Magnetic levitation

 ( Magnetism ) 

The driving platform PCB was built with multiple layers of wire traces like a voice coil actuation. Shown in figure [8] there are four layers of wires in the PCB board which represents two sets placed perpendicular to each other that stand for X and Y direction movement. From top to bottom, the order comes in XYXY that cross each other evenly and same axis were interlaced to control actuation. Since the force created by every layer must be the same on the circuit, deeper layers need higher current to transmit the same magnetic force to the robots on top. Set of currents with 0.25A, 0.33A, 0.5A, and 0.7A were used at SRI. One square of the above 4-layer system acts as a zone on the driving platform. This enables the circuit to control multiple robots in the same zone easily, but each robot cannot move separately. However, the platform can be divided into multiple zones which enable the separate control of robots in different zones.

Finally, a thin layer of pyrolytic graphite (500 um) acts as diamagnetic layer, placed on the top to provide stable levitation. Thin copper (15 um) placed above the graphite was used in earlier versions of the system for eddy current damping.

On moving from 1 to 2, the first trace path is turned off while the second is turned on. This causes the magnets to move to their new equilibrium, toward the higher magnetic flux density.

Repeating this procedure with opposite currents on the same trace paths, a movement in the desired direction is produced.

To find the velocity, the forces on the microrobot must be analyzed (fig. [9]). The microrobot is supposed to levitate and so no friction forced is produced, other than the air drag which is also not considered.

The force produced by the interaction of the magnetic moments of the microrobot and the flux density of the serpentine traces is:

$F=\backslash nabla(\backslash overrightarrow\{m\}\backslash cdot\backslash overrightarrow\{B\})$

The magnetic moment vector, given the orientation requirement for the diamagnetic levitation, is:

Meanwhile, the contribution to the B field by the 2 closest traces is:

$\backslash vec\{B\}=\{\backslash vec\{B\}\}\_1+\{\backslash vec\{B\}\}\_2=\backslash \; \backslash mu\_0\backslash frac\{\backslash vec\{I\_1\}\backslash times\{\backslash vec\{r\}\}\_1\}\{2\backslash pi\backslash left|\{\backslash vec\{r\}\}\_1\backslash right|^2\}+\backslash mu\_0\backslash frac\{\backslash vec\{I\_2\}\backslash times\{\backslash vec\{r\}\}\_2\}\{2\backslash pi\backslash left|\{\backslash vec\{r\}\}\_2\backslash right|^2\}$

Since for this approximation $\backslash overrightarrow\{m\}\backslash cdot\backslash overrightarrow\{B\}$ is not dependent on y or z, their derivatives are zero and only force in the x direction is produced:

$F\_x=\; \backslash frac\{B\_rV\_mI\}\{2\backslash pi\}\backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; \backslash left\; (\; \backslash frac\{x-\backslash Delta\; x\_0\}\{(x-\backslash Delta\; x\_0)^2+d^2\}\; +\; \backslash frac\{x+\backslash Delta\; x\_0\}\{(x+\backslash Delta\; x\_0)^2+d^2\}\; \backslash right\; )$

This is the only force applied on the magnet, and it can be equated to the robot's mass multiplied by its acceleration. This equation can be integrated to find the velocity of the microrobot:

$F\_x\; =\; mv\backslash frac\{dv\}\{dx\}$

$\backslash frac\{B\_r\; V\_m\; I\}\{2\backslash pi\; m\}\; \backslash left\; (\backslash frac\{2\backslash Delta\; x\_0\}\{4\backslash Delta\; x\_0^2+d^2\}\backslash right)=\backslash frac\{1\}\{2\}v\_\{max\}^2$

Introducing the relation between magnet volume, mass, and density $V\_m=\backslash frac\{m\}\{\backslash rho\_m\}$ in the previous equation cancels out the mass, which means that if more magnets are added (N number of magnets), force will increase linearly:

$v\_\{max\}=\backslash sqrt\{N\backslash frac\{B\_r\; I\}\{\backslash pi\; \backslash rho\_m\}\; \backslash left\; (\backslash frac\{2\backslash Delta\; x\_0\}\{4\backslash Delta\; x\_0^2+d^2\}\backslash right)\}$

This is the expression for the robot speed as a function of the current.

For a second DOF, more traces must be added. Two more intertwined serpentine traces must be added below the existing ones, rotated 90 degrees, to generate forces in the Y direction. Intensity on these traces will have to be higher to account for the higher distance.

Diamagnetic levitation can produce two effects on a micro robot. The first is reducing the sliding friction and the second is fully levitating the micro robot. The fully levitation system will be the focus. To produce passive levitation a diamagnetic layer (such as graphite) must exist in the presence of a ferromagnet (such as NdFeB). Diamagnetic materials are characterized by having negative susceptibility, induced magnetic moment opposite to the external magnetic field. For that reason, they are repelled by an external magnetic field and tend to move toward the field minimum. This repulsive force is a result of the diamagnets having a magnetization direction antiparallel to the external magnetic fields.

The magnetizations of diamagnetic materials vary with an applied magnetic field which can be given as:

$M\; =\; \backslash chi\_p\; H$

Where $H$ is the magnetic field strength and $\backslash chi\_p$ is the dimensionless susceptibility. For an object with volume $V\_p$, the induced magnetic moment m can be given by:

$\backslash overrightarrow\{m\}\; =\; \backslash chi\_p\backslash frac\{V\_p\}\{\backslash mu\_0\}\backslash overrightarrow\{B\}$

The magnetic force acting on the object is there for described as:

$F\_m\; =\; \backslash frac\{\backslash chi\_pV\_p\}\{\backslash mu\_0\}\backslash nabla(\; \backslash overrightarrow\{B\}\backslash cdot\; \backslash overrightarrow\{B\}\; )$

If the object has density $\backslash rho\_p$ and is levitating in a medium with density $\backslash rho\_m$ and magnetic susceptibility $\backslash chi\_m$ the total energy of the object, with a magnetic and gravitational term, is:

$E\; =\; \backslash frac\{\backslash chi\_p-\backslash chi\_m\}\{2\backslash mu\_0\}V\_p\; (\; \backslash overrightarrow\{B\}\backslash cdot\; \backslash overrightarrow\{B\}\; )+\; (\backslash rho\_p-\backslash rho\_m)V\_p\; gz$

Such that the resulting force becomes:

$\backslash sum\; F\; =\; \backslash frac\{\backslash chi\_p-\backslash chi\_m\}\{2\backslash mu\_0\}V\_p\; \backslash nabla\; (\; \backslash overrightarrow\{B\}\backslash cdot\; \backslash overrightarrow\{B\}\; )+\; (\backslash rho\_p-\backslash rho\_m)V\_p\; g\; \backslash overrightarrow\{k\}$

The necessary condition for stability is:

To calculate the whole diamagnetic force acting on the levitated materials, each single dipole of the diamagnetic material must be considered. The diamagnetic force for the entire volume can be expressed as:

$F\_m\; =\; \backslash frac\{\backslash chi\_p\}\{\backslash mu\_0\}\backslash iiint\_\{V\_p\}\backslash nabla(\; \backslash overrightarrow\{B\}\backslash cdot\; \backslash overrightarrow\{B\}\; )dv$

The diamagnetic repulsion force is proportional to the magnetic susceptibility of diamagnetic materials. To counteract gravity in the magnetic field, materials with strong diamagnetism and lightweight properties are preferred.

$I\text{'}\; =\; \backslash frac\{dI\}\{ds\}\; =\; \backslash frac\{B\_r\}\{\backslash mu\_0\}$

Here, I' is the equivalent surface current density, B_r is the remanent magnetic field of the magnet, and μ_0 is the permeability of free space, given by: $\backslash mu\_0\; =\; 4\backslash pi\; \backslash times\; 10^\{-7\}\; \backslash ,\; \backslash text\{(A/Tm)\}$

The surface current density I^' is directly proportional to the remanent magnetic field B_r, which is a measure of the magnet's residual magnetization after an external magnetic field is removed. This property is crucial for the magnet's performance in applications such as magnetic levitation, where maintaining a stable and strong magnetic field is essential. To calculate the magnetic field generated by permanent magnets, we can use an approach based on the Biot-Savart law applied to finite-size rectangular current sheets. This method involves modeling the magnets as an assembly of such sheets, allowing for the calculation of the three components of the magnetic field B ⃗_z at any point in space. By applying this law to a finite-size rectangular current sheet, we can compute the magnetic field by integrating the contributions from all current elements within the sheet. For a rectangular sheet carrying a surface current density I', the magnetic field at a point z ⃗ can be determined by summing the contributions from each infinitesimal segment of the sheet. To model a permanent magnet, we consider it as a stack of such current sheets. The total magnetic field B _z at any point is the superposition of the fields generated by each sheet. This superposition is expressed mathematically as:

$\backslash vec\{B\}\_z\; =\; \backslash sum\_i\; \backslash vec\{B\}\_\{z,i\}$

where $\backslash vec\{B\}\_\{z,i\}$ represents the magnetic field contribution from the i-th current sheet. This method provides a comprehensive way to calculate the magnetic field components of permanent magnets, enabling precise modeling of their magnetic behavior in various applications, such as magnetic levitation, where accurate field distribution is crucial for stability and performance.

$\backslash vec\{B\}(\backslash vec\{r\}\_2)\; =\; \backslash frac\{\backslash mu\_0\; I\}\{4\backslash pi\}\; \backslash int\; \backslash frac\{d\backslash vec\{r\}\_1\; \backslash times\; (\backslash vec\{r\}\_2\; -\; \backslash vec\{r\}\_1)\}\{|\backslash vec\{r\}\_2\; -\; \backslash vec\{r\}\_1|^3\}$

where $\backslash vec\{r\}\_1\; =\; (1\; -\; t)\backslash vec\{r\}\_a\; +\; t\backslash vec\{r\}\_b$ (a point on the filament 0\frac{d\vec{r}_1}{dt} = -\vec{r}_a + \vec{r}_b. The equation can be organized as:

$\backslash vec\{B\}(\backslash vec\{r\}\_2)\; =\; \backslash frac\{\backslash mu\_0\; I\}\{4\backslash pi\}\; \backslash int\; \backslash frac\{d\backslash vec\{r\}\_1\; \backslash times\; (\backslash vec\{r\}\_2\; -\; \backslash vec\{r\}\_1)\}\{|\backslash vec\{r\}\_2\; -\; \backslash vec\{r\}\_1|^3\}$

Such that we can write the $(\backslash vec\{B\}(\backslash vec\{r\}\_2)$as:

$\backslash vec\{B\}(\backslash vec\{r\}\_2)\; =\; \backslash frac\{\backslash mu\_0\; I\}\{4\backslash pi\}\; (\backslash vec\{r\}\_b\; -\; \backslash vec\{r\}\_a)\; \backslash times\; (\backslash vec\{r\}\_2\; -\; \backslash vec\{r\}\_a)\; \backslash int\_\{0\}^\{1\}\; \backslash frac\{1\}\{A^\{3/2\}\}\; \backslash ,\; dt$

with $A\; =\; (\backslash vec\{r\}\_2\; -\; \backslash vec\{r\}\_a)^2$, $B\; =\; (\backslash vec\{r\}\_2\; -\; \backslash vec\{r\}\_a)\; \backslash cdot\; (\backslash vec\{r\}\_b\; -\; \backslash vec\{r\}\_a)$, $C\; =\; (\backslash vec\{r\}\_b\; -\; \backslash vec\{r\}\_a)^2$

Finally, the magnetic field of a straight current-carrying filament is derived as: $\{B\}(\backslash vec\{r\}\_2)\; =\; \backslash frac\{\backslash mu\_0\; I\}\{4\backslash pi\}\; \backslash frac\{\backslash vec\{r\}\_b\; -\; \backslash vec\{r\}\_a\; \backslash times\; \backslash vec\{r\}\_2\; -\; \backslash vec\{r\}\_a\}\{AC\; -\; B^2\}\; (C\; -\; B\; -\; A\; -\; 2B\; +\; C\; +\; B\; -\; A)$

$\backslash vec\{p\}\; =\; \backslash alpha\; (\backslash vec\{r\}\_b\; -\; \backslash vec\{r\}\_a)\; \backslash times\; \backslash vec\{n\}$

$\backslash vec\{p\}$ is perpendicular to the filament and α is the aspect ratio of the sheet. The width of the sheet is: $w\; =\; \backslash alpha\; (\backslash vec\{r\}\_b\; -\; \backslash vec\{r\}\_a)\; =\; \backslash alpha\; C$

The field from the displaced filament is then given by replacing the parameters A, B and C by their s-dependent counterparts given by:

$A(s)\; =\; (\backslash vec\{r\}\_2\; -\; \backslash vec\{r\}\_a\; -\; s\backslash vec\{p\})^2$

$B\_s\; =\; (\backslash vec\{r\}\_2\; -\; \backslash vec\{r\}\_a\; -\; s\backslash vec\{p\})\; \backslash cdot\; (\backslash vec\{r\}\_b\; -\; \backslash vec\{r\}\_a)$

$C\_s\; =\; (\backslash vec\{r\}\_b\; +\; s\backslash vec\{p\}\; -\; \backslash vec\{r\}\_a\; -\; s\backslash vec\{p\})^2\; =\; C$

Then, we spread out the current evenly across the sheet width:

$I\; =\; I\text{'}\; w\; =\; \backslash frac$

{B_r}{\mu_0} \alpha C

The magnetic field of a finite current sheet can be derived as

$\backslash vec\{B\}(\backslash vec\{r\}\_2)\; =\; \backslash frac\{B\_r\; \backslash alpha\; C\}\{4\backslash pi\}\; \backslash int\_\{-1/2\}^\{1/2\}\; \backslash frac\{(\backslash vec\{r\}\_b\; -\; \backslash vec\{r\}\_a)\; \backslash times\; (\backslash vec\{r\}\_2\; -\; \backslash vec\{r\}\_a\; -\; s\backslash vec\{p\})\}\{A(s)C^\{3/2\}\}\; ds$

Such that we can write the $\backslash vec\{B\}(\backslash vec\{r\}\_2)$ as:

$\backslash vec\{B\}(\backslash vec\{r\}\_2)\; =\; \backslash frac\{B\_r\; \backslash alpha\; C\}\{4\backslash pi\}\; \backslash int\_\{-1/2\}^\{1/2\}\; \backslash left\; ds$

where $R\_1\; =\; a\_1\; +\; b\_1\; s\; +\; c\_1\; s^2$, $a\_1\; =\; AC\; -\; 2BC\; +\; C^2$, $b\_1\; =\; -2CD$, $c\_1\; =\; \backslash alpha^2\; C^2$, $R\_2\; =\; a\_2\; +\; b\_2\; s\; +\; c\_2\; s^2$, $a\_2\; =\; AC\; ,\; b\_2\; =\; -2CD\; =\; b\_1,\; c\_2\; =\; \backslash alpha^2\; C^2\; =\; c\_1,\; P\_1\; =\; -(B\; -\; C)^2,\; P\_2\; =\; -B^2$

Finally, the magnetic field of a finite Current Sheet is organized as:

$\backslash vec\{B\}(\backslash vec\{r\}\_2)\; =\; \backslash frac\{B\_r\; \backslash alpha\; C\}\{4\backslash pi\}\; \backslash left\; +\; \backslash frac\{B\_r\; \backslash alpha^2\; C^2\}\{4\backslash pi\}\; \backslash left\; \backslash vec\{n\}$

where $J\_1\; =\; \backslash int\_\{-1/2\}^\{1/2\}\; \backslash frac\{1\}\{R\_1\; +\; P\_1\}\; ds,\; J\_2\; =\; \backslash int\_\{-1/2\}^\{1/2\}\; \backslash frac\{s\}\{R\_1\; +\; P\_1\}\; ds,\; J\_3\; =\; \backslash int\_\{-1/2\}^\{1/2\}\; \backslash frac\{1\}\{R\_2\; +\; P\_2\}\; ds,J\_4\; =\; \backslash int\_\{-1/2\}^\{1/2\}\; \backslash frac\{s\}\{R\_2\; +\; P\_2\}\; ds$

Having the magnetic field extracted from the magnet, we now try to use it in the levitation application. Diamagnetic materials possess a unique property: they repel magnetic fields, seeking regions of minimal field intensity. This behavior enables the levitation of diamagnetic materials above strong magnetic fields, a phenomenon demonstrated vividly in the Levitated Frog Experiment. Understanding these fundamental principles has paved the way for innovative technologies such as magnetic levitation in Microrobots. Diamagnetically levitated milli- and micro-robots offer precise control and minimal force noise, ensuring intrinsic stability and efficient zone control. Leveraging diamagnetic levitation, these robots experience reduced sliding friction and can achieve full levitation when paired with a diamagnetic layer, such as graphite, in the presence of a ferromagnet like NdFeB. Diamagnetic materials, characterized by negative susceptibility and an induced magnetic moment opposing the external field, are repelled by magnetic fields, naturally gravitating towards field minima. This repulsion arises from diamagnets' magnetization direction being antiparallel to the external field, enabling passive levitation and facilitating advanced control strategies.

For levitating a magnet, the net forces acting on it in Z direction should be zero. The free body diagram is shown below:

For levitating a diamagnetic particle the below equation should apply:

$\backslash \; F\; =\; \backslash frac\{\backslash chi\_p\; -\; \backslash chi\_m\}\{2\backslash mu\_0\}\; V\_p\; \backslash nabla\; (\backslash vec\{B\}\; \backslash cdot\; \backslash vec\{B\})\; +\; (\backslash rho\_p\; -\; \backslash rho\_m)\; V\_p\; g\; z$

The diamagnetic particle with density $\backslash rho\_p$ and magnetic susceptibility $\backslash chi\_p$ is levitating in a medium with density $\backslash rho\_m$ and magnetic susceptibility $\backslash chi\_m$. In this case the only way the net force is going to be zero is when: . The difference of our problem is we are trying to levitate a magnet on a diamagnetic surface sheet and not the other way around. That’s why there should be justification in the formulation above:

$F\; =\; \backslash frac\{\backslash chi\_\{\backslash text\{dia\}\}\}\{2\backslash mu\_0\}\; V\_\{\backslash text\{dia\}\}\; \backslash nabla\; (\backslash vec\{B\}\; \backslash cdot\; \backslash vec\{B\})\; +\; (\backslash rho\_\{\backslash text\{mag\}\}\; -\; \backslash rho\_\{\backslash text\{air\}\})\; V\_\{\backslash text\{mag\}\}\; g\; z$

The parameter $V\_\{\backslash text\{dia\}\}$ is debatable and further investigation is needed to assign the actual value for it, because when the magnet is the object levitating, the magnetic field produced by it is not going to cover the whole volume of the diamagnetic surface sheet. The depth of penetration and the effective surface should be calculated. (note that χ_dia is negative so there’s at least one solution for the equation).

In the free body diagram above, two other forces are visible Fx and Fy. In order to explain that lets go through the term ∇(B ⃗⋅B ⃗ ) and expand it. Magnetic field density is a vector function displayed like below:

$\backslash vec\{B\}(x,y,z)\; =\; B\_x(x,y,z)\; \backslash hat\{x\}\; +\; B\_y(x,y,z)\; \backslash hat\{y\}\; +\; B\_z(x,y,z)\; \backslash hat\{z\}$

$\backslash vec\{B\}\; \backslash cdot\; \backslash vec\{B\}\; =\; B\_x^2(x,y,z)\; +\; B\_y^2(x,y,z)\; +\; B\_z^2(x,y,z)$

$\backslash nabla\; (\backslash vec\{B\}\; \backslash cdot\; \backslash vec\{B\})\; =\; \backslash left(\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\}\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; z\}\; \backslash right)\; (B\_x^2(x,y,z)\; +\; B\_y^2(x,y,z)\; +\; B\_z^2(x,y,z))$

$=\; 2\; \backslash left(\; B\_x\; \backslash frac\{\backslash partial\; B\_x\}\{\backslash partial\; x\}\; +\; B\_y\; \backslash frac\{\backslash partial\; B\_y\}\{\backslash partial\; x\}\; +\; B\_z\; \backslash frac\{\backslash partial\; B\_z\}\{\backslash partial\; x\}\; \backslash right)\; \backslash hat\{x\}\; +\; \backslash left(\; B\_x\; \backslash frac\{\backslash partial\; B\_x\}\{\backslash partial\; y\}\; +\; B\_y\; \backslash frac\{\backslash partial\; B\_y\}\{\backslash partial\; y\}\; +\; B\_z\; \backslash frac\{\backslash partial\; B\_z\}\{\backslash partial\; y\}\; \backslash right)\; \backslash hat\{y\}\; +\; \backslash left(\; B\_x\; \backslash frac\{\backslash partial\; B\_x\}\{\backslash partial\; z\}\; +\; B\_y\; \backslash frac\{\backslash partial\; B\_y\}\{\backslash partial\; z\}\; +\; B\_z\; \backslash frac\{\backslash partial\; B\_z\}\{\backslash partial\; z\}\; \backslash right)\; \backslash hat\{z\}$

The third term which is $B\_x\; \backslash frac\{\backslash partial\; B\_x\}\{\backslash partial\; z\}\; +\; B\_y\; \backslash frac\{\backslash partial\; B\_y\}\{\backslash partial\; z\}\; +\; B\_z\; \backslash frac\{\backslash partial\; B\_z\}\{\backslash partial\; z\}$ should be compared with the gravity force. The first and second term should be zero for the magnet to have a stable levitation. Further investigation is needed.

To find the distance the magnet is being levitated from, the equation below should be solved:

$\backslash left|\; \backslash frac\{\backslash chi\_\{\backslash text\{dia\}\}\}\{2\backslash mu\_0\}\; V\_\{\backslash text\{dia\}\}\; \backslash left(\; B\_x\; \backslash frac\{\backslash partial\; B\_x\}\{\backslash partial\; z\}\; +\; B\_y\; \backslash frac\{\backslash partial\; B\_y\}\{\backslash partial\; z\}\; +\; B\_z\; \backslash frac\{\backslash partial\; B\_z\}\{\backslash partial\; z\}\; \backslash right)\; \backslash right|\; =\; (\backslash rho\_\{\backslash text\{mag\}\}\; -\; \backslash rho\_\{\backslash text\{air\}\})\; V\_\{\backslash text\{mag\}\}\; g$

Based on the magnet we choose, we can derive the behavior of the Magnetic field in space $\backslash vec\{B\}(x,y,z)$. For solving the equation numerically, and using magnetic field as a vector function (which it is), there’s going to be a z=d which:

$\backslash left(\; B\_x\; \backslash frac\{\backslash partial\; B\_x\}\{\backslash partial\; z\}\; +\; B\_y\; \backslash frac\{\backslash partial\; B\_y\}\{\backslash partial\; z\}\; +\; B\_z\; \backslash frac\{\backslash partial\; B\_z\}\{\backslash partial\; z\}\; \backslash right)\_\{x,y,d\}\; =\; (\backslash rho\_\{\backslash text\{mag\}\}\; -\; \backslash rho\_\{\; text\{air\}\})\; V\_\{\backslash text\{mag\}\}\; g$

For simplifying this |B_(x,y) |≪|B_z |,|(∂B_z)/∂x|≪ |(∂B_z)/∂z| ,|(∂B_z)/∂y|≪ |(∂B_z)/∂z|, problem before solving this, a paper (R. Engel–Herbert and T. Hesjedal , “ Calculation of the magnetic stray field of a uniaxial magnetic domain,” reported:

Which will make the calculations lighter:

$\backslash left|\; \{\backslash chi\_\{\backslash text\{dia\}\}\}\{V\_\{\backslash text\{dia\}\}\}\backslash right|\; B\_z\; \backslash frac\{\backslash partial\; B\_z\}\{\backslash partial\; z\}\_\{x,y,d\}\; =\; (\backslash rho\_\{\backslash text\{mag\}\}\; -\; \backslash rho\_\{\backslash text\{air\}\})\; V\_\{\backslash text\{mag\}\}\; g$

At the end by calculating the Bz by equations explained earlier and put it into this, we’ll can solve the problem of levitation by founding out in which height the equation above works.

Simulation Parameters and Results

Consider a 3x3 array of NdFeB magnets, each with a pole size of 1 mm and a thickness of 0.4 mm. The simulations of the diamagnetic force as a function of the distance from the magnet surface are illustrated in Fig. 5. These simulations provide critical insights into the force profile experienced by the micro robot at varying heights above the magnet array.

Additionally, Fig. 6 shows the $B\_z$ component (the magnetic flux density in the z-direction) at the magnet's surface. This component is to understand the magnetic field distribution, which directly influences the levitation and stability of the micro robot.

• 1839 Earnshaw's theorem showed electrostatic levitation cannot be stable; later theorem was extended to magnetostatic levitation by others
• 1913 Emile Bachelet awarded a patent in March 1912 for his "levitating transmitting apparatus" (patent no. 1,020,942) for electromagnetic suspension system
• 1933 Superdiamagnetism Walther Meissner and Robert Ochsenfeld (the Meissner effect)
• 1934 Hermann Kemper "monorail vehicle with no wheels attached." Reich Patent number 643316
• 1939 Werner Braunbeck's extension showed that magnetic levitation is possible with diamagnetic materials
• 1939 Bedford, Peer, and Tonks aluminum plate placed on two concentric cylindrical coils shows 6-axis stable levitation.
• 1961 James R. Powell and BNL colleague Gordon Danby electrodynamic levitation using superconducting magnets and "Null flux" figure-8 coils
• 1970s Spin stabilized magnetic levitation Roy M. Harrigan
• 1974 Magnetic river Eric Laithwaite and others
• 1979 transrapid train carried passengers
• 1981 First single-tether magnetic levitation system exhibited publicly (Tom Shannon, Compass of Love, collection Musee d'Art Moderne de la Ville de Paris)
• 1984 Low-speed maglev shuttle in Birmingham Eric Laithwaite and others
• 1997 Diamagnetically levitated live frog Andre Geim
• 1999 Inductrack permanent magnet electrodynamic levitation (General Atomics)
• 2000 The first man-loading HTS maglev test vehicle "Century" in the world was successfully developed in China.
• 2024 The first passive maglev train was unveiled and demonstrated in Verona.

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