Resistance of a infinite ladder
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Naveen Sharma 8 years, 3 months ago
We want to compute the input resistance to a resistor ladder network of the form in given fiqure.
This network can be thought of as an infinite train of ‘L’ sections, each with a horizontal.
If the input resistance is finite and non-zero, then adding one more section to the front will not change the resistance of the network. Hence, if the resistance is Z ohms, we find that
{tex}Z = R + R \ || Z{/tex}
In other words, taking the original ladder resistance, Z in parallel with ‘R’ and adding another series R will still give the resistance, Z. Using the equation for parallel resistances, we find
{tex}Z = R + {RZ \over R + Z}{/tex}
{tex}=> Z^2 + RZ = 2RZ + R^2 {/tex}
{tex}=> Z^2 − RZ − R^2 = 0{/tex}
This quadratic in Z can be solved for Z, yielding,
{tex}Z ={1 + \sqrt 5 \over 2}R = 1.618 R{/tex}
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