The following circuit is known as the Wheatstone Bridge:
The output voltage W in relation to the input voltage w is given by
Our simple project is the classification of these circuits when we are allowed to shuffle the resistances around. As we mentioned earlier, symmetry arguments are very efficient as classification/sorting tools. We assume here that all four resistance values are distinct, thus avoiding degeneracy. We follow the notation and definitions introduced in previous postings, such as in here.
We will shuffle the (indices in the) resistances in the circuit according to the symmetries of $$ K_4=\{1,(12)(34),(13)(24),(14)(23)\}\equiv \{1,h,v,o\}, $$ and then compare the resulting output voltages
$$W_h, W_v,W_o$$
with the original output voltage $W_1$.
There are, however, several other circuits that can be used as a starting point of the K_4 orbits. So let's redraw the circuit in a simpler form and permute the resistances indexed by {1, 2, 3} according to
$$
D_3=\{1,(123),(132),(12),(13),(23)\},
$$
thus obtaining the following six distinct circuits:
Indicate them, respectively, by
$$
\{\;W^1,\;\;W^{(123)},\;\;W^{(132)},\;\;W^{(12)},\;\;W^{(13)},\;\;W^{(23)}\;\}.
$$
As K_4 shuffle each one of these six generating circuits, we would then obtain the totality of the 24 distinct bridge circuits. This is in analogy to the sorting of the 24 flags used in the flag preference experiment.
Associated with K_4 we have the following table of characters (to be made precise later):
$$
\boxed{ \begin {array}{c|rrrr} &1&h&v&o\\\hline z_1&1&1&1&1
\\ z_2&1&1&-1&-1\\ z_3&1&-1&1&-1
\\ z_4&1&-1&-1&1\end {array} },
$$
$$
W_\tau = z (\tau)\; W_1,
$$
Indicate them, respectively, by
$$
\{\;W^1,\;\;W^{(123)},\;\;W^{(132)},\;\;W^{(12)},\;\;W^{(13)},\;\;W^{(23)}\;\}.
$$
As K_4 shuffle each one of these six generating circuits, we would then obtain the totality of the 24 distinct bridge circuits. This is in analogy to the sorting of the 24 flags used in the flag preference experiment.
Associated with K_4 we have the following table of characters (to be made precise later):
$$
\boxed{ \begin {array}{c|rrrr} &1&h&v&o\\\hline z_1&1&1&1&1
\\ z_2&1&1&-1&-1\\ z_3&1&-1&1&-1
\\ z_4&1&-1&-1&1\end {array} },
$$
Now a really important concept: We say that a circuit W reduces as z if
for all symmetries in K_4.
Direct calculation will show us that:
$$W^1\;\;\text{ and }\;\;W^{(23)} \text{ reduce as }\; z_4;$$
$$W^{(123)}\;\;\text{ and }\;\;W^{(12)} \text{ reduce as }\; z_3;$$
$$W^{(132)}\;\;\text{ and }\;\;W^{(13)} \text{ reduce as }\; z_2.$$
$$W^1\;\;\text{ and }\;\;W^{(23)} \text{ reduce as }\; z_4;$$
$$W^{(123)}\;\;\text{ and }\;\;W^{(12)} \text{ reduce as }\; z_3;$$
$$W^{(132)}\;\;\text{ and }\;\;W^{(13)} \text{ reduce as }\; z_2.$$
We have then obtained a classification of all (distinct- resistance) Wheatstone Bridge circuits according to the character in $$\{z_1,z_2,z_2,z_4\}$$ to which its K_4 orbit belongs to.
Connected Bridges: If $$W^{\sigma_j}$$ reduces as $$z_j$$ and $$W^{\sigma_i}$$ reduces as $$z_i$$ then the iterated bridge circuit, in which one circuit's input voltage is the other's output voltage, reduces as $$z_jz_i.$$
If we define the resonance of two iterated bridges by
$$<z_j,z_i>=\sum_\tau\; z_j(\tau)z\;_i(\tau),$$
then we will find out that two iterated bridges resonate, that is, $$<z_j,z_i>\neq 0,$$ if and only if they reduce according to the same character.
Connected Bridges: If $$W^{\sigma_j}$$ reduces as $$z_j$$ and $$W^{\sigma_i}$$ reduces as $$z_i$$ then the iterated bridge circuit, in which one circuit's input voltage is the other's output voltage, reduces as $$z_jz_i.$$
If we define the resonance of two iterated bridges by
$$<z_j,z_i>=\sum_\tau\; z_j(\tau)z\;_i(\tau),$$
then we will find out that two iterated bridges resonate, that is, $$<z_j,z_i>\neq 0,$$ if and only if they reduce according to the same character.
Only circuits that belong to the same character resonate. This is a simple sort of Selection Rule in that (current) transitions among circuits (states) can occur only when they share the same characters. This is a fundamental principle in Chemistry and Physics.
First posted 02/24/2012
This posting is based on "Symmetry Studies An Introduction to the Analysis of Structured Data in Applications" Cambridge Press (2008)
This posting is based on "Symmetry Studies An Introduction to the Analysis of Structured Data in Applications" Cambridge Press (2008)
