Lumped circuit analysis dealing with electromagnetic propagation in silicon

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Discussion Overview

The discussion revolves around the application of lumped circuit analysis in the context of electromagnetic propagation in silicon, particularly concerning the conditions under which this abstraction can be valid. Participants explore the implications of using this model for a computer chip that is 1 inch square and operates at 2 GHz.

Discussion Character

  • Homework-related
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that the timescale of interest must be significantly greater than the speed of electromagnetic propagation to use the lumped circuit abstraction effectively.
  • One participant calculates the speed of electromagnetic propagation in silicon as approximately 1.5 x 10^8 m/s and questions whether this speed poses a problem for a 2 GHz signal.
  • Another participant emphasizes the importance of including units in relevant equations and raises questions about the wavelength of a 2 GHz waveform in silicon and its comparison to the longest path in the integrated circuit (IC).
  • A participant corrects an earlier calculation, clarifying that the speed of light in silicon should be divided by 2, not 5, and discusses the longest path in the IC as potentially being 4 inches.
  • There is a calculation presented regarding the wavelength of a 2 GHz signal in silicon, suggesting it is 15 cm, and a participant attempts to relate this to the dimensions of the chip.
  • Some participants express uncertainty about whether their calculations indicate that using the lumped circuit abstraction is valid, indicating a lack of confidence in their understanding of the problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the lumped circuit abstraction can be applied in this scenario, as there are competing views and calculations presented without definitive resolution.

Contextual Notes

Participants express uncertainty about the mathematical steps involved and the assumptions underlying their calculations. There is also a discussion about the appropriate distances and dimensions relevant to the IC, which remain unresolved.

Kevin2341
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Homework Statement



One of the conditions that we must obtain for us to use the lumped circuit abstraction is that the timescale of interest in analysis of the circuit must be greater than the speed of electromagnetic propagation. What was discussed in class was how much greater. The answer is very much greater, on the order of a power of 10. Electromagnetic propagation in silicon is about half the speed of light. A priori, is there a problem using the lumped circuit abstraction on a computer chip which is 1 inch square running at 2ghz?

Homework Equations



...?
3.00x10^8/5=1.5x10^8
2ghz = 2,000,000,000hz or 2.00x10^9
1 sq inch = 6.4516 sq cm

The Attempt at a Solution


My only thoughts concerning this is that hertz is the unit of frequency (cycles per seconds, correct?). If the speed of light in silicon is 1.5 x 10^8 and it is within a timescale of 2 billion cycles per second, there shouldn't be any issue. I don't know if there is some kind of mathematical voodoo I need to invoke to have a satisfactory answer though. Any help?
 
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Kevin2341 said:

Homework Statement



One of the conditions that we must obtain for us to use the lumped circuit abstraction is that the timescale of interest in analysis of the circuit must be greater than the speed of electromagnetic propagation. What was discussed in class was how much greater. The answer is very much greater, on the order of a power of 10. Electromagnetic propagation in silicon is about half the speed of light. A priori, is there a problem using the lumped circuit abstraction on a computer chip which is 1 inch square running at 2ghz?

Homework Equations



...?
3.00x10^8/5=1.5x10^8
2ghz = 2,000,000,000hz or 2.00x10^9
1 sq inch = 6.4516 sq cm

The Attempt at a Solution


My only thoughts concerning this is that hertz is the unit of frequency (cycles per seconds, correct?). If the speed of light in silicon is 1.5 x 10^8 and it is within a timescale of 2 billion cycles per second, there shouldn't be any issue. I don't know if there is some kind of mathematical voodoo I need to invoke to have a satisfactory answer though. Any help?

In your Relevant Equations section, you really should be including units on all those quantities.

What is the wavelength of a 2GHz waveform in silicon?

How does this compare to the longest path in the IC size that you are given?

Is it an order of magnitude larger?
 
Why did you divide 3e8 m/s by 5 and not by 2?
 
I didn't intend to divide 3e8 m\s by 5, isn't 3e8 divided by 2 = 1.5e8? (300,000,00 \ 2 = 150,000,000)? Never mind.. I see what I did, I wrote " "\5, when I meant 2 .

as for the longest path in the IC size given, I'd think it would be 4inches (1+1+1+1, 1 per side). Convert 4 inches down to cm or whatever size would be appropriate for this application.

As for wavelength in silicon, I found 2ghz is equal to 15cm\2 (because of silicon halves the speed of light). 4 inches = 10.16cm, 15\2 = 7.5, so if I'm doing things correctly here, then the answer should be that there isn't a problem? Or am I missing something? (By the way, I did not expect to see this kind of problem in my circuits class so early.. I'm seriously reconsidering engineering right now because of the vast pool of knowledge I am expected to remember.)
 
Kevin2341 said:
I didn't intend to divide 3e8 m\s by 5, isn't 3e8 divided by 2 = 1.5e8? (300,000,00 \ 2 = 150,000,000)? Never mind.. I see what I did, I wrote " "\5, when I meant 2 .

as for the longest path in the IC size given, I'd think it would be 4inches (1+1+1+1, 1 per side). Convert 4 inches down to cm or whatever size would be appropriate for this application.

As for wavelength in silicon, I found 2ghz is equal to 15cm\2 (because of silicon halves the speed of light). 4 inches = 10.16cm, 15\2 = 7.5, so if I'm doing things correctly here, then the answer should be that there isn't a problem? Or am I missing something? (By the way, I did not expect to see this kind of problem in my circuits class so early.. I'm seriously reconsidering engineering right now because of the vast pool of knowledge I am expected to remember.)

The longest path would just be across the diagonal of the chip, not all around the edges. So what distance is that? What is the wavelength of a 2GHz signal propagating in silicon?
 

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