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After having learned the methods to calculate the overall resistance of series and parallel circuits, I decided to work out the overall resistance of some circuits I made up. The only one I got stuck at was this one. If the resistance of each $n$th resistor is $R_n$, how can someone calculate the resistance of such a circuit? The main problem I am facing is that, I cannot resolve the circuit into separate parallel and series connections, since the wires are connected wherever they overlap.

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  • $\begingroup$ Are the 2 wires crossing in the center of the drawing connected or not? $\endgroup$ – The Photon Oct 29 '16 at 17:22
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I haven't done a lot of these so there might well be a more efficient solution but one tool you could use is the Y-$\Delta$ transform. It lets you convert from three resitors in a Y configuration to three resistors with different values in a $\Delta$ configuration.

Equivalent resistor networks

The values of reistors in the new network are given by $$R_1=\frac{R_bR_c}{R_a+R_b+R_c}$$ $$R_2=\frac{R_aR_c}{R_a+R_b+R_c}$$ $$R_3=\frac{R_aR_b}{R_a+R_b+R_c}$$

or, for the inverse transformation,

$$R_a=\frac{R_1R_2+R_2R_3+R_3R_1}{R_1}$$ $$R_b=\frac{R_1R_2+R_2R_3+R_3R_1}{R_2}$$ $$R_c=\frac{R_1R_2+R_2R_3+R_3R_1}{R_3}$$

Now you can start simplying you circuit like so:

Drawing of curcuit simplification

and go on to use the Y to $\Delta$ transform again on the middle three resistors etc.

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  • $\begingroup$ Or you could transform the $\Delta$s on the left and and right to $Y$s . This at once reduces the circuit to a bunch of series and parallel ones. $\endgroup$ – symanzik138 Oct 29 '16 at 15:48
  • $\begingroup$ True, that sounds a bit quicker. $\endgroup$ – M. Enns Oct 29 '16 at 15:49
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There are some tricks you can use. Look up Delta<->Star transformations. It should be useful in your case also.

For general circuits, you can get the resistance between two points in the circuit by assuming that a current $I$ travels from the input to the output lead, as you have shown. From Kirchoff's laws you can find the potential drop between the input and output leads. The resistance is just the ratio of the two numbers.

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