11.2 MAGNETIC CIRCUITS 477TABLE 11.2.1Analogy with dc Resistive Circuits
DC Resistive Circuit Magnetic Circuit
CurrentI(A) Fluxφ(Wb)
VoltageV(V) Magnetomotive force (mmf)(At)
ResistanceR=l/σ A () Reluctance=l/μA (H−^1 )
Conductivityσ(S/m) Permeabilityμ( H/m)
ConductanceG= 1 /R(S) PermeanceP= 1 /(H)
I=V/R φ=/
Current densityJ=I/A(A/m^2 ) Flux densityB=φ/A(Wb/m^2 )
Analogous to KVL in electrical circuits, Ampere’s law applied to the analysis of a magnetic
circuit leads to the statement that the algebraic sum of the magnetic potentials around any closed
path is zero. Series, parallel, and series–parallel magnetic circuits can be analyzed by means of
their corresponding electric-circuit analogs. All methods of analysis that are valid for dc resistive
circuits can be effectively utilized in an analogous manner.
The following differences exist between a dc resistive circuit and a magnetic circuit
- Reluctanceis not an energy-loss component like a resistanceR(which leads to an
I^2 Rloss). Energy must be supplied continuously when a direct current is established and
maintained in an electric circuit; but a similar situation does not prevail in the case of a
magnetic circuit, in which a flux is established and maintained constant. - Magnetic fluxes takeleakagepaths [asφlin Figure 11.2.1(b)]; but electric currents flowing
through resistive networks do not. - Fringingor bulging of flux lines [shown in Figure 11.2.1(b)] occurs in theair gapsof
magnetic circuits; but such fringing of currents does not occur in electric circuits. Note that
fringing increases with the length of the air gap and increases the effective area of the air
gap. - There are no magnetic insulators similar to the electrical insulators.
In the case of ferromagnetic systems containing air gaps, a useful approximation for making
quick estimates is to consider the ferromagnetic material to have infinite permeability. The relative
permeability of iron is considered so high that practically all the ampere-turns of the winding are
consumed in the air gaps alone.
Calculating the mmf for simple magnetic circuits is rather straightforward, as shown in the
following examples. However, it is not so simple to determine the flux or flux density when the
mmf is given, because of the nonlinear characteristic of the ferromagnetic material.
EXAMPLE 11.2.1
Consider the magnetic circuit of Figure 11.2.1(b) with an air gap, while neglecting leakage flux.
Correct for fringing by adding the length of the air gaplg= 0 .1 mm to each of the other two
dimensions of the core cross sectionAC= 2 .5cm× 2 .5cm. The mean length of the magnetic
path in the corelCis given to be 10 cm. The core is made of 0.15- mm-thick laminations of M-19
material whose magnetization characteristic is given in Figure 11.1.2. Assume the stacking factor
to be 0.9. Determine the current in the exciting winding, which has 100 turns and produces a core
flux of 0.625 mWb.