2.1. Electrical and Magnetic Analysis
During the electrical and magnetic analysis, the vibrating motor was modeled with several different positions of the vibrating part, and current was applied through the coil. The governing equations are Maxwell’s equations.
where
D is electric flux density,
B is magnetic flux density,
J is current density,
H is permanent magnetic intensity, and
ρ is the charge density.
For each position and current, the flux density was obtained by FEM. After solving Maxwell’s equations, the flux density located at every node point in the model was acquired. The model used for electrical and magnetic analysis is shown in
Figure 2.
After obtaining the results, the flux density was converted into flux linkage, from which it is easy to calculate the inductance and speedance.
The definition of flux linkage is the product of the flux density, coil turns, and the coil’s sectional area. The flux density and flux linkage are shown in
Figure 3.
After calculating the flux linkage, inductance and speedance can be calculated using the following method: Inductance is defined as the derivative of flux linkage with respect to the current, and speedance is defined as the derivative of flux linkage with respect to displacement.
Additionally, flux linkage, inductance, and speedance are all functions of displacement and current and these parameters can be used to calculate the back EMF.
where
i,
y,
N,
S, and
Bx are current, displacement, coil turns, coil sectional areas, and average flux density, respectively.
λ(
y,
i),
L(
y,
i), and
Kemf(
y,
i) are flux linkage, inductance, and speedance, which are described as functions of displacement and current. By using flux density, the force that acts on the vibrating part was calculated using the Maxwell stress tensor method. The Maxwell stress tensor is a second-order tensor used in electromagnetic field analysis to represent the interaction between electromagnetic forces and mechanical momentum,
where
Fm is the magnetic force,
B is the flux density,
ne is the unit normal vector, and
S is the closed surface of the selected part. The electromagnetic force (
Fm) is the force acting on the magnet of the motor and it is expressed as the total force (
Ftotal). Because of the permanent magnet, magnetic material, and current, the total force (
Ftotal) is composed of the cogging force (
Fcogging) and the current force (
Fcurrent).
The cogging force is a purely attractive force between the magnetic material and the permanent magnet depending on the position of the vibrating part when there is no current in the coil.
The current force is calculated by subtracting the cogging force from the total force on the vibrating part. Additionally, the force factor is calculated as the force divided by the current,
where
Fcogging(
y),
Ftotal(
y,
i),
Fcurrent(
y,
i), and
Kf(
y,
i) are the cogging force, current force, total force, and force factor, respectively. These parameters are also functions of current and displacement. The nonlinear parameters, such as inductance, speedance, and force factor, are expressed as functions of displacement and current, as shown in
Figure 4.
The nonlinear parameters are constantly changing with position and current. These parameters are involved in the following voltage equation,
where
V,
R, and
t are voltage, resistance, and time, respectively.
2.2. Mechanical Analysis
In the mechanical analysis, the displacement of the vibrating part was calculated using the FEM with forced vibration. The governing equation can be written as follows:
where [
M], [
C], [
K],
y, and {
Fcurrent} indicate the matrix representations of mass, damping, stiffness, displacement, and the current force vector. The displacement of the vibrating part is obtained using a given input force and boundary condition for the vibrating motor. The model used for the mechanical simulation is shown in
Figure 5.
However, it is important to remember that the force applied in this step is a constant value. In reality, the force of the motor is constantly changing with current and position. Therefore, a transfer function is needed.
Because the mechanical system is linear, a transfer function can be applied between the input force and output displacement. The transfer function is defined as the output divided by the input of the system. The relationship can be described as follows:
where
h(
f) is the transfer function that is shown in
Figure 6,
F0 is the assumed constant force amplitude,
Y0(
f) is the response displacement amplitude, and α(
f) is the response displacement phase.
2.3. Electrical, Magnetic, and Mechanical Coupling Analysis
To perform the coupling analysis, two differential equations, which are the forced vibration equation and the voltage equation, must be solved simultaneously. However, it is difficult to solve these correlated equations by hand. Therefore, numerical analysis is needed to solve these two equations.
The numerical fixed-point iteration method was chosen to solve the electrical, magnetic, and mechanical coupling analysis.
Figure 7 shows a flow chart that describes this method. The fixed-point method was chosen because it is easy to program in an application and the flow of the method can be seen and understood intuitively. Through this procedure, the forced vibration equation and the voltage equation can be solved simultaneously and the simulation displacement and current can be obtained. There are many other numerical methods that can be used to find the solution, such as the Newton–Raphson method or the modified secant method. These may provide the solution, but these methods require a more advanced level of programing skill.
In the electrical and magnetic analysis, the voltage equation is expressed with nonlinear parameters such as inductance, speedance, and force factor as Equation (5). Input voltage and current are waveforms and can be written using Euler’s formula. By dividing the input voltage into the input current, the impedance can be obtained as:
where
y and
i are the initial displacement and current and
Ze is the impedance. Because the transfer function from Equation (7) is maintained, Equation (9) can be obtained:
The actual force depending on current and displacement can be expressed as follows:
By substituting Equations (9) and (10), Equation (11) can be obtained:
Here, the electrical, magnetic, and mechanical coupling effects are derived and, finally, the calculated impedance can be obtained using nonlinear parameters from the electrical and magnetic analysis and the transfer function from the mechanical analysis. By using this calculated impedance, the simulated current is obtained as follows:
The simulated displacement and impedance can also be obtained:
When using the numerical iteration method, the simulation was run until the initial values and simulated values converge. Before solving the voltage equation, the initial values of current and displacement, which are determined from the mechanical analysis, were set. For solving the voltage equation, the simulation procedure was as follows. In the beginning, frequency was given as an initial value, and the corresponding voltage and mechanical force were given. The initial values of current and displacement were fed into the simulation. By using the initial values, the initial impedance, which includes nonlinear parameters, was obtained. The value of the calculated current was acquired by dividing the voltage by the initial impedance. The calculated and initial current values were then checked to see whether they fall within the margin of error.
The simulation was run until the current converged. After the value of the calculated current was obtained, the displacement iteration, which includes current information, was started using the calculated current. Because of the electrical, magnetic, and mechanical coupling effects, both the current and displacement iterations should be considered at the same time. When the iteration was complete, the simulated impedance, current, and displacement were attained. Through the iterations, the current in the electrical and magnetic domains and the displacement in the mechanical domain were considered at the same time. At the beginning of the simulation, the initial values of current and displacement were assumed values. At the end of the simulation, the obtained values of current and displacement were very close to the experimental values.