The self-similarity trick

An infinite ladder circuit is a periodic chain of identical cells: on each cell a resistor $R_A$ in series along the upper wire and a resistor $R_B$ in shunt (vertical, between the two wires). The network cannot be simplified group by group, because the groups are infinite. The winning strategy exploits self-similarity: the sub-circuit starting from the second node is identical to the whole circuit.

Infinite ladder: $R_A$ in series on the upper wire, $R_B$ in shunt. The sub-circuit from the second node onward is identical to the whole, hence the self-similarity equation.

Let $Z_\infty$ be the equivalent resistance seen at the terminals. If a cell is added at the front, the rest of the ladder remains identical and is still worth $Z_\infty$. But the new cell is: $R_A$ in series, then $R_B$ in parallel with everything else ($Z_\infty$). So $Z_\infty$ must satisfy the equation in which it appears on both sides:

$Z_\infty = R_A + \frac{R_B\,Z_\infty}{R_B + Z_\infty}$

Multiplying everything by $(R_B + Z_\infty)$ and expanding:

$Z_\infty(R_B + Z_\infty) = R_A(R_B + Z_\infty) + R_B\,Z_\infty$

$Z_\infty^2 + R_B\,Z_\infty = R_A\,R_B + R_A\,Z_\infty + R_B\,Z_\infty$

The term $R_B Z_\infty$ cancels, leaving a second-degree equation:

Key formula

$\ev{Z_\infty^2 - R_A\,Z_\infty - R_A\,R_B = 0}$

The quadratic formula gives two roots; only the positive one is physically acceptable (a resistance cannot be negative):

$\ev{Z_\infty = \frac{R_A + \sqrt{R_A^2 + 4\,R_A R_B}}{2}}$

Principle — Self-similarity

An infinite periodic circuit is identical to itself with the first cell removed. Imposing this equality translates the infinite structure into a single algebraic equation for $Z_\infty$.

Links

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

Related exercises: Worked exercise — mixed series-parallel network · Problem — true or false on series and parallel · Problem — series-parallel equivalent resistance