FINITE ELEMENT ANALYIS OF AN AC SOLENOID
INCLUDING THE EFFECTS OF HARMONIC LOADING
AND MAGNETIC SATURATION
A. Bax, C. Andersen, DRD Technology Corporation
M. Yaksh, Vectra Technologies, Inc.
D. Ostergaard, Swanson Analysis Systems, Inc.
Finite element analysis can be used as a basis for design of AC solenoids. This paper shows how an analysis of an AC solenoid can be performed before the physical prototype is built to understand the device behavior. Linear AC harmonic, linear transient, and nonlinear transient analyses are performed. The analysis determines magnetic flux density, power loss due to eddy current, instantaneous and time average armature forces, and inductance of the AC solenoid. The effect of eddy currents in the solenoid shading ring is shown to improve the design by preventing the armature force, coil current, and inductance. This paper describes how the understanding of the device behavior gained from finite element analysis before a physical prototype is constructed.
Practical modeling considerations are also addressed in this paper. Assumptions are made to reduce an important 3D component of the magnetic circuit to axisymmetric geometry so that the model size can be signnificantly reduced. The model is constructed from an IGES file and then partially reconstructed using parameters so that the armature position can easily be changed to examine solenoid behavior for different air gap sizes. The permeability of air in the air gap is modified so that very small air gaps can be modeled efficiently without excessively small elements.
INTRODUCTION
The solenoid which was analyzed is used to open and close valves in hydraulic systems by applying an AC voltage. A good solenoid design is one which consumes the minimum amount of power while providing fast response and high armature force. It is desirable to minimize eddy currents and magnetic saturation in the solenoid to minimize power loss. It is also desirable to minimize the inductance of the device so that the mechanical response to turning the voltage on or off is fast.
A 3D representation of the solenoid assembly is given in Figure 1. Key components including the armature, the copper coil, the face, shading ring, AC frame, and end cap are labeled. The magnetic circuit is made up of the AC frame, the armature, and the pole face.
The geometry was simplified to be 2D axisymmetric for computational efficiency. Initially, linear harmonic AC analysis was performed to determine the magnetic flux distribution throughout the device in addition to real and imaginary components or armature force, eddy currents, power loss, and inductance. The linear AC harmonic analysis indicates that the device saturates to a large extent. Since the linear analysis neglects saturation effects, those results are nor expected to accurately match the actual performance of the device. Subsequently, nonlinear transient analysis which includes the effect of magnetic saturation was performed.
Figure 1. 3D Representation of Solenoid
FINITE ELEMENT MODELING
Geometry and Mesh Construction
Figure 2. 3D AC Frame Representations
Figure 3. IGES 3.0 Geometry from CAD Package
A 2D IGES file representing a slice of the solenoid was used as a starting point for constructing the model, and the geometry read into ANSYS from this file is shown in Figure 3. The slice of the AC frame was modified as described in the preceding paragraph. The IGES geometry defining the armature and air around the armature was deleted and then regenerated using parameters so that the armature could be easily moved to any air gap position for the analysis. This approach allowed the model to be run at several different air gap sizes with a small amount of work. The finite element mesh was created for each version of the model using the ANSYS triangular/quadrilateral mixed meshing in combination with automatic and manual mesh control.
A small copper washer called shading ting sits in a circular groove of the pole face. The high conductivity of the shading ring allows large eddy currents to be generated in the ring, and these eddy currents are out of phase with the applied coil current. The purpose of the shading ring is to provide a magnetic flux in the armature and pole face with sufficient strength to keep the armature closed when the flux due directly to the coil current goes to its minimum valve. When the shading ring does not accomplish its purpose, the armature can chatter at a frequency of twice the voltage frequency. The chattering frequency is twice the coil voltage frequency since the sinusoidal voltage has two zero values during each cycle.
The skin depth of the magnetic flux was calculated to use as a guide to determine the mesh density throught the armature and pole face, especially near the shading ring. The skin depth was calculated as follows.
Approximately two elements were used through the skin depth at the pole face and near the shading ring.
Air was modeled to a distance of approximately 2 inches from the solenoid. Figure 4 shows one of the 2D finite element meshes used in the analysis, and the key solenoid components are labeled. Figure 5 shows the refined mesh in the region of the shading ring.
Figure 4. One of the Finite Element Meshes
Figure 5. Refined Mesh in Shading Ring Region
Material Properties
The permeabilities of all other materials such as air, plastic, stainless steel and copper were set to 1. The electrical conductivity of the shading ring and coil were set to be that of copper.
Figure 6. BH Curves Used in Nonlinear Analysis
Loading
Three types of analysis were performed:
Linear AC Harmonic Analysis
Linear Transient Dynamic Analysis
Nonlinear Transient Dynamic Analysis
The actual solenoid coil is subjected to a voltage of 120V RMS, and the corresponding coil current density is not known. ANSYS , on th other hand, requires that current density be specified and does not allow voltage to be specified as a boundary condition in a time varying magnetics analysis. In the linear AC harmonic analysis a guess was made for the value of current density, and the corresponding voltage was calculated using the ANSYS WHB2D.MAC macro and the finite element results to initially calculate stored energy. Voltage was then calculated using the following relationships.
The linear results were then scaled to correspond to an applied coil voltage of 120V RMW. The same current densities were used in the linear and nonlinear transient analyses.
In the linear AC harmonic analysis the coil current density amplitude and frequency were specified t apply the load. In the linear and nonlinear transient dynamic analysis, the current density was specified explicitly at each instant in time, and the analysis was continued for approximately three cycles of the sinusoidal load. This was accomplished easily using the ANSYS parametric trigonometric functions. The current density amplitude determined from the linear AC harmonic analysis and the one step correction procedure described above were used to determine the coil current load in the linear and nonlinear transient analyses.
Table 1 provides a summary of the analysis results of the linear AC harmonic models including time average armature force, instantaneous armature force of the imaginary solution, inductance, and power loss at 0.5, 1.0, and 3.5 mm air gap sizes with and without the shading ring. These results confirm that the shading ring provides substantial additional armature force when the coil current is zero and the armature force is at a minimum. these results also exhibit the correct trends as the air gap size changes and the shading ring is added and removed.
| Gap Size (mm) |
Shading Ring |
Coil Current (AmpRMS) |
Inductance (mh) |
Power Loss (Watts) |
Rep. Air Gap Flux (Tesla) |
Time Avg Force (Lbf) |
| 3.5 | Yes | 1.647 | 194 | 28.9 | 0.71 | 10.34 |
| 3.5 | No | 1.637 | 195 | 27.3 | 0.71 | 10.26 |
| 1.0 | Yes | 0.960 | 332 | 35.42 | 0.85 | 17.50 |
| 1.0 | No | 0.944 | 337 | 32.78 | 0.85 | 17.18 |
| 0.5 | Yes | 0.832 | 383 | 41.44 | 1.0 | 22.24 |
| 0.5 | No | 0.810 | 393 | 37.98 | 1.0 | 21.46 |
Table 1. Summary of Linear AC Harmonic Results
Linear Transient Dynamic Analysis Results
Linear transient dyanmic analyses were performed as a first step toward
the goal of performing nonlinear transient dynamic analyses. As in the linear AC harmonic
analyses, saturation was neflected, and results nearly identical to the linear AC harmonic
results were obtained.
Figure 7. Magnetic Potential Contours
in Solenoid
Figure 8. Real Components of Magnetic Flux
at 3.5 MM Gap in AC HarmonicAnalysis
Figure 9. Real Component of Magnetic Flux at 3.5 MM Gap in AC Harmonic Analysis
Figure 10. Path used to Calculate Armature
Force with MAXF2D.MAC
Nonlinear Transient Dynamic Analysis Results
Only a single nonlinear
transient run was performed, and the air gap chosen was 3.5 mm. The magnetic flux
distribution for the magnetic materials at the instant that the coil current is maximum is
shown in Figure 11.
Figure 11. Maxiumum Magnetic Flux for
Nonlinear Transient Analysis
Figure 12. Source (JS) and Eddy (JE) Current Densities for Nonlinear Transient
Analysis
Figure 12 shows the coil current density and shading ring eddy current
density as a function of time during the transient analysis. It is important the eddy
current density is significantly out of phase with the coil current density. This
phenomenon means that the eddy current density has a high value when the coil current
density is neat zero, and confirms that the shading ring will exert a closing force on the
armature when the armature force due to the coil current density is at its minimum.
Comparison of Linear Transient, Nonlinear Transient and Experimetnal Results
Ideally, the magnetic flux density contours for the nonlinear transient solution in Figure 11 would exactly match the magnetic flux density contours for the linear AC harmonic solution in Figure 8. The maximum magnetic flux density in Figure 8 for the linear AC harmonic solution is however, more than 6 Tesla, while the maximum magnetic flux density for the nonlinear transient solution in Figure 11 is approximately 1.6 Tesla. The large discrepancy in results is due to the fact that magnetic saturation is neglected in the linear analysis . Since the magnetic flux increases as the air gap size decreases, the linear results will become even less accurate as the air gap size is decreased.
Table 2 gives RMS coil current and solenoid inductance for a 3.5 mm air gap size for the linear AC harmonic analysis and compares these with experimentally measured values. The model and experimental data compare favorably.
| Measured Value | ANSYS Results | |
| RMS Coil Current | 1.4 Amps | 1.65 Amps |
| Inductance | 185 mh | 194 mh |
Table 2 Calculated and Measured Coil Current and
Inductance for 3.5 mm Air Gap Size
Table 3 gives time average force for the linear AC harmonic and nonlinear transient dynamic analyses compared to the experimentally measured time average force at various air gaps. The nonlinear transient force compares well with the measured force at 3.5 mm. As expected, the linear AC harmonic force does not compare as well with the measured force at 3.5 mm because of magnetic saturation, and the comparison becomes worse as the air gap size decreases.
| Gap Size (mm) |
Measured Force (Newtons) |
ANSYS Linear AC Harmonic Force (Newtons) |
ANSYS Nonlinear Transient Force (Newtons) |
| 3.5 | 27 | 46 | 29 |
| 1.0 | 45 | 78 | - |
| 0.5 | 65 | 94.5 | - |
Table 3. Calculated and
Measured Time Average Armature Force vs Air Gap
Table 4 gives maximum and minimum instantaneous armature force and power loss for the linear AC harmonic and nonlinear transient dynamic analyses. As expected, the maximum armature force for the linear results is much higher than that for the nonlinear results because the linear model neglects saturation. The nonlinear results are expected to be more accurate that the linear results.
| Analysis Type | Max Force (Newtons) |
Min Force (Newtons) |
Max Power Loss (Watts) |
Min Power Loss (Watts) |
| Linear | 189.4 | 3.15 | 80.0 | 4.1 |
| Nonlinear | 61.4 | 1.63 | 75.0 | 1.6 |
Table 4. Comparison of
Force and Power Loss from Linear and Nonlinear ANSYS Solutions
The analyses were performed at a variety of air gap sizes. For extremely small air gap sizes, the appropriate reluctance of the iar gap was modeled by increasing the permeability of air in the fap instead of actually modeling the tiny air gap size. This technique avoided unreasonably fine finite element meshes.
For all of the analyses performed the magnetic vector potential was set to zero at the boundaries of the model as illustrated by Figure 4.
Key Assumptions
- The solenoid is assumed to be axisymmetric. This assumption is especially significant is regard to the AC frame whose shape is not nearly axisymmetric.
- It was assumed that there is a small amount of leadkage from the circuit , and little effort was devoted to making sure that enough surrounding air was modeled.
- The armature was assumed to not be moving at each air gap position.
- The coil current correspomding the 120 VRMS for the 3.5 mm air gap size was calculated using the single iteration correction procedure described in the Loading section of this paper, and this same coil current was used in the nonlinear transient analysis with 3.5 mm air gap size. The coil current corresponding to 120 VRMS will change when nonlinearities are accounted for, and this change was neglected.
PRESENTATION AND DISCUSSION OF RESULTS
Linear AC Harmonic Analysis Results
Real and imaginary components of the magnetic flux, eddy current density, and vector magnetic potential are a available for the linear AC harmonic solutions. Figure 7 shows magnetic potential contours in the solenoid and illustrates the magnetic flux at air gaps of 3.5 mm and 0.5 mm are given in Figures 8 and 9, respectively. The magnitudes of magnetic flux in Figures 8 and 9 indicate that there is a small amount of saturation in the solenoid at an air gap of 3.5 mm, while there is extensive saturation at the 0.5 mm air gap. Since the linear AC harmonic solution neglects saturation, we cannot expect the solution to be accurate for small air gaps.
ANSYS directly calculates magnetic flux, eddy current density, and magnetic potential at the time the solution is performed, and they are readily available for post processing. Other performance characteristics such as instantaneous armature force, power loss, and inductance are calculated using macros.
The armature force is calculated by defining a path around the armature as shown in Figure 10, and then using the MAXF2D.MAC macro to integrate the Maxwell stress tensor along this path to calculate force.
Inductance is calculated by using the WHB2D.MAC macro to calculate the device stored energy and subsequent hand calculations as described in the previous section of this paper on Loading.
The solenoid power loss based on eddy currents is calculated using the POW2D.MAC macro.
CONCLUSIONS
- The performance of the solenoid can be determined before a physical prototype is made by using the ANSYS program's magnetic analysis capabilities.
- The axisymmetric modeling approach provides reasonable solutions to this problem even though the AC frame component of the solenoid assembly is not nearly axisymmetric. The ability to assume axisymmetry greatly decreases the time and computational resources to perform the analysis.
- The results indicate that the linear AC harmonic analysis approach can be used to calculate reasonable inductance, coil current, and time average armature force for large air gap sizes. As the air gap sizes vecome smaller, however, the linear AC harmonic results besome increasingly inaccurate.
- Nonlinear transient analysis of the solenoid is required to obtain an accurate solution over the entire range of air gap sizes.
- The analysis results confirm that the shading ring functions as intended by increasing the minimum instantaneous armature force.
ADDITIONAL WORK TO BE PERFORMED
A great deal of additional work should be performed to determine the usefulness of linear AC harmonic analysis for this application, and to gain further confidence in the ability of nonlinear transient analysis to predict solenoid behavior.
Since the coil current in the nonlinear transient analysis is based on the linear AC harmonic results, additional work should determine the correct coil current that should be applied in the nonlinear transient analysis. We expect ANSYS to make this additional work unnecessary in the future when voltage is allowed to be applied directly in this type of analysis.
The model should be modified to account for slots cut in the armature parallel to the armature axis that are used to reduce eddy currents and associated power loss.
REFERENCE
1. ANSYS-PC/MAGNETIC Reference Manual Revision 4.4 A, Swanson Analysis Systems, Inc., September1, 1991.
2. Lowther, D.A., Silvester, P.P., Computer-Aided Design in Magentics, Springer-Verlag, New York, 1986.