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What exactly are electrical circuits as mathematical objects?

It seems quite intuitive to me, that they are geometric realization of some graph with some additional structure.

Another thing I notice is that application of KVL and KCL together turns this graph to a system of linear equations.

$$\text{circuit} \to^{KVL}_{KCL} \to \text{linear eqtns}$$

All sites I look online eg: Wikipedia give me undetailed answers.


Edit: I see that many people are like there is no singular view.... sure that's understandable... but sometimes it is very clear that some matters are much more clearer and understandable in certain views than others. For example, consider how differential forms give clarity on the whole bunch of vector calculus theorems, and whose mathematical properties quite directly suggest effects like that of Aharonov-Bohm, which would have been otherwise difficult to arrive at mathematically with standard theories of vector calculus.

Also, for the camp of people there who say it's just differential equations and see the right side of the above equation. I think they've not seen circuits enough. There are very many cases where the circuit itself offers quick ways to arrive at simplified equations.

Consider for example things like the Source Transformation Theorem's or simplifying complicated circuits using symmetry arguements

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    $\begingroup$ A newly emergent view is that electrical circuits are string diagrams in certain hypercategories (see for example ocw.mit.edu/courses/… for an introduction). $\endgroup$ Commented Aug 10, 2023 at 18:06
  • $\begingroup$ Note that they are only linear if the branch currents are linear functions of the node potentials. In the real world, it is often difficult to choose components and topologies so that this is an adequate approximation for linear models to usefully apply. $\endgroup$
    – John Doty
    Commented Aug 10, 2023 at 21:51
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    $\begingroup$ Follow up question: what exactly are humans as mathematical objects? $\endgroup$
    – JimmyJames
    Commented Aug 11, 2023 at 17:25
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    $\begingroup$ I think an issue with this question, as you can see from the variety of answers, is that circuits are not a mathematical object. Mathematical physics is all about modeling physical situations with a mathematical formalism, and in practice no model or formalism is a perfect representation of an actual physical situation. Therefore you end up with different formalisms with different goals that apply in different situations, as opposed to a single "best" answer. $\endgroup$
    – Andrew
    Commented Aug 11, 2023 at 19:41
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    $\begingroup$ Even with your edit, I think "circuits" is too broad a topic to get a single answer. Do you mean circuits with only linear elements behaving in an ideal way? Do you want to consider what happens when you push linear components outside their nominal operating range? Temperature effects? Non-linear components like transistors or diodes or op-amps? Are you happy with DC or equilibrium AC or do you want to look at what happens at high frequencies? Transmission lines? $\endgroup$
    – Andrew
    Commented Aug 12, 2023 at 14:40

9 Answers 9

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The current-voltage relation between the three common circuit elements (resistor, capacitor, inductor) are, $$V=IR,\quad I=C\frac{\mathrm{d}V}{\mathrm{d}t},\quad V=L\frac{\mathrm{d}I}{\mathrm{d}t}\tag{1}$$ which suggests that mathematically, circuits are differential equations. The KVL and KCL rules you indicate then provide you with addition rules for the series and parallel circuits to generate the differential equations needed to be solved to determine the voltage or current as a function of time.

As an example, if you combine a resistor and an inductor in series, you obtain, $$RI+L\frac{\mathrm{d}I}{\mathrm{d}t}=V\to I(t)=\frac{V}{R}\left(1-\mathrm{e}^{-Rt/L}\right).$$ Replacing the inductor or resistor with the capacitor should be straight-forward.

When you start adding more complex circuit elements, I think some of the simple linear ODE features of circuits vanish, but it's been a while since I've had a circuits course.

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    $\begingroup$ I/V/R in what point in the circuit? The functions can be different at different places so how is the locality determined? $\endgroup$ Commented Aug 10, 2023 at 18:50
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    $\begingroup$ @Ken-EnoughaboutMonica I'm not sure I follow. For the simple series circuit I describe, $\sum V_k=0$ is for the whole circuit, not a particular point. No? $\endgroup$
    – Kyle Kanos
    Commented Aug 10, 2023 at 19:39
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    $\begingroup$ I have 3 degrees in EE, and I find this much better than the probabilistic interpretation of a resistor network. $\endgroup$ Commented Aug 10, 2023 at 20:46
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    $\begingroup$ @JosephDoggie You might (as an electrical engineer). However, the interest of the probabilistic interpretation is not so much for electric networks, but for Markov chains. It allows to import both intuition and various techniques developed for the former in order to better analyze the latter. So, if you're a probabilist (or a statistical physicist), the probabilistic interpretation turns out to be very useful. $\endgroup$ Commented Aug 11, 2023 at 7:25
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    $\begingroup$ @Nabla You are completely misunderstanding my claims. Everything is explained in detail in my links. Please read instead of writing many wrong and irrelevant comments… $\endgroup$ Commented Aug 11, 2023 at 21:42
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If you only allow resistors and voltage sources, then you get a resistor network. The latter are equivalent to reversible Markov chains and have thus been studied in detail by probabilists. In particular, current and voltage acquire natural probabilistic interpretations. A highly pedagogical, elementary introduction can be found in the book Random Walks and Electric Networks by Doyle and Snell. You can find a lot of additional information in many books on Markov chains, for instance:

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    $\begingroup$ "this book", "here", "here", and "here" won't be very useful when those links go away, especially as the links themselves don't provide any useful information about title or author. $\endgroup$ Commented Aug 10, 2023 at 18:12
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    $\begingroup$ @RayButterworth You're right, that was lazy. I have updated my answer. $\endgroup$ Commented Aug 10, 2023 at 19:50
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    $\begingroup$ @And where exactly have I said that it was about EE? My point was, and still is, that resistor networks can be reinterpreted in terms of Markov chains, which is one of the possible answers to the OP. That's all. If you're not interested in it, so be it, but the answer is still perfectly valid in my opinion. $\endgroup$ Commented Aug 12, 2023 at 7:53
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    $\begingroup$ @Nabla The tags in the original version of the OP included Mathematical-Physics. This includes the theoretical aspects of equilibrium statistical physics, where reversible Markov chains and this particular relation between the latter and resistor networks are important. In fact, there are many papers in Communications in Mathematical Physics on this topic. $\endgroup$ Commented Aug 12, 2023 at 7:57
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    $\begingroup$ Still you are Not getting what I say but I really have No time to discuss it further. My only advice to you would be, try telling the same theory here: electronics.stackexchange.com :) Have a nice day and thanks for deep discussion! $\endgroup$
    – Nabla
    Commented Aug 12, 2023 at 14:19
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A better Wikipedia page to address the graph theoretic concepts might be https://en.wikipedia.org/wiki/Circuit_topology_(electrical)

What exactly are electrical circuits as mathematical objects?

An "exact" answer likely depends on how deep into the subject you wish to look.


You might find these resources useful:

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    $\begingroup$ Bamberg and Sternberg is indeed a very nice reference. They use the intuitive setting of electric circuits to motivate algebraic topology; it’s all a nice interplay of physical ideas, mathematical ideas, and the bridge between the two, that too at a rather elementary level. $\endgroup$
    – peek-a-boo
    Commented Aug 10, 2023 at 21:31
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There is no single answer to this question, but of course make note: circuits aren't anything mathematical, they are abstractions, and thus the appropriate mathematical representation is one which represents the behavior of the desired system. This is just a small remark, in response to anyone tempted into saying "circuits are differential equations" or "circuits are signal flow graphs."

A very good treatment which combines practical circuit analysis with rigorous mathematical foundations is Chua, Kuh, and Desoer's Linear and Nonlinear Circuits (excerpts online). This book is notable for explicitly handling time-dependent, nonlinear, and distributed behavior (finite propagation delay), and uses digraphs and linear algebra to represent linearized circuits. You will find an excellent practical treatment of the graph-theoretic approach (the graph model being perhaps the most intuitively satisfying abstract model of a circuit) in the first chapter of Linear and Nonlinear Circuits.

To answer the question, then, in one sense we can represent a network as a collection of nodes $\mathcal N$ and a collection of directed, weighted edges $\mathcal E$ between those nodes, where the direction defines the reference direction for the current between the nodes (where current is taken as a fundamental concept which obeys KVL and KCL, the axioms of this theory). Further, in a connected network (i.e. one where every node is reachable from any other via edges) we define a potential $e$ at each node relative to an arbitrary reference node $n$ such that $e_n = 0$ and $e_a = 0$ iff $a$ and $n$ are the same node; for all lumped connected circuits for all times $t$ and any pairs of nodes $a,b$, for all choices of reference node $n$, the difference in potential is always well-defined and given by $$ v_{ab} = e_a - e_b \qquad\qquad\text{(KVL)} $$ The second axiom of this theory is the node law, which states that for all lumped circuits, for all times $t$, the algebraic sum of currents leaving any node is equal to zero (where leaving is defined in relation to the direction of each edge connected to the node).

The definitions and axioms given above are clearly not enough to uniquely determine the voltage and currents of a given network, for which we must specify the role of the weights of the edges as regards the voltages and currents in the network. A familiar example is a linear two-terminal resistor which applies the constraint $v_{ab} = r i$ where $i$ is the current through an edge of weight $r$ connected between nodes $a$ and $b$. This weight is independent of time and is independent of the voltages and currents in any other part of the circuit. Other components impose different constraints, e.g. sources act as negative resistances, inductors and capacitors impose time-dependent constraints (via differential equations), nonlinear components impose nonlinear constraints, etc.

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  • $\begingroup$ In "they are abstractions" in your first paragraph, I assume that "they" refers to "anything mathematical". The first time I read it, it looked as if it referred to "circuits". $\endgroup$ Commented Aug 11, 2023 at 17:45
  • $\begingroup$ Maybe it's a bit too philosophical, but I meant that our concept of "circuits" is abstract, there's no such thing as a circuit, it's just the concept and the way we model it for our particular purposes. Therefore, there's no use in saying "mathematically, a circuit is X" and instead one might say "mathematically, a circuit can be represented by X". The former implies some sort of "correctness" of the representation while the latter openly admits to being an interpretation. But anyway, it's not really too important a point. $\endgroup$ Commented Aug 11, 2023 at 19:20
  • $\begingroup$ Beyond the issue of nonlinear components, at high frequencies, particularly multiple GHz, it becomes difficult to design circuits as networks of discrete, independent components with minimal "parasitic" characteristics (the lumped-element model), and possible to design circuits that use the electromagnetic properties of the substrate and conductive traces. It can be non-obvious that these distributed circuits actually perform a function (they can appear to be just decorative, disconnected fragments of traces), and impractical to analyze them as a network of components. $\endgroup$ Commented Aug 12, 2023 at 15:12
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Perhaps the use of electrical impedance may be managed in a graph. Specifically, a weighted graph containing complex-valued weights, where each edge is an electrical component and each node is a split in the circuit. In the model, you could specify the rules for merging nodes and edges as they exist normally for electrical impedance (adding for two in a row, inverse of sum of inverses for two in parallel).

I'm not sure if this sort of graph-with-rules has a name, but a weighted graph is certainly a thing.

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Google 'signal-flow graph'

"A signal-flow graph is a diagram that represents a set of simultaneous algebraic equations. When applying the signal flow graph method to analysis of control systems, we must first transform linear differential equations into algebraic equations in [the Laplace transform variable] s."

They are most commonly use in control system analysis but can be used for circuits in general.

What exactly are eletrical circuits as mathematical objects?

So circuits can be a graphical representation of differential equations.

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The resistors, capacitors, and inductors, are associated with vector quantities that capture their respective electrical properties. These vectors give relationships between voltage, current, and charge, serving as elemental building blocks for the circuit's mathematical description.

Volts and amps manifest as coefficients within the Kirchhoff equations. The electrons, encapsulating the physical flow of charge, contribute to the matrix of coefficients that defines the system of linear equations. The currents, guided by Kirchhoff's laws, determine the relationships between voltages and currents.

Solving this system of equations furnishes a comprehensive quantitative depiction of voltage distributions and current magnitudes across the circuit's intricate topography.

Other fields to represent circuit maths are:

Complex Impedances: In alternating current (AC) analysis, components are characterized using complex impedances. Circuits are analyzed using phasors, complex numbers that incorporate both magnitude and phase information, simplifying AC analysis.

Matrix Formulation: Circuits can be represented using matrices, where nodes and branches are defined as rows and columns. The matrix equations encapsulate the relationships between voltages and currents, allowing for systematic analysis using matrix manipulation techniques.

Graph Theory: Graph theory provides a structural representation of circuits, with nodes and edges corresponding to circuit elements. This graph-based approach aids in understanding connectivity, paths, and loops within the circuit, facilitating analysis and optimization.

State-Space Modeling: State-space representation captures the dynamic behavior of circuits using differential equations. Circuits are described in terms of state variables, inputs, outputs, and matrices representing component relationships. This method is particularly suited for systems with energy storage elements.

Mesh and Nodal Analysis: These techniques involve formulating equations based on loop currents (mesh analysis) or node voltages (nodal analysis). These methods provide a systematic approach to solving circuits, particularly when the number of components is significant.

Transfer Function Representation: Common in control systems analysis, this representation uses Laplace transforms to analyze circuits in the frequency domain. Transfer functions relate input and output signals, aiding in stability and response analysis.

Network Analysis Software: Advanced software tools can automatically generate and solve circuit equations, providing rapid analysis and design capabilities. https://electronicsguruji.com/best-circuit-simulation-software/

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    $\begingroup$ This doesn't answer what the circuit is, just methods in solving it $\endgroup$ Commented Aug 11, 2023 at 9:37
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    $\begingroup$ Solving circuit goes along way, especially since mathematics is mentioned in the title, after all, mathematics is an abstraction, but very useful and accurate for many situations. Indeed, knowing which 'regime' it applies to is very much part of EE. For example, a real voltage source can often be modeled by an idela voltage source with a resistor in series; often the real situation can be much more complex (yes, a pun! see Euler's theorem)! $\endgroup$ Commented Aug 14, 2023 at 14:26
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Also, for the camp of people there who say it's just differential equations and see the right side of the above equation. I think they've not seen circuits enough. There are very many cases where the circuit itself offers quick ways to arrive at simplified equations.

The section I bolded out of your question (actually, and edit to your question) is probably the most useful for understanding the variety of answers you are receiving. You ask "What exactly are electrical circuits as mathematical objects?" On its own, this is a easy question to answer in the negative. Electrical circuits are not mathematical objects. They're physical objects. But most likely what you're really asking is what mathematical objects are used to model electrical circuits. And when we come to modeling, it's all about intended use. A model cannot be applied meaningfully to a real system unless one considers the intent behind using the model.

Some would like a model which truly captures the behavior of any electronic circuit. For those people, Maxwell's Equations are the definitive model of electronic circuits. And I use "definitive" properly here -- we define electronic circuits to be a subset of the things governed by Maxwell's Equations. So in that sense, differential equations over a field is the best mathematical object to model a circuit.

But "there are very much cases where the circuit in itself gives quick ways to arrive at simplified equations." This is absolutely true. In many cases, we are comfortable assuming some details don't matter. In the hierarchy of mathematical objects used to model electronic systems, the "distributed element model" is the next most common model. In this model, we assume that many of the interactions between parts of the field do not matter. In particular, we assume that electrical fields closely follow conductors, and magnetic fields do not "leak." For instance, if we have two inductors side by side, Maxwell's equations says that they will magnetically couple, affecting each other. But, in practice, we can often design circuits where this effect is negligible.

This is the first model where Kirchoff's laws apply, because we can speak to "wires" as individual discrete entities, rather than regions with a given permittivity and permeability. This is the first point where a "graph" is a meaningful model. However, it is an infinite graph. When modeling systems as distributed elements, we come across interesting concepts like transmission lines, which are modeled as an infinite series of infinitesimal inductors, resistors and capacitors, and we apply calculus to it.

The next most common big leap in modeling assumptions is to assume that some properties are evenly distributed across regions. For example, we can assume that the capacitance of all points in a capacitor can be treated as "the same." This is the lumped element model., another very well accepted way of modeling electronic circuits. Once we accept these assumptions we can develop the finite graphs that you are most likely familiar with, which are solved directly via KVL and KCL (rather than having to apply those laws over infinite sums using calculus).

Beyond this, many circuits admit even more useful models. Consider linear circuits. Not all circuits are linear -- circuits with transistors in particular are famously non-linear. But those using just resistors, capacitors, and inductors are linear. For these, the most useful models are often the Laplace transformation of the circuit's behavior. The Laplace transform has the wonderful property of converting all linear systems (including linear differential equations) into algebraic objects. If you work in control theory, these are the bees' knees because it is so much easier to work with algebra than differential equations!

And some non-linear circuits admit convenient mathematical models too. By far the most famous is that of saturated transistors, which form the basis of every computer you've ever seen. These are extremely well modeled by boolean logic. I find this interesting because it shows that this series of models isn't even a simple progression from complex to simple. There are many branches of intended uses for electronic circuits, each of which has their own mathematical objects which are most convenient for modeling.

And sometimes its not even the circuit that matters, but your intended use for the circuit. Amplifiers involve decidedly non-linear circuits (large transistors), but we drive them in such a way that we find a "linearized" model is applicable. This is the so called small-signal model, which does not fully characterize the circuit's behavior, but it does characterize the part of the behavior we care about (the amplification of an input signal).

So every one of these models is applied to electronic circuits on a regular basis. Which model one uses depends on the intended use of the model, and the assumptions one is willing to bake in. One will find this is true for all modeling, not just that of circuits. So pick your mathematical objects wisely. Consider the intended use of the model, and go from there.

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    $\begingroup$ An important addition here: Maxwell's equations are, by themselves, insufficient as a model. You need to couple them to some sort of model for the behavior of the electric "fluid" involved (i.e. the charge density, $\rho_q$, whose flow becomes the current density, $\mathbf{J}$.). This is needed as soon as you want to try and derive, say, Ohm's law, or even Kirchhoff's junction law (which is in effect a statement that the electric fluid is incompressible as well as conserved, since a compressible electric fluid could "pile up" in, or "dissipate from", a point). The rest is on point, though. $\endgroup$ Commented Aug 12, 2023 at 20:40
  • $\begingroup$ @The_Sympathizer Indeed. Maxwell's equations tell you very little about what goes on in a transistor. A BSIM3 model of a transistor has ~150 parameters, and it's far from mathematically exact. And then, a circuit may contain billions of transistors. So, how do we deal with this mathematically? Well, we manage by choosing the modeling approaches that approximately answer the questions we have. No "universal" model can do this. $\endgroup$
    – John Doty
    Commented Aug 15, 2023 at 15:12
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Electrical circuits are not mathematical objects, as Sam Gallagher points out. They are instead real physical processes - with all the imperfections that this implies. We can however use mathematics to generally model the behavior of such things.

Electrical circuits are an example of an energy dissipation system involving 3 modifier elements, i.e. resistors, capacitors and inductors, whose steady state (as opposed to transient) behavior is generally modellable via a second order ordinary differential equation.

In the case of a circuit with the 3 modifiers connected in series:

$ $

$$ E_{app}(t) = L \frac{d^2q(t)}{dt^2} + R \frac{dq(t)}{dt} + \frac{q(t)}{C}$$

where:

$E_{app}(t)$ is the applied e.m.f.;

$L$ is the inductance of the circuit;

$R$ is its resistance; and

$C$ is the capacitance.

$ $

Putting it the other way round, second order ordinary differential equations can describe the steady-state behavior of energy dissipation systems in the case of electrical energy.

Analogous behavior and modelling may be done in the case of hydraulic and mechanical energy dissipation systems.

Obviously the above 3 modifiers in series circuit is an unrealistically simple situation. In real circuits you may well have other RCL subcircuits across or in series with each of the modifiers shown, further subcircuits off each of the subcircuit modifiers and so on and on.

Alternate means of analysis to differential equations may be used, e.g. those employing phasor diagrams to add the separate but out of phase sub-voltage/sub-current components.

I think I am getting at what your original question really is. You want to know if a system such as a complex electric power dissapation system (or indeed its mechanical, hydraulic, structural, tranportational, etc analogues) can be abstracted so as to be viewed as a mathematical object in itself and its behavior explored so as to get a clear understanding of its strengths and limits ? I should think so. Engineers of all types are great believers in using existing knowledge - the do-it-once or keep-it-simple-stupid principle: it saves repetitive work.

Read into here.

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