Parallel Co-Winding in Superconducting Magnets

• Parallel co-winding is a winding architecture where multiple superconducting conductors are co-wound to maximize mutual inductance and enable controlled current redistribution.
• It is applied in fast quench protection and charging delay reduction by exploiting deliberate inter-conductor coupling and precise voltage sharing.
• Key design challenges include ensuring conductor symmetry, balancing terminal resistances, and mitigating circulating currents to reduce losses and enhance stability.

Searching arXiv for the cited papers and closely related work on parallel co-winding / bifilar coil winding. arxiv_search.query({"^^^^^^^^5search_query5 Coil Winding for Fast Quench Protection\" OR ^^^^^^^^5all:\5 and numerical study on current distribution in parallel co-wound no-insulation coils\" OR ^^^^^^^^5all:\5 conformal REBCO dipole for a next hadron collider\" OR ^^^^^^^^5all:\5 Currents in Windings: Fundamental Property\"","start":^^^^5search_query5^^^^,"max_results":^^^^5all:\5search_query5 arxiv_search.query({"^^^^^^^^5search_query5 co-wound\" OR ^^^^^^^^5all:\5 coil winding\" OR ^^^^^^^^5all:\5 superconducting magnet","start":^^^^5search_query5^^^^,"max_results":^^^^5all:\5search_query5 arxiv_search.query({"^^^^^^^^5search_query5 OR id:(&&&^^^^5all:\5^^^^&&&) OR id:(&&&^^^^5 OR all:\5^^^^&&&) OR id:(&&&^^^^5 OR all:\5^^^^&&&)","start":^^^^5search_query5^^^^,"max_results": Parallel co-winding denotes a class of winding architectures in which two or more conductors are wound side-by-side, turn-for-turn, or assembled as a tightly coupled tape stack so that their electromagnetic states remain strongly linked. In superconducting magnets, the concept appears in at least three distinct but related forms: bifilar or “two-in-hand” co-winding for fast quench protection, parallel co-wound no-insulation REBCO coils for reduced charging delay, and conformal multi-tape REBCO cables that exploit controlled inter-tape current sharing while maintaining favorable field orientation (&&&^^^^5search_query5^^^^&&&). Across these variants, the central technical problem is the same: parallel conductors can either be used deliberately to obtain near-zero differential inductance, dynamic current redistribution, and reduced charging delay, or they can develop non-uniform current distribution and circulating currents that increase losses and create thermal or mechanical stability issues (&&&^^^^5all:\5^^^^&&&).

^^^^5all:\5^^^^. Conceptual scope and terminology

In the bifilar protection formulation, two superconducting cables are wound side-by-side, turn-for-turn, so that their fields are very tightly coupled, with coupling coefficient PRESERVED_PLACEHOLDER_^^^^5search_query5^^^^–PRESERVED_PLACEHOLDER_^^^^5all:\5 and with self-inductances PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^ (&&&^^^^5search_query5^^^^&&&). In normal operation the two windings are connected in series so that their amp-turns add; in protection mode they are re-bussed so that they share the same end-points but currents flow in opposite senses around the turns. The same geometric co-location that gives a large mutual inductance in normal operation is therefore repurposed to produce a very small differential inductance in protection mode.

In parallel co-wound no-insulation REBCO coils, the conductors are not merely magnetically coupled; they are also electrically coupled through turn-to-turn contacts. The ^^^^5 OR all:\5search_query5 OR all:\55^^^^ study distinguishes a single-tape no-insulation coil, a dual-tape co-wound coil with ^^^^5 OR all:\5^^^^ tapes in parallel per turn, and a quad-tape co-wound coil with ^^^^5 OR all:\5^^^^ tapes in parallel per turn, all immersed in liquid nitrogen at 77 K (&&&^^^^5all:\5^^^^&&&). Here the motivation is not quench triggering but reduction of charging delay while maintaining thermal stability.

In the conformal REBCO dipole design, each cable turn is built from ^^^^5 OR all:\55^^^^ commercially available 6 mm-wide REBCO tapes pressed face-to-face into a rectangular stack. The tapes are arranged so that the plane of the tapes follows a level-surface of the dipole’s ^^^^5 OR all:\5^^^^D field lines, and the cable is surrounded by a thin laminar spring that applies a uniform radial compression of PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^ MPa (&&&^^^^5 OR all:\5^^^^&&&). This is a co-winding architecture in a broader sense: the tapes are parallel-connected within a shared mechanical and electromagnetic envelope, and current is intended to redistribute dynamically among them.

These implementations differ in purpose, but they share two defining attributes. First, multiple conductors are placed in close proximity so that mutual coupling or inter-conductor contact is deliberately large. Second, the system-level behavior depends critically on how the parallel branches share voltage, current, and inductive energy. This suggests that “parallel co-winding” is best understood as an architectural family rather than a single circuit topology.

^^^^5 OR all:\5^^^^. Electromagnetic basis: coupled inductance, redistribution, and loss

For two tightly coupled coils, the full two-coil inductance matrix yields two natural modes (&&&^^^^5search_query5^^^^&&&). In the series-additive mode,

PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^

where $M = k\sqrt{L_1L_2}$. In the differential or anti-parallel mode,

$L_{\mathrm{diff}} = L_1 + L_2 - 2M.$

When $L_1 \approx L_2 \equiv L$ and $k \to 1$, one has $M \to L$, and therefore

PRESERVED_PLACEHOLDER_^^^^5all:\5search_query5^^^^

In practice the remaining leakage inductance is

PRESERVED_PLACEHOLDER_^^^^5all:\5all:\5^^^^

which can be on the order of ^^^^5all:\5^^^^–^^^^5all:\5search_query5 of PRESERVED_PLACEHOLDER_^^^^5all:\5 OR all:\5^^^^.

The same general logic reappears in parallel-connected windings outside superconducting magnets. For PRESERVED_PLACEHOLDER_^^^^5all:\5 OR all:\5^^^^ conductors connected in parallel between two common end-nodes, if PRESERVED_PLACEHOLDER_^^^^5all:\5 OR all:\5^^^^ is the total injected current, then

PRESERVED_PLACEHOLDER_^^^^5all:\55^^^^

Writing each branch current as

PRESERVED_PLACEHOLDER_^^^^5all:\5^^^^6

the formal definition given for circulating currents is that they occur if and only if there exists some branch for which PRESERVED_PLACEHOLDER_^^^^5all:\5^^^^7 (&&&^^^^5 OR all:\5^^^^&&&). Under common end-node voltage, any impedance asymmetry PRESERVED_PLACEHOLDER_^^^^5all:\5^^^^8 implies PRESERVED_PLACEHOLDER_^^^^5all:\5^^^^9, and therefore circulating currents must flow.

The loss consequence is explicit. If PRESERVED_PLACEHOLDER_^^^^5 OR all:\5search_query5^^^^ denotes branch resistance, then the total ohmic loss is

PRESERVED_PLACEHOLDER_^^^^5 OR all:\5all:\5^^^^

while the ideal equal-share loss is

PRESERVED_PLACEHOLDER_^^^^5 OR all:\5 OR all:\5^^^^

The excess loss due solely to current unbalance is

PRESERVED_PLACEHOLDER_^^^^5 OR all:\5 OR all:\5^^^^

For equal resistances PRESERVED_PLACEHOLDER_^^^^5 OR all:\5 OR all:\5^^^^,

PRESERVED_PLACEHOLDER_^^^^5 OR all:\55^^^^

A common misconception is that parallelization alone guarantees benign current sharing. The general winding theory shows the opposite: symmetry is a requirement, not a consequence, of parallel connection. The superconducting studies provide concrete manifestations of this principle in regimes where mutual inductance, contact resistance, and nonlinear PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^6–PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^7 behavior dominate.

^^^^5 OR all:\5^^^^. Bifilar co-winding for fast quench protection

The ^^^^5 OR all:\5search_query5 OR all:\5 OR all:\5^^^^ bifilar protection concept was introduced as a response to the fact that, as superconducting magnet technology is pushed towards higher performance, energy density and total stored energy follow exponentially, and protecting magnets becomes substantially more challenging with traditional methods being stretched to their limits (&&&^^^^5search_query5^^^^&&&). New technologies such as CLIQ, or Coupling Loss Induced Quench, promise a robust method to protect advanced magnets, but become inductance limited in large magnet strings or at low field. The bifilar approach addresses that limitation by winding coils in a bifilar fashion and connecting them in series for typical operation, while providing an anti-parallel connection for quasi-zero-inductance in a protection case.

In protection mode, the circuit is essentially a low-PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^8, low-PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^9 loop driven by the quench-trigger capacitor or other source, with governing equation

PRESERVED_PLACEHOLDER_^^^^5 OR all:\5search_query5^^^^

If PRESERVED_PLACEHOLDER_^^^^5 OR all:\5all:\5^^^^ is tiny, then

PRESERVED_PLACEHOLDER_^^^^5 OR all:\5 OR all:\5^^^^

or more generally,

PRESERVED_PLACEHOLDER_^^^^5 OR all:\5 OR all:\5^^^^

From energy conservation for a capacitor PRESERVED_PLACEHOLDER_^^^^5 OR all:\5 OR all:\5^^^^ charged to PRESERVED_PLACEHOLDER_^^^^5 OR all:\55^^^^ and discharging into PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^6,

PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^7

In the anti-parallel hookup each coil sees half the capacitor current.

The stated advantages over a single-coil quench trigger are specific: ultra-fast PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^8 for a given voltage, very low effective inductance, no net field change in protection mode so that mechanical forces on the bulk coil cancel, large PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^9 in each winding producing high AC and coupling losses, only one power-supply lead pair plus one extra protection lead, and uniform over-current in all turns promoting simultaneous quench and avoiding local hotspots (&&&^^^^5search_query5^^^^&&&). The phrase “no net field change,” however, should not be interpreted as the elimination of all electromechanical effects. The same source notes that adjacent turns still see local attractive or repulsive forces proportional to PRESERVED_PLACEHOLDER_^^^^5 OR all:\5search_query5^^^^ as the protection current rises.

The experimental demonstration used a small REBCO bifilar solenoid with ^^^^5 OR all:\5search_query5^^^^ total turns, ^^^^5all:\5search_query5^^^^ turns in each half-coil, inner radius ^^^^5all:\5search_query5^^^^ mm, measured half-coil inductances PRESERVED_PLACEHOLDER_^^^^5 OR all:\5all:\5^^^^, series inductance PRESERVED_PLACEHOLDER_^^^^5 OR all:\5 OR all:\5^^^^, anti-parallel inductance PRESERVED_PLACEHOLDER_^^^^5 OR all:\5 OR all:\5^^^^, and REBCO critical current PRESERVED_PLACEHOLDER_^^^^5 OR all:\5 OR all:\5^^^^ at 77 K of 96 A per conductor (&&&^^^^5search_query5^^^^&&&). With a ^^^^5 OR all:\5search_query5search_query5^^^^ PRESERVED_PLACEHOLDER_^^^^5 OR all:\55^^^^F film capacitor charged to 5^^^^5search_query5^^^^ V, the oscillation frequency was PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^6 kHz, peak PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^7 MA/s, and peak coil boost PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^8 A in PRESERVED_PLACEHOLDER_^^^^5 OR all:\5^^^^9s. With a ^^^^5 OR all:\5search_query5^^^^ mF capacitor at 5^^^^5search_query5^^^^ V, the discharge was non-oscillatory, the initial peak $M = k\sqrt{L_1L_2}$^^^^5search_query5^^^^ was $M = k\sqrt{L_1L_2}$^^^^5all:\5^^^^ MA/s, the peak boost reached $M = k\sqrt{L_1L_2}$^^^^5 OR all:\5^^^^ A in $M = k\sqrt{L_1L_2}$^^^^5 OR all:\5^^^^s, and coil resistance developed to $M = k\sqrt{L_1L_2}$^^^^5 OR all:\5^^^^ in a few $M = k\sqrt{L_1L_2}$5s before decaying as current fell. Simulations reported quenches in $M = k\sqrt{L_1L_2}$6s from the combination of large AC losses and instantaneous over-current of $M = k\sqrt{L_1L_2}$7 locally, and the coil recovered fully once the protection current decayed below the short-sample limit.

^^^^5 OR all:\5^^^^. Parallel co-wound no-insulation coils and current-distribution physics

The ^^^^5 OR all:\5search_query5 OR all:\55^^^^ experimental and numerical study on current distribution in parallel co-wound no-insulation coils addresses a different operating regime: charging rather than triggered protection (&&&^^^^5all:\5^^^^&&&). The test articles were small REBCO coils: Coil A with a single ^^^^5 OR all:\5^^^^ mm tape and 6^^^^5search_query5^^^^ turns, Coil B with ^^^^5 OR all:\5^^^^ tapes in parallel per turn and ^^^^5 OR all:\5search_query5^^^^ turns, and Coil C with ^^^^5 OR all:\5^^^^ tapes in parallel per turn and ^^^^5all:\55^^^^ turns. All had inner diameter ^^^^5all:\5search_query5search_query5^^^^ mm and outer diameter ^^^^5all:\5all:\5all:\5^^^^.^^^^5 OR all:\5^^^^ mm, were wound on G^^^^5all:\5search_query5^^^^ formers, and were immersed in liquid nitrogen at 77 K. To measure individual tape currents, the tapes were separated only in the input and output sections, and Rogowski coils were used to recover each tape’s instantaneous current through

$M = k\sqrt{L_1L_2}$8

The coupled modeling framework combined a ^^^^5 OR all:\5^^^^D axisymmetric T–A formulation with an equivalent circuit model. In the superconducting sheets,

$M = k\sqrt{L_1L_2}$9

with nonlinear resistivity from the $L_{\mathrm{diff}} = L_1 + L_2 - 2M.$^^^^5search_query5^^^^–Ldiff=L1+L2−2M.L_{\mathrm{diff}} = L_1 + L_2 - 2M.^^^^5all:\5^^^^ power law

$L_{\mathrm{diff}} = L_1 + L_2 - 2M.$^^^^5 OR all:\5^^^^

where $L_{\mathrm{diff}} = L_1 + L_2 - 2M.$^^^^5 OR all:\5^^^^, and field-dependent $L_{\mathrm{diff}} = L_1 + L_2 - 2M.$^^^^5 OR all:\5^^^^ from the Kim model,

$L_{\mathrm{diff}} = L_1 + L_2 - 2M.$5

Boundary conditions along each superconducting tape took the form

$L_{\mathrm{diff}} = L_1 + L_2 - 2M.$6

In the equivalent circuit, each tape is one branch, terminal resistances $L_{\mathrm{diff}} = L_1 + L_2 - 2M.$7 and $L_{\mathrm{diff}} = L_1 + L_2 - 2M.$8 connect each tape to the common current source, and turn-to-turn contact resistance $L_{\mathrm{diff}} = L_1 + L_2 - 2M.$9 allows radial sharing between adjacent turns.

The principal result was that current distribution during ramping was highly non-uniform, with some tapes carrying reverse currents (&&&^^^^5all:\5^^^^&&&). For Coil B at ^^^^5 OR all:\5^^^^7 A/s, the reported peak imbalance was $L_1 \approx L_2 \equiv L$^^^^5search_query5^^^^ A and $L_1 \approx L_2 \equiv L$^^^^5all:\5^^^^ A, and one tape briefly carried reverse current of several tens of amperes. For Coil C at ^^^^5 OR all:\5^^^^7 A/s, the inner and outer tapes showed the largest swings, while the central tapes were smoother, and reverse currents up to $L_1 \approx L_2 \equiv L$^^^^5 OR all:\5^^^^ A appeared in outer tapes at the coil ends. Agreement between measurements and model was reported within $L_1 \approx L_2 \equiv L$^^^^5 OR all:\5^^^^–L1≈L2≡LL_1 \approx L_2 \equiv L^^^^5 OR all:\5^^^^ for peak currents, time constants, and voltage waveforms.

The study also compared four insulation schemes for a dual-tape coil: NI, MI, INS, and F-INS. NI, with direct turn-to-turn and tape-to-tape contacts, produced the largest reverse currents at the innermost and outermost turns. INS, where tapes within a turn remain in contact but turns are insulated, gave moderate reverse currents. F-INS, where all tapes are insulated from each other, produced only a small initial reverse current due to mutual inductance imbalance and then monotonic positive current (&&&^^^^5all:\5^^^^&&&). The interpretation offered for NI coils is mechanistic: large radial currents between turns force systematic current redistribution from the outer tape of turn $L_1 \approx L_2 \equiv L$5 to the inner tape of turn $L_1 \approx L_2 \equiv L$6, yielding opposite-sign currents at coil ends.

Steady-state current distribution was dominated by branch resistances $L_1 \approx L_2 \equiv L$7, because the superconducting layer is negligible at low current. The example given for Coil B used optimized terminal resistances $L_1 \approx L_2 \equiv L$8 n$L_1 \approx L_2 \equiv L$9 and $k \to 1$^^^^5search_query5^^^^ n$k \to 1$^^^^5all:\5^^^^, producing a predicted steady current split of approximately ^^^^5 OR all:\5all:\5^^^^ A and ^^^^5 OR all:\5^^^^9 A versus measured values of ^^^^5 OR all:\5 OR all:\5^^^^.6 A and ^^^^5 OR all:\5^^^^6.8 A (&&&^^^^5all:\5^^^^&&&). Higher uniform terminal resistance reduced $k \to 1$^^^^5 OR all:\5^^^^ during ramping by enlarging the resistive share relative to the inductive share, whereas non-uniform terminals amplified imbalances, especially at low ramp rates where inductive impedance is low.

5. Conformal REBCO co-winding and dynamic current sharing

The hybrid conformal REBCO dipole extends parallel co-winding to a high-field accelerator-magnet design in which geometry, contact resistance, and current redistribution are co-optimized (&&&^^^^5 OR all:\5^^^^&&&). Each conformal insert cable turn is built from ^^^^5 OR all:\55^^^^ REBCO tapes pressed face-to-face into a rectangular stack ^^^^5all:\5^^^^ mm thick and 6 mm wide. A thin laminar spring applies approximately ^^^^5all:\5^^^^ MPa radial compression. Under this compression, the contact resistivity between neighboring tapes was measured as

$k \to 1$^^^^5 OR all:\5^^^^

The effective inter-tape parallel resistance per unit length is then

$k \to 1$^^^^5 OR all:\5^^^^

The winding is “conformal” in the sense that each turn is bent so that the plane of the tapes follows a level-surface of the dipole’s ^^^^5 OR all:\5^^^^D field lines, keeping the tape face nearly parallel to $k \to 1$5 throughout the insert. In the body of the dipole, the insert is arranged in three nested shells of ^^^^5all:\5^^^^6 such cables each, and Sampson’s twisted-flare method is used in the ends so that no tape sees a hard bending condition. Maximum misalignment is reported as less than $k \to 1$6 even in the highly flared end turns (&&&^^^^5 OR all:\5^^^^&&&).

The dynamic current-sharing model follows Noguchi (^^^^5 OR all:\5search_query5all:\5^^^^9). For tape $k \to 1$7,

$k \to 1$8

Each tape loop has self-inductance per unit length

$k \to 1$9

and the homogenization time constant between adjacent tapes is

$M \to L$^^^^5search_query5^^^^

Over the approximately 5^^^^5search_query5^^^^–^^^^5all:\5search_query5search_query5^^^^ m of one turn, this was described as fast enough that, as cable current ramps, the outer tape first approaches its own $M \to L$^^^^5all:\5^^^^, then shares into the next tape, and so on, without local hot-spots.

The principal electromagnetic payoff is anisotropy management. At ^^^^5 OR all:\5search_query5^^^^ K and ^^^^5all:\5search_query5^^^^ T, the reported critical current density is approximately $M \to L$^^^^5 OR all:\5^^^^ for $M \to L$^^^^5 OR all:\5^^^^ and approximately $M \to L$^^^^5 OR all:\5^^^^ for $M \to L$5, giving a ratio of about $M \to L$6 (&&&^^^^5 OR all:\5^^^^&&&). By maintaining $M \to L$7 everywhere, the design aims to use the full peak $M \to L$8 of the tape. The field solution places ^^^^5 OR all:\5^^^^8 body cables plus a sextupole correction turn so that the total bore field is ^^^^5all:\5^^^^8.^^^^5 OR all:\5^^^^ T at ^^^^5 OR all:\5^^^^.^^^^5 OR all:\5^^^^ K, with the insert contributing 8 T and the outsert ^^^^5all:\5search_query5^^^^ T. This is a markedly different objective from bifilar protection or NI charging-delay reduction, yet it relies on the same underlying co-winding premise: controlled interaction among parallel superconducting elements.

6. Design constraints, misconceptions, and broader significance

The design constraints that recur across the literature are highly consistent. In bifilar protection, conductor placement must maximize $M \to L$9 by minimizing spacing between the two windings, the coupling coefficient should target PRESERVED_PLACEHOLDER_^^^^5all:\5search_query5search_query5^^^^, lead routing should minimize loop area outside the winding, and diode or switch losses must be included in PRESERVED_PLACEHOLDER_^^^^5all:\5search_query5all:\5^^^^ when estimating PRESERVED_PLACEHOLDER_^^^^5all:\5search_query5 OR all:\5^^^^ and peak current (&&&^^^^5search_query5^^^^&&&). Even when net Lorentz force on the coil is near zero, the protection lead can carry large currents for a few milliseconds, and adjacent turns can experience local attractive or repulsive forces as the protection current rises.

In parallel co-wound NI coils, the principal trade-off is explicit: increasing the number of parallel tapes lowers coil inductance in proportion to PRESERVED_PLACEHOLDER_^^^^5all:\5search_query5 OR all:\5^^^^ and therefore accelerates ramping, but more tapes amplify sensitivity to uneven terminal or joint resistances, risking thermal hot spots and reverse currents (&&&^^^^5all:\5^^^^&&&). The practical guidelines given are correspondingly specific: use uniform terminal solder joints, target moderately higher but uniform terminal or joint resistance of a few hundred nPRESERVED_PLACEHOLDER_^^^^5all:\5search_query5 OR all:\5^^^^, consider MI rather than full NI for large coils, optimize ramp rate, insulate tapes within turns to break reverse-loop paths, and use T–A field-circuit coupled modeling with small-scale mock-ups and Rogowski measurements before full-scale fabrication.

The more general winding literature sharpens the interpretation of these findings. Parallel-connected windings reduce DC resistance and improve current-carrying capacity, but any imperfection in DC resistance, inductance, capacitance, or imposed potential forces circulating currents that unbalance the current share and strictly increase the bundle’s losses (&&&^^^^5 OR all:\5^^^^&&&). In a case study of a surface-PM synchronous machine, strand currents deviated by PRESERVED_PLACEHOLDER_^^^^5all:\5search_query55^^^^–PRESERVED_PLACEHOLDER_^^^^5all:\5search_query5 around PRESERVED_PLACEHOLDER_^^^^5all:\5search_query5^^^^7, local loop currents reached up to ^^^^5all:\55^^^^% of the phase-current amplitude, total losses increased by approximately ^^^^5all:\5 OR all:\5^^^^%, and local hotspots of 5–^^^^5all:\5search_query5^^^^ K were predicted. Although that study concerns conventional electrical machines rather than superconducting magnets, the formal result that asymmetry implies circulating currents is directly relevant to co-wound superconducting systems.

A common misconception is that parallel co-winding is inherently self-equalizing. The available evidence supports a narrower statement. Dynamic sharing can be beneficial when contact resistivity, geometry, and terminal resistances are intentionally controlled, as in the conformal REBCO cable or in optimized NI designs (&&&^^^^5 OR all:\5^^^^&&&). Near-zero differential inductance can be highly advantageous when the explicit objective is fast quench triggering, as in the bifilar protection scheme (&&&^^^^5search_query5^^^^&&&). But neither outcome is automatic. A plausible implication is that parallel co-winding should be treated less as a single technique than as a design space whose success depends on whether the desired current redistribution pathways are the same pathways that the actual electromagnetic and resistive asymmetries will excite.