Worked exercise — infinite ladder (R_A=1, R_B=3)

Problem

An infinite ladder network is a periodic chain of identical cells: on each cell a resistor $R_A$ in series on the top wire and a resistor $R_B$ shunted between the two wires. With $R_A = 1\;\Omega$ and $R_B = 3\;\Omega$, find the equivalent resistance $Z_\infty$ seen at the input terminals.

$R_A$ in series on the top wire, $R_B$ shunted; the cell repeats to infinity.

Solution

Self-similarity trick. Adding one more cell leaves the ladder unchanged, so $Z_\infty$ satisfies $Z_\infty = R_A + \frac{R_B\,Z_\infty}{R_B + Z_\infty}$ which, multiplying by $(R_B + Z_\infty)$ and simplifying, becomes the quadratic equation $Z_\infty^2 - R_A\,Z_\infty - R_A R_B = 0$

Substituting the values $R_A = 1\;\Omega$, $R_B = 3\;\Omega$: $Z_\infty^2 - 1\cdot Z_\infty - 1\cdot 3 = 0 \;\Rightarrow\; Z_\infty^2 - Z_\infty - 3 = 0$

Discriminant and positive solution (the other root is negative, physically impossible): $\Delta = 1 + 12 = 13$ $Z_\infty = \frac{1 + \sqrt{13}}{2} \approx \frac{1 + 3{,}61}{2} \approx 2{,}30\;\Omega$

$\ev{Z_\infty = \dfrac{1 + \sqrt{13}}{2} \approx 2{,}30\;\Omega}$

Collegamenti

Argomenti: Circuiti elettrici Concetti: Resistenze in serie e parallelo Competenze: Impostazione simbolica