Comparison of properties of the new electro-mechanical model and circumferential model of the inductive-dynamic drive

Abstract

The article presents a new electromechanical model of inductive-dynamic drive. The model is implemented in two environments: Maxwell for analyzing electrodynamic phenomena including moving of the disc and Ansys which allows for stress analysis on the basis of the volume forces imported from Maxwell. Additionally, a model validation by comparing with the existing circumferential model was carried out.

Keywords

Actuator electromechanical model stress

1. Introduction

The development of hybrid circuit breakers tends towards arc-less turning off of the ever increasing short-circuit currents [1–3]. Growth in the ability of limitation of short-circuit currents is achieved by using faster and faster electrodynamic actuators (Fig. 1), more accurately called inductive dynamic drives (IDD). These drives must be characterized by very large acceleration (a (t) > 100000 g). Such high dynamics may be achieved by using, on the one hand, a sufficiently large energy (0. 5CU²) of the capacitor bank which powers the actuator coil and, on the other hand, smallest possible mass of the disc [4–9]. However, the drive must maintain absolute repeatability, hence any plastic deformations of the disc are unacceptable.

In Fig. 1a the dynamic inductive drive (IDD) is shown in the measuring system where the primary task of the drive is to separate (by sliding) the contacts of the hybrid circuit breaker (Fig. 1b) at the appropriate speed and at the appropriate moment i.e., when the short circuit current exceeds the detector setting (CDS). At the same time, the power electronics element (EE) of the branch in which the current commutates, is switched on. When the gap between the contacts (CS) reaches the suitable distance value (3 mm) the current will commutate to the branch with PE element limiting the current. Proper determining of the parameters of all the hybrid circuit breaker elements and their synchronous operation allows for arcless switching off of short-circuit currents.

Fig. 1.

(a) Measurement system of the drive. (b) Drive (IDD) in hybrid circuit breaker.

Existing electromagnetic models of IDD can be subdivided into circumferential [4] field-circumferential [5,6] an d field models [7–12]. All these models were characterized by cylindrical symmetry. Among the field and circumferential models of IDD not only electromagnetic analysis was carried out, but also thermal and mechanical with, however, a number of simplifications. The major simplifications were based on the assumption that in both the coil and the disc the currents flow only in the skin layer [12,13]. In turn, in compact multi-physics programs, it was necessary to apply current extortion, i.e., a priori knowledge of the distribution of current density in the disc was assumed [5,7–11]. In fact, the system of the coil supplied through the bank of capacitors is of a voltage-forced nature. Therefore in the studies conducted so far, the authors were using their own circumferential model of actuator constituting a coil circuit and, coupled with it, circuits representing the so-called disc filaments onto which it was partitioned [4]. The basic assumption in the circumferential model of the drive was also the adoption of cylindrical symmetry of the whole system so that known analytical formulas could be used. Such analytical-numerical approach significantly reduced computational time. The fact is that the circumferential model included disc movement, but did not carry out the stress analysis that had to be performed separately by solving the equation of vibration [14].

Therefore, the main goal of the authors was to build and test a new complex 2D&3D electro-mechanical model (hereinafter called EL_M), consisting of an electromagnetic circumferential-field model, (in which the circumferential part performs the voltage extortion from the capacitor banks) and the coupled mechanical solver. The new model will be able to test both linear and nonlinear drives (e.g., with ferromagnetic core).

The above goal in the article was pursued as follows: in chapter 3, the authors presented new 2D and 3D electromagnetic models realized in Maxwell environment. The 2D Maxwell model is characterized by cylindrical symmetry, so the circumferential model [4] (Chapter 2) is used for its validation and comparison of properties. The realized 3D model is also characterized by cylindrical symmetry and has been compared to the 2D model for considerably higher initial capacitor voltage at which significant deformations and stresses in the disc may occur (Section 3.2). The 3D model can of course be easily adapted to any geometry without any symmetry. Then the electromagnetic models were coupled to the Transient Structural solver via the Workbench environment to create a comprehensive Electro-Mechanical model - EL_M (chapter 4). The new EL_M model allows for stress analysis of the drive so that the limit parameters of the IDD can be determined to ensure its repetitive operation in the hybrid circuit breaker.

2. Analytic-numerical model of IDD-short description

If an inductive-dynamic drive is characterized by cylindrical symmetry (Fig. 2a), then analytical formulas can be used to determine the self and mutual inductances between the circular rings, that is, the individual coil turns and disc filaments into which the disc is partitioned (Fig. 2a,b). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle M=\frac{{\Phi}_{12}=\oint Adl}{i}={\mu}_{0}\sqrt{r\cdot R}\left[\left(\frac{2}{k}-k\right)K(k)-\frac{2}{k}E(k)\right]\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle L_{j}={\mu}_{0}R\left[\left(\frac{2}{k}-k\right)K(k)-\frac{2}{k}E(k)\right]\quad L_{coil}=\mathop{\sum }_{j=1}^{N}L_{j}+\mathop{\sum }_{j=1}^{N}\mathop{\sum }_{n=1}^{N}M_{nj}+L_{inner}\end{eqnarray}$ (2) where: K and E are the elliptic integral of the first and second type respectively. $\begin{eqnarray}\displaystyle k=f(r,R,z). & & \displaystyle \nonumber\end{eqnarray}$ The formula (1) defines the mutual inductance between such turns. The first formula of (2) determines the self inductance L _j of such turns, and second formula of (2) determines total inductance L _coil of the coil which is equal to the sum of the self and mutual inductances with taking into account the inner inductance of its turns [15]. Then in the circumferential model, the coil and disc system can be modeled as a multi-loops circuit with variable mutual inductances which depend on the change in distance (z = x) of the disc from the coil (Fig. 3a). The circumferential model can determine the constant-current resistance of the coil R _c, but in reality such resistance is useless due to the strong skin phenomenon and the proximity effect that occurs in the coil-disc system. Hence, equivalent resistance of RLC circuit is most often determined on the base of recording the coil current without the disc. When determining the maxima of the first pulses and the period of the coil current one can precisely calculate the parameters L and R (assuming the knowledge of C) using known formulas. With such a large dynamics of discharge of the capacitor in the real system, the so-defined equivalent resistance is by an order of magnitude greater than the DC current resistance.

Fig. 2.

(a) Coil turns and Disc filaments. (b) Vector potential around ring.

Fig. 3.

(a) Multi-loop IDD model. (b) Comparison of experimental and simulative coil current waveforms (Uc = 800 V C = 105 μF).

In the circumferential model on the basis of the pattern (3) which uses known mutual inductance functions (1), the resulting axial force which displaces the disc is determined. The (4) is a general formula for total electrodynamical force which is applied at field-approach (Maxwell). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F(t)=i_{coil}\cdot \mathop{\sum }_{i=1}^{n}\left(i_{disc\text{\_}fil_{i}}\mathop{\sum }_{j=1}^{k}\frac{dM_{ij}}{dz}\right)\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F(t)=\iiint _{V_{disc}}\mathbf{J}(t)\times \mathbf{B}(t)dV_{disc}.\end{eqnarray}$ (4) Finally, the circumferential model consisted of the system of differential equations describing the circuit from Fig. 3 and the equation of mass motion under the influence of force (3) and braking forces. Validation of the circumferential model was performed on the basis of comparison of simulation current coil with the coil current obtained from the measuring system for data as in Table 1 specified in a line labeled Low energy. As can be seen in Fig. 3b, very good compatibility of the coil currents from the model and the experiment was obtained. Equally good compatibility was obtained for the displacement waveforms, because the elements of drive (disc and coil) only slightly deform for these parameters (Low energy in Table 1). This compatibility deteriorates as the ratio of the initial capacitor energy to the mass of the disc increases, because of the deformation of the disc and the vibration of the entire measurement system. For additional details regarding the circumferential model the readers may refer to [4].

Table 1

IDD parameters

Low energy: Capacitor voltage: Uc(0) = 800 V, Disc dimension: R _{o _disc} = 7 cm, R _{i _disc} = 4.5 cm, h _disc = 1 cm,

Coil dimension: R _{o _coil} = 7 cm, R _{i _coil} = 3 cm, h__coil = 1 cm, Turns of coil N =24

High energy: Capacitor voltage: Uc(0) = 6 kV, Disc dimension: R _{o _disc} = 7 cm, R _{i _disc} = 3 cm, h__disc = 0.5 cm,

Coil dimension: R _{o _coil} = 6 cm, R _{i _coil} = 4 cm, h__coil = 1 cm, Turns of coil N =8

Common parameters: Capacitance C = 105 μF, gap (initial distance coil-disc) = 1 mm, parasitic inductance Lo = 0,

Coil material: copper, disc material: duraluminum, core material: air

3. IDD models in Maxwell environments

3.1. 2D model

In Maxwell, the 2D model (Fig. 4) is also built with the assumption of cylindrical symmetry. In the project, coil of the stranded type was used for which its resistance must be specified (Maxwell does not calculate even the constant-current resistance in the case of a stranded coil). In order to account for disc movement, a band area is defined in the model window in which the disc moves under the effect of the electrodynamic force determined by Maxwell. The program also enables to determine parameters of the forces that damp disc movement (e.g., spring). Around the band an additional area of air was created which improves meshing in the area of movement. An important stage in the design of the model is to select the appropriate size of the Region relative to the coil-disc system as it affects the coil inductance calculated by Maxwell. Conformity of inductance (with the value determined by formula (2)) was obtained only when the region was increased to more than 300% in relation to other objects. As can be seen in Fig. 4, the new electromagnetic model contains a core type object, although the presented research was carried out without a core (core: air, Table 1). However, unlike the existing circumferential and other models, the new electromagnetic model in Maxwell allows to research IDD with the nonlinear core.

Fig. 4.

Screen with elements of 2D model of IDD in Maxwell environment.

Then the coil is coupled with RLC circuit with use of the Maxwell_Circuit_Editor subprogram - therefore the model of IDD in Maxwell could be called a circumferential-field model (Fig. 4). Convergence study was performed by sequential simulations in which the numerical step was reduced by half. It was assumed that the obtained numerical step is satisfactory if the relative error between the last two iterations is lesser than 0.25%. It turned out that the step at which the above convergence was achieved in the Maxwell model must be as many as 64 times smaller than in the circumferential model. For the determined step, Maxwell’s simulations were compared to the circumferential model. Figures 5a and 5b show an excellent compatibility of the coil currents and displacements of the disc from both models. An equally excellent compatibility was obtained for the resulting axial force waveform.

Fig. 5.

(a) Comparison of coil current (b) and disc displacement from circumferential and Maxwell models (Uc(0) = 800 V, C =105 μF).

Of course, despite this sole advantage of the circumferential model resulting from faster convergence, the 2D model in the Maxwell environment is obviously superior to the circumferential, as it can be used in further research with discs which are also characterized by cylindrical symmetry but with any random cross-sectional shape. In addition, the Maxwell model makes it much easier to determine and visualize important quantities such as flux density distribution or forces acting on any volume element of a disc (Fig. 6). The distribution of forces acting on the elements of the disc at the selected time moment confirms strong advantage of axial forces in relation to radial forces (Fig. 6b). This fact is important in analytical stress analysis, in which the effect of disc stretching can be omitted [13,14].

Fig. 6.

(a) Flux density in initial time (contour and vector picture). (b) Forces distribution along radius (Uc(0) = 6 kV, C = 105 μF).

The existence of slight radial components of forces explains the vector distribution of flux density, where in turn radial components prevail (Fig. 6a). It is true that in a further period of motion the field diffuses into the disc, thus increasing the axial components of flux density, but their values are insignificant due to the transient nature of the phenomenon. This may explain the occurrence of opposing (braking) forces that have been observed in animation at later time moments.

3.2. 3D Model

The Maxwell 3D model is easy to build following a 2D model if it is characterized by cylindrical symmetry. However, due to the longer calculation time, it is advisable to optimize the mesh. Hence, taking into account the skin effect of field penetration, apart from extra air areas, additional disc division into layers corresponding to skin thickness (whose value was estimated at approx. 1 mm) was applied (Fig. 7a).

Fig. 7.

(a) Optimizing mesh in 3D model. (b) Contour and vector picture of flux density (Uc = 6 kV, C = 105 μF).

Fig. 8.

Comparison of coil current (a) and disc displacement (b) from 2D and 3D models (Uc = 6 kV, C = 105 μF).

For comparison of 2D and 3D models parameters that significantly increase drive dynamics by increasing the capacitor voltage Uc to 6 kV (High Energy line – Table 1) have been used. When determining the simulation parameters, one must also bear in mind the size of the region relative to the other objects and the selection of the step and be aware that the computation time depends on these parameters. As a result of simulation tests, it was found that a sufficient total number of elements is about 200,000. Hence, with such satisfactory compliance as shown in Fig. 8a,b the correctness of the 3D model was recognized. Similar compliance was obtained for the coil current. In the case in question, that is the presence of cylindrical symmetry, in order to shorten the simulation time we could have obviously divided our 3D model layout into, for example, 0.25 of the whole model. However, the main purpose of creating and validating a 3D model is to use it in further studies with discs of completely arbitrary shapes when the system is no longer characterized by cylindrical symmetry. Figure 7b shows an example of magnetic flux density distribution at a selected moment. While observing (during the animation) disc movement without any deformation it should be reminded that in Maxwell environment all elements are treated as rigid bodies. In fact, the disc may be subject to deformations and hence there exists the need to create a mechanical model.

4. Electro-mechanical (EL_M) model of IDD

In the case of cylindrical symmetry and knowing the distribution of forces, one can calculate the electromagnetic pressure p (r, 𝜑, t) acting on the surface of the disc by treating it as dominant, which is justified by strong attenuation of the magnetic field penetrating into the disc and the insignificant contribution of radial forces. If the radial dimensions of the disc are much larger than its thickness, then the displacement of the central surface of the plate (w (r, 𝜑, t)) can be used for stress analysis by solving the known thin plate vibration equation (5). Such analyzes were carried out (for systems with perfect cylindrical symmetry) and presented in [14,16,17], but the assumption that the plate must be thin considerably limits the scope of the research especially since such a condition exposes the disc to permanent deformation. $\begin{eqnarray}\displaystyle \frac{D}{\overline{{\rho}}}{\nabla}^{4}w+\frac{\partial ^{2}w}{\partial t^{2}}=\frac{p(r,{\varphi},t)}{\overline{{\rho}}} & & \displaystyle\end{eqnarray}$ (5) where: $\bar{{\rho}}=h{\rho}$ plate mass per unit area,

$D=\frac{Eh^{3}}{12(1-{\upsilon}^{2})}$ flexural rigidity, E, h, 𝜐 – respectively: Young modulus, plate thickness, Poisson ratio,

Therefore, a stress analysis study was conducted in the Workbench environment using the Transient Structural Solver [19]. In this environment one can import the results obtained in Maxwell and afterwards send them to a mechanical solver. To transfer data from the Maxwell 2D to a mechanical solver, one needs to change the coordinate axes in which the model is drawn, because in Maxwell the surfaces of the model lay in the XZ plane, and Transient Structural requires that they be in the XY plane. In order to do such a swap of these axes relative to the geometry of model, Space_Claim environment was used (block B, Fig. 9a). Thus changed geometry could be transferred to a mechanical solver (block C Fig. 9a). The important thing that will ensure a more accurate transfer of data from Maxwell to Transient Structural is to set up a mesh generation based on the same elements as in Maxwell, (triangular). In addition, this mesh should most often be denser [18,19]. After importing the data (forces acting on the disc), one can compare the values of the forces components in Maxwell with the corresponding forces components mapped to the Transient Structural solver with the shown coefficient. Figure 9b shows the components of three selected Maxwell’s electrodynamic forces (before transfer) and after transfer to Transient Structural along with the transfer factor. If this factor is not yet satisfactory (distinctly different from one) then we should increase the mesh density (in this case the mesh density was 4 times higher than the Maxwell mesh).

Fig. 9.

(a) EL_M-electromagnetic model (Maxwell) with a mechanical model (Transient Structural) in a Workbench environment, (b) the components of three examples of Maxwell’s electrodynamic forces (before transfer) and after transfer to Transient Structural .

Because we consider the case of force acting on a free (non-supported) disc, we do not have to specify any boundary conditions, but we can take into account the force of gravity (the weight of the disc along with the additional mass) and the braking force. Once the forces have been imported, one will still need to specify the time steps for which the simulation will be performed in a mechanical solver (the same number of steps as in Maxwell was used). After the simulation, we can choose the type of results. Figure 10a shows the Transient Structural screens with equivalent (von-Misses) stress distribution in the disc at the selected time and the course of maximum and minimum values of this stress in a function of time. In the case under consideration, a duraluminum with a permissible value of von-Misses stress 500 MPa was used as the disc material. Therefore, the obtained result (Figs. 10, 12) should serve as a warning for the designer to reduce, for example, the capacitor energy. Figure 10b shows the position and deformation of the disc at the selected moment against the background of the maximum and minimum displacement waveforms. Animation in time allows one to precisely determine where and when stresses exceed the permissible value for the material. Figure 11 shows the selected frames of such animation for the oscillating motion of the disc against the background of forces acting at selected moments. In order to emphasize the disc deformation in the Transient structural environment, one can use the scale factor (3.8) when visualizing. In the Transient Structural analysis, linear material was considered because we assume that the applied parameters are appropriate if the deformation of the disc does not exceed the yield point. Otherwise nonlinear material should be considered.

Fig. 10.

Transient Structural screens - 2D model, (a) Equivalent Stress, (b) total disc deformation (half disc) .

Fig. 11.

Animation frames of vibrating disc movement (half) and forces acting on half the disc at selected moments.

In order to connect Maxwell _3D & Transient_Structural_3D solvers we proceed in the same way as with the 2D model, i.e., the mesh in Transient Structural should have tetrahedral elements and be denser than in Maxwell so that the forces are transferred with a coefficient as close as possible to unity. In contrast, in preparation for the transfer of forces in this case, we do not need to employ the geometry converting block using Space_Claim application. Figure 12 displays the result obtained in the 3D model showing the oscillation of the disc at successive time intervals. In order to emphasize visually the change of the disc shape a scale factor of 1.8 was also used.

Fig. 12.

Stress distribution and change of shapes of disc for selected moments.

5. Conclusion

The realized EL_M model of the drive is devised to quickly design parameters (appropriate for a particular application) ensuring synchronous operation with the remaining elements of the hybrid circuit breaker, while warranting no permanent deformation of the disc. Often, the feature of actuators used in technical systems such as the above switches is their cylindrical symmetry, and then the optimal (due to simulation time) is the 2D approach. The earlier two-dimensional circumferential IDD model, as shown in the study, obtains convergence faster than the field models due to the analytical formulas which have been used in it thanks to the drive’s cylindrical symmetry, but it is the only advantage of this model. However, the EL_M model (in 2D version) compared to the existing circumferential model additionally allows to seek an optimal (with regard to the drive dynamics while maintaining the elasticity of the disc) cross-sectional shape of the disc (not necessary rectangular). The 3D model in the Maxwell environment was presented for a cylindrical symmetry system with a stranded coil to demonstrate its correctness by comparing its results with the 2D model. However, the 3D EL_M model realized by the authors can be applied directly to systems that do not have any symmetry (after changing the shape of a disc or a coil) using at the same time a solid coil. This approach was used to study the thermal phenomena presented in the accompanying article [20]. It should be noted, however, that the simulation results for the coils of solid or stranded types did not differ significantly.

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