What are the equations for calculating equivalent resistance in a circuit?

In summary, the conversation discusses calculating the equivalent resistance of an infinite circuit with identical resistors. The equations for series and parallel equivalent resistance are mentioned, and the method for solving the problem is explained. A link to a paper with further explanation is provided.
  • #1
rbwang1225
118
0

Homework Statement


A circuit extends forever to the right, and all the resistors have the same value ##R##. Calculate the equivalent resistance measured across the two terminals at left.
circuit.jpg



Homework Equations


The series and parallel equivalent resistance equation.


The Attempt at a Solution


I have no idea to begin, could someone give some suggestions for me.
Sincerely.
 
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  • #2
There is a paper on this on the web but if I post a link to it that will give you too much help.

If the chain is infinite then adding another "stage" to the input won't change the resistance because it's equivalent to shifting the whole chain physically one place to the left.

So call the existing resistance Z and write an equation for the resistance with an additional stage added...

Z = Z//R + 2R

Over to you..
 
Last edited:
  • #3
Dear CWatters:

Could you post the link to me?
I want to understand it more clearly.

Regards
 
  • #4
The forum rules discourage us giving away the whole solution so before I do that which bit are you struggling with?

Here is a diagram explaning of why adding a stage doesn't change the resistance. It works because ∞ + 1 = ∞.

Although you are adding a stage to the left the result is indistinguishable from adding a stage to the right hand side which is at infinity.
 

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  • #5
Here is how you derive the equation for Z that you need to solve...
 

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  • #6
I think I understand the argument, so the equations are as follows:
##Z=Z//R+2R=(\frac{1}{Z}+\frac{1}{R})^{-1}+2R=(\frac{R+Z}{RZ})^{-1}+2R=\frac{ZR}{Z+R}+2R##
##ZR+Z^2=ZR+2R^2+2ZR##
##Z^2-2RZ-2R^2=0##
##Z=\frac{2R±√(4R^2+8R^2)}{2}=(1±√3)R##
There is a problem for choosing the sign.
Is there any argument that one of which is right?
If my calculations are correct, could you please let me see the paper?
Thank you for kind reply!
 
  • #7
I think that Z should be positive, therefore, ##Z=(1+\sqrt 3)R##.
 
  • #9
Dear CWatters:

Thank you for kind help!

Regards.
 

Related to What are the equations for calculating equivalent resistance in a circuit?

1. What is Equivalent Resistance?

Equivalent Resistance is the overall resistance of a circuit or network that is equivalent to a single resistor. It is the resistance value that would produce the same effect as the combination of resistors in the circuit.

2. How is Equivalent Resistance calculated?

Equivalent Resistance can be calculated using two different methods:
1. Series Resistance - Simply add all the individual resistances together to get the equivalent resistance.
2. Parallel Resistance - Take the reciprocal of each individual resistance, add them together, and then take the reciprocal of the sum to get the equivalent resistance.

3. Why is Equivalent Resistance important?

Equivalent Resistance is important because it helps us understand and analyze how electricity flows in a circuit. It also allows us to simplify complex circuits into simpler, equivalent circuits that are easier to understand and work with.

4. What factors affect Equivalent Resistance?

The two main factors that affect Equivalent Resistance are the number of resistors in a circuit and how they are connected. In a series circuit, the equivalent resistance increases as more resistors are added, while in a parallel circuit, the equivalent resistance decreases as more resistors are added. The individual resistance values and the type of material used for the resistors can also affect the equivalent resistance.

5. How is Equivalent Resistance used in real-world applications?

Equivalent Resistance is used in many real-world applications, such as in designing electronic circuits, calculating power and current in a circuit, and determining the overall resistance in an electrical system. It is also used in industries such as telecommunications, automotive, and aerospace to optimize and improve the efficiency and performance of electronic devices.

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