"Is there any abstract theory of electrical networks? Designing electrical networks is among the highly mathematical engineering disciplines, which uses a vast scope of techniques from Fourier analysis and complex function theory, to logic, combinatorics and topology. But, at least to me, with my minor knowledge of electrical engineering, it seems that at the end of the day, what is physically built out of these theories--I mean in a manufacturer laboratory-- is always a finite graphs with nodes labeled with ""simple functions"", in a way that the whole thing is again a function, with desired characteristics. But, from a mathematical point of view, it is customary to investigate such structured functions, in a categorical context and exploit the language and power of category theory, much

c0nman56 2022-10-23 Answered
Is there any abstract theory of electrical networks?
Designing electrical networks is among the highly mathematical engineering disciplines, which uses a vast scope of techniques from Fourier analysis and complex function theory, to logic, combinatorics and topology. But, at least to me, with my minor knowledge of electrical engineering, it seems that at the end of the day, what is physically built out of these theories--I mean in a manufacturer laboratory-- is always a finite graphs with nodes labeled with "simple functions", in a way that the whole thing is again a function, with desired characteristics. But, from a mathematical point of view, it is customary to investigate such structured functions, in a categorical context and exploit the language and power of category theory, much like what programmers do.
Here, I do not dare to further these vague ideas and pose my question:
What is the right and fruitful mathematical foundation for the theory of electrical networks? Is there any purely axiomatic approach to the subject, accessible to a mathematics enthusiast with minor background in electrical engineering.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

spornya1
Answered 2022-10-24 Author has 18 answers
The theory of electrical circuits consists of several sub-theories. Predominant mathematical disciplines that arise in the study of electrical circuits are linear algebra, differential equations, functional analysis (Fourier Transform, Laplace Transform) and graph theory. A circuit is a physical system which implements a mathematical function. Thus a circuit can be abstracted from its physics into its mathematical behavior and one can choose to study the latter.
Then the mathematical behavior is captured by what is known as a "system", which there are various ways to define mathematically.
Some authors begin by defining the "input space" and the "output space" and then a system is a particular kind of "morphism" between those two spaces. Another abstract approach is to define input and output spaces and then take a system to be a subset of the cartesian product of the input and output space, i.e. the set of all input-output signal pairs that can occur in the system. This is known as the behavioral approach. Another approach is the algebraic approach.
As an example, Rudolf Kalman, the giant of mathematical systems theory, about 50 years ago, wrote a paper saying that a linear system is actually a module over a principal ideal domain. This opened the road for algebraic systems theory and if you like categories, you will find many interesting things there. But if you want to make a start, on the textbook level, i recommend any good book on Signals and Systems (e.g. Oppenheim's) or on Control Theory e.g. William Brogan's "Modern Control Theory". Warning: the further you go on this road, the less relevant your study will be with actual circuit design and analysis practices used in industry.
This is because there is a huge distance to be covered from having a meaningful and useful theory to actually using this theory and modifying it locally in order to obtain something that actually works. Let me give you a simple example: there is only one system model of the BJT transistor (and a couple more equivalent) but there are hundreds of various types of BJT transistors, very different from each other. These differences are not captured in the system theory level, but they are crucial for implementation.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-01-31
The centers for Disease Control reported the percentage of people 18 years of age and older who smoke (CDC website, December 14, 2014). Suppose that a study designed to collect new data on smokers and Questions Navigation Menu preliminary estimate of the proportion who smoke of .26.
a) How large a sample should be taken to estimate the proportion of smokers in the population with a margin of error of .02?(to the nearest whole number) Use 95% confidence.
b) Assume that the study uses your sample size recommendation in part (a) and finds 520 smokers. What is the point estimate of the proportion of smokers in the population (to 4 decimals)?
c) What is the 95% confidence interval for the proportion of smokers in the population?(to 4 decimals)?
asked 2022-09-03
Optimization Problem : Dumpster
I am trying to help my friend. This is his problem related to constructing a dumpster, so it can minimize construction cost :
For this project we locate a trash dumpster in order to study its shape and construction. We then attempt to determine the dimensions of a container of similar design that minimize construction cost."
(Already located, measured, and described a dumpster found).
"While maintaining the general shape and method of construction, determine the dimensions such a container of the same volume should have in order to minimize the cost of construction. Use the following assumptions in your analysis:
The sides, back, and front are to be made from 12-gauge (0.1046 inch thick) steel sheets, which cost $0.70 per square foot (including any required cuts or bends).
The base is to be made from a 10-gauge (0.1345 inch thick) steel sheet, which costs $0.90 per square foot.
Lids cost approximately $50.00 each, regardless of dimensions.
Welding costs approximately $0.18 per square foot for material and labor combined.
Give justification of any further assumptions or simplifications made of the details of construction.
Describe how any of your assumptions or simplifications may affect the actual result.
If you were hired as a consultant on this investigation, what would your conclusion be? Would you recommend altering the design of the dumpster? If so, describe the savings that would result."
asked 2022-10-17
Finding a basis of an infinite-dimensional vector space?
the opposite day, my trainer become speakme limitless-dimensional vector spaces and headaches that arise when trying to find a foundation for the ones. He referred to that it is been demonstrated that a few (or all, do no longer pretty remember) infinite-dimensional vector areas have a foundation (the end result uses an Axiom of choice, if I recall efficiently), that is, an endless listing of linearly independent vectors, such that any detail within the area can be written as a finite linear aggregate of them. however, my instructor stated that honestly finding one is simply complicated, and i were given a experience that it changed into essentially not possible, which jogged my memory of Banach-Tarski paradox, where it is technically 'viable' to decompose the sector in a given paradoxical way, however this can not be truly exhibited. So my question is, is the basis state of affairs analogous to that, or is it surely viable to explicitly discover a basis for endless-dimensional vector areas?
asked 2022-09-01
Computing the fixed point for cos x
While studying about Compiler Design I came with the term 'fixed point'.I looked in wikipedia and got the definition of fixed point but couldn't get how fixed point is computed for cos x as said in fixed point.
It says that the fixed point for cos x=x using Intermediate Value Theorem.But I couldn't get how they computed the fixed point for cos x.Do anyone know how they computed this?
asked 2020-11-20
In 2014, the Centers for Disearse reported the percentage of people 18 years of age and older who smoke. Suppose that a study designed to collect new data on smokers and nonsmokers uses a preliminary estimate of the proportion who smoke of 0.30.
a) How large a sample should be taken to estimate the proportion of smokers in the population with a margin of error of 0.02? Use 95% confidence.(Round your answer up to the nearest integer.)
b)Assume that the study uses your sample size recommendation in part (a) and finds 470 smokers. What is the point estimate of the proportion of smokers in the population? (Round your answer to four decimal places.)
c) What is the 95% confidence interval for the proportion of smokers in the population? (Round your answer to four decimal places.)
asked 2022-09-06
Though there are several posts discussing the reference books for topology, for example best book for topology. But as far as I looked up to, all of them are for the purpose of learning topology or rather on introductory level.
I am wondering if there is a book or a set of books on topology like Rudin's analysis books, S.Lang's algebra, Halmos' measure (or to be more updated, Bogachev's measure theory), etc, serving as standard references.
Probably, such a book should hold characteristics such as being self-contained, covering the most of classical results, and other good properties you can name.
In the end, I only have the interest on general-topology (topological space, metrization, compactification...), and optionally differential topology (manifolds). So please don't divert into algebraic context.
asked 2022-09-09
Fourier and Laplace transforms together, is this possible?
Answering on some posts on MSE about Laplace transform and Fourier transform I stumbled upon a question to which I cannot answer myself (not having a good ground in pure mathematics).
The question is the following:
Is there some mathematical constraint that doesn't let us use both Fourier and Laplace transform on the same equation?
I'm not saying that it would be useful in any case, I was just wondering if it's feasible! Just as an example I could use both transforms to solve the one dimensional (or three, doesn't change much) wave equation with some external force
{ t 2 u ( x , t ) c 2 x 2 u ( x , t ) = f ( x , t ) u ( x , 0 ) = t u ( x , t ) | t = 0 = 0 < x < t > 0

New questions

i'm seeking out thoughts for a 15-hour mathematical enrichment course in a chinese language high faculty. What (pretty) simple concern would you advocate as a subject for any such course?
historical past/issues:
My students are generally pretty good at math, but many of them have no longer been uncovered to rigorous or summary mathematical reasoning. an amazing topic would be one that could not be impossibly hard for students who have by no means written or study proofs in English.
i have taught this magnificence three times earlier than. (a part of the purpose that i'm posting that is that i have used up all my thoughts!) the primary semester I taught an introductory range theory elegance (which meandered its way toward a proof of quadratic reciprocity, though I think this become in the end too advanced/abstract for some of the students). the second one semester I taught fundamental graph idea and packages (with a focal point on planarity and coloring). The 1/3 semester I taught a class at the Rubik's dice.
the students' math backgrounds are pretty numerous: a number of them take part in contest math competitions, and so are familiar with IMO-fashion techniques, however many aren't. a number of them may additionally realize some calculus, however I cannot assume it. all of them are superb at what in the united states is on occasion termed "pre-calculus": trigonometry, conic sections, systems of linear equations (though, shockingly, no matrices), and the like. They realize what a binomial coefficient is.
So, any ideas? preferably, i'd like to find some thing a bit "sexy" (like the Rubik's cube) -- tries to encourage wide variety theory through cryptography seemed to fall on deaf ears, however being capable of "see" institution idea on the cube became pretty popular.
(Responses specifically welcome from folks who grew up in the percent -- any mathematical subjects you desire were protected within the excessive college curriculum?)