Control Reading in Tesla Gravity Flux Values

Incident Electromagnetic Field Dosimetry

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

iii.1.ii.2 The Magnetic Field

The magnetic flux density at an capricious bespeak due to a current chemical element shown in Fig. 3.17 is adamant by the Biot–Savart'southward constabulary:

Fig. iii.17. Straight current element.

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

where μ is permeability, i denotes the current along the segment, and $R=|\overrightarrow{r}-\overrightarrow{r}\prime |$ is the altitude from the source $i(t)\overrightarrow{dl}$ to the observation betoken P.

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

(three.26) $\overrightarrow{B}={\hat{e}}_{\phi }\frac{\mu i(t)}{\mathrm{four}\pi \rho }\underset{{\theta }_{\mathrm{one}}}{\overset{{\theta }_2}{\int }}\cos \theta d\theta ={\hat{e}}_{\phi }\frac{\mu i(t)}{\mathrm{iv}\pi \rho }(\sin {\theta }_1+\sin {\theta }_2),$

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

The ELF magnetic field value at an capricious point can exist assessed by assembling the contributions of all conductors divided in a certain number of straight segments. The yardth straight segment carrying current $i_g$ in Cartesian three-dimensional coordinate system is shown in Fig. three.18.

Fig. 3.18. Straight segment in Cartesian coordinate system.

Using the Biot–Savart's police force, the magnetic field value at point C due to the considered conductor segment tin be written as follows [6]:

(3.27) $B_m(t)=\frac{\mu i_k(t)}{4\pi R_{\text{RS}}}(\frac{R_{\text{PS}}}{R_{\text{PR}}}+\frac{R_{\text{SQ}}}{R_{\text{QR}}}),$

where the corresponding distances R are assigned as in Fig. 3.18.

Total components of the magnetic flux density generated by N segments are assembled from the contributions of all segments. Therefore, the total value of the magnetic flux density at a given indicate of infinite can be expressed equally

(3.28) $B(t)=\begin{array}{c}\hfill \sqrt{{(\sum _{i=1}^{\mathrm{Northward}}{B}_{\mathrm{ten},i}(t))}^2+{(\sum _{i=1}^{\mathrm{North}}{B}_{y,i}(t))}^2+{(\sum _{i=1}^{\mathrm{North}}{B}_{z,i}(t))}^{\mathrm{ii}}}\end{array},$

where $B_{\mathrm{10},i}(t)$ , $B_{y,i}(t)$ and $B_{z,i}(t)$ are the components of the magnetic flux density due to the ith segment.

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

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

If the human body is exposed to an ELF magnetic field, the round electric current density is induced within the body due to the beingness of the normal component of the magnetic flux density.

Once the magnetic flux density is determined, the internal current density can be calculated using the deejay model of the human torso.

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Sensors and Actuators

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

Electrical Motor Actuators

Perhaps the nigh important electromechanical actuator in automobiles is an electric motor. Electric motors have long been used on automobiles first with the starter motor, which uses electric power supplied by a storage bombardment to rotate the engine at sufficient RPM that the engine can exist made to start running. Motors take as well been employed to raise or lower windows, position seats equally well equally for actuators on airflow control at idle (come across Affiliate 7). In recent times, electric motors have been used to provide the vehicle primary motive power in hybrid or electric vehicles.

There are a bang-up number of electric motor types that are classified by the blazon of excitation (i.east., dc or air conditioning), the physical structure (e.g., shine air gap or salient pole), and by the type of magnet structure for the rotating chemical element (rotor) which can exist either a permanent magnet or an electromagnet. However, there are sure cardinal similarities between all electric motors, which are discussed below. However another stardom betwixt types of electrical motors is based upon whether the rotor receives electric excitation from sliding mechanical switch (i.e., commutator and brush) or by induction. Regardless of motor configuration, each is capable of producing mechanical power due to the torque applied to the rotor by the interaction of the magnetic fields between the rotor and the stationary structure (stator) that supports the rotor along its axis of rotation.

It is beyond the telescopic of this book to consider a detailed theory of all motor types. Rather, we introduce basic physical construction and develop belittling models that can exist practical to all rotating electromechanical machines. Furthermore, nosotros limit our word to linear, fourth dimension-invariant models, which are sufficient to permit performance analysis advisable for most automotive applications.

We innovate the structures of various electric motors with Figure 6.34, which is a highly simplified sketch depicting just the near basic features of the motor.

Figure vi.34. Schematic representation of electric motor.

This motor has coils wound around both the stator (having N _ane turns) and the rotor (having Due north _ii turns), which are placed in slots effectually the periphery in an otherwise compatible gap auto. In this simplified drawing, just two coils are depicted. In practice, at that place are more than two 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 material having a very high permeability (see discussion higher up on ferromagnetism). Information technology is worthwhile to develop a model for this simplified arcadian motor to provide the basis for an understanding of the relatively complex structure of a practical motor. In Figure 6.34, the stator is a cylinder of length ℓ and the rotor is a smaller cylinder supported coaxially with the stator such that information technology tin can rotate nigh the mutual axis. The angle between the planes of the two coils is denoted θ and the angular variable nearly the axis measured from the aeroplane of the stator coil is denoted α. The radial air gap between rotor and stator is denoted chiliad. It is of import in the design of whatever rotating electrical machine (including motors) to maintain this air gap every bit small as is practically feasible since the forcefulness of the associated magnetic fields varies inversely with g. The terminal voltages of these two coils are denoted v₁ and five₂. The currents are denoted i ₁ and i ₂ and the magnetic flux linkage for each is denoted λ ₁ and λ ₂, respectively. Assuming for simplification purposes that the slots carrying 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 concluding excitation voltages are given past:

$\begin{array}{c}{v}_{\mathrm{ane}}={\dot{\lambda }}_1\\ v_2={\dot{\lambda }}_2\end{array}$

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

(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 material.

The contour integral along whatever path (e.k., contour C of Figure 6.34) that encloses the 2 coils is given by

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

The magnetic flux density B_r (α) is also directed radially and is given by

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

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

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

The magnetic flux linkage for the two coils λ ₁ and λ ₂ are given by

${\lambda }_{\mathrm{i}}=N_1{\int }_o^{\pi }{B}_r(\alpha )\ell R_{r}d\alpha$

(87) ${\lambda }_{\mathrm{ii}}={\mathrm{Northward}}_{\mathrm{two}}{\int }_{\theta }^{\pi +\theta }{B}_r(\alpha )\ell R_{r}d\alpha$

where R_r is the rotor radius.

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

(88) ${\lambda }_1=L_1{i}_1+L_m{i}_2$

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

where

(ninety) $L_1={\mathrm{Due\; north}}_1^2{\mathrm{Fifty}}_o=\text{self inductance of coil}1$

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

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

The parameter L_thou is the common inductance for the two coils which is defined equally the flux linkage induced in each coil due to the electric current in the other divided by that current and is given past

$\begin{array}{c}\begin{array}{cc}{\mathrm{50}}_m=L_o{\mathrm{Due\; north}}_{\mathrm{ane}}{\mathrm{North}}_{\mathrm{two}}\left(1-\frac{\mathrm{ii}\theta }{\pi }\right)& 0<\theta <\pi \\ =L_o{\mathrm{North}}_1{N}_{\mathrm{two}}\left(\mathrm{one}+\frac{2\theta }{\pi }\right)& -\pi <\theta <0\end{array}\\ \end{array}$

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

(93) $L_m(\theta )={\mathrm{1000}}_{\mathrm{i}}\cos (\theta )+{\mathrm{Grand}}_{\mathrm{iii}}\cos (3\theta )+K_{\mathrm{five}}\cos (5\theta )+\dots$

In any applied motor, at that place will be a distribution of windings such that the fundamental component Chiliad ₁ predominates; that is, the mutual inductance is given approximately by

(94) ${\mathrm{50}}_m\simeq \mathrm{Yard}\cos (\theta )$

For notational convenience, the subscript 1 on M _one is dropped. Any motor made upwards of multiple matching pairs of coils in the stator and rotor will take a set of terminal relations in the flux linkages for the stator and rotor λ_s and λ_r , respectively, given by

${\lambda }_{\mathrm{due\; south}}=L_s{i}_s+M{i}_r\cos \theta$

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

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

$T_e=\frac{\partial {\mathrm{West}}_{m\mathrm{Thou}}}{\partial \theta }$

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

$W_{\mathrm{thou}\mathrm{Chiliad}}=i_s{i}_r{L}_m(\theta )$

The torque T_east is given by

$T_e=-i_s{i}_{r}M\sin \theta$

The mechanical dynamics for the motor are given by

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

where J_r is the rotor moment of inertia about its axis, 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 past calculating the motor mechanical power P_m for a given excitation. Let the excitation of the stator and rotor be from ideal current sources such that

(95) $\begin{array}{l}{i}_{\mathrm{due\; south}}=I_{\mathrm{southward}}\sin ({\omega }_{s}t)\\ i_r=I_r\sin ({\omega }_{r}t)\\ \theta (t)={\omega }_{m}t+\gamma \end{array}$

where ω_k is the rotor rotational frequency (rad/sec) and γ expresses an capricious fourth dimension stage parameter. The motor ability is given by

(96) $P_{\mathrm{grand}}=T_{\mathrm{east}}{\omega }_{\mathrm{chiliad}}$

(97) $=-{\omega }_m{I}_{\mathrm{due\; south}}{I}_{r}M\sin ({\omega }_{s}t)\sin ({\omega }_{r}t)\sin ({\omega }_{m}t+\gamma )$

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

(98) $P_m=-\frac{{\omega }_m{I}_s{I}_r\mathrm{Chiliad}}{\mathrm{four}}\left\{\sin \left[\left({\omega }_{\mathrm{thou}}+{\omega }_{\mathrm{due\; south}}-{\omega }_r\right)t+\gamma \right]+\sin \left[({\omega }_m-{\omega }_s+{\omega }_r)t+\gamma \right]-\sin \left[\left({\omega }_m+{\omega }_{\mathrm{due\; south}}+{\omega }_r\right)t+\gamma \right]-\sin \left[\left({\omega }_m-{\omega }_{\mathrm{southward}}-{\omega }_r\right)t+\gamma \right]\right\}$

The fourth dimension average value of whatsoever sinusoidal function of fourth dimension is zero. The but weather condition under which the motor can produce a nonzero average power are given by the frequency relationships below:

(99) ${\omega }_{\mathrm{one\; thousand}}=\pm {\omega }_{\mathrm{due\; south}}\pm {\omega }_r$

For example, whenever ω_m   = ω_{due south}   + ω_r , the motor time average power $P_{m_{a\text{v}}}$ is given by

(100) $P_{{\mathrm{1000}}_{a\text{v}}}=\frac{{\omega }_m{I}_s{I}_{r}M}{4}\sin \gamma$

In such a motor, an equilibrium operation will be achieved when $P_{m_{a\text{v}}}=P_L$ where P_L   =   load power. Thus, the phase between rotor and stator fields is given by

(101) $\sin \gamma =\frac{\mathrm{four}{P}_L}{{\omega }_{\mathrm{1000}}{I}_s{I}_{r}M}$

provided

(102) $P_L\le \frac{{\omega }_{\mathrm{one\; thousand}}{I}_{\mathrm{due\; south}}{I}_{r}G}{\mathrm{iv}}$

The higher up frequency weather (Eqn (99)) are key to all rotating machines and are required to be satisfied for whatsoever nonzero average mechanical output power. Each dissimilar type of motor has a unique way of satisfying the frequency conditions. We illustrate with a specific example, which has been employed in sure hybrid vehicles. This example is the induction motor. All the same, before proceeding with this case, information technology is important to consider an issue in motor operation. Normally, electrical motors that are intended to produce substantial amounts of power (e.g., for hybrid vehicle awarding) are polyphase machines; that is, in addition to the windings associated with stator excitation, a polyphase machine will have one or more than additional sets of windings that are excited past the same frequency but at different phases. Although 3-phase motors are in mutual use, the analysis of a ii-phase induction motor illustrates the bones principles of polyphase motors with a relatively simplified model and is assumed in the following discussion.

A two-stage motor has ii sets of windings displaced at 90° in the θ direction and excited by currents with a xc° phase for both stator and rotor. A so-called balanced 2-stage motor will have its roll excited by currents i_as , i_bs for phases a and b, respectively, where

(103) $i_{a\mathrm{south}}=I_s\cos ({\omega }_{s}t)$

$i_{bs}=I_{\mathrm{southward}}\sin ({\omega }_{\mathrm{southward}}t)$

The rotor is also synthetic with two sets of windings displaced physically by 90° and excited with currents i_ar and i_br having ninety° phase shift:

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

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

A two-stage induction motor is i in which the stator windings are excited by currents given to a higher place (i.e., i_as and i_bs ). The rotor circuits are curt-circuited such that v _ar   =   v _br   =   0, where v _ar is the final voltage for windings of stage a and v _br is the terminal voltage for the b phase. The currents in the rotor are obtained by induction from the stator fields. By extension of the assay of the unmarried-phase excitation, the last flux linkages are given past

(105) $\begin{array}{l}{\lambda }_{a\mathrm{south}}=L_{\mathrm{south}}{i}_{as}+M{i}_{ar}\cos \theta -M{i}_{br}\sin \theta \\ {\lambda }_{bs}={\mathrm{50}}_{\mathrm{south}}{i}_{b\mathrm{south}}+M{i}_{ar}\sin \theta +M{i}_{br}\cos \theta \\ {\lambda }_{ar}=L_r{i}_{ar}+G{i}_{as}\cos \theta +M{i}_{b\mathrm{southward}}\sin \theta \\ {\lambda }_{br}={\mathrm{Fifty}}_r{i}_{br}-M{i}_{a\mathrm{south}}\sin \theta -M{i}_{b\mathrm{south}}\cos \theta \end{array}$

The torque T_e and instantaneous power P_m for the ii-phase induction motor are given by

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

$P_m={\omega }_{\mathrm{thousand}}M{I}_{\mathrm{southward}}{I}_r\sin [({\omega }_m-{\omega }_{\mathrm{southward}}+{\omega }_r)t+\gamma ]$

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

$P_a={\omega }_{\mathrm{one\; thousand}}M{I}_s{I}_r\sin \gamma$

Since the rotor terminals are brusque-circuited, we have

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

The two rotor currents, thus, satisfy the following equations:

(108) $0=R_r{i}_{ar}+{\mathrm{50}}_r\frac{d{i}_{ar}}{dt}+M{I}_s\frac{d}{dt}[\cos ({\omega }_{s}t)\cos ({\omega }_{\mathrm{one\; thousand}}t+\gamma )+\sin ({\omega }_{\mathrm{southward}}t)\sin ({\omega }_{m}t+\gamma )]$

(109) $0=R_r{i}_{br}+{\mathrm{50}}_r\frac{d{i}_{br}}{dt}+M{I}_s\frac{d}{dt}[-\cos ({\omega }_{\mathrm{south}}t)\sin ({\omega }_{\mathrm{chiliad}}t+\gamma )+\sin ({\omega }_{s}t)\cos ({\omega }_{m}t+\gamma )]$

where R_r and L_r are the resistance and self-inductance of the ii sets of (presumed) identical construction). These equations can exist rewritten as

(110) $L_r\frac{d{i}_{ar}}{dt}+R_r{i}_{ar}=\mathrm{Grand}{I}_s({\omega }_{\mathrm{south}}-{\omega }_m)\sin [({\omega }_s-{\omega }_{\mathrm{thousand}})t-\gamma ]$

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

The current i_ab is identical to i_ar except for a 90° stage shift as tin exist seen from Eqn (111). Notation that the current for both phases are at frequency ω_r where

${\omega }_r=({\omega }_s-{\omega }_m)$

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

(112) $i_{ar}=\frac{({\omega }_{\mathrm{southward}}-{\omega }_g)M{I}_s}{\sqrt{R_r^2+{({\omega }_s-{\omega }_{\mathrm{thou}})}^2{L}_r^2}}\cos [({\omega }_s-{\omega }_m)t-\alpha ]$

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 stage b is identical except for a xc° stage shift. Substituting the currents for rotor and stator into the equation for torque T_e yields the remarkable consequence that the this torque is independent of θ and is given by

(114) $T_e=\frac{({\omega }_{\mathrm{due\; south}}-{\omega }_{\mathrm{thou}})M^{\mathrm{two}}{R}_r{I}_s^2}{R_r^2+{({\omega }_s-{\omega }_k)}^2{\mathrm{50}}_r^2}$

The mechanical output ability P_g is given by

$\begin{array}{c}{P}_{\mathrm{1000}}={\omega }_k{T}_{\mathrm{due\; east}}\\ =\left[\frac{{\omega }_s^{\mathrm{ii}}{\mathrm{Thousand}}^{\mathrm{two}}{I}_{\mathrm{due\; south}}^2}{{\left(R_r/\mathrm{south}\right)}^{\mathrm{ii}}+{\omega }_s^{\mathrm{two}}{L}_r^2}\right]\left(\frac{1-s}{s}\right)R_r\end{array}$

where s is called slip and is given by

(115) $s=\frac{{\omega }_s-{\omega }_{\mathrm{chiliad}}}{{\omega }_s}$

The induction car has 3 modes of functioning as characterized by values of southward. For 0   < s  <   i information technology acts as a motor and produces mechanical ability. For −1   < south  <   0 it acts like a generator and mechanical input power to the rotor is converted to output electrical power. For s  >   1, the induction machine acts like a brake with both electrical input and mechanical input power dissipated in rotor i_r ² R_r losses. Considering of its versatility, the induction motor has slap-up potential in hybrid/electric vehicle propulsion applications. Nevertheless, it does crave that the control system incorporates solid-state power switching electronics to be able to handle the necessary currents. Moreover, it requires precise control of the excitation current.

The awarding of an induction motor to provide the necessary torque to motility 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_grand   − ω_s ). The torque of an consecration motor initially increases from its value at ω_g   =   0 reaches a maximum torque (T _max) at a speed ${\omega }_{\mathrm{thousand}}>{\omega }_{\mathrm{thou}}^{\ast }$ when

$0\le {\omega }_{\mathrm{one\; thousand}}^{\ast }\le {\omega }_s$

The torque has a negative slope given by

$\begin{array}{cc}\frac{d{T}_e}{d{\omega }_{\mathrm{thou}}}<0& {\omega }_m>{\omega }_{\mathrm{1000}}^{\ast }\end{array}$

Normally, an induction motor is operated in the negative slope region of T_{one thousand} (ω_g ) (i.e., ${\omega }_m^{\ast }>{\omega }_m<{\omega }_s$ ) for stable operation. Equilibrium is reached at a motor rotational speed ω_m at which the motor torque T_e and load torque T_L 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) $T_L=K_L{\omega }_m$

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

Figure 6.35. Normalized torque T_m vs. normalized load torques T_{L 1} T_{L 2}.

For convenience of presentation, Figure half-dozen.35 presents normalized motor torque and load torque normalized to the maximum torque T _max where

(117) $T_{\max }=\underset{{\omega }_m}{\max }(T_e({\omega }_{\mathrm{one\; thousand}}))$

This maximum occurs at ${\omega }_{\mathrm{thou}}={\omega }_{\mathrm{thou}}^{\ast }$ , which, for the present hypothetical normalized example, is given by

$\frac{{\omega }_{\mathrm{1000}}^{\ast }}{{\omega }_{\mathrm{south}}}\cong .68$

Effigy 6.33 likewise presents ii load torques normalized to T _max:

$\begin{array}{l}{T}_{L\mathrm{ane}}={\mathrm{Yard}}_{L1}{\omega }_{\mathrm{thousand}}/T_{\max }\\ T_{\mathrm{Fifty}2}={\mathrm{Thou}}_{L2}{\omega }_{\mathrm{thou}}/T_{\max }\end{array}$

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

$\begin{array}{l}{T}_{\mathrm{thou}}({\omega }_{01})=T_{\mathrm{50}1}({\omega }_{01})\\ T_m({\omega }_{02})=T_{\mathrm{Fifty}2}({\omega }_{02})\end{array}$

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

Chapter 7 discusses the control of an induction motor that is used in a hybrid electric vehicle. There the model for load torque vs. vehicle operating atmospheric condition is adult.

Brushless DC Motors

Next, we consider a relatively new type of electrical motor known as a brushless DC motor. A brushless DC motor is not a DC motor at all in that the excitation for the stator is AC. Withal, it derives its name from physical and functioning similarity to a shunt-connected DC motor with a constant field current. This blazon of motor incorporates a permanent magnet in the rotor and electromagnet poles in the stator every bit depicted in Figure 6.36. Traditionally, permanent magnet rotor motors were generally only useful in relatively low-power applications. Recent development of some relatively powerful rare earth magnets and the development of high-power switching solid-state devices have substantially raised the power capability of such machines.

Figure 6.36. Brushless DC motor.

The stator poles are excited such that they take magnetic N and S poles with polarity as shown in Effigy vi.36 past 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 system that includes an angular position sensor fastened to the rotor. This switching is done so that the magnetic field produced by the stator electromagnets e'er applies a torque on the rotor in the direction 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) ${\overline{T}}_m=\gamma (\overline{G}\times \overline{H})$

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

The management 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_m is given past

$T_m=\gamma \mathrm{1000}H\sin (\theta )$

where M = magnitude of $\overline{\mathrm{1000}}$ , H = magnitude of $\overline{H}$ and θ = angle between $\overline{\mathrm{Grand}}$ and $\overline{H}$ .

If the permanent magnet rotor were allowed to rotate in a static magnetic field, information technology would simply plough 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 society for this motor to continue to have a nonzero torque applied, the stator windings must be continuously switched synchronous with rotor rotation. Although only two sets of stator windings are shown in Effigy 6.36 (i.e., two-pole machine), normally at that place would be multiple sets of windings, each driven separately and synchronously with rotor rotation. In effect, the sequential application 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 6.36a and b is shown in Figure half dozen.36c. A sensor Due south 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 duration. The switching times are adamant such that a torque is applied to the rotor in the direction of rotation.

At the advisable fourth dimension, transistor A is switched on, and electric power from the on-lath DC source (e.1000., bombardment pack) is supplied to the poles A of the motor. The duration of this electric current is regulated past controller C to produce the desired ability (equally commanded by the driver). Later on rotating approximately xc°, current I_b is switched on past activating transistor B via a signal sent by controller C.

The rotor permanent magnet is equivalent to an electromagnet with d-c excitation (i.due east., ω_r   =   0). The frequency at which the currents to the stator coils are switched is ever synchronous with the speed of rotation. Thus, the frequency condition for the motor is satisfied since ω_s   = ω_m . This speed is determined by the mechanical load on the motor and the power commanded by the controller. As the ability command is increased, the controller responds by increasing the duration of the current pulse supplied to each stator gyre. The power delivered by the motor is proportional to the fraction of each bicycle that the current is on (i.due east., the so-called duty cycle).

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Electrical Machines, Design

Enrico Levi , in Encyclopedia of Physical Science and Technology (3rd Edition), 2003

2.C Magnetic and Electric Loadings

The magnetic flux density B, which is relevant to the electromechanical power conversion procedure, is the effective or rms value of the radial component of B at the air gap. Except for special cases, such as superconducting field excitation and printed windings, this value is adamant past the characteristics of the ferromagnetic structure into which the conductors are embedded. This consists of a cadre or yoke, which provides both physical integrity and a path for the magnetic flux, and a slotted portion adjacent to the air gap, which accommodates the active conductors. The slot dimensions represent a compromise between the conflicting requirements of conduction of current in the copper and of flux in the iron teeth. It turns out that there is a value of the ratio between the slot and molar widths that minimizes the volume and weight of the ferromagnetic structure. This value is unity, so that the tooth width should exist one-half the slot pitch. If one 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 equally

(fifteen) $\text{B}=\frac{1}{2}{\text{B}}_{\text{t}}.$

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 deformed. Also, part of the flux is diverted to the slot, which causes an increase in the additional copper losses and the transfer of the strength from the tooth to the conductor. As a consequence the electric insulation is stressed mechanically. For these reasons rms values of B _t in excess of 1.four   T are non recommended and B is practically limited to about 0.vii   T.

In dissimilarity to the magnetic loading, at that place is no optimal value for the electric loading. Its value is determined primarily past 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 Engineering science (Eighth Edition), 2016

thirteen.3 Equations of Magnetism

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

The magnetizing strength, 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   m^−one) (Bennett et al., 1978).

The intensity of magnetization or the magnetization (M, A   thou⁻¹) of a material relates to the magnetization induced in the textile and can likewise be thought of every bit the volumetric density of induced magnetic dipoles in the fabric. The magnetic induction, B, field intensity, H, and magnetization, M, are related by the equation:

(xiii.one) $B={\mu }_0(H+M)$

where μ ₀ is the permeability of free space and has the value of 4π×ten^−vii  Northward   A^−two. In a vacuum, M=0, and M is extremely low in air and h2o, such that for mineral processing purposes Eq. (13.i) may exist simplified to:

(13.2) $B={\mu }_{0}H$

so 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 and so frequently loosely used for both the H-field and the B-field. However, when dealing with the magnetic field inside materials, especially ferromagnetic materials that concentrate the lines of force, the value of the induced flux density volition be much higher than the field intensity. This relationship is used in high-gradient magnetic separation (discussed farther in Section 13.iv.i). For clarity information technology must be specified which field is existence referred to.

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

(thirteen.3) $\chi =\frac{M}{H}$

Combining Eqs. (13.i) and (xiii.iii) we get:

(13.4) $B={\mu }_{0}H(\mathrm{i}+\chi )$

If nosotros and then define the dimensionless relative permeability, μ, as:

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

nosotros can combine Eqs. (xiii.4) and (13.5) to yield:

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

For paramagnetic materials, χ is a small positive constant, and for diamagnetic materials information technology is a much smaller negative constant. As examples, from Figure 13.one the slope representing the magnetic susceptibility of the material, χ, is virtually 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 as the material becomes saturated. Figure thirteen.2 shows a plot of K versus H for magnetite, showing that at an applied field of lxxx   kA   m⁻¹, or 0.1   T, the magnetic susceptibility is about i.7, and saturation occurs at an practical magnetic field force of almost 500   kA   m⁻¹ or 0.63   T. Many high-intensity magnetic separators apply iron cores and frames to produce the desired magnetic flux concentrations and field strengths. Iron saturates magnetically at nigh 2–2.v   T, and its nonlinear ferromagnetic relationship between inducing field forcefulness and magnetization intensity necessitates the apply of very large currents in the energizing coils, sometimes up to hundreds of amperes.

The magnetic strength felt by a mineral particle is dependent not only on the value of the field intensity, but also on the field gradient (the rate at which the field intensity increases across the particle toward the magnet surface). As paramagnetic minerals take college (relative) magnetic permeabilities than the surrounding media, usually air or water, they concentrate the lines of force of an external magnetic field. The higher the magnetic susceptibility, the higher the induced field density in the particle and the greater is the attraction up the field gradient toward increasing field force. Diamagnetic minerals have lower magnetic susceptibility than their surrounding medium and hence expel the lines of force of the external field. This causes their expulsion downwardly the gradient of the field in the direction of the decreasing field strength.

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 slope. This equation, when considered in only the ten-management, may be expressed as (Oberteuffer, 1974):

(13.7) $F_x=V({\chi }_{\mathrm{p}}-{\chi }_{\mathrm{one\; thousand}})H\frac{\mathrm{d}B}{\mathrm{d}\mathrm{ten}}$

where F _ten is the magnetic force on the particle (N), 5 the particle book (mⁱⁱⁱ), χ _p the magnetic susceptibility of the particle, χ _m the magnetic susceptibility of the fluid medium, H the practical magnetic field strength (A   m⁻ⁱ), and dB/dten the magnetic field gradient (T   g^−ane=N   A⁻¹  m^−two). The product of H and dB/dx is sometimes referred to as the "strength factor."

Production of a high field gradient every bit well equally high intensity is therefore an important attribute of separator design. To generate a given bonny force, there are an infinite number of combinations of field and gradient which volition requite the same effect. Another important cistron is the particle size, as the magnetic force experienced by a particle must compete with diverse other forces such as hydrodynamic elevate (in wet magnetic separations) and the force of gravity. In one case, considering only these two competing forces, Oberteuffer (1974) has shown that the range of particle size where the magnetic force predominates is from nearly 5   μm to 1   mm.

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

M. El-Hakim , ... A. El-Shafei , in 10th International Conference on Vibrations in Rotating Mechanism, 2012

1 Nomenclature

B:

magnetic flux density (T)

Φ:

magnetic flux (Tm)

μ_o:

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

μ_r:

relative permeability

H:

magnetomotance (A/thousand)

l:

magnetic flux iron path length (m)

l_thou :

magnetic gap (m)

F:

forcefulness (Due north)

J:

current density (A/1000²)

I:

current (A)

α:

pole inclination angle

N:

coils number of turns

A_g:

magnetic pole surface area (m^two)

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Awarding of Evolutionary Algorithm for Multiobjective Transformer Design Optimization

Due south. Tamil Selvi , ... Due south. Rajasekar , in Classical and Recent Aspects of Power Arrangement Optimization, 2018

Pick of Electric 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 electric current density is increased, cross-exclusive area of the windings is reduced and hence, the book and in plow copper weight are reduced. On the other hand, copper loss, which varies as a square of current density, is increased causing efficiency to reduce. Moreover, temperature rise will increase and hurt the insulation [7].

The choice of the current density must be done in such a way that the maximum temperature of the transformer due to losses is beneath the insulation class temperature. Current density chosen should guarantee the level of losses and cooling atmospheric condition required. However, a designer must compare the increased cost due to the improved cooling method required with the economy in material due to the choice of increased value of current density. In brusque, current density is governed by load losses, temperature class of insulation, and short circuit current withstanding ability. Maximum limit for electric current density is calculated as

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

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

(7) ${\theta }_{\mathrm{ii}}={\theta }_0+j^2.y.t1{\mathrm{.ten}}^{-\mathrm{three}}°\mathrm{C}$

where

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

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

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

y  =   Function of $\frac{1}{2}\left({\theta }_2+{\theta }_0\right)$ , in accord with (Clause nine.15: Table six, IS2026 Office I).

It is therefore necessary to give considerations in choosing value for 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, development interval; B _P = magnetic flux density, preparation interval; G _i (τ) = dipolar autocorrelation function; J ⁽ⁱ⁾(ω) = intensity function of the Larmor frequency; M _E = Curie magnetization, development interval; K _P = Curie magnetization, preparation interval; S/Due north = signal-to-noise ratio; T ₁ = spin–lattice relaxation time; T _d = dipolar-order relaxation time;T _1ρ = rotating-frame relaxation time; T _two = transverse relaxation fourth dimension; γ = gyromagnetic ratio; μ₀ = magnetic field constant.

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Solid-State 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 abiding; I = spin- $\frac{1}{2}$ nucleus; J = coupling constant; q = field gradient tensor; Q = nuclear quadrupole moment; south′ = doublet splitting; Southward = quadrupolar nucleus; γ = magnetogyric ratio; θ = angle between main tensor axes; δ = chemical shift; τ _one = relaxation time; ξ = angle between chief axes of interaction tensors and sample spinning axis; χ = quadrapole coupling constant.

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Biomedical Applications of Electromagnetic Fields

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

7.one.3.3 Magnetic Flux Density

The results for the magnetic flux density, obtained using (7.viii) and (7.9), were compared to the analytical results. The results for the maximum values are given in Tabular array 7.seven.

Table vii.7. Comparing of maximum magnetic flux density. From CVETKOVIĆ, Mario; POLJAK, Dragan; HAUEISEN, Jens. Assay of transcranial magnetic stimulation based on the surface integral equation conception. IEEE Transactions on Biomedical Applied science, 2015, 62.6: 1535–1545 [28].

		Circular	8-coil	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. 7.6. Comparison of magnetic flux density in the homo brain (coronal cantankerous-section). The results on the left are obtained via analytical expressions for (A) circular, (B) eight-ringlet, and (C) butterfly scroll, while the results on the correct are obtained via proposed model for (D) circular, (East) 8-coil, and (F) butterfly coil. From [28].

The results from Table 7.7 and Fig. vii.6 indicate that the brain itself does not significantly disturb the magnetic field of the coil, 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 cantankerous-section obtained using the SIE model shows some discontinuities, which can be related to the interpolation method used. This numerical artifact could be overcome by calculating the field at more points before interpolating results in the neighboring area.

The magnetic flux density B on the brain surface tin be clearly seen on Fig. 7.7.

Fig. 7.7. Magnetic flux density on the brain surface due to: (A) circular curl, (B) effigy-of-eight whorl, and (C) butterfly scroll. All coils are placed i cm over the primary motor cortex. From [28].

It is interesting to notice the dependence of the induced electrical field Due east and magnetic flux density B on the distance from the brain surface, as shown on Fig. 7.eight.

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

From Fig. 7.8 the rapid decrease of both E and B fields direct under the geometric center of the stimulation coil is clearly evident in all three cases. For the circular coil, the maximum value is much lower compared to the other 2 coils as the maximum field will be induced nether the curlicue windings, equally shown on Fig. 7.4.

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Development Tools

Edward Ramsden , in Hall-Effect Sensors (Second Edition), 2006

Gaussmeter

A gaussmeter measures magnetic flux density ( B) at a given point in space. Well-nigh gaussmeters employ Hall-event sensor elements equally the magnetic probe. In its simplest class, a gaussmeter is a linear Hall-effect sensor with a meter readout. Indeed, it is possible to build a simple gaussmeter from a linear Hall-effect sensor IC, a small corporeality of interface electronics, and a DMM, but the result would not provide anywhere almost the capabilities of a modern gaussmeter. A few of the features to look for in a gaussmeter are:

•

Range – How pocket-sized a field can it measure, and how large a field can it measure?

•

Accurateness – To what degree does the reading reflect reality?

•

Interface options – In addition to a front-panel display, tin can it communicate with PCs or other instruments?

Range is important because in that location are times when you volition want to measure fields of a few gauss, and others where you will want to measure fields of several kilogauss. Low ranges are often important in sensor work. Even though almost Hall-consequence sensor ICs aren't useful for discriminating field differences much beneath 1 gauss, yous will typically want an instrument with an guild of magnitude finer resolution than what you need to measure.

The need for accuracy requires picayune if any elaboration. Inaccurate instruments can make your life vastly more difficult. Accurate instruments, regularly calibrated, tin make development work go more smoothly, by reducing one potential source of errors. Notation that accurateness is a key specification for gaussmeters and is often the merely difference between ii instrument models of differing price.

While interface options may not seem that important, they enable one to hook the gaussmeter to a PC and automate many elementary tasks. Pop interface standards include RS-232, IEEE-488, and analog outputs.

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