# Hybrid Multi-Physics Modeling of an Ultra-Fast Electro-Mechanical Actuator

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## Abstract

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## 1. Introduction

**Figure 1.**A sketch of a DC breaker showing the current carrying contacts (2), the pull rod (4), the armature (5), the coils (6) and the bistables (7).

## 2. Mechanical Challenges of an HVDC Breaker

**Figure 3.**Mushroom-shaped conductive armature with threads on the stem’s extremity to screw in a pull rod.

## 3. Electromagnetic Modeling

**Figure 4.**A SPICE circuit coupled to an FEM model for a Thomson coil. The coil and the armature are modeled using FEM, since the resistance and inductance of the coil and armature, ${R}_{\mathrm{TC}}$ and ${L}_{\mathrm{TC}}$, are nonlinear and changing dynamically as the armature moves away. The capacitor, diode, thyristor and cables are modeled by lumped parameters.

**Figure 5.**The current density 100 $\mathsf{\mu}\mathrm{s}$ after the discharge of the capacitor bank is distributed in only small portions of the geometry. Positive current densities appear in the top part of the coil conductors, while negative currents are induced in a piece of the armature situated directly above the coil.

**Figure 6.**The magnetic flux density 100 $\mathsf{\mu}\mathrm{s}$ after the discharge of the capacitor bank reaches 5 $\mathrm{T}$ and is highest in the air gap separating the coil and the armature.

**Figure 7.**Force density 100 $\mathsf{\mu}\mathrm{s}$ after the discharge of the capacitor bank. The coil is subjected to compressive forces, while the armature is subjected to an axially-directed body force that is mostly concentrated in the first few millimeters closest to the coil.

## 4. Mechanical Modeling

#### 4.1. Model 1: Full Multi-Physics Model

#### 4.2. Model 2: Hybrid Multi-Physics First Order Model

#### 4.3. Model 3: Pull Rod Assumed Infinitely Stiff

#### 4.4. Model 4: Armature and Pull Rod Assumed Infinitely Stiff

**Figure 8.**A sketch showing the modeling of the armature (in silver), the pull rod (in green) and the contacts (in dark red): by a full multi-physics model (

**A**); by a first order hybrid model (

**B**); and by a generalized n segment hybrid model (

**C**). The pull rod’s total mass is represented by m and its stiffness along elongation by k. The mass of the copper contact is represented by M.

## 5. Model Validation by Experiments

**Figure 9.**A picture showing the slim and large mushroom-shaped armatures. Although both are designed to withstand the mechanical stresses, one is flexible and prone to bending, while the other is more robust and stiffer.

**Figure 10.**A picture showing the experimental setup. The slim mushroom armature is sitting on a flat spiral coil that is connected with large cables to a capacitor bank. It is mounted with a 3.6 $\mathrm{k}\mathrm{g}$ steel mass.

**Figure 11.**A picture showing the measured currents (solid lines) and the simulated currents (dashed lines) of the slim armature for increasing charging voltages in steps of 100 V.

**Figure 12.**Comparison of the measured and simulated bending of the mushroom armature upon the discharge of a capacitor bank charged with 500 V. The bending cannot be measured for longer time scales, since the picture loses focus with large displacements, rendering the tracking unreliable.

**Figure 13.**A 3D picture showing the velocity profile of the system in (m/s) after 150 $\mathsf{\mu}\mathrm{s}$. (

**B**) Zoom in of the entire actuator shown in (

**A**). The armature that is situated directly on top of the coil is threaded into a pull rod and attached firmly. Point A is located at the outermost extremity of the mushroom armature. Point B is located at the top of the stem of the armature, i.e., just below the rounded corner joining the head of the mushroom to its stem. Point C is located on the bottom of the pull rod, i.e., where the copper contacts are attached.

**Figure 14.**A picture showing the measured velocities (solid lines) and the simulated velocities (dashed lines) of the slim armature for increasing charging voltages in steps of 100 V.

**Figure 15.**A picture showing the measured currents (solid lines) and the simulated currents (dashed lines) of the large armature for increasing charging voltages in steps of 100 V.

**Figure 16.**A picture showing the measured velocities (solid lines) and the simulated velocities (dashed lines) of the large armature for increasing charging voltages in steps of 100 V. The simulation error increases with increasing impulsive forces, since even the large armature is elastic and will therefore bend eventually.

## 6. Results and Discussion

**Figure 17.**The displacements of three different points characterizing the dynamic motion of the breaker. The bending of the head of the mushroom can be computed by subtracting the axial displacement of ptB from that of pt A. Similarly, the elongation of the pull rod and the stem of the mushroom armature can be computed by subtracting the displacement of pt C from that of pt B.

**Figure 18.**The arising current pulse in the four models following the discharge of the capacitor bank in the spirally-shaped coil.

**Figure 19.**The arising force impulse in the four models following the discharge of the capacitor bank in the spirally-shaped coil.

#### Investigation of Using Models with Finer Segmentations

**Figure 21.**The displacement of the load with respect to each model. Model 2: first order hybrid model. Model 5: second order hybrid model. Model 6: third order hybrid model. Model 7: Tenth order hybrid model.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Bissal, A.; Eriksson, A.; Magnusson, J.; Engdahl, G. Hybrid Multi-Physics Modeling of an Ultra-Fast Electro-Mechanical Actuator. *Actuators* **2015**, *4*, 314-335.
https://doi.org/10.3390/act4040314

**AMA Style**

Bissal A, Eriksson A, Magnusson J, Engdahl G. Hybrid Multi-Physics Modeling of an Ultra-Fast Electro-Mechanical Actuator. *Actuators*. 2015; 4(4):314-335.
https://doi.org/10.3390/act4040314

**Chicago/Turabian Style**

Bissal, Ara, Anders Eriksson, Jesper Magnusson, and Göran Engdahl. 2015. "Hybrid Multi-Physics Modeling of an Ultra-Fast Electro-Mechanical Actuator" *Actuators* 4, no. 4: 314-335.
https://doi.org/10.3390/act4040314