Infinite ladder of resistors: general case

Deriving an expression for the effective resistance of an infinite ladder-like resistor network, in the general case where the repeating unit contains three arbitrary resistors. We finish with some special cases, and see how the golden ratio turns up in the solution.
About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard.
My website: benyelverton.com/
#physics #mathematics #electricity #circuits #resistance #effectiveresistance #seriescircuit #parallelcircuit #voltage #current #quadraticequation #quadratic #infinity #goldenratio #reciprocal #math #science #education

[@mikoajmetelski18]

Note: (1+sqrt(5))/2 is the golden ratio and corresponds to the case on an infinitely repeated voltage divider - anlogously to a continued fraction [1,1,1,1,1,...]

[@kuzu2104]

So exciting!

[@alexthe15]

Great video!

[@DrBenYelverton]

Thanks!

[@santiospina4504]

Great video Dr Ben... As always. Btw, what software/program do you use to take your annotations?

[@DrBenYelverton]

Thanks! The software is Xournal++.

[@santiospina4504]

@@DrBenYelverton And do you use it on a Table or PC?

[@DrBenYelverton]

On a PC, but with a graphics tablet connected for input.

[@edcoad4930]

R can't be negative? Tunnel diode exhibits negative resistance if R = dV/dI.

[@LuizPoublan]

Now it's time to do the same with capacitors, since inductor would just have the same result as the resistive problem

[@DrBenYelverton]

The solution for capacitors would be very similar, but with each resistance R replaced with the reciprocal of the corresponding capacitance. This means that for example in the last special case we considered, the constant of proportionality would be 1/φ instead of φ, where φ is the golden ratio.