Electronic – Closed form expression for a resistor ladder network

circuit analysisresistor-ladder

Is it possible to find a closed form expression for the resistance between A and B for the general n-section ladder network, below, where all resistors are \$\small 1\Omega\$? An iterative expression is quite easy to derive (\$\small R_n =2 + 1//R_{n-1})\$
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Best Answer

By inspection,the maximum possible value of resistance is \$\small3\Omega\$ (for \$\small n=1\$), and for \$\small n>1\$ the resistance must be \$\small >2\Omega\$, hence \$\small 2< R_n\le 3\$

Calculating the effective resistance values, \$\small R_n\$, for \$\small n=1, \:2,\: 3,\: 4,\: 5,\: 6\:...\$ gives:

$$\small[3,\:2\frac{3}{4},\: 2\frac{11}{15},\: 2\frac{41}{56},\: 2\frac{153}{209},\: 2\frac{571}{780}\:... ]$$

Initially considering just the fractional parts, and noting that \$\small 3= 2\frac{1}{1}\$, we may write the sequence:$$\small [1,\:1,\:3,\:4,\:11,\:15,\:41,\:56,\:153,\:209,\:571,\:780\:...]$$

Searching a catalogue of z-transform generating functions for this particular sequence (http://www.lacim.uqam.ca/~plouffe/articles/MasterThesis.pdf) gives:

$$\small F(z)=\frac{z^2+z-1}{z^4-4z^2+1}$$

The denominator factorises to a very convenient form (what luck!), and the generating function may be expressed in partial fractions as:

$$\small F(z)=\frac{A}{z-a}+\frac{B}{z+a}+\frac{C}{z-b}+\frac{D}{z+b}$$

where the constants: a, b, A, B, C, D are:

\$\small a=\sqrt {2+\sqrt3}\$,

\$\small b= \sqrt{2-\sqrt3}\$,

\$\small A=\frac{a^2+a-1}{2a(a^2-b^2)}\$,

\$\small B=\frac{a^2-a-1}{2a(b^2-a^2)}\$,

\$\small C=\frac{b^2+b-1}{2b(b^2-a^2)}\$,

\$\small D=\frac{b^2-b-1}{2b(a^2-b^2)}\$

Inverse z-transforming \$\small F(z)\$ gives the closed form expression for the sequence:

$$\small f(k)=A(a)^k +B(-a)^k+C(b)^k+D(-b)^k$$

The resistance value, \$\small R_n\$, for n sections can now be obtained by evaluating the last equation with \$\small k=2n\$ and \$\small k=2n-1\$, to form the denominator and numerator of the fractional part of \$\small R_n\$; and then adding \$\small 2\Omega\$:

$$\small R_n = 2+\frac{A(a)^{2n-1}+B(-a)^{2n-1}+C(b)^{2n-1}+D(-b)^{2n-1}}{A(a)^{2n}+B(-a)^{2n}+C(b)^{2n}+D(-b)^{2n}}$$

This is the required closed form expression.

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