Keywords continuum electromechnics; electrical education; electrodynamics teaching; electromechanical systems; magnetic levitation
(ProQuest: ... denotes formulae omitted.)
In any advanced course in electromechanical systems, students usually study the distributed parameters modeling of electromagnetic systems, including electrodynamic forces on conducting materials.1-4 In this context, we propose in this paper two experiments where the electrodynamic forces significantly affect the systems' dynamics. The first experiment takes a pendulum system made of a conducting plate, which is subjected to a uniform and constant magnetic field. Due to the electrodynamic forces induced in the conducting plate, the pendulum damping can be controlled. The system is first modelled by the use of Maxwell's equations,5 assuming a set of assumptions coming from the system's geometry and its materials. Experimental results obtained with the pendulum system are compared with results estimated from the model to show the consistency and validity of the model assumptions and thus validate the model.
In a second experiment, two permanent magnets are located above a conducting plate which moves with a speed that can be controlled. The relative motion between the magnets and plate induce electric currents in the conducting sheet, producing a repulsive force that causes the lifting of the permanent magnets. The magnetic field distribution in the system is studied first to explain the appearance of drag and lifting forces. Following this, the co-dependence of these forces with speed of the conducting plate and the lifting height is discussed taking into account the experimental results obtained.
Experiment 1
Experimental setup
Figure 1(a) shows the experimental apparatus where a rectangular conducting plate made of aluminum is suspended by a pendulum with length r, and located in a gap (see this detail in Fig. 1(b)) where a constant magnetic flux density B0 of 70.2 mT is imposed by a magnetic circuit. Magnetic drag forces appearing in the conducting plate cause damping when it swings in the perpendicular direction to B0 (see Fig. 1(a)). The potential and kinetic energies of the pendulum are dissipated in the form of power losses in the aluminium plate.
The magnetic circuit shown in Fig. 1(b) is composed of two parts with an E-form. Also, one coil with 400 turns and located in the central leg was used to energise the magnetic circuit. The pendulum position was measured using an ultrasonic sensor.
Experiment 1 - distributed parameter modelling
Figure 2 is a schematic diagram of the experimental system shown in Fig. 1 but seen from the upper side. The conducting sheet shown in Fig. 2 has the shape of a long plate with uniform thickness d and conductivity σ, moving in the positive x direction with constant speed vx. This experiment considers a uniform and constant magnetic field B = -B0ey located in a rectangular region on the conducting plate with area lz0 which is imposed by an electromagnet not represented in the figure.
With the assumptions of a symmetric system and neglecting border effects, the system is studied in two dimensions only, -x and -y directions, with the z component of B assumed to be zero. The system's magnetic flux density becomes, thus, defined by:
... (1)
with the conducting plate moving with a linear speed defined by:
... (2)
The electric field E induced in the conducting plate is given by:
... (3)
which results in an induced density current according to Ohm's law J = σE of:
... (4)
Using (4) with the following two Maxwell equations
... (5)
and
... (6)
the equation establishing field B in the conducting plate becomes defined as:
... (7)
The -x and -y components of (7) can be obtained considering the earlier assumptions used in the definition of the magnetic flux density in (1), resulting in:
... (8)
... (9)
Considering that the thickness of the conducting plate is very small when compared with the distance between the magnetic poles (d
- the -x component of B inside the plate can be ignored, B^sub x^ [congruent with] 0;
- and it can be considered that the vertical component B^sub y^ only varies with the x direction, B^sub y^(x,y) [congruent with] B^sub y^(x).
Using these approximations (9) can be rewritten as:
... (10)
The solution of (10) can assume the form given by (11). In this equation, parameter R^sub m^ = σμv^sub x^l is usually called the magnetic Reynolds number, and parameters C1 and C2 are constants to be determined from two boundary conditions.
... (11)
First boundary condition
As indicated in Fig. 3, the first boundary condition is obtained by integrating Ampere's Law over the ABCD way yielding:
... (12)
where Jz is the -z component of the eddy current density induced in the conducting plate in z direction, and linked by the contour ABCD. Due to the plate's extreme thinness, we assume that the magnetic field component Hy does not vary appreciably over its width d and, from our previous assumptions, that the magnetic field component Hx becomes zero. Therefore (12) results in:
... (13)
where J^sub z^ = σv^sub x^B^sub y^.
Second boundary condition
This condition is obtained using the magnetic flux conservation through the plate surface that is located between the magnetic poles.B^sub 0^ being the magnetic flux density, imposed by the magnetic circuit it can be written:
... (14)
Resolving (12) and (13) for the constants C1 and C2, we obtain the analytic expression for the flux density distribution inside the conducting plate given by:
... (15)
The drag force appearing in the conducting plate can thus be determined by integrating the force density (16) through plate volume, yielding the result shown in (17). This result shows that the drag force depends on the integral of B^sub y^^sup 2^, being thus proportional to the power dissipated by the eddy currents.
... (16)
... (17)
Drag force
Because the aluminium plate is very thin and the induced currents are not considered to be high enough, in order for their magnetic field to distort the applied one B0, it has been assumed that the value of the flux density in the plate is constant:
... (18)
Substituting eqn (18) for eqn (17), the drag force becomes given by:
... (19)
The pendulum linear velocity v^sub x^ can be approximated by expression (20) for small pendulum angular displacements.
... (20)
The motion equation of the pendulum given by (21) results then in the approximated (22), where θ is the angular position, m is the pendulum mass, r its radius and I is the moment of inertia of the pendulum, being equal to I = mr^sub 2^.
... (21)
... (22)
Substituting (19) and (20) in eqn (22) and rearranging the equation terms, this results in the second-order position equation (23) for the pendulum system where the parameter Kθ becomes given by (24).
... (23)
... (24)
The roots of the characteristic equation for the differential equation (23) are:
... (25)
Where, because the pendulum is a damped oscillating system, relation (26) must be satisfied.
... (26)
The main dimensions of the pendulum system are: z^sub 0^ = 4 cm, d = 1 mm, l = 4 cm and r = 10 cm. The aluminium conductivity is equal to σ = 3 × 10^sup 7^ S*m^sup -1^ resulting in a mass to the conducting plate equal to m = 24.3 g. Substituting these parameter values in (26), this becomes satisfied, presenting a value of about 6 × 10^sup -4^, which is less than 1 as required for an oscillatory damping system.
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