The gyrator–capacitor model - sometimes also the capacitor-permeance model - is a lumped-element model for magnetic circuits, that can be used in place of the more common resistance–reluctance model. The model makes permeance elements analogous to electrical capacitance (see § Magnetic capacitance) rather than electrical resistance (see § Magnetic reluctance). Windings are represented as gyrators, interfacing between the electrical circuit and the magnetic model. The primary advantage of the gyrator–capacitor model compared to the magnetic reluctance model is that the model preserves the correct values of energy flow, storage and dissipation. The gyrator–capacitor model is an example of a group of analogies that preserve energy flow across energy domains by making power conjugate pairs of variables in the various domains analogous. It fills the same role as the impedance analogy for the mechanical domain.
Nomenclature Magnetic circuit may refer to either the physical magnetic circuit or the model magnetic circuit. Elements and dynamical variables that are part of the model magnetic circuit have names that start with the adjective magnetic, although this convention is not strictly followed. Elements or dynamical variables in the model magnetic circuit may not have a one to one correspondence with components in the physical magnetic circuit. Symbols for elements and variables that are part of the model magnetic circuit may be written with a subscript of "M". For example, C M {\displaystyle C_{\mathrm {M} }} would be a magnetic capacitor in the model circuit. Electrical elements in an associated electrical circuit may be brought into the magnetic model for ease of analysis. Model elements in the magnetic circuit that represent electrical elements are typically the electrical dual of the electrical elements. This is because transducers between the electrical and magnetic domains in this model are usually represented by gyrators. A gyrator will transform an element into its dual. For example, a magnetic inductance may represent an electrical capacitance.
Summary of analogy between magnetic circuits and electrical circuits The following table summarizes the mathematical analogy between electrical circuit theory and magnetic circuit theory.
Gyrator
A gyrator is a two-port element used in network analysis. The gyrator is the complement of the transformer; whereas in a transformer, a voltage on one port will transform to a proportional voltage on the other port, in a gyrator, a voltage on one port will transform to a current on the other port, and vice versa. The role gyrators play in the gyrator–capacitor model is as transducers between the electrical energy domain and the magnetic energy domain. An emf in the electrical domain is analogous to an mmf in the magnetic domain, and a transducer doing such a conversion would be represented as a transformer. However, real electro-magnetic transducers usually behave as gyrators. A transducer from the magnetic domain to the electrical domain will obey Faraday's law of induction, that is, a rate of change of magnetic flux (a magnetic current in this analogy) produces a proportional emf in the electrical domain. Similarly, a transducer from the electrical domain to the magnetic domain will obey Ampère's circuital law, that is, an electric current will produce a mmf. A winding of N turns is modeled by a gyrator with a gyration resistance of N ohms. Transducers that are not based on magnetic induction may not be represented by a gyrator. For instance, a Hall effect sensor is modelled by a transformer.
Magnetic voltage Magnetic voltage, v M {\displaystyle v_{\mathrm {M} }} , is an alternate name for magnetomotive force (mmf), F {\displaystyle {\mathcal {F}}} (SI unit: A, or ampere-turn), which is analogous to electrical voltage in an electric circuit. Not all authors use the term magnetic voltage. The magnetomotive force applied to an element between point A {\displaystyle A} and point B {\displaystyle B} is equal to the line integral through the component of the magnetic field strength, H {\displaystyle \mathbf {H} } :
v M = F = − ∫ A B H ⋅
d ℓ . {\displaystyle v_{\mathrm {M} }={\mathcal {F}}=-\int _{A}^{B}\mathbf {H} \cdot \mathop {} \!\mathrm {d} {\boldsymbol {\ell }}.}
The resistance–reluctance model uses the same equivalence between magnetic voltage and magnetomotive force.
Magnetic current
Magnetic current, i M {\displaystyle i_{\mathrm {M} }} , is an alternate name for the time rate of change of flux, Φ ˙ {\displaystyle {\dot {\Phi }}} (SI unit: Wb/sec or volts), which is analogous to electrical current in an electric circuit. In the physical circuit, Φ ˙ {\displaystyle {\dot {\Phi }}} , is magnetic displacement current. The magnetic current flowing through an element of cross section, S {\displaystyle S} , is the area integral of the magnetic flux density B {\displaystyle \mathbf {B} } :
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