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Battershell, Minder, Wachtveitl
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Magnetic Actuator modeling in 2­D Ansoft Maxwell
Jon Battershell, Spencer Minder., and Zachary Wachtveitl
Abstract​— ​Our project was to model the Magnetic Actuator described in section 6.5 of Senturia’s ​Microsystem Design[1]​. We
modeled the system in Ansoft Maxwell to allow us to perform a force analysis of the magnetic field on the armature. The first
experiment conducted was to apply 1 A of current to the coil around the core of the actuator and examine the forces acting on the
armature over a total displacement of 10µm. After the initial experiment, two different spring constants were chosen to be
hypothetically modeled in the system. With these spring constants, we examined the amount of current and force required to move the
armature over the same total displacement of 10µm. In addition, the B fields and B vector lines within the system were examined at
different points along the armatures path through the gap. After we tested and verified our model, we produced graphical analysis of
our results and compared the experimental results versus the theoretical equations from the textbook.
Index Terms​— Ansoft Maxwell, Magnetic Actuator, Magnetic Flux Density, MEMS
I. I​NTRODUCTION
T
HE principles behind energy conserving transducers in MEMS can be used in many different applications. The uses range from
piezoresistive sensors to electrostatic and magnetostatic actuators. These types of transducers can be useful in wireless
technology, automotive and biomedical fields as well. For our purposes, we chose to model a magnetic actuator. We chose this
type of actuator because we had already worked with electrostatic actuators and wanted to try a different energy domain. After
modeling the magnetic actuator, we wanted to compare the differences from electro­mechanical to magneto­mechanical designs.
We chose to model the system Ansoft Maxwell 2­D MEMS software package. This type of software is great for modeling
electrostatic or magnetostatic components of MEMS type devices that are relatively simple. It provides an easy user interface,
quick set­up and small learning curve. The downfalls are that it is relatively limited in the types of solvers that can be used.
Only one type of domain can be analyzed at a time such as electrical, in the form of electrostatics, or magnetic, in the form of
magnetostatics. A more robust MEMS software could provide a multi­physics based solution using a magneto­mechanical
solver. This would allow for modeling the entire magnetic actuator system. However, these sophisticated software packages
have a much steeper learning curve and were not feasible in the time allowed. Two experiments were conducted using Maxwell.
The first experiment was to look at the forces applied to the armature as it traversed the gap. The second experiment used two
different assumed spring constants and looked at the current required to move the armature through the gap. Both experiments
assumed the armature started at the same initial displacement and travelled the same total displacement.
The main focus of this paper is to compare the experimental results of these modeled experiments in Ansoft Maxwell to
the theoretical results derived in section 6.5 of the textbook ​Microsystem Design​[1].
II. MODEL DEFINITION AND DESCRIPTION
The magnetic behavior of the magnetic actuator was modeled in Ansoft Maxwell 2D. The model was designed as a flat
structure in the x­y plane since the actuator isn’t axisymmetric. It was designed using a micrometer scale as this actuator was
intended to be in a MEMS device. The model consists of the three necessary magnetic components of a magnetic actuator: the
core, the coil, and the armature, see Figure 1. The core is the blue rectangular shape shown in Figure 1. The core is designed to
have a gap for the armature to pass freely through. The gap is designed to be slightly larger than the width of the armature. To be
a magnetic actuator the core must be able to support the formation of a magnetic field within itself. Therefore, the core must be
use a material with a relatively large permeability. For this design, iron was the chosen material.
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Figure 1: Ansoft Maxwell Magnetic Actuator Model.
The armature is the purple rectangular shape in Figure 1. The armature was designed to have an arbitrary length but the width
was defined by the size of the gap in the core. The width of the armature needed to be slightly smaller than the gap in the core,
see Figure 2 and Table 1 for width and gap dimensions. Similar to the core, the armature must be a permeable material so again
iron was chosen. The coil consists of the red and green rectangles shown in Figure 1. The red portion of the coil model is
defined to carry current in one direction (positive current) while the green portion is defined to carry a current in the opposite
direction (negative current). The coil’s dimensions were arbitrarily chosen for this design but it represents only a single turn of
wire around the core. The coil needed to be a material with good electrical conductivity copper was a natural choice to use in
this model. Since this is intended to be a magnetic actuator, the position or displacement of the armature would change
depending on the force applied by the magnetic actuator and the mechanical stiffness of the spring, the spring isn’t included in
this model. So when creating this model the armature was placed at an arbitrary position or displacement as shown in Figure 1.
When using the model for simulations, the armature must manually be moved to different positions to represent different
displacements; this process is illustrated in section IV.
III. THEORETICAL SOLUTION
To obtain the theoretical values for the system we took the equation from Senturia’s ​Microsystem Design​ 6.68[1]
(1)
This provides the force applied on the armature through the magnetomotive force. For the system to be in equilibrium at a
point, the force from the coil must be equal to the force applied by the spring. If we assume that we are operating at a point
where the system is at equilibrium, then the equation below describes the forces acting on the armature.
(2)
This equation can be rewritten to solve for the current required to produce a displacement (x) for a given spring constant (k).
Our simulation has a single turn coil (n=1). Therefore, . The resulting equation defines a positive current as one whichFMM = I
creates a force that pulls the armature in.

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(3)
The required current is based on the physical characteristics of the simulated system. Figure 2 and Table 1 below describe the
system as implemented.
Figure 2: Actuator Model Physical Dimensions
We defined full actuation as displacement of 8μm for our system based on our analysis of the range where the simulation and
calculated results are similar. Solving the equation yielded a necessary current of 24mA to fully actuate a system with a 100 N/m
spring.
The two values of k produce different responses with respect to current. Solving for displacement produces the following
equation.
(4)
Plotting this equation over a range of current values shows the difference in the response of the systems caused by the spring
constant. The graph below shows displacement vs current for the two values of k.

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Figure 3: Displacement for Two Spring Constants
The lower value of k results in more movement in the armature for a given current. This or an even lower spring constant
would be advantageous in an application where power efficiency is the primary objective.
The higher value of k requires more current to reach full actuation. However, this has the benefit of allowing more precise
control over the movement of the actuator. Changing the current by 1mA in this case results in a much smaller movement in the
actuator than the k=100 N/m case. If the primary objective is to move the system to precise positions, a high spring constant
would be preferred.
This apparent tradeoff between accuracy and efficiency can be approached using the TIPS methods. Abstracting the two
factors when using the lower spring constant, the accuracy of measurement is improved (28) but the energy spent moving the
object gets worse (19). The suggested approaches are 3, 6, or 32. TIPS principle #3 is the principle of local quality. Applying
this principle suggests that perhaps a device could be created that utilizes the best of both spring constants. If the spring with the
higher constant could be detached, it would allow the system to move more easily until it reached the range near the target
position. If then the spring could be reattached, it would give finer control over the actuator without requiring as much energy to
move the system.
IV. MODEL SIMULATION RESULTS
The next step after computing theoretical results was to simulate the model. Simulating the model required a current to be
defined for the coil. To begin simulations, a current of 1A was assigned to the red rectangle of the coil and a current of ­1A was
assigned to the green rectangle of the coil. The goal of this initial simulation was to examine the consistency of the force over
the actuator’s range of motion. To properly simulate the magnetic actuator, the armature must be located at some displacement.
This means the direction of displacement and the location of zero displacement must be defined in the model. For this design,
the displacement is occurring along the x axis with increasing displacement being in the –x direction, see Figure 2. The location
of x = 0 was defined as the left edge of the armature being located 2µm from the opening of the gap in the core. The armature
was initially placed at a displacement of 5µm as shown in Figure 1. When simulated with a current a 1A and a displacement of
5µm, Maxwell computed the net force applied to the armature as 1.37194 N. This computed force was very close to the
theoretical force of 1.372 N calculated using the equations described in section III.
The force applied by the magnetic actuator on the armature is supposed to be independent of displacement and therefore
constant. To see how constant the force was over armature displacement the simulation was performed 11 times at
displacements from x = 0µm to x = 10µm, this means the armature was moved 1µm between each simulation. Again, 1A of
current was used.

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Figure 4: Plot of Force vs. Displacement
From the graph shown in Figure 4, it is clear the force computed with Maxwell isn’t constant across displacement of the
armature. However, once the left edge of the armature hits the right edge of the gap in the core (displacement of 2µm) the force
dramatically increases and remains fairly constant over displacement until 9µm. This result actually makes sense given the
armature will change the magnetic flux density as it enters the gap and continues to proceed through the gap. This is because the
armature itself is a magnetically permeable material. This effect of the armature is what actually causes the force from actuator
to change slightly as the displacement of the armature changes. Figure 5 through Figure 8 illustrate the armature’s effect on the
magnetic flux density.
To show how the magnetic flux density changes as the armature approaches the gap and moves through gap, the simulation
was performed four times, each with a different armature displacement. The results of these four simulations are shown in
Figure 5 through Figure 8. The current applied to the coil in each simulation was the current computed to achieve the desired
displacement. This current value was computed assuming there was a force applied by a spring with a stiffness of k = 100. This
is the force the actuator must overcome to move the armature into the gap.
Figure 5 shows magnetic flux density with the armature displaced to 1µm and an applied current of 20.1mA.
Figure 5: Magnitude of Magnetic Flux (with Vectors), k = 100, Armature Displaced 1µm
Figure 6 shows magnetic flux density with the armature displaced to 4µm and an applied current of 17mA.

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Figure 6: Magnitude of Magnetic Flux (with Vectors), k = 100, Armature Displaced 4µm
Figure 7 shows magnetic flux density with the armature displaced to 8µm and an applied current of 24.7mA.
Figure 7: Magnitude of Magnetic Flux (with Vectors), k = 100, Armature Displaced 8µm
Figure 8 shows magnetic flux density with the armature displaced to 10µm and an applied current of 39.3mA.

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Figure 8: Magnitude of Magnetic Flux (with Vectors), k = 100, Armature Displaced 10µm
V. THEORETICAL VS SIMULATED RESULTS
Our next objective was to analyze the simulated behavior of the system and compare it to the behavior modeled in the formula.
Figure 9 below plots the calculated and simulated results overlaid.
Figure 9: Simulated and Calculated Response
In the middle of its displacement range, the simulation closely follows the predicted behavior. As the system reaches the limits
of its displacement, the simulation diverges from the model. This aligns with our analysis of the displacement range where the
force remaining basically constant for a given current. The equations assume that the force is constant irrespective of position,
so this correlation is expected.
VI. CONCLUSION
The theoretical values calculated did indeed matchup with the experimental results found in Maxwell. An interesting thing to
be noted is the force on the armature as it traverses the gap of the core. In theory the magnetic force on the armature throughout
the gap should be constant, but in our experimental results we saw a slight variation in force as the armature traveled farther into
the gap. We believe this to be because the armature itself is a magnetic material and as it reaches the gap it starts to have some
effect on the shape of the B­field in the system thus changing the magnetic force on the armature. One of our theories is that this
effect depends on the width of the core relative to the size of the gap. If the gap was made smaller relative to the core width, we
expect the force to be more constant over the range of actuation by limiting the fringing effects. We also found that the
analytical model requires that the armature be inside but not completely through the gap in the core. Outside of these conditions,
the mathematical model diverges from simulations.
For future extensions of the project, the magnetic actuator could be modeled in a more complex MEMS software such as
CoventorWare. The spring could be modeled by itself and any nonlinearities in the spring could be incorporated into the
magnetic simulations. Alternatively, we could physically model the spring and actuator and run a magneto­mechanical solver.
We could then set the spring constants to 100 and 1000 and see if the Maxwell results agree.
VII. REFERENCES
[1] Senturia, S.D,2000, ​Microsystem Design​, Springer,New York, NY, 139­142