Circuit Theory/Simultaneous Equations

Solving Equations
The goal of this section is to illustrate how complicated circuits can get very quickly. This sets up the need for learning circuit simplification techniques described later. The goal here is to use math tools to design circuits.

There are several issues that need to be reviewed while solving simultaneous linear equations: After quick reviews, the goal is to start designing/reverse engineering circuits.
 * Numeric or Symbolic answers
 * Algebra
 * Symbolic Computation
 * Numeric Solutions (Linear Algebra)

Numerical or Symbolic
You are going to need to solve linear simultaneous equations two ways in this course: You are going to need to recognize the difference between the two.
 * result in numerical answers when perfectly constrained
 * result in variable symbol answers when under constrained

Algebra
Algebra can be used for simple circuits. Even more complicated circuits can be designed or reversed engineered using algebra, but not in a way that others can check. The solutions become very complicated. Better, more standard, techniques exist. You need to learn when to switch to these more respectable techniques.

Symbolic Computation
In its research context, "symbolic computation" has many goals, of which two concern us: numeric computing solutions that approximate with floating point numbers, and symbolic computing that focuses on exact solutions using symbols and formulas. You are going to need both to succeed. There are lots of packages that try to do both.

This text focuses on MathWork's MuPAD and MatLab plus Mathematica's Wolfram Alpha. MatLab has a huge engineering community growing it's toolboxes including the symbolic toolbox MuPAD. Mathematica represents how people think, learn and communicate math. Mathematica is more attractive to scientists.

You need to invest time in three types of solutions: All you have ever done with a calculator are numeric approximations. All you have ever done in algebra is form solutions with variable names. The middle solution is called a "symbolic solution" where answers are expressed in whole number ratios instead of decimal places.
 * solutions using variable names
 * symbolic solutions using fractions with whole numbers and constants like $$\pi$$
 * numeric approximations

Numeric Solution
Symbolic solutions (MathWorks MuPAD or Mathematica) require a lot more typing. They stress computers more and are more likely to fail. But they are a lot better at exploring theories, pointing out concepts, and creating transparency. This transparency enables conversations, corrections, and growth of intuition.

Numeric solutions hide details, make consideration of design alternatives harder, and make troubleshooting/harmonizing/checking engineering work more difficult. But they are a lot faster both setting up, executing and in exploring alternatives interatively. This is what MathWorks MatLab focuses on.

Numeric solutions require organizing all solutions into matrices before going to the computer. Linear Algebra is a course typically taken after calculus that explains how to do this. But lots of engineering classes require using some of Linear Algebra's concepts before you take the course. The same is true here. Read the examples below and you will get the hang of it.

Solution Summary
From the previous two pages, the solution can be outlined as follows:
 * 1) Label currents and voltages
 * 2) Identify loops, add + - to each voltage
 * 3) Identify Junctions, put arrow heads on currents
 * 4) Count knowns and unknowns
 * 5) Write terminal equations
 * 6) Write loop equations
 * 7) Write junction equations
 * 8) Solve equations:
 * 9) Algebra
 * 10) Differential Equations
 * 11) Symbolic Solution
 * 12) Numeric Solution
 * 13) Simulate (circuitlab everycircuit icircuit falstad animations ecstudio)
 * 14) Build Intuition/Check/Compare