## 2007-Myers, Joseph and Brian Winkel - Modeling a Mystery Circuit in Mathematics Class Using Differential Equations, Laplace Transforms, and Transfer Functions

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From Myers, Joseph and Brian Winkel. 2007. Modeling a Mystery Circuit in Mathematics Class Using Differential Equations, Laplace Transforms, and Transfer Functions. *The UMAP Journal*, 28(1): 15-26.

We discuss the introduction, student discovery, and teaching of a system of differential equations which model a circuit. In an engineering mathematics course we provide sufficient conceptual framework (Conservation of Charge and Kirchhoff's Circuit Laws) and background in differential equations theory and solutions using Mathematica and Laplace Transforms to enable students to build a system of differential equations for a given two loop circuit and solve the system for various values of parameters of the input voltage to the system. We give each student a unique input voltage frequency parameter and ask for system response to that input as a voltage over one of the resistors in the second loop of the circuit. Students collect the data, plot their results, and discover a cascaded high-low filter circuit is the object of their study. We offer a further enhancement of the modeling perspective by interpreting the system with Laplace Transforms and the transfer function for the system. This path will permit us to compute the gain in an electrical circuit by easy use of the transfer function of the circuit. In this way the students see the meaning of the transfer function and the power of mathematics and technology in their study of a system of differential equations to model a circuit.

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