# Understanding equilibrium equations

I initially asked this question on math.stackexchange.com, but nobody answer so I thought I could try to ask here, this at least to some degree related to electrical engineering.

$$C^{-1}y+Ax=b \\ A^Ty =f$$

This pair is equilibrium equations, $C^{-1}+Ax=b$ represents Ohm's law, derived from:

$$e=b-Ax$$

Vector $x$ represent potential on each node in a graph, on each node present abstract force which repel abstract flow. Flow goes to lower potential (which repel less). Act of multiplication $Ax$ produces potential difference. I just add all columns vectors adjusted by corresponding potential on each node, this is certainly should give me potential differences on each edge. But formula telling me:

$$Ax=b-e$$

This is my problem. What $e$ stands for? I know $b$ - potential differences, I can find potential at each node, I know $x$, I can find differences. What does $e$ mean? In another words if I know potential at each node, then $C^{-1}y=Ax$?

• You have to tell more to get some useful help. Where did you get those equations from? Normally you should find there also a definition of the symbols used. BTW Just by looking at the dimensions and assuming that $A$ is not dimensionless if $x$ represents potential $b$ cannot mean potential differences. – Curd May 25 '17 at 14:16

Just a guess: it could be that you are dealing with some vector/matrix notation of nodal analysis or mesh analysis.

Nodal analysis:

• $A$ would represent conductivities (usually denoted by $G$),
• $x$ would be node potentials (usually denoted by $v$) and
• $b$ and $e$ would mean branch currents and (independent) current sources or vice versa (usually denoted by $i$).

Mesh analysis:

• $A$ would represent resistances (usually denoted by $R$),
• $x$ would be mesh currents (usually denoted by $i$) and
• $b$ and $e$ would mean branch voltages and (independent) voltage sources or vice versa (usually denoted by $v$).

In both cases $Ax = b - e$ would in fact represent Ohm's law.

• Hello Curd thank you for response! I get these equations from book of prof.Gilbert Strand introduction to Applied Mathematics, i think this is about nodal analysis, could you please clarify, as you telling in comment "if x represents potential b cannot mean potential differences", why? when we multiply this would be potential on node a - potential on node b, what else potential difference could mean? – Anatoly Strashkevich May 25 '17 at 14:46
• as I wrot above: just from looking at dimensions (units): Elements of $A$ probably have unit Ohm (resistance) or Siemens (conductivity); i.e. they are not just pure numbers. Therefore if you multiply a quantity $x$ by $A$ the result can not have the same dimension as $x$. I.e. $x$ and $Ax$ can not have both dimension potential (unit Volt) which is what you suggested. – Curd May 25 '17 at 16:22