Domain_3 Of Magnetic Field Formula

Incident Electromagnetic Field Dosimetry

Dragan Poljak PhD , Mario Cvetković PhD , in Human Interaction with Electromagnetic Fields, 2019

3.1.2.ii The Magnetic Field

The magnetic flux density at an arbitrary point due to a current element shown in Fig. 3.17 is determined past the Biot–Savart's law:

Fig. 3.17. Straight current element.

(iii.25) $d\overrightarrow{B}=\frac{\mu }{4\pi }\frac{i(t)\overrightarrow{dl}\times (\overrightarrow{r}-\overrightarrow{r}\prime )}{{|\overrightarrow{r}-\overrightarrow{r}\prime |}^3}=\frac{\mu }{\mathrm{four}\pi }\frac{i(t)\overrightarrow{d\mathrm{50}}\times \overrightarrow{R}}{R^3},$

where μ is permeability, i denotes the electric current along the segment, and $R=|\overrightarrow{r}-\overrightarrow{r}\prime |$ is the distance from the source $i(t)\overrightarrow{d\mathrm{fifty}}$ to the observation point P.

Performing some mathematical manipulation and integrating the contributions along the entire length of a conductor, we get

(three.26) $\overrightarrow{B}={\hat{e}}_{\phi }\frac{\mu i(t)}{4\pi \rho }\underset{{\theta }_{\mathrm{i}}}{\overset{{\theta }_{\mathrm{ii}}}{\int }}\cos \theta d\theta ={\hat{e}}_{\phi }\frac{\mu i(t)}{4\pi \rho }(\sin {\theta }_{\mathrm{i}}+\sin {\theta }_2),$

where ρ and θ are the variables in the cylindrical coordinate organization.

The ELF magnetic field value at an capricious point can be assessed past assembling the contributions of all conductors divided in a sure number of straight segments. The kth straight segment conveying electric current $i_{\mathrm{thousand}}$ in Cartesian 3-dimensional coordinate arrangement is shown in Fig. iii.18.

Fig. three.18. Straight segment in Cartesian coordinate organisation.

Using the Biot–Savart's law, the magnetic field value at point C due to the considered conductor segment can exist written as follows [vi]:

(3.27) $B_k(t)=\frac{\mu i_{\mathrm{thousand}}(t)}{\mathrm{iv}\pi R_{\text{RS}}}(\frac{R_{\text{PS}}}{R_{\text{PR}}}+\frac{R_{\text{SQ}}}{R_{\text{QR}}}),$

where the corresponding distances R are assigned equally in Fig. three.xviii.

Full components of the magnetic flux density generated by N segments are assembled from the contributions of all segments. Therefore, the full value of the magnetic flux density at a given indicate of space tin exist expressed every bit

(three.28) $B(t)=\begin{array}{c}\hfill \sqrt{{(\sum _{i=\mathrm{ane}}^N{B}_{\mathrm{10},i}(t))}^{\mathrm{two}}+{(\sum _{i=1}^{\mathrm{Northward}}{B}_{y,i}(t))}^2+{(\sum _{i=\mathrm{i}}^{\mathrm{North}}{B}_{z,i}(t))}^{\mathrm{two}}}\end{array},$

where $B_{x,i}(t)$ , $B_{y,i}(t)$ and $B_{z,i}(t)$ are the components of the magnetic flux density due to the ithursday segment.

A computational instance is related to the 110/10 kV/kV transmission substation of GIS (Gas-Insulated Substation) type. A simplified 2-dimensional layout of the substation is shown in Fig. 3.15. The calculation domains 1 to 5, in which higher field values are expected, are assigned as in Fig. 3.fifteen. The spatial distribution of the magnetic field over domain 3, where the highest field value is captured, is shown in Fig. 3.19.

Fig. 3.nineteen. Spatial distribution of the magnetic field over domain 3.

If the man body is exposed to an ELF magnetic field, the circular electric current density is induced inside the body due to the existence of the normal component of the magnetic flux density.

Once the magnetic flux density is determined, the internal current density tin can exist calculated using the disk model of the human body.

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Sensors and Actuators

William B. Ribbens , in Agreement Automotive Electronics (7th Edition), 2013

Electric Motor Actuators

Possibly the most of import electromechanical actuator in automobiles is an electric motor. Electrical motors have long been used on automobiles commencement with the starter motor, which uses electric ability supplied past a storage battery to rotate the engine at sufficient RPM that the engine tin be fabricated to start running. Motors have as well been employed to heighten or lower windows, position seats too equally for actuators on airflow control at idle (see Affiliate vii). In recent times, electric motors have been used to provide the vehicle primary motive power in hybrid or electrical vehicles.

In that location are a great number of electric motor types that are classified past the type of excitation (i.east., dc or air-conditioning), the concrete structure (e.thou., smooth air gap or salient pole), and by the type of magnet structure for the rotating element (rotor) which can exist either a permanent magnet or an electromagnet. However, there are certain fundamental similarities between all electric motors, which are discussed beneath. Withal another stardom between types of electric motors is based upon whether the rotor receives electrical excitation from sliding mechanical switch (i.eastward., commutator and brush) or by consecration. Regardless of motor configuration, each is capable of producing mechanical ability due to the torque practical to the rotor by the interaction of the magnetic fields between the rotor and the stationary structure (stator) that supports the rotor forth its axis of rotation.

It is beyond the scope of this book to consider a detailed theory of all motor types. Rather, nosotros introduce basic physical structure and develop analytical models that can exist applied to all rotating electromechanical machines. Furthermore, we limit our discussion to linear, time-invariant models, which are sufficient to permit performance analysis appropriate for near automotive applications.

We introduce the structures of various electric motors with Figure 6.34, which is a highly simplified sketch depicting merely the about basic features of the motor.

Effigy 6.34. Schematic representation of electric motor.

This motor has coils wound effectually both the stator (having N ₁ turns) and the rotor (having Due north _ii turns), which are placed in slots effectually the periphery in an otherwise uniform gap machine. In this simplified drawing, merely two coils are depicted. In exercise, there are more than 2 with an equal number in both the stator and rotor. Each winding in either stator or rotor is termed a "pole" of the motor. Both stator and rotor are made from ferromagnetic fabric having a very loftier permeability (see give-and-take above on ferromagnetism). It is worthwhile to develop a model for this simplified idealized motor to provide the basis for an understanding of the relatively circuitous structure of a practical motor. In Figure vi.34, the stator is a cylinder of length ℓ and the rotor is a smaller cylinder supported coaxially with the stator such that it can rotate about the mutual axis. The bending between the planes of the two coils is denoted θ and the angular variable about the axis measured from the plane of the stator coil is denoted α. The radial air gap between rotor and stator is denoted g. Information technology is important in the design of whatsoever rotating electric machine (including motors) to maintain this air gap as small as is practically feasible since the strength of the associated magnetic fields varies inversely with one thousand. The terminal voltages of these two coils are denoted v_i and five_two. The currents are denoted i _ane and i _ii and the magnetic flux linkage for each is denoted λ ₁ and λ _ii, respectively. Bold for simplification purposes that the slots conveying the coils are negligibly small, the magnetic field intensity H is directed radially and is positive when directed outward and negative when directed inward.

The final excitation voltages are given past:

$\begin{array}{c}{v}_{\mathrm{ane}}={\dot{\lambda }}_1\\ {\mathrm{five}}_2={\dot{\lambda }}_2\end{array}$

The magnetic flux density in the air gap B_r is also radially directed and is given past

(85) $B_r={\mu }_o{H}_r$

where μ_o is the permeability of air.

This magnetic flux density is continuous through the ferromagnetic structure, but because the permeability of the stator and rotor (μ) is very large compared with that of air, the magnetic field intensity inside both the rotor and stator is negligibly small:

H  ≃   0   inside ferromagnetic cloth.

The contour integral along any path (e.g., contour C of Figure 6.34) that encloses the two coils is given past

(86) $I_T={\oint }_C\overline{H}·\overline{\mathrm{d}}\overline{\ell }=2\mathrm{yard}{H}_{\text{r}}(\alpha )$

The magnetic flux density B_r (α) is as well directed radially and is given past

$B_r(\alpha )={\mu }_o{H}_r(\alpha )$

This magnetic field intensity is a piecewise continuous function of α as given below:

$\begin{array}{c}\begin{array}{cc}\mathrm{ii}g{H}_r(\alpha )={\mathrm{Northward}}_1{i}_1-{\mathrm{Northward}}_2{i}_2& 0\le \alpha <\theta \\ =N_1{i}_1+N_2{i}_2& \theta <\alpha <\pi \\ =-{\mathrm{Northward}}_1{i}_1+{\mathrm{Due\; north}}_{\mathrm{two}}{i}_{\mathrm{two}}& \pi <\alpha <\pi +\theta \\ =-N_{\mathrm{one}}{i}_1-{\mathrm{Northward}}_2{i}_2& \pi +\theta <\alpha <2\pi \end{array}\\ \end{array}$

The magnetic flux linkage for the two coils λ _one and λ ₂ are given by

${\lambda }_{\mathrm{one}}={\mathrm{North}}_1{\int }_o^{\pi }{B}_r(\alpha )\ell R_{r}d\alpha$

(87) ${\lambda }_2=N_2{\int }_{\theta }^{\pi +\theta }{B}_r(\alpha )\ell R_{r}d\alpha$

where R_r is the rotor radius.

It is assumed in the integrals for λ ₁ and λ ₂ that the then-called fringing magnetic flux exterior of the centric length ℓ of the rotor/stator is negligible. Using the concept of inductance for each coil equally introduced in the discussion nearly solenoids, this flux linkage can be written as a linear combination of the contributions from i ₁ and i ₂:

(88) ${\lambda }_1=L_1{i}_{\mathrm{one}}+{\mathrm{50}}_{\mathrm{thou}}{i}_{\mathrm{ii}}$

(89) ${\lambda }_2=L_m{i}_{\mathrm{i}}+L_{\mathrm{two}}{i}_2$

where

(90) $L_1=N_1^2{L}_o=\text{self inductance of curlicue}\mathrm{ane}$

(91) ${\mathrm{Fifty}}_2=N_2^2{L}_o=\text{self inductance of coil 2}$

(92) $L_o=\frac{{\mu }_o\ell R_r\pi }{2g}$

The parameter L_yard is the mutual inductance for the ii coils which is defined as the flux linkage induced in each whorl due to the electric current in the other divided by that current and is given by

$\begin{array}{c}\begin{array}{cc}{\mathrm{50}}_m=L_o{N}_1{\mathrm{North}}_2\left(\mathrm{one}-\frac{2\theta }{\pi }\right)& 0<\theta <\pi \\ ={\mathrm{50}}_o{N}_1{N}_{\mathrm{two}}\left(\mathrm{one}+\frac{2\theta }{\pi }\right)& -\pi <\theta <0\end{array}\\ \end{array}$

The higher up formulas for these inductances provide a sufficient model to derive the terminal voltage/current relationships besides as the electromechanical models for motor performance calculations. The self-inductances for each coil are independent of θ, but the mutual inductance varies with θ such that Fifty_k (θ) is a symmetric part of θ. Information technology tin be formally expanded in a Fourier series in θ having only cosine terms in odd harmonics as given below:

(93) ${\mathrm{Fifty}}_{\mathrm{chiliad}}(\theta )=M_1\cos (\theta )+{\mathrm{Grand}}_{\mathrm{three}}\cos (3\theta )+M_{\mathrm{v}}\cos (5\theta )+\dots$

In whatsoever practical motor, there will be a distribution of windings such that the fundamental component One thousand _ane predominates; that is, the mutual inductance is given approximately past

(94) ${\mathrm{Fifty}}_g\simeq M\cos (\theta )$

For notational convenience, the subscript ane on M _ane is dropped. Whatever motor fabricated up of multiple matching pairs of coils in the stator and rotor volition have a set of final relations in the flux linkages for the stator and rotor λ_s and λ_r , respectively, given by

${\lambda }_{\mathrm{southward}}={\mathrm{50}}_{\mathrm{due\; south}}{i}_{\mathrm{due\; south}}+\mathrm{1000}{i}_r\cos \theta$

${\lambda }_r=L_r{i}_r+M{i}_s\cos \theta$

The torque of electrical origin acting on the rotor T_e is given past

$T_{\mathrm{eastward}}=\frac{\partial {\mathrm{West}}_{kM}}{\partial \theta }$

where, for a linear lossless system, the common coupling energy W_mM is

$W_{m\mathrm{Thousand}}=i_s{i}_r{\mathrm{Fifty}}_m(\theta )$

The torque T_e is given by

$T_e=-i_{\mathrm{south}}{i}_{r}M\sin \theta$

The mechanical dynamics for the motor are given by

$T_{\mathrm{eastward}}=J_r\frac{d^2\theta }{d{t}^{\mathrm{two}}}+B_{\mathrm{v}}\frac{d\theta }{dt}+C_c\mathrm{sgn}\left(\frac{d\theta }{dt}\right)$

where J_r is the rotor moment of inertia nearly its centrality, B _v is the rotational damping coefficient due to rotational viscous friction, and C_c is the coulomb friction coefficient.

Information technology is of interest to evaluate the motor performance by computing the motor mechanical power P_{one thousand} for a given excitation. Permit the excitation of the stator and rotor exist from ideal current sources such that

(95) $\begin{array}{l}{i}_s=I_s\sin ({\omega }_{s}t)\\ i_r=I_r\sin ({\omega }_{r}t)\\ \theta (t)={\omega }_{m}t+\gamma \end{array}$

where ω₁₀₀₀ is the rotor rotational frequency (rad/sec) and γ expresses an arbitrary time phase parameter. The motor ability is given by

(96) $P_m=T_e{\omega }_m$

(97) $=-{\omega }_{\mathrm{1000}}{I}_s{I}_{r}M\sin ({\omega }_{\mathrm{due\; south}}t)\sin ({\omega }_{r}t)\sin ({\omega }_{\mathrm{yard}}t+\gamma )$

This equation can be rewritten using well-known trigonometric identities in the form

(98) $P_{\mathrm{chiliad}}=-\frac{{\omega }_m{I}_{\mathrm{south}}{I}_r\mathrm{Grand}}{4}\left\{\sin \left[\left({\omega }_m+{\omega }_{\mathrm{south}}-{\omega }_r\right)t+\gamma \right]+\sin \left[({\omega }_{\mathrm{chiliad}}-{\omega }_{\mathrm{due\; south}}+{\omega }_r)t+\gamma \right]-\sin \left[\left({\omega }_m+{\omega }_{\mathrm{southward}}+{\omega }_r\right)t+\gamma \right]-\sin \left[\left({\omega }_m-{\omega }_{\mathrm{southward}}-{\omega }_r\right)t+\gamma \right]\right\}$

The time average value of whatever sinusoidal function of time is zilch. The merely conditions under which the motor tin can produce a nonzero average ability are given by the frequency relationships below:

(99) ${\omega }_{\mathrm{thou}}=\pm {\omega }_s\pm {\omega }_r$

For example, whenever ω_m   = ω_s   + ω_r , the motor time average power $P_{{\mathrm{grand}}_{a\text{v}}}$ is given by

(100) $P_{m_{a\text{5}}}=\frac{{\omega }_{\mathrm{thou}}{I}_s{I}_{r}M}{4}\sin \gamma$

In such a motor, an equilibrium functioning will be achieved when $P_{{\mathrm{grand}}_{a\text{v}}}=P_{\mathrm{Fifty}}$ where P_L   =   load ability. Thus, the stage between rotor and stator fields is given by

(101) $\sin \gamma =\frac{\mathrm{iv}{P}_L}{{\omega }_m{I}_s{I}_{r}M}$

provided

(102) $P_L\le \frac{{\omega }_{\mathrm{thousand}}{I}_{\mathrm{southward}}{I}_r\mathrm{Thou}}{\mathrm{four}}$

The to a higher place frequency conditions (Eqn (99)) are cardinal to all rotating machines and are required to be satisfied for any nonzero boilerplate mechanical output power. Each unlike type of motor has a unique way of satisfying the frequency atmospheric condition. Nosotros illustrate with a specific example, which has been employed in certain hybrid vehicles. This example is the induction motor. Still, earlier proceeding with this instance, information technology is important to consider an upshot in motor functioning. Normally, electric motors that are intended to produce substantial amounts of power (e.chiliad., for hybrid vehicle application) are polyphase machines; that is, in addition to the windings associated with stator excitation, a polyphase machine will take ane or more additional sets of windings that are excited by the same frequency only at dissimilar phases. Although iii-phase motors are in mutual use, the assay of a two-stage induction motor illustrates the basic principles of polyphase motors with a relatively simplified model and is causeless in the following discussion.

A ii-phase motor has two sets of windings displaced at 90° in the θ direction and excited by currents with a ninety° phase for both stator and rotor. A so-chosen balanced two-phase motor will have its roll excited by currents i_as , i_bs for phases a and b, respectively, where

(103) $i_{as}=I_{\mathrm{due\; south}}\cos ({\omega }_{s}t)$

$i_{bs}=I_s\sin ({\omega }_{s}t)$

The rotor is also constructed with ii sets of windings displaced physically by ninety° and excited with currents i_ar and i_br having xc° phase shift:

(104) $i_{ar}=I_r\cos ({\omega }_{r}t)$

$i_{br}=I_r\sin ({\omega }_{r}t)$

A two-phase induction motor is one in which the stator windings are excited by currents given above (i.due east., i_equally and i_bs ). The rotor circuits are curt-circuited such that v _ar   =   v _br   =   0, where v _ar is the last voltage for windings of phase a and v _br is the last voltage for the b phase. The currents in the rotor are obtained by induction from the stator fields. By extension of the analysis of the single-phase excitation, the final flux linkages are given past

(105) $\begin{array}{l}{\lambda }_{as}={\mathrm{Fifty}}_{\mathrm{south}}{i}_{as}+\mathrm{Thou}{i}_{ar}\cos \theta -M{i}_{br}\sin \theta \\ {\lambda }_{bs}={\mathrm{Fifty}}_s{i}_{bs}+M{i}_{ar}\sin \theta +M{i}_{br}\cos \theta \\ {\lambda }_{ar}={\mathrm{50}}_r{i}_{ar}+M{i}_{a\mathrm{due\; south}}\cos \theta +\mathrm{One\; thousand}{i}_{b\mathrm{south}}\sin \theta \\ {\lambda }_{br}={\mathrm{50}}_r{i}_{br}-M{i}_{as}\sin \theta -M{i}_{bs}\cos \theta \end{array}$

The torque T_e and instantaneous power P_m for the two-stage induction motor are given by

(106) $T_e=\mathrm{Yard}[(i_{ar}{i}_{b\mathrm{due\; south}}-i_{br}{i}_{as})\cos \theta -(i_{ar}{i}_{a\mathrm{south}}+i_{br}{i}_{bs})\sin \theta ]$

$P_{\mathrm{thou}}={\omega }_{m}M{I}_s{I}_r\sin [({\omega }_m-{\omega }_s+{\omega }_r)t+\gamma ]$

The average power P_av is nonzero when ω_m   = ω_south   − ω_r and is given by

$P_a={\omega }_{\mathrm{yard}}M{I}_s{I}_r\sin \gamma$

Since the rotor terminals are brusque-circuited, nosotros have

(107) $\frac{d{\lambda }_{ar}}{dt}=\frac{d{\lambda }_{br}}{dt}=0$

The two rotor currents, thus, satisfy the post-obit equations:

(108) $0=R_r{i}_{ar}+L_r\frac{d{i}_{ar}}{dt}+M{I}_{\mathrm{southward}}\frac{d}{dt}[\cos ({\omega }_{\mathrm{southward}}t)\cos ({\omega }_{\mathrm{grand}}t+\gamma )+\sin ({\omega }_{s}t)\sin ({\omega }_{m}t+\gamma )]$

(109) $0=R_r{i}_{br}+L_r\frac{d{i}_{br}}{dt}+\mathrm{1000}{I}_s\frac{d}{dt}[-\cos ({\omega }_{s}t)\sin ({\omega }_{\mathrm{one\; thousand}}t+\gamma )+\sin ({\omega }_{s}t)\cos ({\omega }_{m}t+\gamma )]$

where R_r and L_r are the resistance and self-inductance of the two sets of (presumed) identical structure). These equations tin can be rewritten as

(110) ${\mathrm{50}}_r\frac{d{i}_{ar}}{dt}+R_r{i}_{ar}=K{I}_s({\omega }_s-{\omega }_{\mathrm{thou}})\sin [({\omega }_s-{\omega }_m)t-\gamma ]$

(111) $L_r\frac{d{i}_{br}}{dt}+R_r{i}_{br}=-\mathrm{1000}{I}_s({\omega }_s-{\omega }_{\mathrm{one\; thousand}})\cos [({\omega }_s-{\omega }_g)t-\gamma ]$

The current i_ab is identical to i_ar except for a 90° phase shift every bit can exist seen from Eqn (111). Note that the electric current for both phases are at frequency ω_r where

${\omega }_r=({\omega }_s-{\omega }_{\mathrm{grand}})$

Thus, the induction motor satisfies the frequency condition by having currents at the difference between excitations and rotor rotational frequency. The current i_ar is given by

(112) $i_{ar}=\frac{({\omega }_s-{\omega }_{\mathrm{one\; thousand}})\mathrm{Yard}{I}_s}{\sqrt{R_r^2+{({\omega }_s-{\omega }_m)}^2{L}_r^2}}\cos [({\omega }_s-{\omega }_{\mathrm{chiliad}})t-\alpha ]$

where

$\alpha =-\left(\frac{\pi }{2}+\gamma +\beta \right)$

and

(113) $\beta ={\tan }^{-1}\left[\frac{\left({\omega }_s-{\omega }_m\right)}{R_r}{L}_r\right]$

The current in phase b is identical except for a ninety° stage shift. Substituting the currents for rotor and stator into the equation for torque T_east yields the remarkable result that the this torque is independent of θ and is given past

(114) $T_e=\frac{({\omega }_s-{\omega }_m)M^2{R}_r{I}_{\mathrm{south}}^{\mathrm{ii}}}{R_r^2+{({\omega }_s-{\omega }_m)}^{\mathrm{two}}{L}_r^{\mathrm{ii}}}$

The mechanical output ability P_m is given by

$\begin{array}{c}{P}_{\mathrm{one\; thousand}}={\omega }_m{T}_{\mathrm{east}}\\ =\left[\frac{{\omega }_s^2{\mathrm{Grand}}^2{I}_s^2}{{\left(R_r/\mathrm{due\; south}\right)}^{\mathrm{two}}+{\omega }_s^2{L}_r^2}\right]\left(\frac{1-s}{s}\right)R_r\end{array}$

where s is called slip and is given past

(115) $s=\frac{{\omega }_s-{\omega }_k}{{\omega }_s}$

The induction automobile has iii modes of operation every bit characterized past values of s. For 0   < s  <   1 information technology acts every bit a motor and produces mechanical power. For −one   < s  <   0 information technology acts like a generator and mechanical input ability to the rotor is converted to output electrical power. For s  >   ane, the induction machine acts similar a brake with both electric input and mechanical input power dissipated in rotor i_r ² R_r losses. Because of its versatility, the induction motor has great potential in hybrid/electric vehicle propulsion applications. All the same, information technology does require that the control system incorporates solid-state power switching electronics to exist able to handle the necessary currents. Moreover, information technology requires precise command of the excitation current.

The application of an induction motor to provide the necessary torque to move a hybrid or electric vehicle is influenced by the variation in torque with rotor speed. Examination of Eqn (114) reveals that the motor produces zero torque at synchronous speed (i.e., ω_thousand   − ω_south ). The torque of an induction motor initially increases from its value at ω_thousand   =   0 reaches a maximum torque (T _max) at a speed ${\omega }_m>{\omega }_m^{\ast }$ when

$0\le {\omega }_{\mathrm{thou}}^{\ast }\le {\omega }_s$

The torque has a negative slope given past

$\begin{array}{cc}\frac{d{T}_{\mathrm{east}}}{d{\omega }_k}<0& {\omega }_m>{\omega }_m^{\ast }\end{array}$

Normally, an induction motor is operated in the negative slope region of T_m (ω_grand ) (i.e., ${\omega }_m^{\ast }>{\omega }_m<{\omega }_{\mathrm{south}}$ ) for stable performance. Equilibrium is reached at a motor rotational speed ω_grand at which the motor torque T_east and load torque T_Fifty are equal, i.e. T_e (ω_{one thousand} )   = T_L (ω_m ).

This point is illustrated for a hypothetical load torque that is a linear function of motor speed such that the load torque is given by

(116) $T_L={\mathrm{Chiliad}}_L{\omega }_{\mathrm{grand}}$

Figure 6.35 illustrates the motor and load torques for a load that varies linearly with ω_grand .

Figure 6.35. Normalized torque T_thou vs. normalized load torques T_{Fifty 1} T_{L ii}.

For convenience of presentation, Figure 6.35 presents normalized motor torque and load torque normalized to the maximum torque T _max where

(117) $T_{\max }=\underset{{\omega }_{\mathrm{yard}}}{\max }(T_e({\omega }_{\mathrm{thou}}))$

This maximum occurs at ${\omega }_m={\omega }_m^{\ast }$ , which, for the present hypothetical normalized instance, is given by

$\frac{{\omega }_m^{\ast }}{{\omega }_s}\cong .68$

Effigy 6.33 also presents two load torques normalized to T _max:

$\begin{array}{l}{T}_{L1}=K_{\mathrm{Fifty}\mathrm{ane}}{\omega }_{\mathrm{grand}}/T_{\max }\\ T_{L2}={\mathrm{Thousand}}_{L2}{\omega }_m/T_{\max }\end{array}$

where

$K_{\mathrm{50}2}>K_{L\mathrm{i}}$

The operating motor speed for these two load torques are the two intersection points ω ₀₁ and ω ₀₂ where

$\begin{array}{l}{T}_m({\omega }_{01})=T_{L1}({\omega }_{01})\\ T_m({\omega }_{02})=T_{L\mathrm{ii}}({\omega }_{02})\end{array}$

These 2 intersection points are the steady-land operating conditions for the two load torques. The higher of the two loads has a steady-state operating indicate lower than the showtime (i.e., ω ₀₂  < ω ₀₁).

Chapter 7 discusses the control of an induction motor that is used in a hybrid electric vehicle. At that place the model for load torque vs. vehicle operating conditions is developed.

Brushless DC Motors

Next, we consider a relatively new blazon of electric motor known every bit a brushless DC motor. A brushless DC motor is non a DC motor at all in that the excitation for the stator is Ac. However, it derives its name from physical and performance similarity to a shunt-connected DC motor with a abiding field electric current. This type of motor incorporates a permanent magnet in the rotor and electromagnet poles in the stator every bit depicted in Effigy 6.36. Traditionally, permanent magnet rotor motors were generally but useful in relatively low-ability applications. Contempo evolution of some relatively powerful rare earth magnets and the development of high-ability switching solid-state devices accept essentially raised the power capability of such machines.

Figure six.36. Brushless DC motor.

The stator poles are excited such that they have magnetic N and S poles with polarity as shown in Figure six.36 by currents I_a and I_b . These currents are alternately switched on and off from a DC source at a frequency that matches the speed of rotation. The switching is done electronically with a organization that includes an athwart position sensor attached to the rotor. This switching is done so that the magnetic field produced by the stator electromagnets always applies a torque on the rotor in the management of its rotation.

The torque ${\overline{T}}_m$ applied to the rotor by the magnetic field intensity vector $\overline{H}$ created by the stator windings is given by the following vector production

(118) ${\overline{T}}_g=\gamma (\overline{M}\times \overline{H})$

where $\overline{M}$ is the magnetization vector for the permanent magnet and γ is the constant for the configuration.

The direction of this torque is such as to crusade the permanent magnet to rotate toward parallel alignment with the driving field $\overline{H}$ (which is proportional to the excitation current). The magnitude of the torque T_k is given by

$T_g=\gamma MH\sin (\theta )$

where M = magnitude of $\overline{M}$ , H = magnitude of $\overline{H}$ and θ = angle between $\overline{M}$ and $\overline{H}$ .

If the permanent magnet rotor were allowed to rotate in a static magnetic field, it would only turn until θ  =   0 (i.e., alignment).

In a brushless DC motor, however, the excitation fields are alternately switched electronically such that a torque is continuously applied to the rotor magnet. In lodge for this motor to continue to have a nonzero torque practical, the stator windings must be continuously switched synchronous with rotor rotation. Although only two sets of stator windings are shown in Figure six.36 (i.e., two-pole motorcar), usually at that place would be multiple sets of windings, each driven separately and synchronously with rotor rotation. In effect, the sequential awarding of stator currents creates a rotating magnetic field which rotates at rotor frequency (ω_r ).

A simplified block diagram of the two-pole motor control system for the motor of Figure half dozen.36a and b is shown in Figure vi.36c. A sensor S measures the angular position θ of the rotor relative to the axes of the magnetic poles of the stator. A controller determines the time for switching currents I_a and I_b on as well as the elapsing. The switching times are determined such that a torque is applied to the rotor in the management of rotation.

At the appropriate time, transistor A is switched on, and electric power from the on-board DC source (e.chiliad., battery pack) is supplied to the poles A of the motor. The elapsing of this current is regulated by controller C to produce the desired power (as commanded by the commuter). After rotating approximately ninety°, current I_b is switched on by activating transistor B via a signal sent by controller C.

The rotor permanent magnet is equivalent to an electromagnet with d-c excitation (i.e., ω_r   =   0). The frequency at which the currents to the stator coils are switched is always synchronous with the speed of rotation. Thus, the frequency condition for the motor is satisfied since ω_s   = ω_thousand . This speed is determined by the mechanical load on the motor and the ability allowable by the controller. Equally the power control is increased, the controller responds past increasing the duration of the current pulse supplied to each stator curlicue. The power delivered by the motor is proportional to the fraction of each bicycle that the electric current is on (i.e., the so-called duty wheel).

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Electrical Machines, Design

Enrico Levi , in Encyclopedia of Concrete Science and Engineering (3rd Edition), 2003

II.C Magnetic and Electric Loadings

The magnetic flux density B, which is relevant to the electromechanical power conversion process, is the constructive or rms value of the radial component of B at the air gap. Except for special cases, such every bit superconducting field excitation and printed windings, this value is determined by the characteristics of the ferromagnetic construction into which the conductors are embedded. This consists of a core or yoke, which provides both physical integrity and a path for the magnetic flux, and a slotted portion side by side to the air gap, which accommodates the active conductors. The slot dimensions represent a compromise between the alien requirements of conduction of current in the copper and of flux in the iron teeth. It turns out that in that location is a value of the ratio between the slot and molar widths that minimizes the book and weight of the ferromagnetic construction. This value is unity, so that the tooth width should be half the slot pitch. If ane assumes that all the flux crossing the air gap passes through the iron teeth, the flux density B _t in the teeth is related to B as

(15) $\text{B}=\frac{\mathrm{i}}{2}{\text{B}}_{\text{t}}.$

When the tooth is driven too far into saturation, the magnetizing current and the iron loss rise steeply to unacceptable values and the wave shape of the flux distribution is plain-featured. Also, part of the flux is diverted to the slot, which causes an increase in the boosted copper losses and the transfer of the force from the tooth to the conductor. As a upshot the electrical insulation is stressed mechanically. For these reasons rms values of B _t in backlog of 1.four   T are non recommended and B is practically express to about 0.7   T.

In contrast to the magnetic loading, there is no optimal value for the electric loading. Its value is adamant primarily by thermal considerations, which are related to the losses in the machine.

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Magnetic and Electrical Separation

Barry A. Wills , James A. Finch FRSC, FCIM, P.Eng. , in Wills' Mineral Processing Technology (8th Edition), 2016

13.three Equations of Magnetism

The magnetic flux density or magnetic induction is the number of lines of force passing through a unit area of material, B. The unit of magnetic induction is the tesla (T).

The magnetizing forcefulness, which induces the lines of force through a cloth, is called the field intensity, H (or H-field), and by convention has the units ampere per meter (A   grand⁻¹) (Bennett et al., 1978).

The intensity of magnetization or the magnetization (1000, A   m^−ane) of a material relates to the magnetization induced in the material and can also be thought of as the volumetric density of induced magnetic dipoles in the material. The magnetic induction, B, field intensity, H, and magnetization, M, are related by the equation:

(13.1) $B={\mu }_0(H+\mathrm{One\; thousand})$

where μ ₀ is the permeability of free infinite and has the value of 4π×ten⁻⁷  N   A⁻². In a vacuum, M=0, and M is extremely depression in air and water, such that for mineral processing purposes Eq. (13.ane) may be simplified to:

(thirteen.2) $B={\mu }_{0}H$

and then that the value of the field intensity, H, is straight proportional to the value of induced flux density, B (or B-field), and the term "magnetic field intensity" is then ofttimes loosely used for both the H-field and the B-field. However, when dealing with the magnetic field inside materials, peculiarly ferromagnetic materials that concentrate the lines of strength, the value of the induced flux density will exist much higher than the field intensity. This relationship is used in loftier-gradient magnetic separation (discussed further in Section 13.4.1). For clarity information technology must be specified which field is being referred to.

Magnetic susceptibility (χ) is the ratio of the intensity of magnetization produced in the fabric over the applied magnetic field that produces the magnetization:

(xiii.3) $\chi =\frac{\mathrm{Grand}}{H}$

Combining Eqs. (13.1) and (13.3) we become:

(thirteen.4) $B={\mu }_{0}H(1+\chi )$

If we then define the dimensionless relative permeability, μ, as:

(13.5) $\mu =1+\chi$

nosotros tin combine Eqs. (thirteen.4) and (thirteen.v) to yield:

(thirteen.6) $B=\mu {\mu }_{0}H$

For paramagnetic materials, χ is a modest positive constant, and for diamagnetic materials it is a much smaller negative abiding. As examples, from Figure 13.i the slope representing the magnetic susceptibility of the fabric, χ, is about 0.001 for chromite and −0.0001 for quartz.

The magnetic susceptibility of a ferromagnetic material is dependent on the magnetic field, decreasing with field strength equally the textile becomes saturated. Figure 13.2 shows a plot of M versus H for magnetite, showing that at an applied field of 80   kA   m⁻¹, or 0.1   T, the magnetic susceptibility is well-nigh 1.7, and saturation occurs at an practical magnetic field force of about 500   kA   thou⁻¹ or 0.63   T. Many high-intensity magnetic separators use fe cores and frames to produce the desired magnetic flux concentrations and field strengths. Iron saturates magnetically at about two–2.5   T, and its nonlinear ferromagnetic relationship between inducing field strength and magnetization intensity necessitates the use of very large currents in the energizing coils, sometimes upwardly to hundreds of amperes.

The magnetic forcefulness felt by a mineral particle is dependent not merely on the value of the field intensity, but also on the field gradient (the rate at which the field intensity increases beyond the particle toward the magnet surface). As paramagnetic minerals take higher (relative) magnetic permeabilities than the surrounding media, commonly air or h2o, they concentrate the lines of strength of an external magnetic field. The higher the magnetic susceptibility, the college the induced field density in the particle and the greater is the attraction upward the field gradient toward increasing field strength. Diamagnetic minerals accept lower magnetic susceptibility than their surrounding medium and hence miscarry the lines of force of the external field. This causes their expulsion down the slope of the field in the direction of the decreasing field force.

The equation for the magnetic force on a particle in a magnetic separator depends on the magnetic susceptibility of the particle and fluid medium, the practical magnetic field and the magnetic field gradient. This equation, when considered in only the x-direction, may be expressed as (Oberteuffer, 1974):

(13.7) $F_x=5({\chi }_{\mathrm{p}}-{\chi }_{\mathrm{m}})H\frac{\mathrm{d}B}{\mathrm{d}\mathrm{10}}$

where F _x is the magnetic force on the particle (N), V the particle book (k^three), χ _p the magnetic susceptibility of the particle, χ _yard the magnetic susceptibility of the fluid medium, H the applied magnetic field strength (A   thou^−one), and dB/dx the magnetic field gradient (T   chiliad⁻¹=North   A⁻¹  grand⁻²). The product of H and dB/dx is sometimes referred to as the "forcefulness cistron."

Production of a loftier field slope as well as high intensity is therefore an important attribute of separator blueprint. To generate a given bonny forcefulness, in that location are an infinite number of combinations of field and gradient which will requite the same consequence. Another important factor is the particle size, every bit the magnetic force experienced by a particle must compete with various other forces such equally hydrodynamic drag (in moisture magnetic separations) and the force of gravity. In i example, considering only these two competing forces, Oberteuffer (1974) has shown that the range of particle size where the magnetic force predominates is from nigh five   μm to one   mm.

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Numerical and experimental identification of the static characteristics of a combined Journal-Magnetic begetting: Smart Integrated Begetting

1000. El-Hakim , ... A. El-Shafei , in tenth International Briefing on Vibrations in Rotating Machinery, 2012

i NOMENCLATURE

B:

magnetic flux density (T)

Φ:

magnetic flux (Tm)

μ_o:

permeability (Tm/A), permeability of free space: 4π × 10^{-   vii}

μ_r:

relative permeability

H:

magnetomotance (A/m)

l:

magnetic flux iron path length (m)

l_thousand :

magnetic gap (yard)

F:

force (North)

J:

current density (A/mⁱⁱ)

I:

current (A)

α:

pole inclination angle

N:

coils number of turns

A_chiliad:

magnetic pole surface area (thou²)

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Application of Evolutionary Algorithm for Multiobjective Transformer Design Optimization

Due south. Tamil Selvi , ... S. Rajasekar , in Classical and Recent Aspects of Ability System Optimization, 2018

Choice of Current Density

The operating magnetic flux density is the parameter that determines the loss in the magnetic core. Similarly, electric current density in the windings determines the loss in the windings. When the current density is increased, cantankerous-sectional surface area of the windings is reduced and hence, the volume and in turn copper weight are reduced. On the other mitt, copper loss, which varies every bit a square of electric current density, is increased causing efficiency to reduce. Moreover, temperature rise will increase and injure the insulation [7].

The pick of the current density must be done in such a mode that the maximum temperature of the transformer due to losses is beneath the insulation grade temperature. Current density chosen should guarantee the level of losses and cooling conditions required. However, a designer must compare the increased price due to the improved cooling method required with the economy in material due to the choice of increased value of current density. In short, current density is governed past load losses, temperature class of insulation, and short circuit electric current withstanding ability. Maximum limit for current density is calculated equally

(6) $\text{Maximum electric current density}=\frac{j}{Z_{\mathit{sc}}}$

where Z _sc is the short circuit impedance in %; j is the brusque circuit electric current density in A/mmⁱⁱ, which can be calculated using,

(7) ${\theta }_2={\theta }_0+j^{\mathrm{ii}}.y.t1{.10}^{-3}°\mathrm{C}$

where

θ ₂  =   Maximum permissible average winding temperature, which is 250°C for copper conductor and 200°C for aluminium conductor;

θ ₀  =   Initial temperature of winding, which is 105°C;

ti   =   Duration of short circuit. It is 2   south;

y  =   Function of $\frac{\mathrm{i}}{\mathrm{ii}}\left({\theta }_2+{\theta }_0\right)$ , in accordance with (Clause ix.15: Table 6, IS2026 Function I).

It is therefore necessary to requite considerations in choosing value for electric current density while designing.

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Relaxometers

Ralf-Oliver Seitter , Rainer Kimmich , in Encyclopedia of Spectroscopy and Spectrometry, 1999

List of symbols

B ₀ = external magnetic flux density; B _D = detection field; B _E = magnetic flux density, evolution interval; B _P = magnetic flux density, preparation interval; Chiliad _i (τ) = dipolar autocorrelation function; J ⁽ⁱ⁾(ω) = intensity office of the Larmor frequency; M _E = Curie magnetization, evolution interval; Grand _P = Curie magnetization, preparation interval; S/N = signal-to-noise ratio; T ₁ = spin–lattice relaxation fourth dimension; T _d = dipolar-lodge relaxation time;T _1ρ = rotating-frame relaxation time; T _two = transverse relaxation time; γ = gyromagnetic ratio; μ₀ = magnetic field constant.

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Solid-Country NMR Using Quadrupolar Nuclei

Alejandro C. Olivieri , in Encyclopedia of Spectroscopy and Spectrometry, 1999

List of symbols

B ₀ = magnetic flux density; D = dipolar coupling constant; D′ = effective dipolar coupling constant; h = Planck'due south constant; I = spin- $\frac{1}{2}$ nucleus; J = coupling abiding; q = field gradient tensor; Q = nuclear quadrupole moment; south′ = doublet splitting; Due south = quadrupolar nucleus; γ = magnetogyric ratio; θ = angle between primary tensor axes; δ = chemic shift; τ ₁ = relaxation time; ξ = angle between primary axes of interaction tensors and sample spinning axis; χ = quadrapole coupling constant.

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Modeling and analysis of forces and finishing spot size in the ball finish magnetorheological finishing (BEMRF) process

Zafar Alam , ... Sunil Jha , in Machining and Tribology, 2022

Exercises

1.

Summate magnetic flux density at the center of a BEMRF tool tip and a mild steel workpiece kept 2  mm apart. Assume all conditions to be same as in Example one. [Respond: B   =   7.87 mT]

2.

Calculate the area of the indented role (A′) of an abrasive particle of bore twenty   μm at which the cutting action volition fail in BEMRF process on a mild steel workpiece. [Answer: A′  =   2.82 × 10 ⁻¹⁶   m ⁱⁱ ]

iii.

Calculate the radius (R) of an MRP fluid hemispherical ball cease rotating at 200   rpm [Reply: R   =   iv.33 mm]

Where required, utilize value of ${\tau }_y$ from Example ii.

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Biomedical Applications of Electromagnetic Fields

Dragan Poljak PhD , Mario Cvetković PhD , in Human Interaction with Electromagnetic Fields, 2019

7.1.iii.3 Magnetic Flux Density

The results for the magnetic flux density, obtained using (7.8) and (7.9), were compared to the belittling results. The results for the maximum values are given in Table vii.7.

Table 7.7. Comparison of maximum magnetic flux density. From CVETKOVIĆ, Mario; POLJAK, Dragan; HAUEISEN, Jens. Analysis of transcranial magnetic stimulation based on the surface integral equation formulation. IEEE Transactions on Biomedical Engineering, 2015, 62.6: 1535–1545 [28].

		Round	eight-whorl	Butterfly
Analytical
Bmax	[T]	0.679	0.672	0.826
SIE model
Bmax	[T]	0.750	0.656	0.792

A comparison of the magnetic flux density in the coronal cross-section of the brain model is shown on Fig. 7.6.

Fig. seven.6. Comparison of magnetic flux density in the human brain (coronal cross-section). The results on the left are obtained via analytical expressions for (A) circular, (B) 8-coil, and (C) butterfly curl, while the results on the correct are obtained via proposed model for (D) circular, (Eastward) 8-coil, and (F) butterfly curlicue. From [28].

The results from Table 7.7 and Fig. 7.6 indicate that the encephalon itself does non significantly disturb the magnetic field of the curlicue, although a lower maximum value of the magnetic flux density was obtained for the 8-coil and butterfly coil. The distribution of the magnetic flux density in the coronal cross-section obtained using the SIE model shows some discontinuities, which can be related to the interpolation method used. This numerical antiquity could be overcome by computing the field at more points before interpolating results in the neighboring area.

The magnetic flux density B on the brain surface can be clearly seen on Fig. vii.7.

Fig. 7.7. Magnetic flux density on the brain surface due to: (A) round coil, (B) figure-of-viii whorl, and (C) butterfly coil. All coils are placed 1 cm over the master motor cortex. From [28].

It is interesting to observe the dependence of the induced electric field E and magnetic flux density B on the altitude from the brain surface, as shown on Fig. 7.8.

Fig. 7.8. Dependence of the induced electric field E and magnetic flux density B on the distance from the brain surface. The values given are on the points directly under the coil geometric center. From [28].

From Fig. seven.8 the rapid decrease of both E and B fields straight under the geometric center of the stimulation curl is clearly evident in all three cases. For the circular ringlet, the maximum value is much lower compared to the other 2 coils as the maximum field volition exist induced under the gyre windings, as shown on Fig. 7.four.

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Domain_3 Of Magnetic Field Formula,

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