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

1. Apply Kirchhoff’s voltage and current laws to set up loop or node equations for R, L, and C circuits

2. Compute Laplace-domain impedances and use voltage and current division to simplify circuit relationships

3. Derive transfer functions relating specified input and output voltages or currents for RL, RC, and RLC circuits

4. Use ideal op-amp constraints to analyze basic op-amp circuits and derive the resulting transfer functions

5. Use Mathematica to solve for transfer functions and verify results using limit checks and physical consistency 

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.