The following problems have little value but might prove interesting as intellectual exercises. (a) Is it possible to construct a network of resistors which cannot be reduced by using the series and parallel formulae together with the delta-Y or Y - delta transformation? (b) An infinite triangular mesh of \(1 \Omega\) resistors is constructed in a plane. What is the equivalent resistance between two adjacent nodes? (c) Nine resistors are of identical appearance but eight are of \(1 \Omega\) while the ninth is of \(2 \Omega\). Can a two-terminal network be constructed out of these resistors of such a form that one measurement of resistance across it would enable the odd resistor to be unambiguously selected?

Resistor network analysis is the method of finding the equivalent resistance of a complex circuit. The network consists of various resistors connected together in series, parallel, or a combination thereof. By understanding the properties of these connections, you can replace the entire network with a single resistor that has the same resistance as the original network. This simplification process relies heavily on series and parallel formulae, delta-Y and Y-delta transformations. For instance, in the context of the provided exercise, a non-reducible network is theoretically impossible as it can always be simplified to an equivalent resistance.

Analyzing resistor networks is fundamental in electrical engineering as it helps in designing circuits with the desired properties and in power management within the circuitry. At a basic level, the goal is to use the formulas for resistors in series and in parallel to simplify the network as much as possible, and for more complex interconnected networks, to use delta-Y and Y-delta transformations.

When resistors are connected end-to-end, it's known as a series circuit where the current flowing through each resistor is the same. The total resistance of a series circuit is the sum of the individual resistances. In contrast, a parallel circuit happens when resistors are connected across the same two points, thus all the resistors have the same voltage across them. The total resistance of resistors in parallel is found by using the formula: \(\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n}\). In practical scenarios, like the one in the textbook exercise, these principles are foundational to determining the equivalent resistance across a network, as every complex network can be broken down into series and parallel components.

Delta-Y (Δ-Y) and Y-Delta (Y-Δ) transformations are techniques used to simplify resistor networks that can't be reduced using just the simple series and parallel arrangements. These transformations allow us to convert a triangle (delta or Δ) configuration of resistors into a Y (star) shape, and vice versa. This is particularly useful for networks where series and parallel rules alone do not apply, such as the infinite triangular mesh in the textbook problem. In this case, a Δ-Y transformation simplifies the recurring pattern within the mesh, ultimately enabling the calculation of equivalent resistance, which would otherwise be a challenging task.

The Wheatstone bridge is a circuit used to measure an unknown electrical resistance by balancing two legs of a bridge circuit. One leg contains the unknown component, while the other leg has resistors with known resistances. When the bridge is balanced, the ratio of the known resistors in one leg is equal to the ratio in the other leg, thus allowing determination of the unknown resistance. This concept was applied in the exercise to identify the resistor with a different value. By creating a network and measuring the resistance across, the odd resistor can be unambiguously selected due to the characteristics of the balanced Wheatstone bridge.

Infinite network analysis involves assessing a resistor network that extends infinitely. The challenge here is that traditional methods for calculating equivalent resistance don't work because there are an infinite number of resistors. However, thanks to principles of symmetry and by observing repeating patterns, you can often find a finite equivalent resistance. This was demonstrated in the textbook exercise, where an infinite triangular mesh was effectively reduced by applying a Y-Δ transformation and identifying a repeating pattern, thus arriving at a solvable equation for equivalent resistance.

The equivalent resistance calculation is the process of determining a simplified resistance for a complex circuit. It is the total resistance that a current would face if the entire network were replaced with a single resistor. This calculation can be straightforward for simple series or parallel circuits but may require iterative techniques and transformations for more complex configurations. As we saw in the problems posed, understanding how to approach the network - whether by series/parallel analysis or by more advanced transformations - is key to finding that single, equivalent resistance value.