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Objectives
Objectives
By the end of this section, you should be able to
Recall what the Laplace transform does and why it is useful for solving linear ordinary differential equations in time.
Use partial fraction expansion to rewrite rational functions so that they match common Laplace transform table entries.
Write and solve simultaneous linear equations in matrix form.
Recognize and interpret basic electronic components and their standard symbols in simple circuits.
These concepts typically come from earlier coursework in
Differential equations and Laplace transforms
Calculus and algebra techniques for partial fractions
Linear algebra and matrix methods
Introductory electrical circuits and physics
Math and Electronics Review is a focused refresher that prepares you to build and analyze dynamic system models in the Modeling unit of Measurement and Dynamic Response. The lessons below revisit four tools that appear constantly in system dynamics, mechanical vibrations, signals, and control systems: the Laplace transform, partial fraction expansion, matrix and linear algebra methods, and the behavior of basic electrical components. You will practice translating time-domain differential equations into algebraic expressions in the Laplace domain, decomposing rational functions so they match standard transform table entries, and organizing simultaneous linear equations in matrix form so multi-equation models stay readable. You will also review common circuit symbols and component behavior so that later work involving sensors, actuators, and simple measurement circuits feels natural. The goal of this section is confidence, not speed. By the time you finish, you should be able to explain why Laplace methods help solve linear ordinary differential equations, perform a straightforward partial fraction expansion, solve small linear systems, and describe how passive components relate voltage and current in qualitative terms. Use the table of contents below to jump directly to the topic you need most, or work through the sequence in order to reset your foundation before moving deeper into modeling.
Laplace Transforms
Laplace Transforms
Laplace transforms are a central tool in system dynamics, mechanical vibrations, signals, and control engineering. The transform turns differential equations in the time domain into algebraic equations in the Laplace domain, which are often easier to solve and interpret.
Partial Fraction Expansion
Partial Fraction Expansion
Partial fraction expansion lets you decompose a rational function into simpler terms that match entries in a Laplace transform table. This step is often what makes it possible to invert a Laplace transform and recover the time-domain solution for a dynamic system.
Matrix/Linear Algebra
Matrix/Linear Algebra
Matrix and linear algebra tools appear throughout system dynamics and control, especially in state-space modeling and multi-degree-of-freedom mechanical systems.
Electrical Components
Electrical Components
Many measurement, actuation, and control problems involve basic circuits. Understanding how standard components behave is essential for building realistic dynamic models.
Before you move on, you should comfortably be able to:
Explain, in a sentence or two, how the Laplace transform helps you solve linear ordinary differential equations.
Take a simple rational function and perform a partial fraction expansion.
Set up and solve a 2 x 2 or 3 x 3 linear system in matrix form.
Identify the main passive electrical components by symbol and describe how they relate voltage and current qualitatively.
If any of these feel uncertain, spend more time with the slides and your notes from your earlier math and circuits courses before starting the next section.
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