Power Notes 6

Power Notes 6 - Unit 05 Magnetic Fields 7& Magnetic...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Unit 05 Magnetic Fields 7& Magnetic Circuits Electro-mechanical energy conversion and energy distribu-tion in power systems is enabled by magnetic fields created by magnetic circuits. In such a sense, these fields and circuits are the main components of electrical power systems. They are essential components of generators, transformer and motors. The creation of magnetic field by electric current was discovered by Oersted in 1820 and the creation of the electric field by a change of magnetic flux was discovered by Faraday in 1931. These two discoveries have created the very physical fundamentals of electrical power systems. The creation of the magnetic field, of intensity I? , is governed by two laws, Biot-Savart Law and Ampere Law. The Biot—Savart Law. When a point of space has a distance 3 fi'om a conductor element oriented in direction :17 with a current i then a magnetic field intensity dfi = Lian—1' =Lidl sin(dl,d)’ 472' d3 472' d2 is created at such a point. Current loop Fig. 1 The vector of the magnetic field intensity is perpendicular to the plane of vectors :1? , and the distance ti". Orientation of this vector can be detemrined by the right-lulu ded screw rule. [if]. I' - / d id? _. displacement d . 51' h Rotation I Fig.2 Ifthe first elementefaaeveeterpreaueufihsremeed on tothe second element (a?) of the product then the result of the vector product is oriented towards the direction of displacement of the right-handed screw (Fig. 2a). This rule can be applied also in a different way. To displace a right-handed screw in the direction of the current flow, it has to be rotated in the direction of the magnetic field intensity, H, (Fig. 2b). Magnetic field intensity, H, has dimension [A/m], but its unit does not have any name. To calculate the magnetic field intensity H created by a current loop at some point of space, integration along the entire current loop is needed. On the condition that the current loop is on a plane and the magnetic field intensity is calculated at same point of the same plane and is is the plane unit vector (a unit vector perpendicular to the current loop plane) a = g m = It; a} Loop Loop d Since the distance d and the angle between the loop element d1i,=Hi,. d1 and the distance vector d changes along the loop, this integral could be difficult for calculation. It is easy to calculate it only if geometry of the loop is simple. Illustration 1. Calculate the magnetic field intensity in the center of a circular current loop of diameter of D = 2R = 0.5m with the current i = 500 A. ' m Fig. 3 When the magnetic field intensity is calculated in the center of a circuit as shown in Fig. 3, then the distance d = D/2 and the angle between the loop element 37 and the distance vector d is constant and equal to 90°. Thus, . ' 0 H: qu=4LSmg° C'Fdlé Loop 7t R Loop =_i_; =_i_=_500:_= 47: Ram 2R 2x025 1000M“ MAMA/\AAAAAAAMAMAAMA The Ampere Law. It says that the integral along a closed path of the magnetic field intensity H is equal to the current flux enclosed by this path. In this integral the symbolfdenotes the vector of current density in [Almz] and ds'is the vector of the surface element. 41%;: may: Path Area Fig. 4 The direction of integration along the path and the direction of the current flux across the area satisfy the right-handed screw rule. When the integration path in the Ampere Law encloses N1 conductors with current i1, flawing in direction that satisfies the right-handed screw rule and N2 conductors with current i; flowing in the opposite direction, then the Ampere Law for any path around these conductors can be written in the form qfidi=NllI1—N2i2 . Path Illustration 2. Calculate the magnetic field intensity H in the distance d = 0.25m from a conductor with the current i = 500 A. r r? Fig. 5 Solution. The vector of magnetic field intensityH at any point of the circle with the current in its center has the same magnitude, H, and it is tangent to the circle. Hence, cjfidi=H cfd1=2mH = i. Path Path Thus .; ___&. = H‘ 2727 _27rx0.25 3184M” AAMMAAAAAAAAAAAAAAAAAAAAA The mechanical F exerted on electric charge q in electric and magnetic fields are governed by the Lorenz Law. The Lorenz Law. If an electric charge dq moves in the electromagnetic field with velocity ii then a force is exerted on such a charge. If electric field is specified by the vector of the electric field intensity E and the magnetiefield is specified by the vector of magnetic flux density, B , then the force is equal to d13=dqa§+r XE). The magnetic flux density is measured in Teslas [T]. It is a vector, tangent to the magnetic flux, ¢, line, with the magnitude M B = g: . The magnetic flux, ¢, is measured in Webers [Wb] and cones- quently, the magnetic flux density, [T] = [Wb/mz]. =l‘he magnetic component ofthe force exerted on ameving charge di ~ dq dfim=dq17x3=quXB=E-dlx§=idlx§. The magnetic flux density li’ in vacuum, and approximately in air and in a great majority of materials, is proportional to the magnetic field intensity, H B=floH. The symbol ,uo denotes magnetic permeability of free space and it is equal to yo = 47r><10'7 [H/m]. The magnetic flux density B in iron, cobalt and nickel and in alloys of these metals with each other, depends on the magnetic field intensity H thousands time stronger than in vacuum. Such materials are called ferromagnetics. For such materials B=y,uoH, where y is referred to as a relative permeability. Its approximate value for some ferromagnetics: - annealed iron: ,u = 5 500 - iron-silicon laminations (96% Fe, 4% Si) ,u = 7 000 - permalloy(55%Fe, 45% Ni) ii = 25 000 - suppermalloy y = 100 000 Due to electron spin and their rotation around the nucleus, individual atoms form elementary magnetic dipoles. In a great majority of materials, these dipoles are oriented randomly and consequently, their net effect is zero. Ferromagnetism is caused be an alignment of elementary magnet dipoles in material microstructine. They form domains. These dipoles in a domain have the same orientation, different from the orientation in neighboring I \\ Mas- Fig. 6 External magnetic field enlarges and rotates individual magnetic domains in the direction of the external field intensity, causing a magnetic flux to occur. This process is strongly nonlinear, specified in terms of hysteresis loop. The reorientation of magnetic domains remains to some degree permanent. Even if the external magnetic field disappears, the ferromagnetic remains magnetized. It is specified by the residual magnetic flux density, Br Negative magnetic field is needed for material re-magnetization. It is specified by the magnetic field intensity coersion, Hc. When all domains are oriented towards the external magnetic field, the ferromagnetic material becomes saturated and the relative permeability p declines to unity. B 5' Fig. 7 Mechanical work is needed for changes of magnetic domains structure. This work at one eyele of ehange of field intensity is proportional to the area of the hysteresis loop. When the field intensity changes periodically, then energy loss in the ferromagnetic material increases with the area of the loop and with fiequency. This energy loss is observed as an increase of the material temperature. Therefore, magnetic circuits used for creation of variable magnetic flux (for example, in transformers) should be built of ferromagnetic materials with the hysteresis loop area as small as possible. Such ferromagnetics are referred to as soft ferromagnetics. The hysteresis loop of a sofi ferromagnetic material is shown in Fig. 7. Ferromagnetic material for a permanent magnet should fulfill different requirements. It should have the residual magnetic flux density, 3,, as high as possible. The higher E, the stronger magnet could be formed. A permanent magnet in an external magnetic field can loose its magnetization, however. This unfavorable feature declines with the increase of the magnetic field intensity coersion, Hc of the ferromagnetic material. The higher coersion intensity the better is the material for a permanent magnet. Ferromagnetics with high residual flux density and high coersion intensity are referred to as hard ferromagnetics. Table l. The values of Br and Hc of ferromagnetic materials used for permanent magnets 0-4 Neod ’ 'um-iron-boron allo 1.25 Tesla 1.25 50 ~o~.—- E UI\IUI : The hysteresis loop of a hard ferromagnetic material is shown in Fig. 8. Fig 8 Magnetic circuits. At the same value of the magnetic field inten— sity H, the flux density B in ferromagnetic materials is p—time higher than in air. Consequently, magnetic flux is mainly con-fined to ferromagnetic material. If this material is formed to create a loop or loops for the magnetic flux, a magnetic circuit is created. Winding with a current is usually the main source of the magnetic field and the flux in such a magnetic circuit. A permanent magnet or external magnetic field could serve as such a source as well. Fig. 9 The magnetic flux a) in the circuit shown in Fig. 9a has the same value along the entire magnetic loop. Consequently, the flux density _ A , does not change along the loop. Since the magnetic permeability changes along the flux loop, changes also the magnetic field intensity. Its value in the ferromagnetic core is Hc=——B , ##o andintheairgap B s #0 The Ampere Law for the average path along the core deI = Hcl+HgA =Ni. path When the magnetic field intensity is expressed in terms of magnetic permeability y no, flux (D and the core cross-section areaA, then 4’ [+3441 =Ni. A ##o A .110 The term I R = —— , c ##o A is referred to as a reluctance of the magnetic core, while A R 3—!!014’ is the reluctance of the air-gap. Consequently, the Ampere Law for the magnetic circuit shown in Fig. 10a can be written in the form d>(Rc +Rg) = Ni. Consequently, the magnetic flux in such a circuit can be calcu-lated from formula Ni (D = —. Rc + Rg Illustration 3. A magnetic core made of silicon-iron laminations with magnetic permeability ,u = 7000 has the length l = 0.5m and the cross-section area A = 100 cm2. The air gap A = 1 mm. The core saturates at the flux density BM = 1.2 T. Calculate the maximum current in the winding with N = 100 turns that do not saturate the core. Solution. To avoid saturation the magnetic flux should not be higher than em = ABsat = 0.01 [m2] x 1.2[Wb/m2] = 12 x10'3 Wb. The reluctance of the core is c=——-=—————-—_7——=5700A/Wb. ##oA 7000x47rx10 x0.01 The reluctance of the air-gap is —3 g=__4 =—__—1X1_9 =79500A/Wb. floA 471x10 x0.01 Observe, that the reluctance of 1 mm long air-gap is almost 14 times higher than the core reluctance. From the Ampere Law -3 ism = ‘13:;ch +le) = Lag—(5.7 +79.5)x1o3 = 10.2A. AMMAAAAAAAAAMAAMAAAM The Faraday Law. Change of the magnetic flux (I) in a loop induces voltage e in such a loop. When the induced voltage e and the flux CD are mutually oriented according to right-hand screw rule as shown in Fig 1 l, then the induced voltage is _ dd! 8 — "' '2‘— . thus, if the flux in the loop increases, then the voltage induced in the loop is oriented towards terminal b, i.e., opposite to the direction marked by the arrow. Orientation of the voitage induced by a change of the magnetic flux could be determined by the Lenz Rule. It says, that the voltage inducedby a change of the magnetic flux has such orientation, that the current caused by this voltage creates a magnetic flux ((1),) that counteract this change. Fig. 12 It means, that orientation of the magnetic flax and the induced voltage satisfy a left—hand screw rule. At such orientation e=— dt ' When a loop with magnetic flux is composed of N turns, as shown in Fig. 13, Fig. 13. then the induced voltage is proportional to the rate of magnetic flux change and the number of turns, d (D e — N -dt— . Observe, that if the current in the coil increases, i.e., di/dt > 0, then the arrow e specifies true orientation of the voltage induced in the coil. Since the magnetic flux could be expressed as Ni ¢—fi-. Thus, _ R dt ' dt’ where N2 =?’ is the inductance of the coil. It increases with the square of turn number N and declines with the core reluctance. For a core without air gap A = ill/:0 N2, thus, it increases with the core magnetic permeability ,u and cross- section area, A. Hysteresis power loss. When a coil with a magnetic core as shown inFig. 14 Fig. 14. is connected in AC circuit then the instantaneous power at the coil terminals is equal to ._ d H114 _ __ p(t)rul—(NE¢)(7) dt(AB)Hl—VH dt, where V=Al is the magnetic core volume. Thus, the instantaneous power is proportional to the change in the magnetic flux density, dB, at magnetic intensity H, i.e., to the area HdB as shown in Fig. 10. Due to hysteresis, magnetic field intensityH at the same flux density B is higher when the core is magnetized than when it is de- magnetized. Censequently, the area of HdB is larger at dB magnetization than at de-magnetization. The difference is released as a heat of the core increase. Fig. 15 The active power at the coil terminals for one cycle T of the current variability, i.e., when B and H change in the whole range shown in Fig. 15, is equal to T T 1 1 1 P=T Jp(t)dt =V? [HdB=?VABH = ABHVf= APh, 0 0 thus it is proportional to the area ABH of the hysteresis loop, the core volume V and the supply voltage frequency, f. This power is referred to hwteresispawer loss. Since soft ferromagnetics have much smaller hysteresis loop area, ABH, than hard ferromagnetics, coils with soft ferromagne-tics have lower power loss than coils with hard ferromagnetics. Therefore, magnetics with a hysteresis loop as narrow as possible are used for AC coils. When sinusoidal voltage u=\/E’Ucoswt is applied to the coil with a magnetic core, then the flux density B in the core has to satisfy F araday’s Law dd) dB u—Nw—NA—dr hence, , _ 1 _J§U _ . _ . —m udt— NA coswtdt—NAwsmwt—Bmxsmmt, then the flux density B in the core has to change as a sinusoidal function, with J3 U NAa) Thus, the RMS value U of the voltage applied to the coil cannot be higher than <Bsa. Bmax = a) J5 N A 3m = 4.44NAme < 4.44NAstat Fig. 16 \ ...
View Full Document

Page1 / 4

Power Notes 6 - Unit 05 Magnetic Fields 7& Magnetic...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online