Problem — resistance of an infinite ladder

Problem

An infinite ladder is made of identical cells: each cell has a resistance $R$ in series followed by a resistance $2R$ in parallel towards the next cell. Using the self-similarity principle, write the equation satisfied by the equivalent resistance $R_\infty$ seen from the input terminals, solve it and check the sign by choosing the physically acceptable root.

Diagram: chain of identical cells ($R$ in series, $2R$ in parallel towards the right) continuing indefinitely.

Solution

Self-similarity. An infinite ladder remains identical to itself if the first cell is removed: what is seen downstream of the first cell is still an infinite ladder, with the same resistance $R_\infty$. Therefore $R_\infty$ is given by: $R$ in series with the parallel of $2R$ and $R_\infty$ itself: $R_\infty = R + \frac{2R\,R_\infty}{2R + R_\infty}$

Solving. Multiply by $(2R + R_\infty)$: $R_\infty(2R + R_\infty) = R(2R + R_\infty) + 2R\,R_\infty$ $2R\,R_\infty + R_\infty^2 = 2R^2 + R\,R_\infty + 2R\,R_\infty$ $R_\infty^2 - R\,R_\infty - 2R^2 = 0$

Solving the quadratic equation in $R_\infty$: $R_\infty = \frac{R \pm \sqrt{R^2 + 8R^2}}{2} = \frac{R \pm 3R}{2}$ The two roots are $R_\infty = 2R$ and $R_\infty = -R$.

Physical choice. A resistance must be positive: the root $-R$ is physically meaningless and is discarded. What remains is $\ev{R_\infty = 2R}$

Links

Topics: Electric circuits Concepts: Resistors in series and parallel Skills: Symbolic problem setup