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anonymous

  • 4 years ago

Could someone explain, step-by-step, how to find the resistance of an infinite ladder circuit with resistors?

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  1. anonymous
    • 4 years ago
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    Something similar to this:

  2. JamesJ
    • 4 years ago
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    This should help: http://www.crbond.com/papers/ent2-3.pdf

  3. anonymous
    • 4 years ago
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    Thanks, I've already been directed to that page before, however. I was wondering if you could explain what they wrote there: Z = R + R || Z

  4. JamesJ
    • 4 years ago
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    That's just a notational short cut for what comes next. If we added just one more resistor, and not in parallel, then the resistance would be Z + R where Z is the original resistance and R is the resistance of one resistor. If we added just one resister in parallel, the new resistance would be \[ \frac{RZ}{R+Z} \] Now, having added one more step in the ladder, the new resistance is \[ R + \frac{RZ}{R+Z} \] This must be equal to the original resistance; hence \[ Z = R + \frac{RZ}{R+Z} \ \ \ \ \ \ --- (*)\] The notation \[ Z = R + R || Z \] is just a short way of writing the equation (*)

  5. anonymous
    • 4 years ago
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    Oh, alright. It took me a bit, but I think I get it now. Thank you very much!

  6. JamesJ
    • 4 years ago
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    Sure. I haven't looked at this problem for a while and it's nice to see it again. I'm particularly happy to see the quantity \[ \frac{1 + \sqrt{5}}{2} \] in the answer. This is the Golden Ratio of mathematics and I'd forgotten that it turns up here as well.

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