The Heaviside Prize

Last weekend I twitted on the following exciting challenge:

The Heaviside Prize:…

$5000 for someone who will explain the physical reality (without using maths!) of the electric current when a digital step propagates in USB-like transmission line. Students, engineers, academics, tackle this challenge!!!

Static vs Dynamic when referring to the electric field in capacitor

I wrote in my paper “Energy current and computing” ( ):

“there is no such a thing as a static electric field in a capacitor. In other words, a capacitor is a form of TL in which a TEM wave moves with a single fixed velocity, which is the speed of light in the medium”.

This statement causes some controversy – Ivor Catt refers to it as “heresy”.

Here I would like to explain what is meant here by static/dynamic:

One of the important aspects of considering the distinction between ‘static’ and ‘dynamic’ is that of what we mean by dynamic/static in the first place.

I think that the notion of dynamic/static, first of all, concerns as to whether a particular value (say, electric field intensity E) changes in time or not, i.e. whether dE/dt is non-zero or not. Another notion of dynamic/static is about the movement of the value in space (and, necessarily in time because movement in space cannot be instantaneous!), so if we talk about the electric field E, we can be talking about dE/dx being non-zero, and here is the critical notion of the link between dE/dt and dE/dx, which MUST be mediated by dx/dt (speed of light in the medium!). The latter MUST BE ALREADY SET UP, ab initio, and that’s what Ivor Catt’s Heaviside signal is about. So, even if we have an impression that something is static – like electric field in a fully charged or fully discharged capacitor, this impression will be viewed in the form of contrapuntal dE/dt=0, we somehow need to retain the notion of c=dx/dt being constant and non-zero. But then the immediate question arises of: what is there that is moving in a longitudal direction at speed c? And the answer is the Heaviside signal! What else? So, my understanding is that THIS MOVING THING is what makes me state that that there is no such a thing as a static electric field in a capacitor!

“Contrapuntal superposition” of Heaviside signals unravelled as a lookalike state coding problem in asynchronous circuit design

This article by Ivor Catt – published (now more than) 40 years ago – proposed looking at transverse electromagnetic (TEM) wave by means of the so-called Heaviside signal. Heaviside signal is basically EM “energy current”, described by Poynting vector ExH (E and H are electric and magnetic field intensities, respectively), that travels and can only travel in space with a speed of light in the medium, fully determined by its fundamental parameters permittivity (epsilon) and permeability (mu) – i.e., c=1/sqrt(mu*epsilon). The key point here, I should again stress, is that ExH cannot stand still – it can only travel with speed of light. One might ask, where does it travel? It travels where the environment – i.e the combination of materials – leads it to, and in practice it predominantly goes where the effective impedance of the medium is smaller. The effective or characteristic impedance of the medium, Z0, is also fully determined by the permittivity (epsilon) and permeability (mu), i.e. Z0=sqrt(mu/epsilon). Moreover, Z0=E/H – this is sometimes called the constant of proportionality of the medium.

Why is this look at the TEM wave more advantageous than some other looks, such as for example, the so called “rolling wave” of the alternating concentrations of magnetic energy 1/2*mu*H^2 and electric energy 1/2*epsilon*E^2 in the direction of propagation? As Catt shows in the above article, this more conventional way is actually meta-physical, because it is based on the assumption of causality between the electric field and magnetic field and vice versa. The latter is a form of tautology because it creates a non-physical, but rather, mathematical or equation-based “feedback mechanism”, which does not make sense in physics.

Another important issue that calls for the use of Heaviside signal is that it retains the notion of the travelling EM “ExH slab” in each direction where it can travel, and hence its change-inducing geometric causality between points in space. As exemplified by the effects of travelling TEM waves in transmission lines (TLs), this look, for example, naturally separates the incident wave from the reflected (of the interface with another medium) wave, or from another wave that may travel in the opposite direction. As a result, the analysis of the behaviour of the TL becomes fuller and can explain the phenomena such as superposition of independent waves in cases such as cross-talk between TLs. Here is another paper by Ivor Catt – published more than 50 years ago – and subsequent clarifications – of the superposition of the even and odd modes (modes of TEM travelling with different speeds of light in the medium due to different epsilon and mu conditions arising between adjacent pairs of metal lines).

As shown in these papers, the view provided by the conventional theory is necessarily contrapuntal – it looks at the combined EM field in every point in space and in time. As a result it simply overlays the travelling ExH signals. And that’s what one can see by measuring voltage and current in points of interest on the TL. Or, equally, what one could see on the oscilloscope’s waveforms at points in space. Interestingly that looking at the same time at a number of points, in a spatially orderly way, leads to a conjecture that there is an interplay of several travelling TEM waves, but the conventional rolling wave approach would not explain the physics behind them properly!

What is remarkable in this for me is that this reminds me the difference between two types of models in asynchronous control circuits and how one of them obscures the information revealed by the other. One type of model that is based on recording purely binary encoded states of the circuit (akin to the contrapuntal notion). The other is based on a truly causal model (say Signal Transition Graph – or STG – called Signal Graph or Signal Petri Net in my early publications: or, where we have the explicit control flow of signal transitions or events running in the circuit. The difference between these two looks is often manifested in the so-called Complete State Coding problem (cf ). If we only look at the contrapuntal notion of the state without knowing the pre-history of the event order we cannot distinguish the semantically different states that map onto the same binary code provided by the signals. To distinguish between such states one needs additional information or memory that should be either provided in the underlying event-based model (the marking of the STG) or by introducing additional (aka internal or invisible) signals (in the process of solving the CSC problem).

I am not claiming that the above-noted analogy leads to a fundamental phenomenon, but it reflects the important epistemic aspect of modelling physical world so that important relationships and knowledge are retained, yet in a minimalist (cf. Occam’s razor) way. Some more investigation into this analogy is needed.

Correction on my previous blog and some interesting implications …

Andrey Mokhov spotted that to satisfy the actual inverse Pythagorean we need to have alpha=1/2 rather than 2. That’s right. Indeed, what happens is that if we have alpha = 1/2 we would have (1/a)^2=(1/a1)^2+(1/a2)^2. This is what the inverse Pythagorean requires. In that case, for instance if a1=a2=2, then a must be sqrt(2). So the ratio between the individual decay a1=a2 and the collective decay is sqrt(2). For our stack decay under alpha = 2, we would have for a1=a2=2, a=1/2, so the ratio between individual decay and collective decay is 4.

It’s actually quite interesting to look at these relations a bit deeper, and see how the “Pythagorean” (geometric) relationship evolves as we change alpha from something like alpha<=1/2 to alpha>=2.

If we take alpha going to 2 and above, we have the effect of much slower collective decay than 4x compared to the individual decay. Physically this corresponds to the situation when the delay of an inverter in the ring becomes strongly inversely proportional to voltage. Geometrically, this is like contracting the height of the triangle in which sides go further apart than 90 degrees – say the triangle is isosceles for simplicity, and say its angle is say 100 degrees.

The case of alpha = 1/2 corresponds to the case where delay is proportional to the square root of Voltage, and here the stack makes the decay rate to follow the inverse Pythagorean! So this is the case of a triangle with sides being at 90 degrees.

But if alpha goes below 1/2, we have the  effect of the collective decay being closer to individual decays, and geometrically the height of the triangle where sides close up to less than 90 degrees!

Incidentally, Andrey Mokhov suggested we may consider a different physical interpretation for inverse Pythagorean. Instead of looking at lengths a, b and h, one can consider volumes Va, Vb and Vh of 4-D cubes with such side lengths. Then these volumes would relate exactly as in our case of alpha=2, i.e. 1/sqrt (Vh)=1/sqrt(Va)+1/sqrt(Vb).


Charge decay in a stack of two digital circuits follows inverse Pythagorean Law!

My last blog about my talk at HDT 2019 on Stacking Asynchronous Circuits contained a link to my slides. I recommend you having a particular look at slide #21. It talks about an interesting fact that a series (stack) discharge rate follows the law of the inverse Pythagorean!

It looks like mother nature caters for a geometric law of the most economic common between two individual sides.

My Talk on Stacked Asynchronous Circuits at HDT 2019

I just attended a Second Workshop on Hardware Design Theory, held in Budapest, collocated with 33rd International Symposium on Distributed Computing

The HDT’19 workshop was organised by Moti Medina and Andrey Mokhov. It had a number of invited talks, and here is the programme:

I gave a talk on Stacked Asynchronous circuits.

Here is the abstract: In this talk we will look at digital circuits from the viewpoint of electrical circuit theory, i.e. as loads to power sources. Such circuits, especially when they are asynchronous can be seen as voltage controlled oscillators. Their switching behaviour, including their operating frequency is modulated by the supply voltage. Interestingly, in the reverse direction if they are driven by external event sources, their switching frequency determines their inherent impedance which itself makes them ideal potentiometers or voltage dividers. Such circuits can be stacked like non-linear resistors in series and parallel, and lend themselves to interesting theoretical and practical results, such as RC circuits with hyperbolic capacitor discharges and designs of dynamic frequency mirrors.

Here is the PDF of my slides:

New book on Carl Adam Petri and my chapter “Living Lattices” in it

A very nice new book “Carl Adam Petri: Ideas, Personality, Impact“, edited by Wolfgang Reisig and Grzegorz Rozenberg, has just been published by Springer:

Newcastle professors, Brian Randell, Maciej Koutny and myself contributed articles for it.

An important aspect of those and other authors’ articles is that they mostly talk about WHY certain models and methods related to Petri nets have been investigated rather than describing the formalisms themselves. Basically, some 30-40 years of history are laid out on 4-5 pages of text and pictures.

My paper  “Living Lattices” provides a personal view of how Petri’s research inspired my own research, including comments on related topics such as lattices, Muller diagrams, complexity, concurrency, and persistence.

The chapter can be downloaded from here:

There is also an interesting chapter by Jordi Cortadella “From Nets to Circuits and from Circuits to Nets”, which reviews the impact of Petri nets in one of the domains in which they have played a predominant role: asynchronous circuits. Jordi also discusses challenges and topics of interest for the future. This chapter can be downloaded from here:


Ultra-ultra-wide-band Electro-Magnetic computing

I envisage a ‘mothball computer’ – a capsule with the case whose outer surface harvests power from the environment and inside the capsule we have the computational electronics.

High-speed clocking can be provided by EM of highest possible frequency – e.g. by visible light, X-rays or ultimately by gamma rays!

Power supply for modulation electronics can be generated by solar cells – Perovskite cells. Because Perovskite cell have lead in them they can insulate gamma rays from propagation outside the compute capsule.

Information will be in the form of time-modulated super-HF signals.

We will represent information in terms of time-averaged pulse bursts.

We will have a ‘continuum’ range of temporal compute which will operate in the range between deterministic one-shot pulse burst (discrete) through deterministic multi-pulse analog averaged signal to stochastic multi-pulse averaged signal (cf. book by Mars & Poppelbaum –

Temporal Computing ( is the right kind of business opportunity for this Odyssey!

Switched electrical circuits as computing systems

We can define computations as processes of working of electrical circuits which are associated with sequences of (meaningful) events. Let’s take these events as discrete, i.e. something that can be enumerated with integer indices.

We can then map sequences of events onto integer numbers, or indices. Events can be associated with the facts of the system reaching certain states. Or, in a more distributed view, individual variables of the system, reaching certain states or levels. Another view is that a component in the system’s model moving from one state to another.

To mark such events and enable them we need sensory or actuating properties in the system. Why not simply consider an element called “switch”:


What we want to achieve is to be able to express the evolution of physical variables as functions of event indices.

Examples of such computing processes are:

  • Discharging capacitance
  • Charging a (capacitive) transmission line
  • Switched cap converter
  • VCO based on inverter ring, modelled by switched parasitic caps.

The goal of modelling is to find a way of solving the behaviour of computational electrical circuits using “switching calculus” (similar to Heaviside’s “operational calculus” used to solev differential equations in an efficient way).

Some of Leonid Rosenblum’s works

L. Ya. Rosenblum and A.V. Yakovlev.
Signal graphs: from self-timed to timed ones,
Proc. of the Int. Workshop on Timed Petri Nets,
Torino, Italy, July 1985, IEEE Computer Society Press, NY, 1985, pp. 199-207.

A paper establishing interesting relationship between the interleaving and true causality semantics
using algebraic lattices. It also identifies an connection between the classes of lattices and the property
of generalisability of concurrency relations (from arity N to arity N+1),
i.e. the conditions for answering the question such as,
if three actions A, B and C are all pairwise concurrent, i.e. ||(A,B), ||(A,C), and ||(B,C), are they concurrent “in three”, i.e. ||(A,B,C)?
L. Rosenblum, A. Yakovlev, and V. Yakovlev.
A look at concurrency semantics through “lattice glasses”.
In Bulletin of the EATCS (European Association for Theoretical Computer Science), volume 37, pages 175-180, 1989.

Paper about the so called symbolic STGs, in which signals can have multiple values (which is often convenient for specifications of control at a more abstract level than dealing with binary signals) and hence in order to implement them in logic gates one needs to solve the problem of binary expansion or encoding, as well as resolve all the state coding issues on the way of synthesis of circuit implementation.

Paper about analysing concurrency semantics using relation-based approach. Similar techniques are now being developed in the domain of business process modelling and work-flow analysis: L.Ya. Rosenblum and A.V. Yakovlev. Analysing semantics of concurrent hardware specifications. Proc. Int. Conf. on Parallel Processing (ICPP89), Pennstate University Press, University Park, PA, July 1989, pp. 211-218, Vol.3

Моделирование параллельных процессов. Сети Петри [Текст] : курс для системных архитекторов, программистов, системных аналитиков, проектировщиков сложных систем управления / Мараховский В. Б., Розенблюм Л. Я., Яковлев А. В. – Санкт-Петербург : Профессиональная литература, 2014. – 398 с. : ил., табл.; 24 см. – (Серия “Избранное Computer Science”).; ISBN 978-5-9905552-0-4
(Серия “Избранное Computer Science”)…/Simulation-of-Concurrent-Processes-Petri-Nets.pdf