Capacitors in Parallel and in Series — Question 10

Question 10

An infinite capacitive ladder satisfies a positive fixed-point equation

An infinite ladder begins with a series capacitor C, followed by a node with a shunt capacitor C to the return terminal. Beyond that node the same complete ladder repeats. All internal nodes are neutral. Let X be the input equivalent capacitance.

1. Derive and solve the self-consistency equation for X.

2. Calculate the first two finite approximations obtained by starting with X0=0 and iterating the network rule; identify the unphysical root.

Question 10 - Solution

An infinite capacitive ladder satisfies a positive fixed-point equation

(a) Method and calculation

$$X=\frac{C(C+X)}{2C+X}\Rightarrow X^2+CX-C^2=0.$$$$X=\frac{\sqrt5-1}{2}C\approx.618034C.$$

(b) Interpretation and further calculation

$$X_1=C/2,\quad X_2=\frac{C(C+C/2)}{2C+C/2}=3C/5.$$$$X_-=-\frac{1+\sqrt5}{2}C<0$$is inadmissible for this passive positive-capacitance network.

Physical check

Finite passive approximations select the positive root and approach the infinite-ladder value from below.