Network Topology Questions and Answers

Answer: B

In circuit theory a branch is a single element (or a series group of elements carrying the same current) between two nodes, and a node is a point where two or more elements join. Reading the figure: the bottom wire is one node, the top left wire where the 18 V source, \(R_A\) and the 6 ohm resistor meet is a second node, and the junction between the 5 ohm resistor and the source \(V_x\) is a third. So there are three nodes.

The elements are the 18 V source, \(R_A\), the 6 ohm resistor, the 5 ohm resistor and \(V_x\), five elements, each in its own branch, so there are five branches. The count is 5 branches and 3 nodes, option B.

Answer: C

The graph has P = 4 trees and Q = 6 cut sets.

The two diagonals cross without a node, so the graph has 4 nodes and 4 branches. Tracing it: top left to top right (1), top right to bottom left (3), bottom left to bottom right (4), and bottom right back to top left (2). It is a single closed loop of four branches.

A tree joins all 4 nodes with 4 - 1 = 3 branches and no loop. Removing any one of the four branches leaves such a tree, so there are 4 trees.

A cut set is a minimal set of branches whose removal splits the graph into two parts. In a single loop, removing one branch leaves it connected, but removing any two branches separates it, so every pair is a cut set: \(\binom{4}{2} = 6\).

Answer: A

The number of mesh equations is B − (N − 1).

For a connected planar network, the number of independent loops (meshes) equals the number of links, the branches not in a tree. A tree joining N nodes has N − 1 branches, so:

\[ \begin{aligned} \text{meshes} &= B - (N - 1) \\ &= B - N + 1 \end{aligned} \]

For example, a network with 5 branches and 4 nodes needs 2 mesh equations. N − 1 itself is the number of independent node equations. B − N − 1 has the wrong sign on the 1, and the other two forms have no basis in network topology.

Answer: C

The number of node pair voltages is 45.

A node pair voltage is the voltage between any two nodes, so the count is the number of ways of choosing 2 nodes out of 10:

\[ \binom{10}{2} = \frac{10 \times 9}{2} = 45 \]

The number of branches does not enter this count. It matters for other quantities: the number of independent loops is b - n + 1 = 17 - 10 + 1 = 8, and the number of independent node voltages is n - 1 = 9. Seven is b - n, 17 is the branch count, and 50 does not follow from either figure.

Answer: C

The key gives The dual representation of a network in terms of nodes and branches.

Two planar networks are duals when the mesh equations of one have the same form as the node equations of the other. In forming the dual, every mesh becomes a node, series connections become parallel, voltage sources become current sources, resistance becomes conductance and inductance becomes capacitance. So duality is a correspondence between the node-and-branch structure of one network and the mesh structure of another.

It is not the input-output relation of a system. It does not let resistors and capacitors be swapped; the dual of an inductor is a capacitor, and of a resistor a conductance. Having both series and parallel elements is simply a property of most networks, not duality.

Answer: D

A cut-set is a set of branches that, if cut, will divide the network into two parts.

In network topology, a cut-set is a minimal set of branches whose removal separates the graph into exactly two connected parts; removing any fewer branches leaves it connected. Each tree branch defines a fundamental cut-set, and Kirchhoff's current law applied to each cut-set gives the independent node-type equations of the network.

A cut-set is defined by the graph, not by the type of element in the branches, so it is not a set of parallel or series resistors or a combination of resistors and capacitors.

Answer: B

A tree is a connected subgraph of a network that contains all its nodes but has no closed path (loop), so the answer is option B. It has one branch fewer than the number of nodes, and the branches that are left out of the tree are the links, which are the ones that close the loops.

A closed path (A) would make it a loop, not a tree. A tree has to touch every node, so it cannot have no nodes (C), and it is made of branches, so it cannot have none (D).

Answer: A

A cut set is a set of branches whose removal separates the network, and the cut set equations are written for the branch voltages, so a matches 2. An RC differentiator passes high frequencies, so it is a high pass filter, and b matches 4. The dual of Kirchhoff's current law is Kirchhoff's voltage law, so c matches 1. The Bode plots show the frequency response, so d matches 3. The code is 2 4 1 3. The codes 3 2 4 1, 4 3 1 2 and 1 2 4 3 each give the cut set a wrong partner. The answer is 2 4 1 3.

Answer: A

The tie-set schedule gives the relation between the branch currents and the link currents, option (A). In graph theory, a tie-set is the set of branches that form a loop containing exactly one link, and the tie-set matrix \(B\) relates the two sets of currents by \(I_{b} = B^{T} I_{l}\), so each branch current is a combination of the link currents of the loops in which that branch lies. Options (B) and (D) mix in branch voltages, and option (C) mixes in link voltages. The relation between voltages and currents of this kind belongs to the cut-set schedule, which links the branch voltages to the twig voltages.