# Circuit Theory/Symmetry

Some very complicated circuits can be solved by seeing symmetry. Three
examples are presented below:

## Example 1

Bridge Balance

<File:Example19a.png%7COriginal> Complicated Circuit
<File:Example19b.png%7CRedrawn> up and down <File:Example19c.png%7CZ>
impedances are all equal, so middle Z has no current going through it,
nodes on either side are at the same EMF <File:Example19d.png%7Cmiddle>
Z can be replaced by short or open, since no current is going through it

## Example 2

Cube

<File:Example20a.png%7Coriginal> flattened cube of resistors
<File:Example20b.png%7Cidentifying> identical EMF locations
<File:Example20d.png%7Credrawing> idential EMF as the same nodes, can
see three groups of parallel resistors

## Example 3

2D-Grid\
Suppose there is an infinite 2 dimensional grid of impedances (Z). What
is the input impedance if connected across (in parallel) with any given
impedance?

<File:Example21a.png%7CCurrent> due to injection at any spot splits in
quarters and then begins to fill the resistive grid like water filling
an infinite bucket. <File:Example21b.png%7CCurrent> removed at another
spot splits in quarters and then begins to drain the resistive grid like
draining an infinite bucket. <File:Example21c.png%7CTogether> the two
create a steady state situtation where the question \"What is the input
impedance\" can be asked. The current i is i/4 + i/4 = i/2. v = i/2\*Z
so v/i = Z/2 = input impedance.

If the grid were three dimensional, the current would split into 6 equal
sections, thus the input impedance would be Z/3.

So what does this mean? It helps us understand that the infinity of
space has an impedance:
376.730\... ohms which is plank\'s
impedance \* 4π and is related to the
speed of light, the permeability of free space and permitivity of free
space.

Perhaps this is related
entanglement and to the trinity
since: $\frac{Z}{3} = \frac{1}{\frac{1}{Z} + \frac{1}{Z} + \frac{1}{Z}}$

