What is the solution for the total resistance of the infinite ladder network?

[strangequark]

Homework Statement

Find the total resistance (each R = 1 ohm) of the infinite ladder network that looks something like this:

!--------!------R-----!------R----!------R-----!----- etc
!        !            !           !            !
V        R            R           R            R
!        !            !           !            !
!--------!------R-----!------R----!------ R----!----- etc

Homework Equations

I assume the revelant equations are:

1/Ra+1/Rb=1/Rab or Rab=(RaRb)/(Ra+Rb) (in parallel)

and Ra+Rb=Rab (in series)

The Attempt at a Solution

Now, I keep wanting to say that the resistors on the top and bottom are in parallel, and that each center resistor is in series with the parallel combo... so I get something like:

1/2 + 1= (3/2) ~ 1.5for the first tier:

Then 3/8+1= 11/8 ~1.375 for the second,

Then 11/30+1= 41/30 ~ 1.36667 for the third

Then 41/112+1= 153/112 ~ 1.36607 for the fourth

and so on...

Now, it is looking like it might converge to some finite value in the quasi-near future... so i might not be completely wrong, but it would be nice to know prior to staying up all night looking for an infinite series to represent it.

Is this correct, or if not, can someone please steer me in the right direction?

Any assistance is much appreciated...

 

[berkeman]

welcome to the PF stragequark.

It's hard to visualize the resistor ladder that you are asking about. Maybe try the "code" keyword in square brackets "[]" to force non-proportional spacing, or just attach a PDF or other document.

With long or infinite ladder configurations, I'll try to see if there is a symmetry that let's me fold up the parallel-series combinations below. Kind of like how an R-2R ladder DAC works.

 

[AlephZero]

Suppose the resistance of the whole ladder is RL.

Then the following two circuits have the same resistance:

!--------!
!        !
V        RL
!        ! 
!--------!

and

!--------!------R-----!
!        !            !
V        R            RL
!        !            !
!--------!------R-----!

This is a common way of analysing transmission lines, etc.

 

[jeroen]

All you need to do is solve this equation:
RL = 1/(1/R + 1/(2*R+RL))

And if R = 1:
RL = 1/(1 + 1/(2+RL))