The Energy Model of the VENDOR.Max Operating Regime
The energy of VENDOR.Max is counted with two balances on two boundaries: the operating core and the complete device. Inside a working regime there is no single quantity that may be called the power of an installation: each process is measured by its own variable, on its own boundary, in its own temporal class.
VENDOR.Max is a nonlinear electrodynamic installation that operates in a controlled discharge-resonance regime. Excitation of the coupled resonant regime is classified, in the functional taxonomy of the project, as discharge-resonance excitation of the Armstrong type; that attribution describes the excitation stage only and is not a classification of the installation as a whole. This page does not reproduce the physical architecture of the installation. It presents the energy model of VENDOR.Max — two balances on two boundaries — and the rules by which the terms of those balances are defined and combined.
This page answers two questions: on which boundaries the energy of the installation is balanced and what each balance contains; and which measurable quantity corresponds to which physical process inside an already established operating regime, and which quantities may not be joined to one another.
It does not answer where the energy comes from, how resonant circulation works, by what mechanism a field transfers energy to the carriers of another ensemble, how local conversions compose into a self-consistent operating point, what the measured balance of the complete device boundary resolves into, or by what instruments active power is physically measured. Each of those questions has its own page.
There are no numerical values here — measured, calculated or illustrative. The subject is rules, not results.
Where a rule has a counterpart in the published literature, the page names the work, states what it established and what kind of evidence it is. None of the cited works examined VENDOR.Max. They are cited to show that the counting rules are not particular to this installation.
The energy model: two balances on two boundaries
The energy model of VENDOR.Max is a pair of balances. Each relates the energy flows crossing one declared boundary to the change of the energy stored inside it. The rules that follow define how every term is obtained.
The engineering control boundary of the operating core, crossed by four electrical ports.
The complete device boundary used for product testing and energy-source attribution.
Frame 1: the balance of the operating core
- \(P_{\mathrm{exc},1}\)
- Input of the excitation path.
- \(P_{\mathrm{bus}\to 13}\)
- Input of the block 13 power stage from the main DC bus.
- \(P_{12\to\mathrm{bus}}\)
- Output of block 12 to the main DC bus.
- \(P_{\mathrm{DC,out},1}\)
- Regulated DC output of the core.
- \(P_{\mathrm{heat},1}\), \(E_1\)
- Heat that has left the boundary of the core, and every store inside that boundary, the thermal one included. Heat is counted once.
The input of the core is sustained by an internal feedback path routed from its output through the main DC bus, which lies outside Frame 1 and inside Frame 0. The ratio of the output of the core to its excitation input is therefore not an efficiency, and a single end-to-end efficiency is not defined: the blocks are not two-ports in series. Every port term is measured; a term defined as the remainder of the others turns the balance into a tautology.
Frame 0: the balance of the complete device
The sums run over every channel that crosses the enclosure, and \(dE_0/dt\) includes the change of every internal store. When the boundary is enlarged from Frame 1 to Frame 0, the internal flows of the ring cancel: a flow that leaves the core and returns to it through the bus appears on neither side of (M2).
One term of the input sum is fixed by design: \(P^{\text{design power port}}_{\mathrm{in},0} = 0\). After start-up, the design provides no continuing normal power-input port. This is a statement about topology; it does not state that \(\sum_k P^{\mathrm{in}}_{k,0} = 0\).
The operating point
The operating point is the state in which average internal stores do not drift while internal flows match current extraction and own losses. Zero drift of the stores is not rest: it is accompanied by intense internal turnover of energy and a continuous flow to the customer.
The residual term
The source attribution residual is the portion of a boundary balance not yet assigned to an identified physical channel. On Frame 0:
Its magnitude and sign characterise the remainder of the balance; its physical nature is established by measurement. The residual has three statuses, and they are not mixed.
- \(P_X^{\mathrm{model}}\)
- The calculated residual of the engineering model at the chosen parameters. Not an observation.
- \(P_X^{\mathrm{meas}}\)
- The residual of a real multichannel measurement on one boundary over one window. A measurement.
- \(P_X^{\mathrm{validated}}\)
- A reproducible residual after independent metrology and an uncertainty budget. A validation result.
At present only \(P_X^{\mathrm{model}}\) exists.
- \(P_X\) is not a source and not a new form of energy.
- \(P_X\) answers the question of how much is not yet attributed, not the question of where it comes from.
- After physical attribution the contribution is counted under its own name, and \(P_X\) disappears.
- \(P_{X,1} \neq P_{X,0}\): a residual on Frame 1 may be explained by a flow that is internal to Frame 0.
The architecture and the mechanism that forms the operating regime are established by engineering. Energy attribution on the complete boundary is a separate measurement task.
The model is applied to a physical installation. Its long run — the connected load, the duration and the internal quantities at key points — is a company-recorded operating observation, documented separately. Observation and model are separate levels, and neither is proved through the other. This page defines how such quantities may enter a balance.
Three validity checks before any arithmetic
Correct energy accounting begins not with a formula but with three validity checks. Regime arithmetic starts after the third one. Before it, any product is a number without physical content.
Which variable is natural to this physical channel — and therefore which operation is defined on it at all.
Whether the chosen boundary forms an electrical port — and therefore whether the product of the chosen voltage and current is lawful.
Whether all quantities of the calculation share one temporal class — and therefore whether the resulting numbers are comparable at all.
Five classes of quantity
The most frequent error in reading systems of this kind is the attempt to describe five different physical processes with one formula. The separation below makes that attempt impossible.
A finite single conversion over a short time.
Repeated exchange of energy between forms inside a loop.
Work of a coupled field on the carriers of another ensemble.
Change of the energy held by a control volume.
Directed transfer through a formed customer port.
Large circulation is not large transfer. Large transfer is not large output. A fast event is not a large energy.
Keeping storage, dissipation and exchange apart is how energy-based circuit theory is built. In the port-Hamiltonian formulation of nonlinear circuits, energy storage, resistive dissipation and the interconnection structure are separate mathematical objects, and exchange with the surroundings is written only through the declared external ports (Gernandt et al., 2021).
In the coupled-mode theory of resonators, the energy stored in a mode and the power exchanged through its ports are different quantities, linked by the decay rate of the mode. The result is derived for a single linear, lossless optical mode (Fan, Suh and Joannopoulos, 2003).
First check: not every boundary is an electrical port
A boundary is not an object of nature but a model defined by the investigator. From the fact that a surface has been drawn it does not follow that power is defined on it.
The boundary follows from the question being asked: a transfer that is external to one chosen system becomes internal once the boundary is enlarged to contain both sides of it.
Every physical channel has a natural variable, and substituting a variable belonging to another physical channel invalidates the result regardless of instrument quality.
- Electrical port
- Synchronous \(u(t)\) and \(i(t)\).
- Charge packet
- \(\int i\,dt\).
- Resonant turnover
- \(\int \lvert i \rvert\,dt\).
- Field transfer
- Poynting flux.
- Storage element
- Change of stored energy.
- Thermal boundary
- Heat flux.
The universal operation of an electrical port:
From this follows a strict distinction between two notations that look alike and are not physically equivalent.
- \(p(t) = u(t)\,i(t)\) is correct always, but only on a strictly defined electrical port and only when both functions are measured synchronously.
- \(P = UI\), where \(U\) and \(I\) are separately obtained effective, mean or peak numbers, is not a universal accounting operation and does not apply to an arbitrary internal boundary.
If the surface passes through a distributed region of electromagnetic field and forms no port with unambiguously defined port voltage and current, the natural accounting quantity becomes the Poynting flux:
From the inapplicability of \(P = UI\) on some internal boundary it does not follow that there is no power inside the system. The physics does not change — the correct accounting quantity of the chosen boundary does.
Three current processes are not one quantity. In some nodes directed charge transfer is essential, in others oscillation without net transfer, in others still the change of the field. All three are called current and require different operations: the first is integrated with sign, the second by modulus, the third is not a conduction current at all and is accounted through the field flux.
The question of which current you are measuring precedes the question of which current you obtained.
Unity of accounting does not mean sameness of measurement. All channels reduce not to identical instrument readings but to energy over one common observation window.
Defining energy exchange on declared ports, and on a field boundary where no port exists, is standard in energy-based modelling. Dissipative-systems theory states the general constraint: over any interval, the energy stored in a dissipative system grows by no more than the energy supplied to it (Willems, 1972).
For distributed fields the port becomes a part of the boundary of the domain. A structure-preserving discretisation reproduces the power balance of linear field systems, Maxwell’s equations among them, at the discrete level (Brugnoli, Rashad and Stramigioli, 2022). An energy-based framework couples eddy-current field models to circuit equations with the power balance preserved at the interface. Its field model is magnetoquasistatic — displacement currents are disregarded — and its stated focus is low-frequency power applications (Altmann et al., 2026).
On a nonlinear discharge element the port integral of equation (1) is the working quantity in practice: the energy per cycle of a dielectric-barrier discharge is taken as the area of its charge–voltage loop, with the discharge itself left as an unmodelled element (Pipa and Brandenburg, 2019).
Second check: the one-boundary rule
The first check establishes which operation is defined. The second establishes whether a particular product is lawful.
The most dangerous chain of reasoning about architectures of this kind looks convincing at every step: measure the charge of a pulse; multiply by frequency and obtain a mean current; take a voltage from another point of the system; multiply; call the product output power. Each individual number here may be measured flawlessly — and the product will be the power of nothing.
Six questions before any calculation of power:
- Where is the boundary physically drawn?
- Which conductors, fields and couplings cross it?
- Where is voltage measured?
- Where is current measured?
- Do they belong to one and the same energy flow?
- Do all quantities of the calculation share one temporal class?
A negative or undetermined answer to any of the six means the calculation is not performed — not that it is performed with a caveat.
You may not take the current of one functional level, the voltage of another and the temporal class of a third, and call the product the power of a fourth.
A separate consequence concerns alternating processes. From the same synchronous functions on a boundary, quantities of different meaning are derived: RMS values characterise the corresponding RMS class of the process; their product on a correctly defined port is apparent power and, without accounting for phase and non-sinusoidal relations, is not active power; the signed integral of current over a period may be close to zero; the integral of the modulus shows charge turnover. Only the synchronous integral \(\int u\,i\,dt\) shows the energy that passed through this port over the observation window.
Zero algebraic charge transfer does not mean zero energy transfer: an alternating circuit transfers energy precisely by oscillating.
Charge accounting and energy accounting are independent. In the port-Hamiltonian formulation of nonlinear circuits, Kirchhoff’s current law and voltage law enter together, as one interconnection structure fixed by the circuit graph, and the power balance is a property of that structure (Gernandt et al., 2021). The current law alone — conservation of charge at the nodes — does not give it.
Computational plasma physics shows the same independence directly. A simulation scheme that conserves charge exactly can still drift in energy, and schemes that conserve energy can violate charge conservation, until each property is enforced by its own construction (Chen et al., 2020). That result concerns a numerical algorithm; it illustrates the independence of the two accounts and is not a measurement on a circuit.
Third check: the temporal-class passport
Two correctly obtained numbers still may not be compared — until it is established that they belong to one temporal class and one window.
Every quantity carries a mandatory passport:
- \(\Gamma\)
- The boundary of measurement.
- \(\mathcal{Q}\)
- The charge ensemble to which the process belongs.
- \(X\), \([X]\)
- The quantity and its unit.
- Status
- Measured, calculated, nominal, limiting or illustrative.
- Type and \(\Delta t\)
- The temporal class together with the averaging window.
- Duty
- The fill factor of a pulsed regime.
- Direction, load, storage
- The conditions under which the quantity was obtained.
- Method and \(\mathcal{U}\)
- The instrument or means of obtaining it, and the uncertainty of the measured quantity.
Without a complete passport, a quantitative comparison of two quantities is void as engineering evidence. Products of quantities belonging to different temporal classes are not power.
The practical consequence for reading any specification: a pair of voltage and current without a stated boundary, ensemble, temporal class and window does not describe an operating point. It describes two numbers.
Mixing temporal classes is a documented source of false energy. In particle-in-cell plasma simulation, the particle-in-cell method and the Monte Carlo collision model each conserve energy, exactly or nearly, on their own; coupled in the standard way they produced a steady artificial energy gain. The cause was traced to quantities taken at different time levels, and the gain was suppressed once the collision step was centred in time (Vay et al., 2025).
The case concerns a numerical algorithm, not a measurement. It shows the mechanism by which two correct parts yield an incorrect whole.
Energy of an event
An event is a finite single conversion occurring over a time short compared with the observation window of the regime. Its energy is defined on the chosen electrical port by integrating the synchronous functions over the duration of the event itself.
- Time compression creates no energy
- Shortening the release of one and the same previously accumulated portion of energy does not by itself increase the energy of the event: it changes its concentration in time and may raise instantaneous power substantially. Event energy is in all cases given by the integral over its actual duration, and if the shapes of \(u(t)\) and \(i(t)\) themselves change, the value of the integral changes with them.
- An event energy belongs to one boundary
- A value obtained on one port does not carry over to another port even inside a single node: between them stand their own conversions, losses and, as a rule, a different charge ensemble.
- Event accounting is not specific to VENDOR.Max
- It applies to any physical channel in which energy is carried by discrete, correctly defined events. What is specific begins not in this arithmetic but in the composition of physical conversions.
- An event energy does not say what paid for it
- It arrives either from outside the control volume under consideration or from its storage. Only a balance of the control volume over a window long enough to distinguish a steady regime from slow depletion of storage can tell the two cases apart.
A switch redistributes energy; it does not add any. In the Hamiltonian model of switching power converters an ideal switch or diode exchanges zero power with the circuit in either of its states: switching changes how the storage elements are interconnected, while the stored-energy function stays the same (Escobar, van der Schaft and Ortega, 1999). The model assumes ideal, lossless switches without jumps of state and does not describe a discharge gap.
That an event energy does not reveal what paid for it has a general form. In dissipative-systems theory, the energy that can be extracted from a system is bounded by what it stores and what is supplied to it, provided the state and the supply are described completely (Willems, 1972). An apparent excess therefore shows that the chosen description is incomplete or inconsistent: a store or a supply is missing from it, or a quantity in it is wrong.
Average power of an event stream
If energy transfer in some channel does consist of repeating events of one class, the average power of that channel follows from the event energy and the repetition frequency. The formula is physically correct. Its domain of validity is narrow, and outside it the formula ceases to describe anything.
The same arithmetic at charge level gives the average current of the same channel, where \(\bar{Q}_{\mathrm{event}}\) is the mean transferred charge per event:
The conditions below govern both forms. The step from event to average flow is lawful in itself — what is not lawful is multiplying the resulting average current by a voltage taken from another boundary or another temporal class.
Conditions of applicability — all of them at once
- The events belong to one defined class and one physical mechanism.
- The event energy is referred to one explicitly named boundary.
- The frequency is the frequency of those events, not any other characteristic frequency of the system.
- Event statistics are stationary over the chosen window — or the averaging accounts for the non-stationarity.
- Omissions, bursting and the fill factor are accounted for explicitly.
- The result refers to this event channel and to no other.
- All quantities of the calculation carry matching temporal-class passports.
What the formula does not do
- It does not determine the power of another port.
- It does not describe resonant circulation, which is not an event stream.
- It does not give the magnitude of the customer output.
- It does not establish the source of the energy: the relation between event energy and repetition frequency holds equally when the events are fed by an external flow and when they draw on internal storage.
Frequency sets the rate at which events repeat. It is not an explanation of the magnitude of useful output.
What may be added, and under what condition
Inside an established regime four operations are admissible, and each carries its condition.
- Integration
- Over one boundary, over synchronous functions, across an explicitly stated window.
- Averaging
- Over an explicitly defined window, sufficient to represent the repeating process and, when a steady regime is being tested, to distinguish a stable state from slow change of internal storage.
- Multiplication
- Only of quantities that have passed the one-boundary rule and carry matching passports.
- Summation
- Only of energies collected over one physical window, on non-overlapping boundaries, accounting for the change of every storage element.
The rule broken last and most often. The correctness of each local quantity does not by itself make the arithmetic between different working planes correct. Local results may be joined only where a physical transition between the planes has been explicitly established and compatibility of boundaries, directions, time windows and regime states is preserved.
Successive summation along a chain of frames is lawful only under a complete non-overlapping partition and with all couplings between frames accounted for. A discrepancy in the result proves a violation of the conditions of joining, not the presence of an unexplained quantity.
The condition of a steady regime is stated through storage: the averaged rate of change of the stored energy of a control volume is close to zero over a long interval. Without that check, any average power may turn out to be the characteristic of slow depletion rather than of a sustained process.
For any internal control volume the balance is written over its own ports: the sum of incoming flows less the sum of outgoing flows equals the rate of change of that volume’s stored energy. The engineering form with an explicit loss term applies only to internal electrical frames. Mixing forms of notation on one boundary is not permitted.
A balance that closes on an enclosing boundary does not by itself certify the transfers between the control volumes inside it: opposite errors at internal junctions can cancel in the total. Each internal transfer is confirmed on its own boundary.
The notation for the input flow of the complete device boundary does not apply to internal boundaries. Internal frames have their own named port flows, and borrowing the external notation creates a false impression that an internal node is externally fed. The measured residual of the complete device boundary and the inventory of channels belong to the verification protocol.
Two results from computational physics bear on joining. The artificial energy gain described under the third check arose exactly at the junction of two individually correct methods (Vay et al., 2025). And global and local conservation are different properties: for a class of fully implicit particle-in-cell schemes whose global energy conservation had already been demonstrated, the cell-by-cell balance had to be established as a separate theorem, and was then checked on a one-dimensional example (Chacón and Chen, 2025). Carrying that distinction over to the internal transfers of an installation is a methodological parallel, not a result of that work.
Energy-based systems theory gives the positive counterpart: when subsystems are joined through an interconnection that neither supplies nor absorbs power, their stored energies add (Willems, 1972; Gernandt et al., 2021).
Where counting ends and a result begins
Between two statements lies the whole difference between an engineering description and a proof.
The first. In a local frame a measurable increase in the organisation of the local process appears — in voltage, in transfer rate, in pulse density, in turnover or in apparent power. This is described by a conversion and follows from it.
The second. An increase in sustained active power. This is a separate experimental result and does not follow from the fact of a conversion.
This page establishes the language and the conditions of correctness of both statements, but does not establish the experimental truth of the second. It defines which quantity corresponds to what and which operations over quantities are lawful. It does not turn correct counting into a confirmed result — that is done by independent measurement under an agreed protocol.
The two places where a local increase arises, and the boundary beyond which there are no further gains, are set out in where is the plus.
Questions this page is asked
Why can voltage and current not simply be multiplied?
Because the product of separately obtained numbers is power only when both quantities are measured synchronously on one electrical port and belong to one temporal class. Outside those conditions, the product cannot be identified with a defined energy flow.
Is event accounting a feature of VENDOR.Max?
No. It applies to any physical channel in which energy is carried by discrete, correctly defined events. What is specific begins not in this arithmetic but in the composition of physical conversions.
Is the formula “event energy times frequency” correct?
Yes, when all seven conditions of applicability are met. It gives the average power of one event channel and does not give the power of other ports, of circulation, or of the customer output.
May correctly measured quantities of different working planes be added together?
Only where a physical transition between the planes has been explicitly established and boundaries, directions, windows and regime states coincide. The correctness of each term taken separately does not replace that condition.
What does the temporal-class passport do?
It fixes the boundary, the ensemble, the type of quantity, the window, the fill factor, the direction, the state of storage, the method and the uncertainty. Without it, two correctly measured quantities remain incomparable.
Is correct counting enough to confirm performance?
No. Correct counting determines which statements are meaningful. Their truth is established by independent measurement under an agreed protocol.
The published work behind the rules
The works below are circuit and field theory, numerical methods of computational physics, and one review of an experimental method. Each is cited for the accounting principle it establishes in its own field.
None of these works examined VENDOR.Max, and none is evidence of its performance.
- Altmann R., Cortes Garcia I., Paakkunainen E., Schulze P., Schöps S. Energy-based modeling for field–circuit coupling. Applied Mathematical Modelling 155 (2026) 116688. doi:10.1016/j.apm.2025.116688
- Brugnoli A., Rashad R., Stramigioli S. Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus. Journal of Computational Physics 471 (2022) 111601. doi:10.1016/j.jcp.2022.111601
- Chacón L., Chen G. Local conservation of energy in fully implicit PIC algorithms. Journal of Computational Physics 529 (2025) 113862. doi:10.1016/j.jcp.2025.113862
- Chen G., Chacón L., Yin L., Albright B. J., Stark D. J., Bird R. F. A semi-implicit, energy- and charge-conserving particle-in-cell algorithm for the relativistic Vlasov–Maxwell equations. Journal of Computational Physics 407 (2020) 109228. doi:10.1016/j.jcp.2020.109228
- Escobar G., van der Schaft A. J., Ortega R. A Hamiltonian viewpoint in the modeling of switching power converters. Automatica 35(3) (1999) 445–452. doi:10.1016/S0005-1098(98)00196-4
- Fan S., Suh W., Joannopoulos J. D. Temporal coupled-mode theory for the Fano resonance in optical resonators. Journal of the Optical Society of America A 20(3) (2003) 569–572. doi:10.1364/JOSAA.20.000569
- Gernandt H., Haller F. E., Reis T., van der Schaft A. J. Port-Hamiltonian formulation of nonlinear electrical circuits. Journal of Geometry and Physics 159 (2021) 103959. doi:10.1016/j.geomphys.2020.103959
- Pipa A. V., Brandenburg R. The equivalent circuit approach for the electrical diagnostics of dielectric barrier discharges: the classical theory and recent developments. Atoms 7(1) (2019) 14. doi:10.3390/atoms7010014
- Vay J.-L., Angus J. R., Shapoval O., Lehe R., Grote D., Huebl A. Energy-preserving coupling of explicit particle-in-cell with Monte Carlo collisions. Physical Review E 111 (2025) 025306. doi:10.1103/PhysRevE.111.025306
- Willems J. C. Dissipative dynamical systems, Part I: General theory. Archive for Rational Mechanics and Analysis 45(5) (1972) 321–351. doi:10.1007/BF00276493
Where this page sits
Understand it, see the path, see the local gains, learn the accounting rules, meet the open question, resolve the circulation misreading, learn how to measure it, read the verification protocol.
The energetic price of transfer
How a charge flow becomes measurable power, and what the source is not.
Explore 02Scientific foundations
The literature on pulsed accumulation, switching and time compression, with a route for verification.
Explore 03Investor room
The route for requesting deeper engineering material.
ExploreCounting rules determine which statements are meaningful. Which of the meaningful statements are true is established by measurement.
Regime-level energy accounting
