Electronics Guide

A Straight Wire

At a point on the perpendicular bisector of a straight segment of length L, a distance r from it, integrating along the segment gives

B = μ₀IL / (2πr√(L² + 4r²))

As L grows without limit, this becomes the field of a long straight wire:

B = μ₀I / (2πr)

The field lines are circles centered on the wire; with the right thumb pointing along the current, the fingers curl in the direction of B. A wire carrying 10 A produces 200 µT at 1 cm and 20 µT at 10 cm, where it is comparable to Earth's field. A segment only twice as long as the distance r already gives 71 percent of the long-wire value.

A Circular Loop

On the axis of a circular loop of radius R, at a distance z from its center, the field components perpendicular to the axis cancel by symmetry and the axial components add:

B = μ₀IR² / [2(R² + z²)^3/2]

The field points along the axis in the direction the right thumb points when the fingers curl with the current. At the center, B = μ₀I/(2R), so 1 A in a loop of 1 cm radius produces about 63 µT, and N closely wound turns multiply that by N.

The Magnetic Dipole

Far from a small current loop, its field depends only on its magnetic dipole moment m = NIA, in ampere square meters, where A is the loop area; m points along the loop's axis by the same right-hand rule. On the axis, at a distance r much greater than the loop's size,

B = μ₀m / (2πr³)

pointing along m. In the plane of the loop the field has half this magnitude and points opposite to m. As with the electric dipole, the field falls as the inverse cube of distance: the 1 cm loop above, carrying 1 A, produces about 63 nT on its axis at 10 cm and 8 nT at 20 cm. Small current loops on a circuit board are dipoles of this kind, so magnetic coupling between them grows with their areas and falls quickly with separation.

The Integral Form

For steady currents, the line integral of H around any closed path C equals the free current passing through any surface bounded by the path:

∫_C H · dl = I_enc

With the fingers of the right hand curling along the direction of travel around C, the thumb points in the direction of positive current. A coil of N turns threading the path contributes NI, which is why magnetomotive force is counted in ampere-turns. The law holds for every path, but it yields H directly only when symmetry provides a path along which H is either tangent with constant magnitude or perpendicular to the path.

For a long straight conductor of radius a carrying current I, a circle of radius r outside it encloses the whole current, so H × 2πr = I and H = I/(2πr), matching the Biot–Savart result. A circle inside encloses the fraction r²/a² of a uniform current, so H = Ir/(2πa²), and that internal field gives the wire its internal inductance.

The Solenoid and the Toroid

A long, tightly wound solenoid with n turns per meter has a nearly uniform axial field inside and almost none outside. A rectangular path with one long side inside the coil, parallel to its axis, and the other outside encloses nI for each meter of its inside side, so inside the coil

B = μ₀nI

independent of position across the bore; 1,000 turns per meter carrying 1 A give about 1.26 mT.

In a toroid of N turns, a circle of radius r within the core encloses the current NI, so

H = NI / (2πr)

and B = μH, where μ is the core's permeability. The field is strongest at the inner radius, where a toroidal core saturates first. A circle outside an ideal toroid encloses no net current, so the field there is zero, which is why toroidal inductors produce little stray field.

The Coaxial Cable

A coaxial cable carries current I on an inner conductor of radius a and returns it on a shield with inner radius b and outer radius c. With the current spread uniformly over each conductor, circular paths give H in four regions:

• Inside the inner conductor, with r less than a: H = Ir/(2πa²).
• Between the conductors: H = I/(2πr).
• Within the shield: H = I(c² − r²) / [2πr(c² − b²)].
• Outside the shield, with r greater than c: the enclosed current is zero, so H = 0.

The field circles the axis in the direction set by the inner conductor's current. When the shield carries the full return current, the cable produces no external magnetic field. If part of the return current flows elsewhere, such as through a chassis, a path around the cable encloses a net current, and the cable produces that current's field.

The Differential Form and Its Limits

Applied to a vanishingly small loop, Ampère's law becomes

curl H = J

where curl H measures the circulation of H per unit area; in Cartesian coordinates its z component, for example, is ∂H_y/∂x − ∂H_x/∂y.

The divergence of any curl is zero, so the law requires div J = 0, the field form of Kirchhoff's current law. Where charge builds up, as on the plates of a charging capacitor, the law fails. James Clerk Maxwell repaired it by adding the displacement current density, the rate of change of the electric flux density D, to give curl H = J + ∂D/∂t.

Faraday's law marks the other boundary: a changing flux induces a circulating electric field, curl E = −∂B/∂t, which couples the electric and magnetic fields. Magnetostatic results stay useful while a structure is much smaller than a wavelength and while eddy currents in nearby conductors are negligible or handled separately. At high frequency, where the skin effect crowds current toward conductor surfaces, inductance can still be found magnetostatically by placing the currents on those surfaces.

Magnetization and the Two Magnetic Fields

Electrons carry magnetic moment from their orbital motion and their spin. In most materials these moments cancel or point randomly, but an applied field partly aligns them. The magnetic dipole moment per unit volume is the magnetization M, in amperes per meter. Aligned atomic moments act like tiny current loops: where M is uniform, their currents cancel inside the material and leave a bound current circulating on the surface, with surface density M × n, where n is the outward normal. Where M varies, a bound current density curl M also flows inside. Flux density responds to free and bound currents alike:

B = μ₀(H + M)

so that curl H = J counts only free current, the current in windings that an engineer controls. The split mirrors D = ε₀E + P in electrostatics: the ampere-turns fix the circulation of H, while B, which exerts force and induces voltage, also depends on the material. In a linear, isotropic material, M = χ_mH, where χ_m is the magnetic susceptibility, and

B = μ₀(1 + χ_m)H = μ₀μ_rH = μH

The relative permeability μ_r = 1 + χ_m is the factor by which a material multiplies the flux, and so the inductance, of a winding whose field it completely fills. Older literature gives H in oersteds; NIST Special Publication 811 lists 1 Oe as about 79.58 A/m, which equals 1,000/(4π) A/m.

Diamagnetic, Paramagnetic, and Ferromagnetic Materials

Most materials respond so weakly that μ_r differs from 1 by parts per hundred thousand. Diamagnetic materials develop moments that oppose the applied field, a response every material has, while paramagnetic materials, whose atoms carry permanent moments, align slightly with it. Georgia State University's HyperPhysics tabulates χ_m at 20 °C as −1.0 × 10⁻⁵ for copper and −0.91 × 10⁻⁵ for water, against +2.2 × 10⁻⁵ for aluminum. For field calculations, copper, aluminum, and ordinary insulators are therefore nonmagnetic. A superconductor, which expels the field from its interior, is the extreme case: a perfect diamagnet, with χ_m = −1.

In iron, nickel, cobalt, and many of their alloys, a quantum-mechanical exchange interaction aligns neighboring atomic moments spontaneously within regions called domains. In an unmagnetized sample the domains largely cancel. An applied field enlarges the domains aligned with it and rotates the others, producing a magnetization many orders of magnitude larger than a paramagnet's. Above the Curie temperature, about 1043 K (770 °C) for iron, thermal agitation destroys the ordering. Ferrites are ferrimagnetic: two sublattices with opposing but unequal moments leave a net magnetization, and their high electrical resistivity suits them to high frequencies.

Hysteresis and Saturation

A ferromagnet's B is neither proportional to H nor a single-valued function of it. Raising H from zero in an unmagnetized sample traces the initial magnetization curve, along which the ratio B/(μ₀H) gives the initial relative permeability near the origin and the maximum relative permeability where that ratio peaks. At high H nearly all the moments align, and B then rises only as fast as μ₀H: the material saturates. Reducing H to zero leaves a remanent flux density B_r, driving H negative to the coercive field H_c returns B to zero, and cycling H traces the closed hysteresis loop.

The table converts HyperPhysics values from oersteds and gauss. Soft magnetic materials, with small coercive fields and narrow loops, suit cores and shields, and their highest permeabilities appear only after careful annealing. Hard materials suit permanent magnets because they resist demagnetization: the coercive field of Alnico 5 is more than 10,000 times that of 78 Permalloy.

The energy supplied per unit volume as B changes is the integral of H dB, so the area of the loop, in joules per cubic meter, is the energy converted to heat in each cycle, the hysteresis part of core loss. Permeability also varies with flux level, frequency, temperature, and mechanical stress, so a single value of μ_r is an idealization. Magnetic Materials covers practical core and magnet materials and their loss and temperature behavior.

Images in Iron and in Conductors

The method of images works here too. A long wire carrying current I at height h, parallel to the flat surface of a material with relative permeability μ_r, produces above the surface the field of itself plus an image current I(μ_r − 1)/(μ_r + 1) at depth h, flowing in the same direction, so the wire is drawn toward the iron. A thick, highly conductive plane does the opposite when the wire carries high-frequency alternating current: induced eddy currents keep the field out, the image current is equal and opposite, and the wire is repelled. That second case describes a circuit-board plane at high frequency, as the applications section shows.

Solenoids and Toroids

For an air-core solenoid of N turns, length l much greater than its radius, and cross-sectional area A, every turn links the flux μ₀NIA/l, so

L = μ₀N²A / l

Inductance grows as the square of the turns because each added turn both adds field and links it. The formula ignores the weaker field near the ends and so overestimates short coils. Harold Wheeler's approximation for single-layer air-core coils, published in the Proceedings of the IRE in 1928, is equivalent to within 0.3 percent to replacing l with l + 0.9r, where r is the coil radius; Wheeler gave its accuracy as about 1 percent for coils longer than 0.8r. For a coil of 100 turns, 5 mm radius, and 50 mm length, the long-solenoid formula gives 19.7 µH and the corrected one 18.1 µH.

For a closely wound toroid of N turns on a linear core of rectangular cross-section with height h, inner radius a, and outer radius b, integrating B = μNI/(2πr) across the section gives

L = μN²h ln(b/a) / (2π)

Transmission Lines and Wire Pairs

For a coaxial line with inner radius a and shield radius b, integrating μI/(2πr) from a to b gives the flux per meter between the conductors, and so the external inductance per unit length:

L′ = (μ/2π) ln(b/a)

The polyethylene-filled line used as an example in Electrostatics, with a = 0.45 mm and b = 1.5 mm, has L′ ≈ 241 nH/m. For any two-conductor line in a uniform medium, with currents on the conductor surfaces, the external inductance and the capacitance per unit length obey

L′C′ = με

so a line's inductance and capacitance are two views of one cross-section. With that line's 104 pF/m, the characteristic impedance √(L′/C′) is about 48 Ω, and the propagation speed 1/√(L′C′) is two-thirds of the speed of light.

Spacing matters logarithmically for wires and linearly for strips. Two wires 0.5 mm in diameter with centers 5 mm apart have about 1.20 µH/m, and moving them to 1 mm apart lowers that only to 0.53 µH/m, while strips 10 mm wide separated by 0.1 mm of insulation have 12.6 nH/m. That is why high-current converters use laminated busbars.

Internal Inductance, Loops, and Partial Inductance

A nonmagnetic round wire carrying uniformly distributed current also links flux inside its own metal, which adds an internal inductance of μ₀/(8π), or 50 nH/m, whatever the radius. The skin effect removes it at high frequency; with uniform current, the bracketed terms in the loop and segment formulas become ln(8R/a) − 7/4 and ln(2l/a) − 3/4.

A single-turn loop 20 mm in diameter, made of wire 0.5 mm in diameter, has about 47 nH at high frequency. Strictly, inductance belongs only to a closed circuit, because only a closed path bounds a surface for the flux. Circuit-board and package designers nonetheless assign inductance to pieces of a loop, such as vias, bond wires, and trace segments, using partial inductance, a method that A. E. Ruehli set out in the IBM Journal of Research and Development in 1972. The partial self-inductances of the pieces, plus the partial mutual inductances between every pair, sum to the loop inductance. A straight segment 10 mm long and 0.5 mm in diameter has a partial self-inductance of about 7 nH, but a loop containing it may have more or less, depending on where the return current flows.

Mutual Inductance as a Field Quantity

The mutual inductance of two circuits is the flux linkage in one per unit current in the other, M₂₁ = Λ₂₁/I₁. For thin circuits in a uniform medium of permeability μ, Franz Neumann's formula gives it as a double line integral around both circuits:

M = (μ/4π) ∫_C1∫_C2 (dl₁ · dl₂) / R

Here R is the distance between the elements dl₁ and dl₂. The formula is symmetric in the two circuits, so M₁₂ = M₂₁. M depends only on geometry and permeability, its sign depends on the current directions chosen as positive, and the coupling coefficient k = M/√(L₁L₂) never exceeds 1 in magnitude. Circuit models of coupled windings and transformers build on this quantity.

Force on a Current-Carrying Conductor

An element dl of a conductor carrying current I feels a force dF = I dl × B. A straight conductor of length l at an angle θ to a uniform flux density B therefore feels BIl sin θ, directed along l × B, where l points along the current. Motor windings and loudspeaker voice coils work by this force; a loudspeaker's force factor is the product Bl, in newtons per ampere, where l is the length of wire in its magnetic gap.

Forces Between Parallel Conductors

Each of two long parallel wires a distance d apart sits in the other's field, B = μ₀I/(2πd), so the force per unit length on each is

F/l = μ₀I₁I₂ / (2πd)

Currents in the same direction attract, and opposite currents repel. Until May 20, 2019, the SI defined the ampere by this force, as the constant current that, maintained in two straight parallel conductors of infinite length and negligible circular cross-section placed 1 m apart in vacuum, would produce between them a force of 2 × 10⁻⁷ N per meter of length. The force grows as the square of the current, which matters during faults: two round conductors 10 cm apart carrying an instantaneous 50 kA in opposite directions push apart with 5 kN per meter, so busbar supports are designed for the peak short-circuit current.

Magnetic Pressure and Virtual Work

Forces on iron parts and on complicated windings follow most easily from energy. For a single circuit in linear materials, if a part moves a distance dx while the current I is held constant, the force on it, positive in the direction of increasing x, is

F = ½I² dL/dx

so magnetic forces act to increase inductance: a relay armature closes its gap, a plunger is drawn into a solenoid, and a coil's turns are pushed outward and pressed together along its axis. Where flux crosses from high-permeability iron into an air gap, the field pulls the facing surfaces together, whichever way the flux points, with a pressure equal to the energy density in the gap:

p = B²/(2μ₀)

At 1 T that is about 398 kPa, nearly four atmospheres. A relay pole face of 20 mm² with 0.5 T in the gap pulls with about 2 N.

Torque on a Current Loop

A flat coil with dipole moment m = NIA in a uniform field feels no net force but a torque

τ = m × B

of magnitude NIAB sin θ, where θ is the angle between the coil's axis and the field. The torque turns the coil to align m with B, the position of lowest potential energy, −m · B. A 100-turn coil of 1 cm² carrying 10 mA in 0.2 T feels at most 20 µN·m. Moving-coil meters balance this torque against a spring.

Planar and Axisymmetric Problems

When all currents flow perpendicular to a cross-section, as in long conductors, A has only the component A_z along that direction. B then lies in the cross-section, tangent to the lines of constant A_z, so those contours are flux lines, and the flux per meter of depth passing between two points equals the difference of their A_z values. In axisymmetric problems A circles the axis, and contours of r times A are the flux lines. FEMM, a finite element program by David Meeker, solves linear and nonlinear magnetostatic problems in this way on two-dimensional planar and axisymmetric domains. Nonlinear B-H curves let it model saturating iron, at the cost of iterating until the permeability everywhere agrees with the local flux density.

Extracting and Checking Inductance

Solvers find inductance from stored energy, L = 2W/I², or from flux linkage computed with A. Interconnect that no single cross-section represents needs three-dimensional extraction; FastHenry, a multipole-accelerated 3-D inductance extraction program described by M. Kamon, M. J. Tsuk, and J. K. White in IEEE Transactions on Microwave Theory and Techniques in 1994, is one such tool.

The checks described in Electrostatics apply. Also confirm that every current has a return path: a planar model of a single conductor with no return has no finite inductance per meter, so its computed value depends on where the outer boundary lies.

Inductor and Transformer Design

Magnetostatics supplies the core equations of wound-component design: inductance from turns and reluctance, energy stored in the gap, and the peak flux density that must stay below saturation. Since flux linkage is both LI and NBA, the flux density in a core of area A is B = LI/(NA), a figure to check at the largest current the winding will carry.

Transformer leakage inductance is a field calculation too. Take two concentric windings on one core leg, each of breadth b_w along the leg, with thicknesses a₁ and a₂, separated by insulation of thickness d, and with a mean turn length MLT. Under load their ampere-turns cancel, so by Ampère's law H is nearly zero outside the pair and equals NI/b_w in the insulation, rising linearly across one winding and falling across the other. Integrating ½μ₀H² over this region and applying L = 2W/I² gives, for a breadth much larger than the total build, the leakage inductance referred to the N-turn winding:

L_lk = μ₀N²(MLT)(d + (a₁ + a₂)/3) / b_w

For 20 turns, a 60 mm mean turn, a 20 mm breadth, 0.5 mm of insulation, and windings 1 mm thick, L_lk is about 1.8 µH. The levers are fewer turns, broader and thinner windings, thinner insulation, and interleaving, which halves the peak field between winding sections.

Current Sensing and Hall-Effect Sensors

Edwin H. Hall reported in the American Journal of Mathematics in 1879 that a magnetic field acting on a current-carrying conductor produces a voltage across it. The Lorentz force pushes the moving carriers toward one edge of the conductor until the charge that builds up there creates an electric field that balances the magnetic force. For a strip of thickness t carrying current I through a perpendicular flux density B, a simple model with one type of carrier gives the Hall voltage

V_H = IB / (nqt)

where n is the carrier density and q the carrier charge; the polarity of the voltage reveals the carriers' sign. In copper, with 8.47 × 10²⁸ free electrons per cubic meter, a strip 0.1 mm thick carrying 1 A in 1 T develops only about 0.74 µV. A semiconductor layer 10 µm thick with 10²² electrons per cubic meter gives 1/(nqt) ≈ 62 V per ampere-tesla, so a 1 mA bias in a 10 mT field produces about 0.6 mV. That is why Hall elements are made from semiconductors and paired with amplifiers. Basic Sensor Components describes linear Hall sensors and Hall switches.

Ampère's law makes a ring-shaped current sensor insensitive to where the conductor passes through its aperture, because the line integral of H around the ring equals the enclosed current wherever the conductor sits. In an open-loop sensor, a gapped ring of high-permeability material concentrates the field onto a Hall element in the gap: by the magnetic-circuit example, 10 A through the aperture of a core with a 1.5 mm gap gives about 8.2 mT. Core remanence, core nonlinearity, and the element's temperature drift limit its accuracy. A closed-loop, or zero-flux, sensor drives a secondary winding so that its ampere-turns cancel the primary's and the Hall element reads zero. The secondary current, smaller than the primary by the turns ratio, becomes the measurement, and the core stays near zero flux.

Coreless sensors place the element beside the conductor instead. The field 1 cm from a 10 A conductor, 200 µT, is only a few times Earth's field, so these sensors sit within millimeters of the current path or subtract the outputs of two elements, which cancels a uniform stray field. A Rogowski coil, an air-cored winding that encircles a conductor, uses the same Ampère's-law geometry, but its output is the voltage induced by the changing current, so it measures only alternating current.

Magnetic Shielding

A static magnetic field passes through copper and aluminum as if they were absent. Shielding at DC therefore relies on high-permeability material, which the boundary conditions make an easy path: flux entering the wall turns to run along it and bypasses the interior. For a long cylinder of diameter D with a thin wall of thickness t, in a field across its axis, the shielding factor, the ratio of the outside field to the inside field, is approximately

S ≈ 1 + μ_rt / D

when μ_r is large. With an assumed effective μ_r of 20,000, a 1 mm wall, and a 100 mm diameter, S is about 200, or 46 dB. Saturation, seams, holes, open ends, and mechanical stress after annealing all reduce it. Two shells separated by a gap attenuate far more than one shell with the same total wall thickness. At higher frequencies, eddy currents let ordinary conductors shield magnetic fields as well, the regime that Electromagnetic Shielding Fundamentals treats.

Canceling a field at its source often works better than shielding it. Two conductors carrying equal and opposite currents a distance d apart produce, at a distance r much greater than d,

B ≈ μ₀Id / (2πr²)

which falls as the inverse square of distance rather than the inverse first power. A pair carrying 10 A with 1 mm spacing produces 0.2 µT at 10 cm, one-hundredth of the 20 µT from a single conductor. Twisting the pair reverses its residual field every half twist.

Loop Inductance on Circuit Boards

Every switching current flows around a loop whose inductance sets the voltage spike L di/dt. Model a trace as a thin wire at height h above a wide, solid plane, at frequencies high enough that the plane keeps the field out. The return current in the plane mirrors the trace, with a surface current density, at a lateral distance x from the point beneath the trace, of

K(x) = Ih / [π(h² + x²)]

Half the return current flows within a distance h on either side of the trace, 80 percent within 3h, and 94 percent within 10h. This distribution minimizes the stored field energy, and with it the loop inductance, which is why high-frequency return current follows the trace; at DC it instead spreads along the paths of least resistance. Several layout rules follow:

• Keep a solid reference plane beneath fast signals. A slot or split forces the return current to detour around it, enlarging the loop and its inductance.
• Keep switching loops small. Place a switching regulator's input capacitor and switching transistors close together, with the return path on the adjacent layer.
• Use thin dielectrics between power and ground planes. Like the parallel strips above, a plane pair has inductance proportional to its spacing.