Circuit Theory/Symmetry

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

Example 1

Bridge Balance

• 
  Original Complicated Circuit
• 
  Redrawn up and down
• 
  Z impedances are all equal, so middle Z has no current going through it, nodes on either side are at the same EMF
• 
  middle Z can be replaced by short or open, since no current is going through it

Example 2

Cube

• 
  original flattened cube of resistors
• 
  identifying identical EMF locations
• 
  redrawing 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?

• 
  Current due to injection at any spot splits in quarters and then begins to fill the resistive grid like water filling an infinite bucket.
• 
  Current removed at another spot splits in quarters and then begins to drain the resistive grid like draining an infinite bucket.
• 
  Together 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: ${\frac {Z}{3}}={\frac {1}{{\frac {1}{Z}}+{\frac {1}{Z}}+{\frac {1}{Z}}}}$