# About the existence and uniqueness of electrical network solutions

• cianfa72
In summary, Z and Y matrices in the equation V=E + ZI and I=J + YV are different when there are no independent current or voltage sources in the network.
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.

cianfa72 said:
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 $G$.

Fisherman199
pasmith said:
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.

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cianfa72 said:
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?

DaveE said:
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 $Y$).

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## 1. What is the existence and uniqueness theorem for electrical networks?

The existence and uniqueness theorem for electrical networks states that for a given electrical network with specified boundary conditions, there exists a unique solution for the voltages and currents in the network. This is typically guaranteed under certain conditions, such as the network being linear and passive, which means it does not contain any active components like transistors or dependent sources.

## 2. What conditions must be met for the existence and uniqueness theorem to apply?

For the existence and uniqueness theorem to apply, the electrical network must be linear, passive, and time-invariant. This means the network components should obey Ohm's Law (linear resistors), and there should be no energy sources that depend on the network's state (passive). Additionally, the network's properties should not change over time (time-invariant).

## 3. How does Kirchhoff's Laws relate to the existence and uniqueness of solutions in electrical networks?

Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) are fundamental in analyzing electrical networks. KVL states that the sum of electrical potential differences around any closed loop is zero, while KCL states that the sum of currents entering a junction must equal the sum of currents leaving. These laws form the basis of the system of equations that describe the network. When combined with the network's component equations, they ensure the existence and uniqueness of the solution, provided the network meets the necessary conditions.

## 4. Can non-linear components affect the existence and uniqueness of solutions in electrical networks?

Yes, non-linear components can significantly affect the existence and uniqueness of solutions in electrical networks. Non-linear components, such as diodes and transistors, introduce complexities that can lead to multiple solutions or no solution at all. The existence and uniqueness theorem generally applies to linear networks, and additional methods or numerical techniques are often required to analyze networks with non-linear components.

## 5. What role do boundary conditions play in determining the existence and uniqueness of network solutions?

Boundary conditions, such as specified voltages or currents at certain points in the network, are crucial for determining the existence and uniqueness of solutions. They provide the necessary constraints to solve the system of equations derived from Kirchhoff's Laws and component characteristics. Without appropriate boundary conditions, the system may be underdetermined (leading to multiple solutions) or overdetermined (leading to no solution).

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