Magnetic circuit
A magnetic circuit is a closed path through which magnetic flux flows, typically consisting of a ferromagnetic core such as iron or steel that confines and directs the flux, much like an electric circuit confines electric current.[1][2][3] The fundamental principles of magnetic circuits draw a direct analogy to electric circuits, where magnetomotive force (MMF) acts like electromotive force (voltage), magnetic flux (Φ) acts like current, and reluctance (ℛ) acts like resistance.[1][2][3] MMF is generated by coils carrying current and is quantified as MMF = N × I, where N is the number of turns and I is the current in amperes, yielding units of ampere-turns.[1][2][3] Flux, measured in webers (Wb), relates to MMF via Ohm's law for magnetic circuits: Φ = MMF / ℛ.[1][2] Reluctance, in ampere-turns per weber (A/Wb), depends on the circuit's geometry and material properties, given by ℛ = l / (μ × A), where l is the mean path length, A is the cross-sectional area, and μ is the permeability of the material.[1][2][3] Permeability μ = μ₀ × μᵣ, with μ₀ as the permeability of free space (4π × 10⁻⁷ H/m) and μᵣ as the relative permeability, which is high (e.g., ~4000 for steel) in ferromagnetic materials to concentrate flux but low (~1) in air gaps that introduce significant reluctance.[2][3] Magnetic flux density B (in teslas, T) is then B = Φ / A = μ × H, where H is the magnetic field intensity in A/m.[1][2][3] Magnetic circuits are essential in numerous engineering applications, including transformers for power transfer, electric motors and generators for converting energy between electrical and mechanical forms, relays and solenoids for actuation, and electromagnets for lifting or control systems.[1][3] These devices often incorporate air gaps to adjust reluctance and prevent saturation, where B reaches a maximum (typically 1.5–2 T in common materials), limiting further flux increases despite higher MMF.[2] Analysis of complex circuits uses Kirchhoff's laws adapted for magnetism—sum of MMFs around a loop equals zero, and flux is conserved at junctions—enabling precise design and performance prediction.[1][3]Fundamental Concepts
Magnetomotive force
In magnetic circuits, the magnetomotive force (MMF), denoted as F, is defined as the line integral of the magnetic field strength \mathbf{H} around a closed path:F = \oint \mathbf{H} \cdot d\mathbf{l}. [4]
This quantity represents the total "driving force" that establishes the magnetic field along the path, analogous to electromotive force in electric circuits. The SI unit of MMF is the ampere-turn (At), which arises from the product of current in amperes and the number of turns in a coil.[5]
Historically, in the CGS electromagnetic system, the unit was the gilbert (Gb), defined such that F = 0.4\pi NI where N is the number of turns and I is the current in abamperes; the conversion is $1 Gb \approx 0.7958 At.[5][6] MMF is generated primarily by electric currents in coils, where for a coil with N turns carrying current I, the MMF is F = NI.[7]
This follows from applying the definition around a path encircling the coil, yielding the enclosed current linkage. In Ampere's circuital law, \oint \mathbf{H} \cdot d\mathbf{l} = I_{\text{enc}}, the MMF serves as the source term equal to the total enclosed current I_{\text{enc}} (or NI for multi-turn coils), driving the circulation of \mathbf{H}.[4][5]
Magnetic flux
Magnetic flux is the fundamental quantity in a magnetic circuit that quantifies the total magnetic field passing through a given surface, serving as the analog to electric current in electrical circuits. It is defined mathematically as the surface integral of the magnetic flux density \mathbf{B} over the surface area: \Phi = \int_S \mathbf{B} \cdot d\mathbf{A}, where d\mathbf{A} is the infinitesimal area vector normal to the surface.[8] This integral accounts for both the magnitude and direction of \mathbf{B} relative to the surface, capturing the effective linkage of the magnetic field through the circuit's cross-section.[8] The SI unit of magnetic flux is the weber (Wb), defined such that one weber is the flux linking a single-turn circuit that induces an electromotive force of one volt when reduced to zero uniformly in one second.[9] Since \mathbf{B} is measured in teslas (T), the weber relates dimensionally as 1 Wb = 1 T ⋅ m², emphasizing flux as the product of field strength and area.[9] In practical magnetic circuits, such as those in transformers or inductors, flux values are typically on the order of millwebers to webers, depending on the applied magnetomotive force and circuit geometry.[10] Magnetic flux in a circuit is driven by the magnetomotive force, which establishes the field responsible for the flux flow.[10] In closed magnetic paths composed of high-permeability materials, flux is conserved, meaning the total flux entering a section equals the flux leaving it, provided leakage is negligible.[10] This conservation principle stems directly from Gauss's law for magnetism, \nabla \cdot \mathbf{B} = 0, which implies no magnetic monopoles exist and the net flux through any closed surface must be zero.[11] Consequently, in idealized magnetic circuits without divergence, the flux remains uniform along the loop, enabling straightforward analysis of field distribution.[10]Magnetic field strength
The magnetic field strength, denoted as H, represents the local magnetizing force or intensity of the magnetic field at a point within a material or space, driven primarily by free currents. It is defined through Ampère's circuital law in its differential form, which states that the curl of H equals the free current density: ∇ × H = Jf, where Jf accounts for conduction currents excluding those induced by material magnetization.[12] This relation highlights H as the portion of the magnetic field attributable to external or free sources, making it a fundamental quantity for analyzing magnetic interactions in circuits and devices.[13] In the International System of Units (SI), the magnetic field strength H is measured in amperes per meter (A/m), reflecting its direct proportionality to current and geometric factors like path length.[14] This unit underscores H's role as a driving force analogous to electric field strength in electrostatics, but tailored to magnetostatics. A key distinction exists between H and the magnetic flux density B: while B encapsulates the total magnetic effect including material responses, H remains independent of the medium's properties and depends solely on free currents. In linear media, B = μ H, where μ is the permeability, but H itself does not vary with the material, allowing it to serve as a universal measure across vacuum, air, or ferromagnetic substances.[15] This independence enables H to be calculated from circuit configurations without prior knowledge of material magnetization. In magnetic circuits, H is particularly significant in generating demagnetizing fields within ferromagnetic cores, where internal field oppositions reduce net magnetization, and in air gaps, where the low permeability of air causes H to intensify dramatically to maintain flux continuity, often dominating the circuit's reluctance.[16] For instance, in gapped inductors or transformers, elevated H in the gap can lead to flux fringing and altered energy storage, necessitating careful design to mitigate losses. The magnetomotive force in such circuits arises as the line integral of H around a closed path.[17]Reluctance Model
Reluctance
In magnetic circuits, reluctance is defined as the opposition to the establishment of magnetic flux, analogous to resistance in electrical circuits. It quantifies how much magnetomotive force (MMF) is required to produce a given magnetic flux in a magnetic path.[18][19] The reluctance R is mathematically expressed as the ratio of the magnetomotive force F to the magnetic flux \Phi: R = \frac{F}{\Phi} where F is measured in ampere-turns and \Phi in webers, yielding reluctance in units of ampere-turns per weber (or inversely, webers per ampere-turn).[20][19] For a uniform magnetic path, the reluctance can be calculated geometrically as R = \frac{l}{\mu A} where l is the mean length of the magnetic path in meters, A is the cross-sectional area in square meters, and \mu is the permeability of the material. Permeability serves as a key parameter influencing reluctance, with higher values reducing R for a given geometry.[20][18][19] In composite magnetic circuits, reluctances combine similarly to resistances: for elements in series, the total reluctance is the sum of individual reluctances, R_\text{total} = R_1 + R_2 + \cdots; for parallel paths, the reciprocal of the total reluctance is the sum of the reciprocals, \frac{1}{R_\text{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots.[18][19] Reluctance is primarily affected by three factors: the material's permeability, which determines how easily flux passes through; the geometry of the path, including length and cross-sectional area; and the presence of air gaps, which introduce high reluctance due to the low permeability of air compared to ferromagnetic materials.[20][18][19]Permeability
Magnetic permeability, denoted as \mu, is defined as the ratio of the magnetic flux density B to the magnetic field strength H, according to the relation \mu = \frac{B}{H}. This constant characterizes the material's response to an applied magnetic field, determining the resulting flux density for a given field strength, and has units of henry per meter (H/m).[21][22] The absolute permeability \mu of a material is often expressed relative to the permeability of free space \mu_0, a fundamental physical constant with the value \mu_0 = 4\pi \times 10^{-7} H/m. The dimensionless relative permeability \mu_r is then defined as \mu_r = \frac{\mu}{\mu_0}, which quantifies the enhancement or reduction of the magnetic field in the material compared to vacuum.[23] In vacuum, \mu = \mu_0 and \mu_r = 1, while air exhibits a permeability nearly identical to vacuum, with \mu_r \approx 1.[22] Ferromagnetic materials, such as iron and certain alloys, possess much higher relative permeabilities, typically ranging from hundreds to several thousand, enabling strong concentration of magnetic flux. For instance, pure iron can achieve \mu_r values exceeding 5,000 under optimal conditions, far surpassing non-magnetic materials.[24][25] The effective permeability of conductive materials is indirectly influenced by their electrical conductivity due to eddy currents induced by time-varying magnetic fields. These currents generate opposing fields that oppose flux changes, reducing the apparent permeability, especially at higher frequencies where skin effects become significant.[26][27]Hopkinson's law
Hopkinson's law, also known as the magnetic analog of Ohm's law, states that in a linear magnetic circuit, the magnetomotive force F equals the product of the magnetic flux \Phi and the reluctance R, mathematically expressed as F = \Phi R where F is measured in ampere-turns (At), \Phi in webers (Wb), and R in At/Wb.[28] This relation holds under the assumption of linear magnetic media, where the magnetization response is proportional and independent of the applied field strength.[29] The law is named after British physicist and engineer John Hopkinson, who developed and applied it in his investigations of magnetic properties during the 1880s, particularly in papers on the behavior of iron and alloys under magnetization.[30] Hopkinson's work built upon foundational experiments by American physicist Henry Augustus Rowland, who first proposed the concept of a magnetic flux law akin to Ohm's law in a 1873 paper on magnetic permeability.[31] This historical progression established the law as a cornerstone for analyzing magnetic circuits in electrical engineering. Hopkinson's law arises directly from the analogy between electric and magnetic circuits in linear media, where the constitutive relation B = \mu H (with \mu as permeability) leads to a proportional relationship between the driving force (MMF) and the resulting flux, mirroring voltage, current, and resistance in Ohm's law V = I R.[28] Reluctance R, which incorporates material permeability and geometric factors, serves as the magnetic counterpart to resistance, enabling the direct application of this proportionality.[29] The units of Hopkinson's law are dimensionally consistent: ampere-turns (At) on the left side match webers (Wb) multiplied by At/Wb on the right, confirming the equation's physical validity without additional conversion factors.[28]Electric circuit analogy
The electric circuit analogy provides a powerful framework for analyzing magnetic circuits by drawing direct parallels between electrical and magnetic quantities, facilitating intuitive design and computation. In this model, magnetomotive force (MMF), denoted as ℱ and measured in ampere-turns, corresponds to electromotive force (voltage) in electric circuits, as both drive the flow through the system. Magnetic flux Φ, in webers, is analogous to electric current I in amperes, representing the quantity that "flows" through the circuit. Reluctance ℛ serves as the magnetic counterpart to electrical resistance R, quantifying opposition to flux, while permeance P, the reciprocal of reluctance, mirrors electrical conductance G = 1/R.[32][33] This analogy is grounded in Hopkinson's law, which establishes the proportional relationship between MMF, flux, and reluctance, akin to Ohm's law. For a magnetic path of length l, cross-sectional area A, and permeability μ, reluctance is given by ℛ = l / (μ A), so permeance P = 1/ℛ = μ A / l.[32][34] The development of this analogy emerged in the late 19th century amid advances in dynamo-electric machinery, with John Hopkinson introducing the key concept of reluctance in papers presented to the Royal Society in 1884 and published in 1886, formalizing the resisted flow image for practical calculations. Contemporaries such as S.P. Thompson in 1884 and Gisbert Kapp in 1885–1886 further refined the model, building on earlier fluid analogies to electricity for magneto-static analysis. By the late 1880s, the analogy had become a standard tool in electrical engineering.[35]| Electric Quantity | Symbol | Unit | Magnetic Analog | Symbol | Unit | Notes |
|---|---|---|---|---|---|---|
| Electromotive force (Voltage) | V | Volts (V) | Magnetomotive force | ℱ | Ampere-turns (A·t) | Drives the flow |
| Current | I | Amperes (A) | Magnetic flux | Φ | Webers (Wb) | Conserved quantity in series |
| Resistance | R | Ohms (Ω) | Reluctance | ℛ | A·t/Wb (H⁻¹) | Opposition to flow; ℛ = l / (μ A) |
| Conductance | G | Siemens (S) | Permeance | P | Wb/(A·t) (H) | Reciprocal; P = μ A / l |