Resonant Circulation
Inside a resonant section, voltage, current and charge turnover can all be very large. That does not mean the same active power passes through the section, and it says nothing about where the energy comes from.
Subject. How a small local charge produces a large internal turnover, what is transferred into the next circuit, and what it costs in energy to hold such a state.
This page does not cover the functional route through the architecture; how a charge flow becomes measurable active power; where local gains appear between working planes; the general rules for counting regime quantities; how local transformations compose into a self-consistent operating point; which instruments measure active power; what the full device boundary balance shows.
No device values appear on this page. The two illustrative examples below are marked as such: their numbers are chosen for round arithmetic and do not describe VENDOR.Max.
Resonance increases the number of repeated cycles of a state — not the amount of energy stored in it.
What this section is and what we examine
The page opens the resonant section as a separate control volume. It does not describe its place in the route.
The architecture of VENDOR.Max — a non-linear electrodynamic installation operating in a controlled discharge-resonant regime — contains a resonant section. Its role is to form a large repeating electromagnetic state and, through electromagnetic coupling, to transfer part of the energy of that state into the next, electrically isolated path.
The route of the installation is not analysed here. Only this section is magnified: what circulates repeatedly within it, why the internal charge turnover becomes large, and what is passed onward from the state.
This page does not offer new resonator physics. It fixes the rules by which the internal quantities of VENDOR.Max may and may not be interpreted.
One place can be crossed many times
Before counting anything, it has to be agreed what exactly is being counted.
Not the amount of charge. Not energy. What is counted is how much charge has passed through a chosen cross-section — including repeated passes of the same charge.
The amount of charge present and the total charge path are different quantities, and in an alternating regime they can diverge radically.
The image that holds the distinction is a vehicle odometer. A car drives from one town to another and back. The odometer shows twice the distance. No second car appeared, and the first one did not disappear halfway through the trip. An odometer counts distance travelled, not final position.
Why one becomes four
Count the path, not the final position.
Illustrative example. The values are chosen purely for round arithmetic and do not describe VENDOR.Max.
Let the amplitude of the oscillating charge be \(1\ \mu\mathrm{C}\). Over one full cycle the charge state travels the path
| Segment | Path travelled |
|---|---|
| from \(+1\) to \(0\) | 1 µC |
| from \(0\) to \(-1\) | 1 µC |
| from \(-1\) to \(0\) | 1 µC |
| from \(0\) to \(+1\) | 1 µC |
| Over one full cycle | 4 µC |
At no moment were four microcoulombs present in the system at once. It was the same oscillating charge of amplitude \(1\ \mu\mathrm{C}\). The figure of four microcoulombs is the total charge path travelled by it during the cycle.
Hence the general expression, whose arithmetic the reader has just carried out unaided:
The equality is independent of waveform as long as the half-period contains no additional extrema. Under ringing and multimode structure the full path exceeds \(4Q\), and turnover is then obtained directly from the integral of the current magnitude.
From four microcoulombs per cycle to four coulombs per second
Only the rate is added. Nothing else changes.
Illustrative example. The values are chosen purely for round arithmetic and do not describe VENDOR.Max.
Let such a cycle occur one million times per second.
At an oscillating charge amplitude of \(1\ \mu\mathrm{C}\), the local ensemble delivers a total absolute passage of \(4\ \mathrm{C}\) through the chosen cross-section every second.
4 C/s does not mean that 4 C were created out of 1 µC. It is the odometer of internal charge motion.
Relation to current readings for a harmonic regime:
Two averages answer two different questions
The unit is the same. The question is not.
The unit of measurement is one and the same: \(1\ \mathrm{C/s} = 1\ \mathrm{A}\). But two averages answer different physical questions, and the answers diverge.
How much charge was ultimately transported in one direction?
What total path did the charge travel back and forth?
The first quantity characterises the algebraic transport of charge through the chosen cross-section over a period. The second characterises the total absolute passage through the same cross-section.
One common objection is worth removing here. In a steady unidirectional circuit the carriers are not consumed either: the same charges move around a closed loop and may cross a chosen cross-section many times. But the direction of motion there is single, and the absolute flow coincides in magnitude with the algebraic one, \(\langle |i| \rangle = |\langle i \rangle|\). In a resonator the local ensemble reverses direction many times per period — which is exactly why the two quantities diverge where a unidirectional regime makes them coincide.
The four coulombs go nowhere. What is transferred onward
The second turning point of the page.
Over a full cycle the charge state has returned to its starting point. The four microcoulombs of absolute path are therefore not four microcoulombs dispatched from the section to anywhere. Asking where they went, in the sense one asks about poured water, is malformed: this is not a quantity that was sent.
But the repeating motion of charge creates a changing electromagnetic state. And that state passes on something else entirely.
Hence the direct conclusion. Over a full cycle the carriers of the first circuit were not transported into the second — which means the coupling of the two circuits is not a transport of those carriers at all. A change in the electromagnetic state of the first circuit produces an electromotive force on the coupled side, and the work is done on charges that already belong to it. Energy therefore passes across the electrical isolation, while the carriers of each circuit remain within their respective circuit.
The next path does not receive those four coulombs. It receives energy through electromagnetic coupling, and what is set in motion are the charges belonging to the next circuit.
The carriers of the next circuit were present in it before the regime started. They need not wait for electrons to arrive — all they need is an organisation of motion, and that is what the changing field supplies.
The specific coupling topology, its geometry and parameters belong to closed technical documentation.
How much is transferred: extraction depth
Since energy is transferred and charge is not, the depth of transfer has to be expressed in energy.
Write \(E_{\text{take,cycle}}\) for the energy that leaves the control volume of the resonant section per cycle through the take-off path. The normalised form of the same quantity is the fraction of the energy of the state transferred per cycle:
Here \(E_{\text{mode}}\) is the stored energy of the resonant regime of the chosen control volume. It is the same physical quantity that appears below, in the definition of the quality factor, as \(E_{\text{res}}\): two notations for one stored energy of the chosen mode. The canonical notation \(E_{\text{mode}}\) is retained in the expression for \(\rho_{\text{take}}\).
The quantity \(\rho_{\text{take}}\) is defined for an explicitly chosen mode, an explicitly stated control volume and a reference cycle of the regime. The metrology of arbitrary measurement windows belongs to the rules of regime-level accounting and is not treated here.
Extraction is not free for the resonant mode. As soon as the ensemble on the receiving side is set in motion, it produces a magnetic field of its own, and by Lenz’s law that field opposes the change that produced it.
Extraction does not transport carriers out of the first circuit. For the resonant mode it appears as an additional external channel of energy outflow and a corresponding reflected load. All else being equal, deeper energy extraction corresponds to a larger reflected load on the mode and to more pump energy required to preserve the same state.
The formal representation of this external outflow channel is given by the loaded quality factor, to which we return after the per-cycle balance has been written.
The cost of preserving the state
Resonance does not answer where the pump energy comes from. It does allow an exact statement of how much of it the regime needs on every cycle.
Balance of the control volume of the resonant section over a cycle of the chosen resonant mode, or over an equivalent periodic window of the regime:
The steady regime is defined by the condition \(\Delta E_{\text{res,cycle}} \approx 0\), taken over a window long enough to distinguish a stable state from slow depletion of the stored energy.
The loss term is expressed through \(Q_u\) — the unloaded quality factor of the resonant state, that is, the measure of how small a fraction of its stored energy the state loses per cycle. Substitution makes its role explicit:
Three distinct physical roles have to be told apart in this expression:
\(E_{\text{res}}\) — how much energy is stored in the chosen mode.
\(2\pi E_{\text{res}} / Q_u\) — dissipated per cycle inside the control volume.
\(E_{\text{take,cycle}}\) — passed onward per cycle through the take-off path.
In this balance Qu enters directly only the internal loss term. Etake,cycle represents a separate channel of energy outflow. Neither quantity, on its own, establishes the origin of Epump,cycle.
The cost of holding grows faster than amplitude. With the equivalent parameters of the chosen mode held fixed, the stored reactive energy is proportional to the square of its amplitude. At unchanged quality factor, therefore, doubling the amplitude quadruples the loss term per cycle.
Under insufficient pumping the regime degrades. If what is supplied per cycle is less than the sum of losses and extraction, \(\Delta E_{\text{res,cycle}}\) is negative, the stored energy of the regime falls and amplitudes decay.
Pumping is an energy problem and a phase problem at once. The energy required per cycle is set by losses and extraction, and resonance does not lift that requirement. A second condition is added to it: the effectiveness of delivery depends on phase agreement with the existing state. A portion delivered out of phase does not sustain the regime, but the energy requirement is not reduced by that.
The condition \(\Delta E_{\text{res,cycle}} \approx 0\) is a condition for preserving the energy state of the regime. Stability of the operating point does not reduce to it: that is additionally governed by admissible voltages, currents, temperature, insulation and switching constraints.
What each further turn costs: the quality factor
If the state is losing something all the time, why does it not simply disappear?
The answer is given by the very quantity that stood in the denominator of the loss term.
An equivalent record through loss power is \(Q_u = \omega_0 E_{\text{res}} / P_{\text{loss}}\).
The meaning follows immediately from the definition. With pumping switched off, the stored energy of a free circuit falls to approximately \(1/e\) of its initial value after \(N = Q_u/2\pi\) cycles. The quality factor divided by \(2\pi\) is the characteristic number of turns the state can persist without pumping.
Illustrative example. The values are chosen purely for round arithmetic and do not describe VENDOR.Max.
Let the energy stored in the state be \(1\ \mathrm{J}\) and let \(0.01\ \mathrm{J}\) be lost per cycle. Then
One per cent of the stored energy is lost per cycle. In everyday terms, the quality factor answers the question of what each further turn costs.
What the quality factor is not
It is not a gain. The resonant rise of voltage amplitude on a reactive element relative to the excitation amplitude is a redistribution of already invested energy between forms of storage, not an increment of it.
It is not an efficiency of the installation. The quality factor is defined for a single circuit and describes the ratio of that circuit’s stored energy to that circuit’s losses per cycle. An end-to-end coefficient for the installation requires an explicitly stated control volume, physically identified input and output flows, a common measurement window and accounting for the change in all internal stored energies.
It is not a characteristic of a source. The ratio of stored energy to losses is equally well defined for a circuit fed externally and for a circuit spending its own stored energy.
Loaded quality factor
For the circuit, useful extraction is indistinguishable from an additional channel of energy outflow.
For a chosen resonant mode, where the internal and external channels of energy outflow can be represented as additive loss rates, the loaded quality factor is written as
Here \(Q_{\text{ext}}\) describes the take-off channel and \(Q_L\) is the quality factor of the loaded circuit. At finite external extraction \(Q_L < Q_u\); in the limit \(Q_{\text{ext}} \rightarrow \infty\) the loaded quality factor tends to the unloaded one, \(Q_L \rightarrow Q_u\). In general, \(Q_L \leq Q_u\).
The unloaded quality factor does not depend on extraction: extraction changes \(Q_L\), not \(Q_u\). A high unloaded quality factor creates margin against internal losses. The engineering goal is not to maximise \(Q_L\), but to match internal losses to the required external extraction.
Internal amplitudes and instrument readings
The second source of large numbers is a product of amplitudes.
On a correctly defined electrical port, two different quantities follow from the same synchronous functions. Active power is the synchronous mean of the product, \(P = \frac{1}{T}\int_0^T u(t)\,i(t)\,dt\). Apparent power is the product of root-mean-square values obtained separately, \(S = U_{\text{RMS}} I_{\text{RMS}}\). Their ratio is the power factor, \(\lambda = \lvert P \rvert / S\), with \(0 \leq \lambda \leq 1\).
The absolute value in the numerator fixes the sign convention: \(P\) is a directed quantity, and without it the ratio changes sign when the active flow reverses. The power factor describes a fraction, not a direction.
Both quantities are defined only on a single electrical port and only for synchronously measured functions. That restriction matters more than it appears: large internal amplitudes on their own do not constitute power, and the voltage of one reactive element and the current of another do not combine into a product at all — they belong to different boundaries.
Two measurement planes
On the port of an individual reactive element, voltage and current may be close to quadrature: the active mean is small, while \(S\) remains large and characterises the scale of reactive exchange through that port. In a resonant system this exchange is associated with the periodic redistribution of energy between electric and magnetic forms of storage.
On the input or output port of the whole loaded resonator, the phase relation is set by tuning, coupling and load. In a tuned resonant regime the reactive components on such a port may largely compensate; the power factor of the external port is therefore under no obligation to be small and may approach unity.
Hence a consequence that is easily lost: the origin of large internal voltages and currents cannot be inferred from the power factor of an external port. These are different measurement planes, and the one-boundary rule forbids carrying a conclusion from either of them to the other.
For non-sinusoidal regimes the difference is amplified. The power factor separates into a phase component and a distortion component. Harmonic components raise the root-mean-square current and may therefore raise \(S\); their contribution to \(P\) is determined only by the corresponding synchronous components of voltage and current.
The practical consequence for reading any internal number. A product of internal kilovolts and amperes is not megawatts of useful power. It becomes apparent power only when both values are root-mean-square and taken on one port. Such a product must not be reported as active power.
URMS and IRMS determine the corresponding electrical and current duties of components; their product S characterises the volt-ampere scale of that port. P characterises the mean active transfer of energy through it. P cannot be determined from URMS and IRMS alone: the synchronous relation between voltage and current on the same port has to be preserved.
Loss map
The loss term is not a coefficient. It is a sum of named physical mechanisms.
| Line | Mechanism | Where the energy goes | Principal dependence |
|---|---|---|---|
| Conductors | Joule dissipation on the carriers of windings and joints | Conductor heating | Rises with frequency through skin and proximity effects |
| Magnetic material, where present | Hysteresis of remagnetisation | Core heating | Rises with frequency and field amplitude |
| Eddy currents | Induced circulating currents in conductive masses within the field region | Heating of surrounding conductive masses | Rises with frequency and with the size of the conducting mass |
| Dielectrics | Repolarisation of insulation and of capacitive dielectrics | Dielectric heating | Rises with frequency and the square of field strength |
| Radiation | Part of the field leaves the structure | External space | Rises with frequency and the electrical size of the structure |
| Pump input | Dissipation of input elements located inside the boundary of the section | Heating of those elements | Set by the energy and rate of pump events |
The boundary of the lines. Only mechanisms physically located inside the chosen control volume enter \(E_{\text{loss,cycle}}\). Losses of the exciting path upstream of the boundary do not enter this term: they belong to the balance of their own control volume. Without that restriction the balance in section 08 stops being a balance of one volume.
What is not a loss line. Leakage inductance in itself is a reactive parameter, not a dissipative line: the unlinked part of the field stores energy and returns it to the circuit, lowering coupling and releasing no heat. A separate matter is the leakage field as a physical cause: the currents it induces in surrounding conducting structures do form real losses, and those are accounted under the eddy-current line. A reactive parameter and field-induced dissipation are different lines.
Extraction is not a loss line either. Energy passed into the take-off path leaves the control volume exactly as losses do, and is replenished by the same pumping — but it is not dissipated, it is carried onward. For the circuit the two lines add; in accounting they are distinct, and the distinction between them is precisely the distinction between \(Q_u\) and \(Q_L\).
Numerical values of the lines are a matter of measurement on a specific implementation. The construction parameters of the resonant section belong to closed technical documentation.
What this page closes
Four readings of internal quantities, each of which looks natural and each of which is wrong.
| Observation | Erroneous conclusion | What was established above |
|---|---|---|
| Large turnover at a small local oscillating charge | “Charge is multiplying” | The charge is the same; the number of crossings of the cross-section grows. No turnover term appears in the balance |
| Large \(U_{\text{RMS}}\) and \(I_{\text{RMS}}\) on one internal port | “Megawatts inside” | Their product is \(S\), the volt-ampere scale of that port. Active transfer is determined separately, as \(P = \langle ui \rangle\) |
| High quality factor | “The device is energy-efficient” | The quality factor describes the cost of holding one mode, not the efficiency of the installation |
| The per-cycle balance closes | “The source is established” | The balance states how much pumping is required and does not state where it comes from |
The page reduces the risk of misreading internal measurements. It is deliberately not used as evidence of an energy result of the installation.
Three separate questions of engineering assessment
Conflating them produces either inflated claims or false refutations.
Can the required internal state be formed?
A question of the operating point. Not resolved here.
How costly is it to hold at a given extraction?
A question of \(Q_u\), the loss lines and \(E_{\text{take,cycle}}\). Resolved above.
Where does the energy covering that cost come from?
A different control volume and a different experiment. Not resolved here.
In the architecture these three questions are deliberately separated across different sections and different measurement boundaries.
Where this page ends
Everything set out above belongs to one control volume.
The per-cycle balance determines how much pumping the regime requires at a given stored energy, quality factor and extraction depth. It does not determine where that pumping comes from: from an external flow, from an internal energy store of the device, or from a redistribution of an already established flow.
The local balance of a control volume and the origin of energy are questions of different boundaries. The first is settled at the boundaries of the section and has been settled here. The second is settled only at the full device boundary, by an inventory of every channel crossing it and a measured remainder.
Resonance explains how the regime is sustained. It does not explain what pays its energy cost.
Resonant CirculationFrequently asked questions
Does resonance amplify energy?
No. Resonance redistributes already invested energy between electric and magnetic forms of storage and holds that redistribution for many cycles. The rise of amplitude on a reactive element relative to the excitation amplitude is a consequence of exchange between forms of storage, not an increment of energy.
Where does the charge turnover go?
Nowhere. Over a full cycle the state returns to its starting point. Turnover measures the total path of one and the same oscillating charge, not a quantity dispatched anywhere.
Does the next circuit receive the charge of the resonant section?
No. Over a full cycle the carriers of the first circuit are not transported into the second, and the coupling of the two circuits is not a transport of carriers. Energy is transferred; the motion in the next circuit is formed by its own charges, which were present in it before the regime started.
Does \(\rho_{\text{take}}\) mean a fraction of departed carriers?
No. It is the fraction of the energy of the resonant state transferred per cycle. It bears no relation to the number of carriers or to the magnitude of the charge turnover.
What do the kilovolts and amperes observed inside the section mean?
They are amplitudes of the internal high-frequency state. Their product belongs to the circulating volt-ampere scale. It becomes apparent power only for root-mean-square values taken on one correctly defined port; the active flow is determined separately, as the synchronous mean of the product of voltage and current on that same port.
Does a high quality factor mean a high efficiency of the installation?
No. The quality factor is defined for a single circuit and describes the ratio of its stored energy to its losses per cycle. An end-to-end coefficient for the installation requires an explicitly stated control volume, identified input and output flows, a common measurement window and accounting for the change in all internal stores.
Why is the integral of current magnitude large while the algebraic integral is near zero?
Because over a period the same local charge passes through the cross-section in both directions. The absolute flow adds both passes; the algebraic one subtracts them.
Can power be obtained from charge turnover?
No. Turnover is measured in coulombs per second and characterises internal circulation. Multiplying it by a voltage taken from any point of the system is a product of quantities from different boundaries and different time classes, and is not power.
How does deeper extraction affect the energy cost of the regime?
All else being equal, deeper extraction corresponds to a lower loaded quality factor and to more energy that must be compensated by pumping in order to preserve a given state. The dynamics of transition between operating points are not treated by this page.
Does a low power factor mean no energy passes through the circuit?
No. A low power factor means the active component is small relative to the product of root-mean-square values. In a predominantly reactive regime this is associated with energy exchange that returns to the circuit; in a non-sinusoidal regime an additional cause may be distortion of the current waveform.
Continue reading
This page is step six of the route. The steps are read in order, but each stands on its own.
Understanding VENDOR.Max
The plain explanation for a first encounter.
Explore 02How it works
The functional route through the architecture.
Explore 03Where is the plus?
The map of local gains between working planes.
Explore 04Energy model
The rules for counting quantities inside the regime.
Explore 05First open engineering question
Composition of transformations and the self-consistent operating point.
ExploreResonant Circulation
This page.
You are hereActive power metrology
Measuring active power in open systems.
Explore 08Technology validation
The verification protocol and the full device boundary.
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