My Math Forum  

Go Back   My Math Forum > Science Forums > Physics

Physics Physics Forum


Thanks Tree2Thanks
  • 1 Post By Benit13
  • 1 Post By v8archie
Reply
 
LinkBack Thread Tools Display Modes
July 13th, 2015, 11:04 PM   #1
Member
 
Joined: Aug 2014
From: Lithuania

Posts: 60
Thanks: 3

Total resistance of an endless circuit

I need to find the total resistance of this endless circuit when R=100Ω.



I want to solve this problem mathematically. Here's my try.

This circuit is made out of loops:
First loop: R, 2R, 3R;
Second loop: 2R, 4R, 6R;
Third loop: 4R, 8R, 12R;
...

The first loop has a total resistance of 6R, the second- 12R, the third- 24R. So I see a geometrical progression here, which can be written as $\displaystyle R_{loop}=6*2^n$ where n is the number of loops.

I can express the total resistance of two last loops like this:
$\displaystyle \frac{2}{3}R_{n-1}+\frac{\frac{1}{3}R_{n-1}*R_n}{\frac{1}{3}R_{n-1}+R_n}$

$\displaystyle \frac{2}{3}(6*2^{n-1})+\frac{\frac{1}{3}(6*2^{n-1})*(6*2^n)}{\frac{1}{3}(6*2^{n-1})+(6*2^n)}$

Then it would seem that the total resistance of the circuit is equal to (assuming that R=1Ω):

$\displaystyle \lim_{n\rightarrow +\infty} {(\frac{2}{3}(6*2^{n-1})+\frac{\frac{1}{3}(6*2^{n-1})*(6*2^n)}{\frac{1}{3}(6*2^{n-1})+(6*2^n)})}$

I don't know if this is correct. Please show me how to solve this problem and where my mistakes are.
kaspis245 is offline  
 
July 13th, 2015, 11:23 PM   #2
Senior Member
 
Joined: Apr 2014
From: UK

Posts: 914
Thanks: 331

The resistance of the first loop is 4R + whatever the rest is (the 2R in parallel with the other loops)
using // to denote parallel:
the total resistance is 4R + (2R//(8R + (4R//(16R + (8R//(16R + etc... ))))))
weirddave is offline  
July 14th, 2015, 07:32 AM   #3
Member
 
Joined: Aug 2014
From: Lithuania

Posts: 60
Thanks: 3

I see, but how should I make an equation out of it?
kaspis245 is offline  
July 14th, 2015, 10:15 AM   #4
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,649
Thanks: 2630

Math Focus: Mainly analysis and algebra
We have two terminals on the left, yes? So the right hand side is closed.

Let r be the resistance of the whole circuit.
We have a resistance of R+3R that is in series.
Then we have the left branch with resistance 2R.
The rest has resistance 2r.
These last two parte are in parallel.
v8archie is offline  
July 15th, 2015, 01:37 AM   #5
Senior Member
 
Joined: Apr 2014
From: UK

Posts: 914
Thanks: 331

I have no idea how to build a formulae for this, but, you can find an adequate answer by limiting the number of loops. This is how I solve these types of problem in the real world.
I get 5.701562119R with 9 loops, more loops just increases the accuracy. Clearly, 9 loops is way over the top since even the best resistors available won't be that accurate.
Excel makes it very easy to do....
weirddave is offline  
July 15th, 2015, 03:20 AM   #6
Senior Member
 
Joined: Apr 2014
From: Glasgow

Posts: 2,155
Thanks: 731

Math Focus: Physics, mathematical modelling, numerical and computational solutions
https://xkcd.com/356/
Thanks from weirddave
Benit13 is offline  
July 15th, 2015, 05:00 AM   #7
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,649
Thanks: 2630

Math Focus: Mainly analysis and algebra
Quote:
Originally Posted by v8archie View Post
We have two terminals on the left, yes? So the right hand side is closed.

Let r be the resistance of the whole circuit.
We have a resistance of R+3R that is in series.
Then we have the left branch with resistance 2R.
The rest has resistance 2r.
These last two parts are in parallel.
It seems that the end of my post got lost.

The parallel part of the circuit has resistance $r_p$ given by$${1 \over r_p} = {1 \over 2R} + {1 \over 2r}$$

Then the total resistance is $$r = 4R + r_p$$
v8archie is offline  
July 17th, 2015, 06:44 AM   #8
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,649
Thanks: 2630

Math Focus: Mainly analysis and algebra
Following the above method yields a resistance of$${R \over 2}\left(5 + \sqrt{41}\right)=5.70156\,R\;\text{to 5 d.p.}$$
Thanks from kaspis245
v8archie is offline  
July 24th, 2015, 12:57 AM   #9
Senior Member
 
Joined: Apr 2014
From: UK

Posts: 914
Thanks: 331

I couldn't follow the method at all
weirddave is offline  
Reply

  My Math Forum > Science Forums > Physics

Tags
circuit, endless, resistance, total



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
A Natural Number as an Endless String of Digits. tomr Real Analysis 7 November 12th, 2012 05:21 PM
Resistance as a function of Temperature dave daverson Algebra 1 October 28th, 2012 06:35 PM
Air resistance arron1990 Physics 5 March 28th, 2012 05:52 AM
Endless fraction thebroker Number Theory 4 November 13th, 2011 08:32 PM
Air resistance oswaler Physics 1 March 19th, 2008 07:36 PM





Copyright © 2019 My Math Forum. All rights reserved.