About the existence and uniqueness of electrical network solutions

[cianfa72]

TL;DR Summary

About the existence and uniqueness of network solutions as explained in the book Applied Graph Theory W.K. Chen

Hi,
I've a question about a proof found in the book Applied Graph Theory from Wai-Kai Chen. My point is that basis circuit matrix B and basis cut matrix Q employed in the proof actually refer to two different networks.

B should be the basic circuit matrix of the initial network with current sources removed (opened) while Q should be the basis cut matrix of the initial network with voltage sources shorted.

Since B and Q refer actually to different networks to me it does not make sense the following part of the proof to show that the solution is unique.

Can you help me ? Thanks.

 

[pasmith]

Since B and Q refer actually to different networks

$B^*$ and $Q^*$ relate to the same network, $G^*$, as stated in the first line of the proof. The definitions of $B$ and $Q$ are not given in this extract - presumably they are defined earlier in the text - but the notation strongly suggests that $B$ and $Q$ both relate to the single network $G$.

 

[cianfa72]

but the notation strongly suggests that $B$ and $Q$ both relate to the single network $G$.

The point I was trying to make is that from earlier in the book $Z$ hence $BZB'$ should be defined only for networks with no indipendent current sources while $Y$ hence $QYQ'$ only for networks with no indipendent voltage sources. So it seems network $G$ is actually two different networks.

 

[DaveE]

The point I was trying to make is that from earlier in the book $Z$ hence $BZB'$ should be defined only for networks with no indipendent current sources while $Y$ hence $QYQ'$ only for networks with no indipendent voltage sources. So it seems network $G$ is actually two different networks.

Are you sure? That doesn't sound right to me. The difference should only be whether you choose to use impedance or admittance matrices. Anyway, how can we comment about text we can't see?

 

[cianfa72]

The difference should only be whether you choose to use impedance or admittance matrices.

In the book branch-impedance matrix $Z$ enters in the equation $V=E + ZI$ while branch-admittance $Y$ in the equation $I=J + YV$.

The text insists that for loop system of equations there are no independent current sources while for cut system of equations there are no independent voltage sources:

So I believe the subscript s in the partitioning of $Z$ and $Y$ matrices actually reflects, respectively, one type of source only (voltage sources for $Z$ and current sources for $Y$).