How to find the equivalent resistance of this infinite circuit?

  • #1
zenterix
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Thread moved from the technical forums to the schoolwork forums
TL;DR Summary: I am following the course 6.002 "Circuit and Electronics" on MIT OCW. There are no solutions to the problem sets. I would like to check my solution to one particular problem.

We are asked to find the equivalent resistance of the network

1695598655023.png

as viewed from its ports.

I simplified the network as

1695598932640.png

Where ##R_{eq}## is not only the equivalent resistance of the branch shown above, it is also the equivalent resistance of the entire network shown above (because the network shown above repeats itself infinitely).

Then

$$R_{eq}=R+\frac{R_{eq}R}{R+R_{eq}}$$

$$R_{eq}(R+R_{eq})=R(R+R_{eq})+RR_{eq}$$

$$R_{eq}^2=R^2+RR_{eq}$$

$$R_{eq}^2-RR_{eq}-R^2=0$$

$$\Delta = R^2+4R^2=5R^2$$

$$R_{eq}=\frac{R\pm R\sqrt{5}}{2}$$

$$=R\frac{1\pm\sqrt{5}}{2}$$

Is this result correct?

Here is the problem set in full if it is useful.
 
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  • #2
zenterix said:
TL;DR Summary: I am following the course 6.002 "Circuit and Electronics" on MIT OCW. There are no solutions to the problem sets. I would like to check my solution to one particular problem.

We are asked to find the equivalent resistance of the network

View attachment 332529
as viewed from its ports.

I simplified the network as

View attachment 332530
Where ##R_{eq}## is not only the equivalent resistance of the branch shown above, it is also the equivalent resistance of the entire network shown above (because the network shown above repeats itself infinitely).Here is the problem set in full if it is useful.
Seems like a clear approach. If your algebra is correct I'd say you have it correct.
 
  • #3
##R=\frac{1-\sqrt{5}}{2}## certainly isn't correct.

Have you checked your own work? How would you use that (positive) value to check?
 
  • #4
Your approach is fine. I think the circuit you want is the one shown below in which the resistance between points A and B is to be found is clearer about what you are replacing with what in view of the drawing provided by the problem.

InfiniteLadder.png
 
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  • #5
Note that: (1±√5)/2 ; is the golden ratio, or its reciprocal.

An infinite ladder can be written as an infinite recurring continued fraction, and solved in the same equivalent way.
 
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  • #7
Babadag said:
According to (Endreny 1967)

Endrenyi, J., “Analysis of transmission tower potentials during ground faults" for a transmission line R1∞=R/2+sqrt(R^2+R^2/4)
If not already clear, note that "R/2+sqrt(R^2+R^2/4)" simplifies to ##R \frac {1+\sqrt 5}2##.
 
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  • #8
The second addend in the expression ## R _ { eq } = R \frac { 1 } { 2 } + ( \pm R \frac { \sqrt 5 } { 2 } ) ## is resistance and must be positive. This is why ## R _ { eq } = R \frac { 1 + \sqrt 5 } { 2 } ## is the only acceptable solution.
 
  • #9
zenterix said:
TL;DR Summary: I am following the course 6.002 "Circuit and Electronics" on MIT OCW. There are no solutions to the problem sets. I would like to check my solution to one particular problem.

We are asked to find the equivalent resistance of the network

View attachment 332529
as viewed from its ports.

I simplified the network as

View attachment 332530
Where ##R_{eq}## is not only the equivalent resistance of the branch shown above, it is also the equivalent resistance of the entire network shown above (because the network shown above repeats itself infinitely).

Then

$$R_{eq}=R+\frac{R_{eq}R}{R+R_{eq}}$$

$$R_{eq}(R+R_{eq})=R(R+R_{eq})+RR_{eq}$$

$$R_{eq}^2=R^2+RR_{eq}$$

$$R_{eq}^2-RR_{eq}-R^2=0$$

$$\Delta = R^2+4R^2=5R^2$$

$$R_{eq}=\frac{R\pm R\sqrt{5}}{2}$$

$$=R\frac{1\pm\sqrt{5}}{2}$$

Is this result correct?

Here is the problem set in full if it is useful.
It is indeed but the answer can't be in negative so + one is right
 
  • #10
Try puting R adjacent to 2R and 4R then 8R and so on , of resitance increases by a factor of two parallely till infinity. It is a must try
 
  • #11
An infinite ladder is insensitive to the far end resistance, because that approximation is attenuated by the intermediate ladder.
Guess a far end value, say 1.5 ohms.
Then repeatedly take the reciprocal, and add one, until you get a stable value.
Solve; x = 1 + 1/x ; and you get the golden ratio = 1.618033
 

Related to How to find the equivalent resistance of this infinite circuit?

1. What is an infinite circuit?

An infinite circuit is a theoretical electrical circuit that extends indefinitely, often consisting of repeating units of resistors. These circuits are used to explore concepts in electrical engineering and physics, and they present unique challenges in calculating equivalent resistance.

2. How do you approach finding the equivalent resistance of an infinite circuit?

To find the equivalent resistance of an infinite circuit, you typically use the concept of self-similarity. This involves recognizing that the resistance of the infinite circuit can be represented as a finite segment of the circuit plus the resistance of the rest of the infinite circuit. By setting up an equation using this self-similarity, you can solve for the equivalent resistance.

3. What is the role of symmetry in solving infinite circuits?

Symmetry plays a crucial role in solving infinite circuits. If the circuit is symmetric, it simplifies the calculation since the repeating units allow you to set up recursive relationships. This symmetry helps in forming equations that can be solved to find the equivalent resistance.

4. Can you provide an example of solving an infinite resistor ladder?

Consider an infinite resistor ladder where each segment consists of a resistor R in series with a parallel combination of another resistor R and the rest of the ladder. Let the equivalent resistance of the entire ladder be Req. By setting up the equation Req = R + (R * Req) / (R + Req) and solving for Req, you can find the equivalent resistance. This often results in a quadratic equation, which can be solved using standard algebraic methods.

5. Are there any special techniques or approximations used in solving infinite circuits?

Yes, special techniques such as the use of continued fractions, transformations, and network theorems are often employed to solve infinite circuits. Approximations may also be used when exact solutions are difficult to obtain. These techniques help simplify the problem and provide insight into the behavior of the circuit.

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