Solenoid
Modeling
02/05/2023
Electromagnetics 314
Jordan Welsh
25462598
Introduction and Purpose
This practical aims to verify theoretical calculations, done using
laws including that of Biot-Savat and Ampere, with rigorous
simulations done using Altair FEKO. Two models are simulated, the
first being a cylindrical solenoid in free space, and the second being
a toroid which is made with a magnetic core. The solenoid clearly
illustrates the relationship with current flowing in the wires
wrapped in a cylindrical shape making up the solenoid, with the
induced magnetic field from the current in the coil. Biot-Savat
integrals are done taking only internal magnetic fields into account,
whereas Altair FEKO simulates the effects of all magnetic fields due
to the current in the coil.
As seen in the figure to the right,
almost all of the magnetic flux lies
within the bounds of the magnetic
material in the case of the toroid. This
is due to its symmetry and is
advantageous to minimizing the
leakage flux of the toroidal inductor.
In electric circuits this is crucial as it
increases the efficiency of the circuit,
as well as decreases the
electromagnetic interference with
other components in the close vicinity
to the circuit which would otherwise be
subjected to unwanted mutual
inductance.
Much of the magnetic flux generated in
the solenoid continues even outside the
bounds of the coil in the solenoid.
Figure 1: Magnetic Field Intensity of Solenoid and Toroid
Theoretical calculations are expected
to differ when compared to
simulations, seeing that Ampere’s law
neglects external magnetic fields.
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MATLAB is
utilized for the
purpose of
juxtaposing
theoretical
computations
with
simulations
that have been
conducted
within the
Altair FEKO
software
platform.
In the analysis of electromagnetic circuits,
impedance plays a critical role as a
measure of the opposition of a circuit to the
flow of current. In the present study,
impedance has been determined by
measuring the voltage across the coil as
applied at the wire port in CADFEKO.
Upon analysis of the cylindrical solenoid in
POSTFEKO, impedance values of 191Ω and
403Ω were obtained, with the increase in
impedance being attributed to a doubling of
the height and an increase in the number of
turns in the coils. In contrast, the toroidal
shaped inductor yielded an impedance
value of 359Ω, which increased to 1680Ω
upon adjustment of the magnetic material
to one with lower magnetic permeability.
It is important to note that the magnetic
permeability of a material affects its ability
to form magnetic fields around the currentcarrying wires. As the permeability
increases, the formation of magnetic fields
becomes easier, and leakage flux can be
minimized doubling of the height as well as
number of turns being increased in the
coils. For the toroidal shaped inductor, the
impedance was calculated as 359Ω and
later 1680Ω when the magnetic material
was adjusted to one of lower magnetic
permeability. The higher a material’s
permeability, the easier it is for it to form
magnetic fields around the currentcarrying wires. With higher permeability
they are also able to minimize leakage flux.
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Ampere’s Law
Ampere’s Law is a fundamental principle in electromagnetism that
establishes a proportional relationship between the magnetic field generated
by an electric current and the magnitude of that current. This relationship is
defined by a constant proportionality factor equal to the permeability of free
space. In its integral form, as expressed by Ampere’s Law, the magnetic field
density and surface current density are utilized, with the assumption that
external magnetic field effects are negligible. However, it is important to
note that these external magnetic fields still contain stored energy, which
contributes to the total inductance of both toroidal and simple solenoidal
components. As a result, the inductance calculated theoretically is always
lower than that derived from FEKO simulations.
In practice, the accuracy of the inductance calculation is greatly improved
when dealing with toroidal-shaped inductors due to their symmetrical
design, which minimizes leakage flux. This enables the FEKO simulation to
yield an inductance value that more closely aligns with theoretical
calculations.
In our current study as electrical engineering students, the theoretical
approaches utilized to calculate inductances have been found to be
acceptable and accurate. It is important to consider component tolerances
when designing electric circuits, and to design in a robust manner to ensure
that the specific inductor values chosen do not adversely affect the
performance of the circuit. Theoretical calculations provide an adequate
basis for selecting appropriate inductor components in such scenarios.
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MATLAB Plots,
POSTFEKO Near Field
Analysis, Magnetic Field
Intensity Plots and
Impedance Simulations
Figure 3: Magnetic Field Intensity as experienced when moving along the z axis
Figure 2: Strong correlation between theoretical calculations
and simulations in FEKO
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Figure 4: Simple Solenoid Magnetic Field Intensity (Theoretical Calculations approximate all of magnetic field intensity is orientated
in the z direction due to "infinite" windings being used when taking the current in the wires to be exclusively in the phi axis
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Figure 5: Magnetic Flux Leakage observed above and below solenoid due to lacking symmetry about either the x or y axis
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Figure 6: Toroidal Shaped Inductor observed with minimal external magnetic field intensity, while a greater internal magnetic field
intensity is observed due to the high permeability of the magnetic material
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Figure 7: Toroidal Shaped Inductor observed with magnetic material of lower permeability. The magnetic flux is now observed more
in external fields and less intense inside the core of the magnetic material
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Figure 8: Toroidal 3D Plot of high permeability material with increased dimensions' magnitude
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Figure 9: Combined Y and Z Magnetic Flux Intensity of Solenoid. The sharp peaks as observed at the edges of the solenoid are due to
the placement of the excitement wire port. This wire passes through the center of the solenoid and thus adds Y component of [H]
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Figure 10: Workspace setup - MATLAB and the FEKO Platform were utilized
Figure 11: Impedance Plot of Simple Solenoid
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Figure 12: Simple Solenoid Impedance value with smaller dimensions
Figure 13: Impedance of Toroidal Shaped Inductor
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Figure 14: Toroidal Impedance with magnetic material of lower permeability
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