Flynn-C primer: steering a permanent magnet's flux with coils

Project document: US 6,246,561 B1, Charles J. Flynn, "Methods for controlling the path of magnetic flux from a permanent magnet and devices incorporating the same" (filed 31 Jul 1998, granted 12 Jun 2001) [read]. Companion files: arbiter.md (final verdicts), example.py (the worked example, runs cleanly).

Source gap. The patent is the only document in the folder. The PDF has no text layer, so it was read by rendering pages to images and running OCR (optical character recognition). The description (PDF pp.34-45) and claims (pp.45-49) were read that way. For this primer I also looked at drawing sheets PDF p.3 (FIGS. 3-8), p.7 (FIGS. 10G, 10H, 14), p.10 (FIGS. 13A-13C) and p.12 (FIGS. 16B, 16C) [read: page image]; the other drawing sheets were not studied. The patent contains no test data. FIG. 16C is a sketch of flux versus coil current with a current axis in milliamps but no flux values and no statement that it was measured [read: page image p.12]. No test reports or efficiency measurements of Flynn-type devices are in the folder, and the reviewers' web searches for independent tests failed. So every statement below about whether such devices were tested means "no test on this record", not "proven untested".

Page references are to PDF pages; the printed column numbers differ (PDF p.37 is printed columns 7-8).

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1. The idea in one paragraph

A permanent magnet sits between two iron pole pieces that branch into two parallel paths, each closed by an iron plate (an armature). Left alone, the magnet's flux splits between the two paths and both armatures are held weakly. Put a coil on one path and pass a current that just cancels the magnet's push along that path: that path goes dead and its armature drops off, while nearly all of the magnet's flux crowds into the other path, whose armature is now held about four times as hard. Pulse coils on alternate paths and you can make a shaft shuttle back and forth, a rotor turn, or (with the armatures fixed and extra "secondary" coils wound on the paths) make the flux in each path rise and fall so that a voltage is induced, like a transformer. Flynn argues that because the magnet supplies the flux, the coils do this with less electrical effort than an electromagnet alone would need, giving "more work per unit of input electrical energy". The patent does not claim to put out more energy than goes in [read]. The switching principle is sound, ordinary magnetics; the efficiency argument is where the trouble lies.

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2. What you need to know first

Magnetic field B and flux Φ. B (tesla, T) is the magnetic flux density. The flux through a surface of area A with uniform, perpendicular B is Φ = B·A, measured in webers (Wb = T·m²). Flux lines form closed loops (∇·B = 0), which is why flux through a closed iron loop behaves like a current that cannot leak away. Read: Griffiths, Introduction to Electrodynamics, ch. 5 (magnetostatics).

Magnetic materials: H, μ, and soft iron. H (amperes per metre, A/m) is the "magnetising field", the part of the field set up by free currents. In a linear material B = μH with μ = μ_r μ0, where μ0 = 4π×10⁻⁷ T·m/A is the permeability of free space and μ_r is the relative permeability (thousands for soft iron, about 1 for air). Soft iron carries flux easily but saturates near 1.5-2 T [memory: high]. Read: Griffiths ch. 6 (magnetic fields in matter).

Magnetic circuits (Hopkinson's law). For flux confined to iron paths there is an exact analogy with DC circuits: magnetomotive force (MMF, symbol 𝓕, unit ampere-turns, A-t) plays the role of voltage, flux Φ plays current, and reluctance 𝓡 = l/(μA) plays resistance, where l is the length of a path segment and A its cross-section. The law is 𝓕 = Φ𝓡 [web: https://en.wikipedia.org/wiki/Magnetic_circuit]. A coil of N turns carrying current I supplies 𝓕 = NI. Reluctances in series add; in parallel they combine like resistors in parallel. Air gaps dominate: 0.05 mm of air has as much reluctance as roughly 10-50 cm of good iron [memory: high, follows from μ_r of iron ≈ 2000-10000]. Read: Fitzgerald, Kingsley & Umans, Electric Machinery, ch. 1 (magnetic circuits and magnetic materials).

Permanent magnets as sources. A modern neodymium-iron-boron (NdFeB) magnet behaves, to a good approximation, like a fixed MMF 𝓕_m in series with its own internal reluctance 𝓡_m (the magnetic version of a battery with internal resistance). If the magnet has remanence B_r (the flux density it keeps with its poles shorted by perfect iron), length l_m along its magnetisation, area A_m and recoil permeability μ_r ≈ 1.05 (the slope of the magnet's B-versus-H line in its working range, divided by μ0; a value near 1 means extra H changes B in the magnet almost as it would in air), then 𝓕_m = B_r l_m/(μ0 μ_r) and 𝓡_m = l_m/(μ0 μ_r A_m) [memory: high for the model; B_r = 1.0-1.4 T and recoil permeability 1.05 for sintered NdFeB from web: https://en.wikipedia.org/wiki/Neodymium_magnet]. Because μ_r ≈ 1, a magnet's internal reluctance is large, like an air gap as long as the magnet. Changing what the magnet is connected to moves its "operating point" (how much flux it delivers), just as changing the load on a battery changes its current.

Magnetic force (Maxwell stress). Where flux crosses from a pole face into an armature, the pull per unit area is B²/(2μ0), so a face of area A feels F = B²A/(2μ0). Force goes as B squared. Read: Griffiths ch. 8 (conservation laws; Maxwell stress tensor). You do not need the full tensor here, only this one result.

Faraday's law. A coil of N turns linking a changing flux has an induced EMF (electromotive force, a voltage) ℰ = −N dΦ/dt. Only changing flux induces voltage. Read: Griffiths ch. 7.

Energy conservation for a magnet. A permanent magnet is a conservative source: like a spring, it can store and release energy as the armature moves, but over a closed cycle (everything back where it started) it supplies no net energy. Net mechanical work out over a cycle equals electrical energy in minus losses (copper I²R heat, iron eddy-current and hysteresis loss) [memory: high, standard thermodynamics of magnetic systems].

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3. The textbook view

3.1 The two-path circuit

Model the device of FIGS. 7-8 [read: page image p.3] as a circuit with two "nodes": the top pole piece (above the magnet's N face) and the bottom pole piece. Between them sit three branches in parallel:

• the magnet: MMF source 𝓕_m in series with reluctance 𝓡_m;
• leg 1 (left pole-piece arms plus armature 118): reluctance 𝓡;
• leg 2 (right arms plus armature 120): reluctance 𝓡.

Symbols: Φ₀ = flux in each leg with no coil current; Φ₁, Φ₂ = fluxes in legs 1 and 2 when switched; 𝓕₁ = MMF of the coil on leg 1; U = magnetic potential difference between the two pole pieces (the "node voltage").

No coil current. Magnet flux (𝓕_m − U)/𝓡_m equals the sum of leg fluxes 2U/𝓡. Solving,

Φ₀ = 𝓕_m / (𝓡 + 2𝓡_m) in each leg.

Coil on leg 1 opposing, set to "just block". Leg-1 flux is (U − 𝓕₁)/𝓡 = 0, so 𝓕₁ = U. All the magnet's flux now goes through leg 2:

Φ₂ = 𝓕_m / (𝓡 + 𝓡_m), 𝓕₁ = 𝓕_m 𝓡 / (𝓡 + 𝓡_m) = Φ₂𝓡.

Flux gain in the open leg: Φ₂/Φ₀ = (𝓡 + 2𝓡_m)/(𝓡 + 𝓡_m). This is 2 only in the limit 𝓡_m ≫ 𝓡 (magnet reluctance much larger than leg reluctance, i.e. armatures closed tight). At 𝓡_m = 𝓡 it is 1.5.

3.2 Force

Each armature touches two pole faces of area A, so its pull is F = 2 × B²A/(2μ0) = B²A/μ0 with B = Φ/A. Since F ∝ Φ², doubling a leg's flux quadruples its force; halving it quarters the force.

3.3 Induction

With armatures fixed and a secondary coil of N_s turns on each leg, the secondary EMF is ℰ = −N_s dΦ_leg/dt. A load current in the secondary adds its own MMF to the leg, which the control coils must cancel to keep the flux waveform; that is how power flows from the control circuit to the load, exactly as from primary to secondary in a transformer [memory: high; derivation in mathematician.md].

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4. Where Flynn departs

Flynn's starting points are all correct textbook physics [read: PDF pp.36-37]: F = B²A/(2μ0), flux splitting between parallel paths, and a coil MMF opposing or aiding a magnet's MMF. The departures are in what he infers.

The single conceptual slip that drives most of this: Flynn reasons in terms of flux a coil produces ("a magnetic flux equal to but opposing the magnetic flux which travels along flux path 130" [read: PDF p.37]), while the electrical cost of a coil is set by the MMF (ampere-turns) it must supply, and that depends on the whole path its flux has to travel.

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5. Worked example: does the blocking coil really need half the effort?

Reproduced by example.py (run command in its header; output quoted here). The magnet and gap values are my illustrative choices; the patent gives none.

Step 1. Quarter force (Part A). Take B = 1.0 T on face area A = 1 cm² = 1×10⁻⁴ m². One face: F = B²A/(2μ0) = (1.0)²(1×10⁻⁴)/(2 × 1.2566×10⁻⁶) = 39.8 N. One armature (two faces): 79.6 N. Same flux on two paths: B = 0.5 T, each armature 19.9 N (ratio 0.250), both together 39.8 N (ratio 0.500). Flynn's quarter is per armature; the total halves.

Step 2. Build the circuit (Part B). NdFeB magnet, B_r = 1.2 T, μ_r = 1.05, length l_m = 10 mm, area A_m = 1 cm². Each leg: iron (reluctance neglected) plus a residual gap g = 0.05 mm at each of the armature's two faces, face area 1 cm².

• 𝓕_m = B_r l_m/(μ0 μ_r) = 1.2 × 0.010/(1.2566×10⁻⁶ × 1.05) = 9095 A-t
• 𝓡_m = l_m/(μ0 μ_r A_m) = 0.010/(1.2566×10⁻⁶ × 1.05 × 10⁻⁴) = 7.58×10⁷ A-t/Wb
• 𝓡 = 2g/(μ0 A) = 2 × 5×10⁻⁵/(1.2566×10⁻⁶ × 10⁻⁴) = 7.96×10⁵ A-t/Wb
• 𝓡_m/𝓡 = 95, close to Flynn's ideal limit.

Step 3. Unswitched. Φ₀ = 𝓕_m/(𝓡 + 2𝓡_m) = 9095/(1.524×10⁸) = 5.97×10⁻⁵ Wb per leg, B = 0.597 T. Force per armature B²A/μ0 = 0.597² × 10⁻⁴/1.2566×10⁻⁶ = 28.3 N.

Step 4. Leg 1 blocked. Φ₂ = 𝓕_m/(𝓡 + 𝓡_m) = 9095/(7.66×10⁷) = 1.188×10⁻⁴ Wb, B = 1.188 T (still below iron saturation). Gain 1.99. Force on armature 120: 112.2 N, 3.96 times the unswitched value. Flynn's "about four times" checks out for a tight armature.

Step 5. What the blocking coil costs (Part C). 𝓕₁ = Φ₂𝓡 = 1.188×10⁻⁴ × 7.96×10⁵ = 94.5 A-t.

Step 6. What a coil-only device costs. FIG. 14 [read: page image p.7] shows the comparison device: a U-shaped iron piece with coils, closed by an armature, the same shape as one leg of FIG. 13A without the magnet. To drive the same flux Φ₂ (hence the same force) through a leg of the same reluctance 𝓡 takes 𝓕 = Φ₂𝓡 = 94.5 A-t. The ratio is exactly 1, and SymPy confirms it symbolically for any 𝓕_m, 𝓡_m, 𝓡:

𝓕₁ = 𝓕_m𝓡/(𝓡 + 𝓡_m) = Φ₂𝓡.

With a 200-turn, 1 Ω coil both need I = 0.473 A and dissipate I²R = 0.223 W. No factor of two.

Step 7. Where Flynn's "half" comes from. His coil must "provide a magnetic flux equal to but opposing" Φ₀. That is true: by superposition, the coil acting alone drives exactly Φ₀ round leg 1 (the script checks this). If that flux only had to cross one leg, the cost would be Φ₀𝓡 = 47.5 A-t, half of 94.5. But the coil's flux has to close somewhere: it returns through leg 2 in parallel with the magnet, and since the magnet's reluctance is huge, it effectively returns through leg 2. So the coil sees 𝓡 + (𝓡 ∥ 𝓡_m) ≈ 2𝓡, and the cost doubles back to 94.5 A-t. Counting flux instead of MMF hides the second leg.

Step 8. Sensitivity (Part D). Flux gain (𝓡+2𝓡_m)/(𝓡+𝓡_m) and force ratio (its square):

Opening the armature gap raises 𝓡 and the "×4" fades fast. In the reciprocating device of FIG. 13B the far armature sits across an air gap (324) [read: PDF p.38], so in operation 𝓡_m/𝓡 for that leg is far from 100.

What this does and does not show. Within the standard linear magnetic-circuit model (no saturation, no leakage, magnet on its straight recoil line), the blocking coil and the coil-only device need the same ampere-turns. The patent says "power" on p.37 and "energy" on p.39 and gives no formula [read: p.37, p.39], so the arbiter's revised verdict (arbiter.md, "QE revision") splits the "twice" claim three ways:

• (a) Coil MMF, or steady I²R holding power: "Exactly 1 for any leg reluctances, any air gap 324" → Refuted by calculation. FIG. 14 is "one U-shaped leg with coils on it" [read: page image p.7], so it is a fair comparison, and example.py gives a blocking-coil / coil-only MMF ratio of exactly 1. Within the linear model, a "twice" saving against FIG. 14 is refuted.
• (b) "Energy to build the switched field (integral of i dλ), equal closed legs (the p.37 assumption)": ratio (𝓡 + 2𝓡_m)/(𝓡 + 𝓡_m) = 1.990 (W_14 = 5.61 mJ, W_13A = 2.82 mJ [calc, arbiter.md]), "tending to exactly 2 for a stiff magnet" → "Reproduced by calculation (idealised; leakage and saturation untested)". Here λ (lambda) is the coil's flux linkage, NΦ, so ∫ i dλ is the energy put into the magnetic field while the current is switched on. This ratio is the same expression as the flux gain in Step 4 (1.990 in example.py). But "Field energy is stored and returns when the current falls; cycle balance is unchanged", so it is not a net power or efficiency gain. "More work per unit input electrical energy" is "Not supported; does not follow."
• (c) Real nonlinear, leaky hardware: untested. Nothing on this record measures it.

The patent offers no calculation or data of its own. FIG. 14, now seen, removes the possibility that the comparison device uses a longer circuit [read: page image p.7]. The arbiter notes one real side benefit: the blocking coil sees about half the inductance, so its current rises faster and less reactive energy is stored per stroke; "That is not less energy consumed" [calc, arbiter.md]. What the magnet genuinely buys is a holding force with zero coil current in the unswitched state, and force that responds more steeply to coil current (because F ∝ (Φ_bias + Φ_coil)²); those are real advantages of polarised and latching actuators [web: https://en.wikipedia.org/wiki/Latching_relay, via arbiter.md].

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6. Evidence table

From arbiter.md; "no test on record" means none in the folder and none found by the reviewers' (incomplete) searches.

Nothing was tested and failed on this record, and no source shows the patent being dismissed without testing or suppressed. Current assignee is QM Power Inc; the patent has expired [web: https://patents.google.com/patent/US6246561B1/en, via physicist.md and contrarian.md].

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7. What a learner should take away

Solid. Flux steering by coils is real and useful; it is how latching relays, polarised solenoids and hybrid stepper motors work. Flynn's force formula and his quarter-force arithmetic are right. The fixed-armature "transformer" really would induce secondary voltage by Faraday's law. A magnet-biased device really does hold its armature with no current at all.

Wrong, and why it is instructive. - Flux is not cost. A coil's electrical cost is set by the MMF it must supply, and MMF depends on the full loop its flux travels. Counting "coil flux = Φ₀" while forgetting that this flux must return through the other leg produces a spurious factor of two. Always close the loop. - Force is not work. A larger force at one position says nothing about energy per cycle. Work is ∫F dx, and a conservative magnet returns zero net energy over a closed cycle. - Flux is not power. In the transformer version the magnet supplies large flux but no net power; the secondary's load is paid for by the control coils. - Hidden idealisation. "Flux doubles, force ×4" requires 𝓡_m ≫ 𝓡, i.e. tight armatures. With an air gap the gain drops toward 1. - Stored energy is not saved energy. The "twice" figure reappears (1.99) as the energy needed to build the field, but that energy comes back when the current falls; against FIG. 14 the coil MMF and I²R ratio is exactly 1 in the linear model. - A magnet is not a flux limiter. Flynn says a magnet is "near saturation B_r" and caps added coil flux at a "B_max" unless a bypass is provided [read: page image p.36]. On the recoil line (μ_r ≈ 1.05) the magnet is just an MMF source plus an air-like reluctance, so an aiding coil pushes B above B_r until the iron saturates. FIG. 16C fits that picture: the opposing-side flux falls linearly, passes through zero at the "critical point" 393, then reverses, while only the aiding side bends over, as iron saturation predicts [read: page image p.12; arbiter.md]. The bypass embodiments (FIGS. 11A-12E, 15-16C; bypass claims 7, 8 and 37) are still good design, because coil flux prefers an iron path to the magnet's air-like reluctance. Arbiter verdict on claim 7: Refuted by calculation as magnet physics, Established as ordinary bypass-path advice. One cost the patent does not discuss: with the coils off, a bypass short-circuits part of the magnet's flux away from the armature.

What the patent actually claims. None of the 42 claims (13 independent) states efficiency, energy gain, output above input, or "twice"; those appear only in the description. The claims cover structure, timing and drive modes (opposing-only, claim 16; aiding plus opposing, claim 18; push-pull, claim 19), and the motor drives of FIGS. 40-44 are ordinary position-synchronised commutation that the patent itself calls "circuitry common in the art" [read: pp.41, 43, 45-49, via arbiter.md Round 2]. What is unusual is the magnetic circuit being switched.

Genuinely open. Whether a well-built Flynn-type actuator or motor, with real nonlinear iron and leakage, saves any coil energy over a good coil-only or conventional design over a full cycle, including holding-phase coil power. No measurement is on record here. Whether QM Power or Magnetic Revolutions ever published efficiency data is unknown (search failed).

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8. Check yourself

1. Same flux, two equal paths instead of one. By what factor does the total pull on both armatures change, and by what factor the pull on each?
2. In the two-leg circuit with 𝓡_m = 𝓡, a coil blocks leg 1. By what factor does leg 2's flux rise, and its force?
3. Why does the blocking coil need 2Φ₀𝓡 (when 𝓡_m ≫ 𝓡) rather than Φ₀𝓡, even though the flux it must cancel is only Φ₀?
4. In the fixed-armature version, a 100-turn secondary sees its leg's flux go from 0 to 1.2×10⁻⁴ Wb in 1 ms. What average EMF appears, and where does the energy delivered to a load come from?
5. A device doubles the holding force for the same coil current. Does that prove it gives more work per joule of electrical input? What would you have to measure?

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9. Further reading

Mainstream: - D. J. Griffiths, Introduction to Electrodynamics: ch. 5 (magnetostatics), ch. 6 (magnetic fields in matter), ch. 7 (Faraday's law, energy in magnetic fields), ch. 8 (Maxwell stress). - A. E. Fitzgerald, C. Kingsley, S. D. Umans, Electric Machinery: ch. 1 (magnetic circuits, permanent magnets), and the chapter on electromechanical energy conversion (force from co-energy, energy over a cycle). - E. P. Furlani, Permanent Magnet and Electromechanical Devices (Academic Press, 2001) [memory: high that it exists]: magnet operating points and magnetic-circuit design of actuators. - Wikipedia, "Magnetic circuit" (https://en.wikipedia.org/wiki/Magnetic_circuit) and "Latching relay" (https://en.wikipedia.org/wiki/Latching_relay): short, reliable introductions.

The researcher: - C. J. Flynn, US 6,246,561 B1 (2001), this folder. Key passages: PDF p.36 (quarter force), p.37 (blocking, "half the power"), p.39 (FIG. 14 comparison, "twice"), p.44 (power conversion). - Earlier Flynn patents cited on its front page: US 5,254,925; 5,304,881; 5,455,474; 5,463,263; 5,710,493; 5,753,990 [read: p.1]; not in the folder. The front page also cites Sturman (US 3,683,239; 3,743,898) and Monroe (US 3,670,189), latching and permanent-magnet actuator patents, as prior art [read: p.1]; they are the earlier "established engineering" behind the flux-switching principle, though none was deep-read here.

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Answers

1. Each armature: B halves, force goes to (½)² = ¼. Total over both: 2 × ¼ = ½.
2. Gain = (𝓡 + 2𝓡_m)/(𝓡 + 𝓡_m) = 3𝓡/2𝓡 = 1.5. Force ratio 1.5² = 2.25, not 4.
3. The coil's flux has to make a closed loop. It goes down leg 1 and must come back through leg 2 (the magnet branch has a much higher reluctance), so the loop reluctance is about 2𝓡 and the MMF is Φ₀ × 2𝓡. Equivalently, it must hold the node potential U = Φ₂𝓡 ≈ 2Φ₀𝓡. That is exactly the MMF a coil alone would need to drive 2Φ₀ through one leg.
4. ℰ = N ΔΦ/Δt = 100 × 1.2×10⁻⁴ / 1×10⁻³ = 12 V average. The energy delivered to the load comes from the control-coil supply: load current in the secondary adds MMF the control coils must cancel. The magnet only provides a static bias and gives no net energy over a cycle.
5. No. Force at one position is not work over a stroke, and a magnet can give back on one stroke only what it takes on another. You would have to measure electrical energy in (including holding-phase power) and mechanical work out over complete cycles, for this device and for a coil-only device of equal force and stroke.