By the end of this section, you should be able to

1. Create transfer functions for input/output voltages/currents for LRC circuits

   • Use current loops, voltage drops and voltage dividers

2. Use impendences to assist in deriving TFs

3. Understand how ideal op-amps work in LRC circuits

4. Continue to use Mathematica to work through algebraic problems to create TFs

This section develops transfer function models for electrical systems, starting with LRC circuits and building toward op-amp based networks. You will use current loops, voltage drops, and voltage divider relationships to write governing equations, then convert them into input-output transfer functions for voltages and currents. A key theme is impedance: by treating resistors, inductors, and capacitors as frequency-domain impedances, you can derive transfer functions more efficiently and see how poles and zeros emerge from circuit structure. The lessons also introduce ideal operational amplifiers and the standard ideal op-amp equations, so you can model common amplifier and filter configurations that appear in measurement circuits and instrumentation. Throughout the section, Mathematica is used to manage the algebra and keep derivations clean, especially when circuits produce coupled equations or higher-order expressions. Code and example files are included so you can validate results, compare time responses, and connect mathematical models to physical behavior. Use the table of contents below to move between LRC circuits, impedance methods, operational amplifiers, and the supporting code materials as you build a repeatable workflow for electrical system modeling, and develop confidence applying these techniques to new circuits and real lab setups.