Basic Magnetics Tlie/iiy

Faradays law v(r) -I- - B{t\ Ф(0

Kg. 13.7 Summary of the steps in duteimiua-tion of the terminal electrical i-v characteristics of a magnetic element,

Terminal characteristics

Amperes law

Core

citaracteristics

mt). m

law. The winiling current i(t) is relateil to the magnetic field strength via Amperes law. The core material characteristics relate В and Я.

We can now determine the electrical terminal characteristics of the simple inductor of Fig. 13.8(a). A winding of n turns is placed on a core having permeability p. Faradays law states that the flux Ф(0 inside the core induces a voltage >, , (/) in each turn of the winding, given hy

(13,U)

Since the same ПихФ(г) passes through each ttirn of the winding, the total winding vohage is

.dФ{t) (13.12)

vit]nv . ,(i)=n-

Equation (13.12) can be expressed in terms of the average flux density B(t) hy substitution of Eq, (13.4):

v(0 = A/-fl 03.13)

Fig. 13,8 Inductoi example: (a) luduclur geometry, (b) application of Amperes law.

t(()

v>it)

n s turns

Core area

Core

permeabiUty )

Magnetic path

length f

where the average flux density B{t) is Ф(г)/Л,.

The tise of Amperes law is illustrated in Fig. 13.S(b). A closed path is chosen which follows an average magnetic field line arottnd the interior of the core. The length of this path is called the wcaiJ rmgni;licpath length If the magnetic field strength Hit) is uniform, then Amperes law states that is equal to the total current passing through the interior of the path, that is, the net current passing through the window in the center of the core. Since there are n turns of wire passing through the window, each carrying current i(i), the net current passing through the window is ni{t). Hence, Amperes law states that

Let us model the core material characteristics hy neglecting hysteresis hut accounting for saturation, as follows:

ВЫИ for/f <ff ,/;i (-

-B. , forff<-B /M

The B-H characteristic saturated slope /i is tnuch smaller than aud is ignored here. A characteristic similar to Fig. 13.6(b) is obtained. The current magnitude at the onset of saturation can be found by substitution of i;, ,/flinto Eq. (13.14). The resuh is

, BJ (13.16)

p.;i

We can now ehminate В and H from Eqs. (13.13) to (13.15), atid solve for the electrical terminal characteristics. ¥or\l\<fj,B = jj,H. Equation (13.13) then becomes

Кг) = цМ, П3,17)

Substitution of Eq. (13.14) into Eq. (13.17) to eliminate H{t) then leads to

.(Oji (13.18)

which IS ofthe form

v(0 = L (13.19)

with

WlK (1X20)

So the device behaves as an inductor for I / I < /jn,.Whcn \1\> /, then the flnx density B(f) = is constant. Faradays law states that the terminal voltage is then

v(0 = nA,

(13.21)

When the core saturates, the magnetic device behavior approaches a short circuit. The device behave.s as an inductor only when the winding current magnitude is less than Practical inductors exhibit some small residual inductance due to their nonzero saturated permeabilities; nonetheless, in saturation the inductor impedance is greatly reduced, and large inductor currents may result.

13.1.2 Magnetic Circuits

Figure 1Э.9(а) illustrates uniibmi flux and magnetic field inside a element having permeability t, length t and cross-sectional area /l.The MMF between the two ends of the clement is

S = HI

SmcaH = li/ft. and B=i/A.,c&n express as

(13.32)

This equation is of the form

(13.23)

(13.24)

with

(13.25)

Equation (13.24) resembles Ohms law. This equation states that the magnetic Ilux through an element is proportional to the MMF actoss the element. The constant of proportionality, or the reluctance .J, is analogous to the resi.stance R ol an electrical conductor. Indeed, we can construct a lumped-element magnetic circuit model that corresponds to Eq. (13.24), as in Fig. i3.9(h). In this magnetic circuit model, voltage and current are replaced by MMF and flux, while the element characteristic, Eq. (13.24), is represented by the analog of a resistor, having reluctance

Complicated magnetic structures, composed of multiple windings and multiple heterogeneous

Length l-MMF ,> -

Flux Ф

-*-ЛЛг-

Core perineability ц

Fig. 13,1 An element containing magnetic flux (a), and its equivalciu nuigiietjt ciivuit (b).