Certain transmission lines can be modeled as a long 3D circuit as shown in the figure below. Here, all the the wires have equal length and electrical resistance R=1 Ω. Determine the equivalent resistance in Ohms between the points A and B. Assume the transmission line to be infinite.
To determine the equivalent resistance between points A and B, you can apply the principle of symmetry.
First, observe that the given circuit can be divided into segments that repeat along the length of the transmission line. Each segment consists of a resistor (R) in series with two parallel conductors.
Let's focus on a single segment, which we will refer to as the "unit cell." The unit cell sequence is as follows:
1. Resistor (R)
2. Conductor
3. Conductor
4. Resistor (R)
Since the transmission line is assumed to be infinite, this unit cell will repeat indefinitely.
Now, consider the electrical properties of this unit cell. The two parallel conductors can be treated as a single equivalent conductor, just like in a standard series-parallel circuit. The equivalent resistance of the unit cell can be calculated using the formula for resistors in series and parallel.
To do this, follow these steps:
1. Calculate the equivalent resistance (Req) of the two parallel conductors (steps 2 and 3). Since both wires are identical and have the same resistance (R=1 Ω), the equivalent resistance of the parallel combination is half of the resistance of each wire, in this case, Req_parallel = 0.5 Ω.
2. Calculate the series combination of the two resistors: Req_series = R + R = 1 Ω + 1 Ω = 2 Ω.
3. Combine the results of steps 1 and 2 to find the equivalent resistance of the unit cell, Req_unitcell:
Req_unitcell = Req_series + Req_parallel = 2 Ω + 0.5 Ω = 2.5 Ω.
Since the unit cell repeats indefinitely in the infinite transmission line, the equivalent resistance between points A and B is just the resistance of one unit cell, i.e., Req_unitcell = 2.5 Ω.
Therefore, the equivalent resistance between points A and B in the given circuit is 2.5 Ω.