User:Eamantang/sandbox/Electrodynamic Wheel

The Electrodynamic wheel (EDW) is a type of wheel proposed for use in electrodynamic levitation maglev (a train transportation that uses magnetic levitation). EDW on the maglev is a practical solution to the innate magnetic drag force. Moving the wheels rotationally, the maglev can get rid of the drag force by flipping its direction, converting it from drag to thrust force instead. There are two types of maglevs - electrodynamic suspension (EDS) and electromagnetic suspension (EMS). In both cases, they require either laminated track or null flux coil track (which eliminates the electric field) to reduce the innate drag force from the translational motion. The construction cost is high in both situations.

Although the main application for EDW is for maglevs, it can also be used in frictionless bearing, contactless gear, and launching systems.

History
Researchers and engineers have been proposing various magnetic levitation methods throughout the late twentieth century. Poor power factor, low lift-to-weight ratio, and the innate magnetic drag force have constantly impeded the actual implementation of the maglevs.

Single-sided Linear Induction Motor (SLIM)
The very first idea that came up regarding magnetic levitation was using the single-sided linear induction motor. In the 1970's and the 1980's, single-sided linear induction motors with aluminum and back iron tracks were introduced by Fredrick Eastham and Eric Laithwaite. Many issues came up with this method of magnetic levitation:


 * The track had to be 5% longer than how much it was actually used because of the end-effects of the guideway.
 * The cost was high because the motor had to use a secondary suspension, an extra pneumatic suspension to aid passengers comfort.
 * The power efficiency calculated by the actual power over the ideal power, or the power factor, was low due to drag losses.

However, there was an obscure advantage if the track composition was changed to iron only. Then, the current could be induced by the track to create both lift and thrust. Unfortunately, the engineers did not realize this advantage because of the low lift-to-weight ratio and poor power factor. Lift-to-weight ratio is the number obtained by dividing the lift force over the weight of the maglev. Ideally the number should be high, because the lift has to be greater than the weight for the maglev to have a secure levitation.

Homopolar Iron Cored Linear Synchronous Motor (HICLSM)
Enrico Levi came up with another passive track design that solved the power factor problem - HICLSM. This motor was used on iron tracks under girders, which allowed the maglev to travel with AC and DC excitation. The AC excitation creates a synchronicity with the pole track forming a propulsion force. The DC excitation creates lift force and a magnetization field. However, similar to SLIM, HICLSM requires high construction cost. The track iron needs to be laminated because of induced drag force from the track. In addition, the irons must be two times thicker than those from SLIM because they have to prevent the iron from saturating the track.

Electrodynamic Wheels (EDW)
The third propulsion idea for maglev was introduced as early at the 1970's. By mechanically rotating the rotor over an aluminum track, the drag force from electrodynamic suspension converts to thrust, allowing the vehicle to have both lift and thrust. However, due to the difficulty in rotating superconductors that are at extremely low temperature (in order for the superconducting levitation property to appear, the temperature has to go as low as 4.2 K), no further research were conducted.

In the early 2000's, rare earth metals were found to replace the superconducting metals. These metals no longer have to be cooled by liquid helium, and they are capable of creating strong magnetic fields as well, making it possible for electrodynamic wheels.

Basics
EDW is a conducting nonmagnetic wheel lined with permanent magnet around the circumference. The permanent magnets are arranged in a Halbach array to concentrate all the magnetic field on the outer rim. This structure allows the wheel to induce magnetic levitation and create a propulsion force.

Halbach Array
The Halbach rotor arranges the magnets in a circular Halbach array. Permanent magnets have both repulsive and attractive forces at the two opposite sides. By arranging them in Halbach array, the inner field cancels out while the outside field gets a stronger magnetic repulsion. This helps alleviate the low lift-to-weight ratio problem.

Function
When maglev trains travel on their nonmagnetic guideway, large amount of magnetic drag force is formed from the fast translation motion of magnetic field over the conducting path. Some methods of solving this problem include putting magnets on the entire guideway; however, the construction cost is extremely expensive. An alternative method is to convert the drag force into a propulsive force, which can be done using the EDW.

Mechanism
The electrodynamic wheel can move both translationally and rotationally, allowing it to create a time-varying magnetic field. This alternating magnetic field creates thrust through the drag force. This rotation allows the wheel to produce a propulsion force, or thrust, that takes care of the large magnetic drag problem.

Induced Current and Thrust Force
Induced current on the conducting guideway:

$$J = \sigma(E + v \times B)$$

Lorentz Force:

$$F = J \times B$$

By substituting the induced current into the Lorentz Force equation:

$$F = \sigma E \times B + \sigma v \times B \times B$$

In a 2D model, considering the current flows in the z axis and the train moves in the x axis, the magnetic field and the thrust will be in the y axis:

$$F=-\sigma(E_z + v_xB_y)B_y\widehat{a}_x + \sigma(E_z + v_xB_y)B_x\widehat{a}_y$$

The direction and magnitude of thrust force, $$F$$, is dependent on $$E_z$$.

Induced Electric Field
The induced electric field on the conducting guideway (Maxwell-Faraday equation):

$$E_z = {\delta A_z \over \delta t} - \nabla V$$ In a linear translation without EDW, the magnetic sources do not change with time, so $$\delta A_z \over \delta t$$is 0 N/C. Since there is also no free charge in the metallic conductor on the guideway, the $$\nabla V$$is also 0 N/C. Without any rotation from the EDW, the induced electric field will remain 0 N/C. $$\delta A_z \over \delta t$$can be easily manipulated if rotational motion is applied. This rotation will change the field, which can eventually change the magnitude and direction of the drag force.

Relative Velocity
$$s = \omega r - v$$

The propulsion force is also dependent on the relative velocity of the maglev. The velocity is no longer just translational; it will be the sum of translation and rotation as shown in the equation above. This relative velocity is called slip. The slip being too high will prevent the thrust from converting to drag force. However, the most it can do is minimizing the drag force, because the thrust is directly proportional to the levitation force.