Continuing the Assumption That No Friction Exists Does the Magnet Reach the Same Peak

Incident Electromagnetic Field Dosimetry

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

3.1.2.2 The Magnetic Field

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

Fig. 3.17. Straight current element.

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

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

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

(3.26) $\overrightarrow{B}={\hat{e}}_{\phi }\frac{\mu i(t)}{4\pi \rho }\underset{{\theta }_1}{\overset{{\theta }_2}{\int }}\cos \theta d\theta ={\hat{e}}_{\phi }\frac{\mu i(t)}{4\pi \rho }(\sin {\theta }_1+\sin {\theta }_2),$

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

The ELF magnetic field value at an arbitrary point can be assessed by assembling the contributions of all conductors divided in a certain number of straight segments. The kth straight segment carrying current $i_k$ in Cartesian three-dimensional coordinate system is shown in Fig. 3.18.

Fig. 3.18. Straight segment in Cartesian coordinate system.

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

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

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 point of space can be expressed as

(3.28) $B(t)=\begin{array}{c}\hfill \sqrt{{(\sum _{i=1}^N{B}_{x,i}(t))}^2+{(\sum _{i=1}^N{B}_{y,i}(t))}^2+{(\sum _{i=1}^N{B}_{z,i}(t))}^2}\end{array},$

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 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 calculation domains 1 to 5, in which higher field values are expected, are assigned as in Fig. 3.15. The spatial distribution of the magnetic field over domain 3, where the highest field value is captured, is shown in Fig. 3.19.

Fig. 3.19. Spatial distribution of the magnetic field over domain 3.

If the human body is exposed to an ELF magnetic field, the circular 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 can be calculated using the disk model of the human body.

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Sensors and Actuators

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

Electric Motor Actuators

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

There are a great number of electric motor types that are classified by the type of excitation (i.e., dc or ac), the physical structure (e.g., smooth air gap or salient pole), and by the type of magnet structure for the rotating element (rotor) which can be either a permanent magnet or an electromagnet. However, there are certain fundamental similarities between all electric motors, which are discussed below. Still another distinction between types of electric motors is based upon whether the rotor receives electrical 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 scope of this book to consider a detailed theory of all motor types. Rather, we introduce basic physical structure and develop analytical models that can be 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 most automotive applications.

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

Figure 6.34. Schematic representation of electric motor.

This motor has coils wound around both the stator (having N ₁ turns) and the rotor (having N ₂ turns), which are placed in slots around the periphery in an otherwise uniform gap machine. In this simplified drawing, only two coils are depicted. In practice, there 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 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 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 it can rotate about the common axis. The angle 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. It is important in the design of any 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 g. The terminal voltages of these two coils are denoted v₁ and v₂. 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 terminal excitation voltages are given by:

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

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

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

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:

$\begin{array}{c}\begin{array}{cc}2g{H}_r(\alpha )=N_1{i}_1-N_2{i}_2& 0\le \alpha <\theta \\ =N_1{i}_1+N_2{i}_2& \theta <\alpha <\pi \\ =-N_1{i}_1+N_2{i}_2& \pi <\alpha <\pi +\theta \\ =-N_1{i}_1-N_2{i}_2& \pi +\theta <\alpha <2\pi \end{array}\\ \end{array}$

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

${\lambda }_1=N_1{\int }_o^{\pi }{B}_r(\alpha )\ell R_{r}d\alpha$

(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 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 about solenoids, this flux linkage can be written as a linear combination of the contributions from i ₁ and i ₂:

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

(89) ${\lambda }_2=L_m{i}_1+L_2{i}_2$

where

(90) $L_1=N_1^2{L}_o=\text{self inductance of coil}1$

(91) $L_2=N_2^2{L}_o=\text{self inductance of coil 2}$

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

The parameter L_m is the mutual inductance for the two coils which is defined as the flux linkage induced in each coil due to the current in the other divided by that current and is given by

$\begin{array}{c}\begin{array}{cc}{L}_m=L_o{N}_1{N}_2\left(1-\frac{2\theta }{\pi }\right)& 0<\theta <\pi \\ =L_o{N}_1{N}_2\left(1+\frac{2\theta }{\pi }\right)& -\pi <\theta <0\end{array}\\ \end{array}$

The above formulas for these inductances provide a sufficient model to derive the terminal voltage/current relationships as well 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 L_m (θ) is a symmetric function of θ. It can be formally expanded in a Fourier series in θ having only cosine terms in odd harmonics as given below:

(93) $L_m(\theta )=M_1\cos (\theta )+M_3\cos (3\theta )+M_5\cos (5\theta )+\dots$

In any practical motor, there will be a distribution of windings such that the fundamental component M ₁ predominates; that is, the mutual inductance is given approximately by

(94) $L_m\simeq M\cos (\theta )$

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

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

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

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

$T_e=\frac{\partial W_{mM}}{\partial \theta }$

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

$W_{mM}=i_s{i}_r{L}_m(\theta )$

The torque T_e is given by

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

The mechanical dynamics for the motor are given by

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

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.

It is of interest to evaluate the motor performance by 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) $\begin{array}{l}{i}_s=I_s\sin ({\omega }_{s}t)\\ i_r=I_r\sin ({\omega }_{r}t)\\ \theta (t)={\omega }_{m}t+\gamma \end{array}$

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

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

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

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

(98) $P_m=-\frac{{\omega }_m{I}_s{I}_{r}M}{4}\left\{\sin \left[\left({\omega }_m+{\omega }_s-{\omega }_r\right)t+\gamma \right]+\sin \left[({\omega }_m-{\omega }_s+{\omega }_r)t+\gamma \right]-\sin \left[\left({\omega }_m+{\omega }_s+{\omega }_r\right)t+\gamma \right]-\sin \left[\left({\omega }_m-{\omega }_s-{\omega }_r\right)t+\gamma \right]\right\}$

The time average value of any sinusoidal function of time is zero. The only conditions under which the motor can produce a nonzero average power are given by the frequency relationships below:

(99) ${\omega }_m=\pm {\omega }_s\pm {\omega }_r$

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

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

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) $\sin \gamma =\frac{4P_L}{{\omega }_m{I}_s{I}_{r}M}$

provided

(102) $P_L\le \frac{{\omega }_m{I}_s{I}_{r}M}{4}$

The above frequency conditions (Eqn (99)) are fundamental to all rotating machines and are required to be satisfied for any nonzero average mechanical output power. Each different type of motor has a unique way of satisfying the frequency conditions. We illustrate with a specific example, which has been employed in certain hybrid vehicles. This example is the induction motor. However, before proceeding with this example, it is important to consider an issue in motor performance. Normally, electric motors that are intended to produce substantial amounts of power (e.g., for hybrid vehicle application) are polyphase machines; that is, in addition to the windings associated with stator excitation, a polyphase machine will have one or more additional sets of windings that are excited by the same frequency but at different phases. Although three-phase motors are in common use, the analysis of a two-phase induction motor illustrates the basic principles of polyphase motors with a relatively simplified model and is assumed in the following discussion.

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

(103) $i_{as}=I_s\cos ({\omega }_{s}t)$

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

The rotor is also constructed with two sets of windings displaced physically by 90° and excited with currents i_ar and i_br having 90° 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.e., i_as and i_bs ). The rotor circuits are short-circuited such that v _ar   =   v _br   =   0, where v _ar is the terminal voltage for windings of phase 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 analysis of the single-phase excitation, the terminal flux linkages are given by

(105) $\begin{array}{l}{\lambda }_{as}=L_s{i}_{as}+M{i}_{ar}\cos \theta -M{i}_{br}\sin \theta \\ {\lambda }_{bs}=L_s{i}_{bs}+M{i}_{ar}\sin \theta +M{i}_{br}\cos \theta \\ {\lambda }_{ar}=L_r{i}_{ar}+M{i}_{as}\cos \theta +M{i}_{bs}\sin \theta \\ {\lambda }_{br}=L_r{i}_{br}-M{i}_{as}\sin \theta -M{i}_{bs}\cos \theta \end{array}$

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

(106) $T_e=M[(i_{ar}{i}_{bs}-i_{br}{i}_{as})\cos \theta -(i_{ar}{i}_{as}+i_{br}{i}_{bs})\sin \theta ]$

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

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

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

Since the rotor terminals are short-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) $0=R_r{i}_{ar}+L_r\frac{d{i}_{ar}}{dt}+M{I}_s\frac{d}{dt}[\cos ({\omega }_{s}t)\cos ({\omega }_{m}t+\gamma )+\sin ({\omega }_{s}t)\sin ({\omega }_{m}t+\gamma )]$

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

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

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

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

The current i_ab is identical to i_ar except for a 90° phase shift as can be seen from Eqn (111). Note 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 by having currents at the difference between excitations and rotor rotational frequency. The current i_ar is given by

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

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 phase b is identical except for a 90° phase shift. Substituting the currents for rotor and stator into the equation for torque T_e yields the remarkable result that the this torque is independent of θ and is given by

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

The mechanical output power P_m is given by

$\begin{array}{c}{P}_m={\omega }_m{T}_e\\ =\left[\frac{{\omega }_s^2{M}^2{I}_s^2}{{\left(R_r/s\right)}^2+{\omega }_s^2{L}_r^2}\right]\left(\frac{1-s}{s}\right)R_r\end{array}$

where s is called slip and is given by

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

The induction machine has three modes of operation as characterized by values of s. For 0   < s  <   1 it acts as a motor and produces mechanical power. For −1   < s  <   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. Because of its versatility, the induction motor has great potential in hybrid/electric vehicle propulsion applications. However, it does require 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 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_m   − ω_s ). The torque of an induction motor initially increases from its value at ω_m   =   0 reaches a maximum torque (T _max) at a speed ${\omega }_m>{\omega }_m^{\ast }$ when

$0\le {\omega }_m^{\ast }\le {\omega }_s$

The torque has a negative slope given by

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

Normally, an induction motor is operated in the negative slope region of T_m (ω_m ) (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_m )   = 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 6.35 presents normalized motor torque and load torque normalized to the maximum torque T _max where

(117) $T_{\max }=\underset{{\omega }_m}{\max }(T_e({\omega }_m))$

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

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

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

$\begin{array}{l}{T}_{L1}=K_{L1}{\omega }_m/T_{\max }\\ T_{L2}=K_{L2}{\omega }_m/T_{\max }\end{array}$

where

$K_{L2}>K_{L1}$

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

$\begin{array}{l}{T}_m({\omega }_{01})=T_{L1}({\omega }_{01})\\ T_m({\omega }_{02})=T_{L2}({\omega }_{02})\end{array}$

These two intersection points are the steady-state operating conditions for the two load torques. The higher of the two loads has a steady-state operating point lower than the first (i.e., 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 conditions is developed.

Brushless DC Motors

Next, we consider a relatively new type of electric 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. However, it derives its name from physical and performance similarity to a shunt-connected DC motor with a constant field current. This type of motor incorporates a permanent magnet in the rotor and electromagnet poles in the stator as 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 have magnetic N and S poles with polarity as shown in Figure 6.36 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 system that includes an angular 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 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 product

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

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 cause 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 by

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

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 order 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 Figure 6.36 (i.e., two-pole machine), normally there 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 6.36c. 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 duration. The switching times are determined such that a torque is applied to the rotor in the direction of rotation.

At the appropriate time, transistor A is switched on, and electric power from the on-board DC source (e.g., battery pack) is supplied to the poles A of the motor. The duration of this current is regulated by controller C to produce the desired power (as commanded by the driver). After rotating approximately 90°, 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_m . This speed is determined by the mechanical load on the motor and the power commanded by the controller. As the power command is increased, the controller responds by increasing the duration of the current pulse supplied to each stator coil. The power delivered by the motor is proportional to the fraction of each cycle that the current is on (i.e., the so-called duty cycle).

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Electric Machines, Design

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

II.C Magnetic and Electric Loadings

The magnetic flux density B, which is relevant to the electromechanical power conversion process, 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 determined by the characteristics of the ferromagnetic structure 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 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 tooth widths that minimizes the volume and weight of the ferromagnetic structure. This value is unity, so that the tooth width should be 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 as

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

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 force from the tooth to the conductor. As a result the electrical insulation is stressed mechanically. For these reasons rms values of B _t in excess of 1.4   T are not recommended and B is practically limited to about 0.7   T.

In contrast to the magnetic loading, there is no optimal value for the electric loading. Its value is determined 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 (Eighth Edition), 2016

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

The intensity of magnetization or the magnetization (M, A   m⁻¹) 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) $B={\mu }_0(H+M)$

where μ ₀ is the permeability of free space and has the value of 4π×10⁻⁷  N   A⁻². In a vacuum, M=0, and M is extremely low in air and water, such that for mineral processing purposes Eq. (13.1) may be simplified to:

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

so that the value of the field intensity, H, is directly proportional to the value of induced flux density, B (or B-field), and the term "magnetic field intensity" is then often loosely used for both the H-field and the B-field. However, when dealing with the magnetic field inside materials, particularly ferromagnetic materials that concentrate the lines of force, the value of the induced flux density will be much higher than the field intensity. This relationship is used in high-gradient magnetic separation (discussed further in Section 13.4.1). For clarity it must be specified which field is being referred to.

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

(13.3) $\chi =\frac{M}{H}$

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

(13.4) $B={\mu }_{0}H(1+\chi )$

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

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

we can combine Eqs. (13.4) and (13.5) to yield:

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

For paramagnetic materials, χ is a small positive constant, and for diamagnetic materials it is a much smaller negative constant. As examples, from Figure 13.1 the slope representing the magnetic susceptibility of the material, 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 as the material 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 about 1.7, and saturation occurs at an applied magnetic field strength of about 500   kA   m⁻¹ or 0.63   T. Many high-intensity magnetic separators use iron cores and frames to produce the desired magnetic flux concentrations and field strengths. Iron saturates magnetically at about 2–2.5   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 up to hundreds of amperes.

The magnetic force 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 have higher (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 strength. 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 down 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 applied magnetic field and the magnetic field gradient. This equation, when considered in only the x-direction, may be expressed as (Oberteuffer, 1974):

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

where F _x is the magnetic force on the particle (N), V the particle volume (m³), χ _p the magnetic susceptibility of the particle, χ _m the magnetic susceptibility of the fluid medium, H the applied magnetic field strength (A   m⁻¹), and dB/dx the magnetic field gradient (T   m⁻¹=N   A⁻¹  m⁻²). The product of H and dB/dx is sometimes referred to as the "force factor."

Production of a high field gradient as well as high intensity is therefore an important aspect of separator design. To generate a given attractive force, there are an infinite number of combinations of field and gradient which will give the same effect. Another important factor is the particle size, as the magnetic force experienced by a particle must compete with various other forces such as hydrodynamic drag (in wet magnetic separations) and the force of gravity. In one example, considering only these two competing forces, Oberteuffer (1974) has shown that the range of particle size where the magnetic force predominates is from about 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 Machinery, 2012

1 NOMENCLATURE

B:

magnetic flux density (T)

Φ:

magnetic flux (Tm)

μ_o:

permeability (Tm/A), permeability of free space: 4π × 10^{-   7}

μ_r:

relative permeability

H:

magnetomotance (A/m)

l:

magnetic flux iron path length (m)

l_g :

magnetic gap (m)

F:

force (N)

J:

current density (A/m²)

I:

current (A)

α:

pole inclination angle

N:

coils number of turns

A_g:

magnetic pole area (m²)

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Application of Evolutionary Algorithm for Multiobjective Transformer Design Optimization

S. Tamil Selvi , ... S. Rajasekar , in Classical and Recent Aspects of Power System Optimization, 2018

Choice of Current Density

The operating magnetic flux density is the parameter that determines the loss in the magnetic core. Similarly, current density in the windings determines the loss in the windings. When the current density is increased, cross-sectional area of the windings is reduced and hence, the volume and in turn 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 injure 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 below the insulation class temperature. Current density chosen should guarantee the level of losses and cooling conditions 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 short, current density is governed by load losses, temperature class of insulation, and short circuit current withstanding ability. Maximum limit for current density is calculated as

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

where Z _sc is the short circuit impedance in %; j is the short circuit current density in A/mm², which can be calculated using,

(7) ${\theta }_2={\theta }_0+j^2.y.t1{.10}^{-3}°\mathrm{C}$

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;

t1   =   Duration of short circuit. It is 2   s;

y  =   Function of $\frac{1}{2}\left({\theta }_2+{\theta }_0\right)$ , in accordance with (Clause 9.15: Table 6, IS2026 Part 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, evolution 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, evolution interval; M _P = Curie magnetization, preparation interval; S/N = signal-to-noise ratio; T ₁ = spin–lattice relaxation time; T _d = dipolar-order relaxation time;T _1ρ = rotating-frame relaxation time; T ₂ = transverse relaxation time; 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's constant; I = spin- $\frac{1}{2}$ nucleus; J = coupling constant; q = field gradient tensor; Q = nuclear quadrupole moment; s′ = doublet splitting; S = quadrupolar nucleus; γ = magnetogyric ratio; θ = angle between main tensor axes; δ = chemical shift; τ ₁ = relaxation time; ξ = angle between main 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 end magnetorheological finishing (BEMRF) process

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

Exercises

1.

Calculate 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 1. [Answer: B   =   7.87 mT]

2.

Calculate the area of the indented part (A′) of an abrasive particle of diameter 20   μm at which the cutting action will fail in BEMRF process on a mild steel workpiece. [Answer: A′  =   2.82 × 10 ⁻¹⁶   m ² ]

3.

Calculate the radius (R) of an MRP fluid hemispherical ball end rotating at 200   rpm [Answer: R   =   4.33 mm]

Where required, use value of ${\tau }_y$ from Example 2.

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Biomedical Applications of Electromagnetic Fields

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

7.1.3.3 Magnetic Flux Density

The results for the magnetic flux density, obtained using (7.8) and (7.9), were compared to the analytical results. The results for the maximum values are given in Table 7.7.

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].

		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 human brain (coronal cross-section). The results on the left are obtained via analytical expressions for (A) circular, (B) 8-coil, and (C) butterfly coil, while the results on the right are obtained via proposed model for (D) circular, (E) 8-coil, and (F) butterfly coil. From [28].

The results from Table 7.7 and Fig. 7.6 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 cross-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 can be clearly seen on Fig. 7.7.

Fig. 7.7. Magnetic flux density on the brain surface due to: (A) circular coil, (B) figure-of-8 coil, and (C) butterfly coil. All coils are placed 1 cm over the primary motor cortex. From [28].

It is interesting to observe the dependence of the induced electric field E and magnetic flux density B on the distance 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. 7.8 the rapid decrease of both E and B fields directly 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 two coils as the maximum field will be induced under the coil windings, as shown on Fig. 7.4.

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