assume each voltage source and each resistor has a value $10\times$ its identifying number. (i.e. $V_2 = 20$ V, $R_3 = 30 \Omega$.)

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### Credits and Licensing

Eric Ayars, "Solving systems of linear equations: resistor networks," Published in the PICUP Collection, August 2016, https://doi.org/10.1119/PICUP.Exercise.linearEquations.

The instructor materials are ©2016 Eric Ayars. 