Transformers

MAGNETIC CIRCUITS

A *magnetic circuit *is a path for magnetic flux, just as an electric circuit provides a path for the flow of electric current. Transformers, electric machines, and numerous other electromechanical devices utilize magnetic circuits. *Magnetic induction *or *magnetic flux density, B, *is defined by the force equation

*F *= *B l *I * *(Formula 1)

where *F *(N) is the force experienced by a straight conductor oflength *t(m) *which carries a current *i *(A) and is oriented at right angles to a uniform magnetic field of flux density *B *(T). In other words, if a conductor is 1 meter long, carries 1 ampere of current, and experiences 1 newton force when located at right angles to parallel magnetic flux lines, then the magnetic flux density is 1 tesla. Actually, the magnetic induction is a vectar, B, of magnitude *B *and tangential to the flux lines. If B is uniform over a surface of area *A *and is everywhere perpendicular to the surface, *magnetic flux, Φ, *is determined by

* *(Formula2)

The unit of magnetic flux is the weber (Wb), and (2) shows that

1 T = 1 Wb/m2.

The source of magnetic flux is either a permanent magnet or an electric current. To measure the

effectiveness of electric current in producing a magnetic field (or flux), the concept of magnetomotive force (mmf), F, defined as

*F *= *NI** *(Formula 3)

is introduced where *I *is the current flowing in an *N-turn *coil. The unit of *F *is the *ampere turn *(At). Schematically, a magnetic circuit with an mmf and magnetic flux is shown in below.

The core material of a magnetic circuit (constituting a transformer or an electric machine) is generally ferromagnetic, and the variation of *B *with *H (magnetic field intensity *measured in units of A/m or At/m) is depicted by *asaturation curve, *as shown in Fig. 2. For region II, where the slope of the curve is nearly constant, we may write

*B *= μ*H ** *(Formula 4)

where μ, (H/m) is defined as the *permeability *of the material. For free space (or air), μ = μo = 4 л x 10^{-7} H/m.

DC Resistive Circuit |
Magnetic Circuit |

current,* I* |
flux,*Φ* |

voltage,* V* |
mmf, F |

conductivity,* σ* |
permeability,μ |

Ohm´s law, *I* =* V* /* R* |
*Φ** = F / R* |

resistance,* R* =*l* /* σ A* |
reluctance,* R* =* l* /μ* A* |

conductance, G = 1/* R* |
permeance,* P* = 1 /* R* |

Table 1 |

The

*dijferences *between a dc resistive circuit and a magnetic circuit are:

(1) there is an

*I2R-loss *in a resistance but no

*Φ*2R-loss in a reluctance;

(2) magnetic fluxes take

*leakage *paths, as

*Φ**l*in Fig. 2, whereas electric currents (flowing through resistances) do not;

(3) Magnetic circuits are non-linear through the

*B-H *characteristic of Fig. 2;

(4) in magnetic circuits with air gaps we encounter

*fringing *(Fig. 2) of flux lines, but we do not have fringing of currents in electric circuits. Fringing increases with the length of the air gap, and increases the effective cross-sectional area of the air gap.