# 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:

-   Numeric or Symbolic answers
-   Algebra
-   Symbolic Computation
-   Numeric Solutions (Linear Algebra)

After quick reviews, the goal is to start designing/reverse engineering
circuits.

#### Numerical or Symbolic

You are going to need to solve linear simultaneous equations two ways in
this course:

-   result in numerical answers when perfectly constrained
-   result in variable symbol answers when under constrained

You are going to need to recognize the difference between the two.

#### 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:

-   solutions using variable names
-   symbolic solutions using fractions with whole numbers and constants
    like $\pi$
-   numeric approximations

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.

#### 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:
    1.  Algebra
    2.  Differential Equations
    3.  Symbolic Solution
    4.  Numeric Solution
9.  Simulate (circuitlab
    everycircuit
    icircuit falstad
    animations
    ecstudio)
10. Build Intuition/Check/Compare

