Tsc Thyristor Switched Capacitor Matlab

Arxiv:2302.03192V2 [Gr-Qc] 2 Jun 2023

Model for an electrostatic capacitor in Einstein–Maxwell theory Ian Khai-Shuen Ng∗, Wei Zheng Choo†, and Yen-Kheng Lim‡

Abstract

A general relativistic model of a parallel-plate electrostatic capacitor is presented. The spacetime is a solution to the Einstein–Maxwell equations and involves class of solution pre- viously studied by Vesel´y and ˇZofka (VˇZ). In particular, the parts containing curvature sin- gularities are cut out and the remaining regular section is glued to asymptotically-Minkowski spacetimes. In essence, this results in a curved electro-vacuum VˇZ spacetime sandwiched on both sides by exterior spacetimes with vanishing electromagnetic fields. Junction conditions require the presence of charged matter on the boundaries. We interpret this configuration as a parallel-plate capacitor with gravitational effects induced by the strong electric fields.

The spherical capacitor is briefly considered.

Introduction

The capacitor is one of the simplest electrical device that is well described by Gauss’ law in elementary electrostatics. However, a sufficiently strong electric field also works as a source of gravity, curving the spacetime in its presence.

As Such, The Main Subject Of This Paper

is to consider a parallel-plate capacitor which takes the curvature of spacetime into account, using Einstein–Maxwell gravity. Previously, other gravitational counterparts of electromagnetic systems have been explored under different contexts. For instance, the cosmic solenoid was studied by Davidson and Karasik in , and the rechargeable black-hole battery by Mai and Yang in .

1

In this paper, the ‘cosmic capacitor’ is obtained from a slight modification of a particular electro-vacuum solution to Einstein–Maxwell equations. The magnetic version of this solution was one of the many cylindrically-symmetric spacetimes obtained in the review . Subsequently it was further studied and generalised to include a cosmological constant by Vesel´y and ˇZofka (VˇZ) . See also for related developments.

Further properties of this spacetime (such as its causal structure, particle motion, and thin shell sources, among others) was studied in further detail in Sec. 2.4 of . Here, let us only review the features that are relevent for our present purposes. Specifically, we are interested in the electric version of this solution with zero cosmological constant, for which the metric and

√

1 −σ2x2 dt.

(1.1B)

The Faraday tensor is obtained by F = dA.

The Domain For The Coordinates Are T, Z ∈R,

x ∈(−1/σ, 1/σ), and w ∈[0, 2π). There are curvature singularities at x →±1/σ . The flat spacetime limit is σ →0, for which A = 0 and the metric becomes ds2 = −dt2 + dx2 + dz2 + dw2.

Note that since w is a periodic coordinate, this flat limit is a three-dimensional Minkowski spacetime times a circle, R2,1 × S1. If we instead consider the solution locally identical to (1.1), but with w ∈R taken to be a non-periodic coordinate, the global nature of the solution is different. For this case, the flat limit σ →0 is four-dimensional Minkowski, R3,1, with no compact dimensions.

In Ref. , Vesel´y considered a shell source which acts as a boundary that separates the VˇZ spacetime with another instance of itself. In this paper, we will also consider shell sources, but by sandwiching the VˇZ spacetime between two exterior, asymptotically-flat spacetimes. More specifically, we consider (1.1) with w ∈R and remove the curvature singularities by cutting the

Spacetime At X = ±A, Where A < 1

σ. Then each side is glued with asymptotically-Minkowski spacetimes with vanishing electromagnetic fields. The junction conditions then requires the presence of oppositely-charged matter at the surfaces x = ±a. We interpret this configuration 1This solution is found in Eq. (3.16) of Ref. , Eq. (13) of , or Eq. (2.176) of .

2

as a model of an electrostatic parallel-plate capacitor in Einstein–Maxwell theory. Indeed, in the limit of weak gravity, the solution reduces to that of a uniform electric field between two oppositely charged planes. We note in passing that charged shell methods have been applied in other contexts, such as in Refs. .

The rest of this paper is organised as follows. In Sec. 2, we describe the spacetime metrics and gauge potentials. In the same section we also investigate the properties of the surface stress tensor as required by the junction conditions.

Physical Orders Of Magnitudes, Including The

‘capacitance’ of the system, are calculated in Sec. 3. The behaviour of charged test particles are investigated in Sec. 4. In Sec. 5 we briefly consider the spherical capacitor. The paper concludes in Sec. 6. We will mainly work in geometric units where G = c = 1, except in Sec. 3 where we briefly convert some physical quantities to SI units. We follow the conventions of Poisson’s book for curvature and stress tensors.

Bulk Solutions And Junction Conditions

Consider a four-dimensional spacetime M partitioned into three parts, M = M−∪M0 ∪M+. Let Σ−be the common boundary between M−and M0. Similarly let Σ+ be the common boundary between M0 and M+, as shown in Fig. 1. Both surfaces Σ± are taken to be time-like. Our convention for the intrinsic and extrinsic geometry is as follows: We will let xµ denote bulk coordinates on Mi, where i stands for i ∈{−, 0, +}, and ya be coordinates on the surfaces ∂Mi.

∂Ya And Gµν Is The Bulk

metric. If nµ is the outward-pointing unit normal at a boundary ∂Mi, the extrinsic curvature is

Aeν

b∇µnν. The trace of the extrinsic curvature is K = Kabhab.

(2.2)

where Li is the Lagrangian for the corresponding matter source in the bulk Mi. Here R is the Ricci scalar and we have used the notation F 2 = FµνF µν, where Fµν = ∇µAν −∇νAµ are

−

−

−

Figure 1: Sketch of the spacetime depicting a cosmic capacitor. The shaded region represents the domain M0 where an electric field points from Σ+ to Σ−. The domains M± are asymptotically- Minkowski spacetimes. The surfaces Σ± carries positive and negative charges, respectively.

the components of the Faraday tensor F = dA, with A = Aµ dxµ being the electromagnetic

(2.3)

where K is the extrinsic curvature and det h is the determinant of the metric induced on Σ±. The term IS is the action describing the ‘capacitor plates’,

√

−det h LS represents the Lagrangian density for the matter source. The equations of motion in the bulk (that is, away from Σ±) are obtained by varying (2.1) with respect to gµν and Aµ, giving the Einstein–Maxwell equations in the bulk M±, M0.

(2.5A)

∇µF µν = 0.

Where Tµν = −2 Δli

δgµν +Ligµν is the stress-energy tensor on Mi. On the surfaces Σ±, we vary (2.1)

With Respect To Hab And Aa = Eµ

aAµ, the latter of which is the electromagnetic four-potential

4

projected onto the surfaces. The resulting equations of motion are

(2.6B)

where the surface stress tensor and surface current are, respectively,

Δaa

.

(2.7)

Here we have used the notation [X] = X(M±)|Σ± −X(M0)|Σ± for the jump of a tensor quantity X across Σ±. We now specify the explicit solution which describes the capacitor. First, the solution on M0, the region between the ‘plates’ is taken to be the electric version of Vesel´y and ˇZofka’s solution . For convenience, let us write it in the form

,

U = arcsin(σx).

(2.9)

This solves the bulk Einstein–Maxwell equations (2.5) for Tµν = 0. The domains of the coordi- nates are taken to be t, z, w ∈R and x ∈(−a, a) for some positive a. There are no curvature

1

|σ|. The boundaries Σ± are approached as x →±a. For regions ‘outside’ the plates, let us take a metric ansatz of the form

Ds2

± = −e−2V±f±(x)dt2 + e2V±f±(x)−1dx2 + e2V±+2U±h±(x)

5

are constants and f±(x) and h±(x) are functions of x. The domain of x is x < −a for M−and x > a for M+. We also take the electric field to vanish outside the plates so χ± are constants. By the Einstein equations, the stress tensor components on M± are respectively



.

(2.12C)

This metric ansatz was so chosen such that the continuity of the metric across the boundaries

Σ± Is Satisfied By The Condition

f±(±a) = h±(±a) = 1.

(2.13)

Let us further suppose that the exterior spacetimes are asymptotically flat, so that

Lim

x→±∞h±(x) = constant.

(2.14)

The continuity of the electric potential across Σ± then fixes χ± to be

Χ± = ∓Ev ±Σa

1 −σ2a2 . For concreteness, we shall view Σ± from the perspective of M0, being the boundary with disconnected pieces ∂M0 = Σ+ ∪Σ−.

We Take Vector Normal To The Boundary, Nµ, To Be

outward-pointing. Therefore, nµ takes the explicit form

X

at Σ−.

Aeν

b , where hab is the inverse of the induced metric. With these ingredients, we can now calculate the stress tensor on the surfaces Σ±, which is

±A

.

(2.18C)

The notation |±a is to remind ourselves that on the surfaces Σ±, the derivatives f ′

± Are

evaluated on x = ±a, respectively. At this stage, our capacitor configuration has not been fully determined yet as the functions f±(x) and h±(x) are yet to be determined. So far, our requirements are the continuity across the boundaries (Eq. (2.13)) and an asymptotically flat exterior (Eq. (2.14)). These fixes the values of {f±(x), h±(x)} at x = ±a and x →±∞, respectively.

The derivatives of these functions at x = ±a determines the surface stress tensors T ±

Ab , And

the functions at the intermediate values of x determines the bulk stress tensors T ±

Μν. We Shall

demand that the metrics on M± is sourced by reasonable stress tensors both on the surfaces Σ± and on the bulk M±. In particular, let us consider the null energy condition (NEC) as a benchmark. Explicitly, we wish to check whether T ±

Ab ℓaℓb ≥0 For Any Null Vector ℓa On Σ±, And

Tµνkµkν ≥0 for any null vector kµ on M±.

−2H′′

± ≥0.

(2.21)

Thus a reasonable matter distribution can be found if functions f±(x) and h±(x) can be chosen to obey Eqs. (2.19)–(2.21). First, we note that if the exterior M± are pure Minkowski spacetimes, f± and h± are constants and (2.19) cannot be satisfied unless σ = 0, which means the entire configuration M−∪M0 ∪M+ is the trivial vacuum Minkowski spacetime. In other words, a capacitor with pure Minkowski exteriors necessarily requires exotic matter to construct. Thus we seek an ap- propriate non-zero exterior matter distribution which leads to non-constant f± and h±. By trial, an explicit configuration that satisfies all conditions are found as:

H±(X) = 1 + E−Αa2 ∓X

ae−αx2.

(2.22B)

For this choice, the surface energy density ρ± is negative, so the weak energy condition is not satisfied. However, the null energy condition can be satisfied for the following choices of

Βa(1−Tanh2(Βa))

satisfies (2.19) by making it an equality. The remaining condition (2.20) can be satisfied by choosing an appropriately small β. For instance, β = 0.0001/a for σ = 0.8/a. Recall that the domain for x is x > +a for the upper signs f+(x) and h+(x) for which they tend to constants as x →+∞, and x < −a for the lower signs f−(x) and h−(x), where they also tend to constants as x →−∞. Therefore this configuration is asymptotically flat in both directions.

Deployment In Out-Of-Position Situations

D. Bendjaballah1, A. Bouchoucha1, M. L. Sahli1,2* and J-C. Gelin2

Abstract

Side-impact collisions represent the second greatest cause of fatality in motor vehicle accidents. Side-impact airbags have been installed in recent model year vehicle due to its effectiveness in reducing passengers’ injuries and fatality rates. In meeting these requirements, simulations of folding and deploying airbags are very useful and are widely used. The paper presents a simulation method for the deploying airbags using three materials in different working conditions. Finite element analysis is primarily used to evaluate this concept. In these simulations, the gas flow is described by the conservation laws of mass, momentum, and energy. The numerical results indicate that the FE method in this paper is capable of capturing airbag deploying process accurately.

Keywords: Airbag simulations, Out-of-position, Crash, Modeling, Out-of-position

Background

The passive safety of cars has become a very high prior- ity issue for the automotive industry. Today, there are not only one or two airbags in a car; certain models have ten times more than that. With the increasing usage of airbags, the number of accidents where the airbag itself can cause an injury to the occupant also increases

(Augenstein Et Al. 2003; Gabauer And Gabler 2010;

Audrey et al. 2011). As is well known, safety belts are also now devices designed to provide protection to the users of vehicles during crash events, minimizing the loads necessary to adapt their movement to the move- ment of the car (Freesmeier and Butler 1999; Schmitt et al. 1997). In general, the seat belt is designed to restrain the occupant in the vehicle and prevent the

Occupant From Having Harsh Contacts With Interior

surfaces of the vehicles. The airbag acts to cushion any impact with vehicle structure and has positive internal pressure, which can exert distributed restraining forces over the head and face. As a safety component of auto- mobile, an airbag decreases occupants’ injury likelihood effectively in case of an accident (Ruff et al. 2007). These safety elements can reduce the death rates on the roads, and its protection effects have been widely approved (Crandall et al. 2001; Teru and Ishikawa 2003). With computational tools such as finite element methods designed for dynamic contact problems, crashworthiness simulations can now be used with reliable accuracy to evaluate occupant protection in various collision condi- tions with safety metric/parameters such as acceleration, head injury criteria, intrusion distance, intrusion vel- ocity, and neck forces (neck injury risk or whiplash).

Thus, new types of airbag products are being developed to handle different collision scenarios.

Become Standard Equipment On Most New Passenger

vehicles (Braver and Kyrychenko 2004; Teng et al. 2007; Yoganandan et al. 2007). The airbag cushion is com- posed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. There- fore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Generally, air bags are designed to deploy in a crash that is equivalent to a vehicle crashing into a solid wall at 8 to 14 mph.

Air bags most often deploy when a vehicle collides with another vehicle or with a solid object like a tree. There are various types of airbags: frontal, side-impact, and curtain airbags. In general, the passenger side airbags are usually larger than the driver airbags (see Fig. 1).

Besançon, France

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Bendjaballah et al. International Journal of Mechanical

Doi 10.1186/S40712-016-0070-2

Extensive studies have shown that the airbag deploy- ment in load cases consists of two occupant loading phases: a punch-out effect where the airbag bursts out of its container with the airbag and airbag module cover accelerating towards the occupant and a second loading phase during which the airbag is taking on its deployed shape and volume (membrane-loading effect). Bankdak et al. (2002) developed an experimental airbag test system to study airbag-occupant interactions during close proximity deployment. The results provided insight for simulating the effect of inflation energy and mass flow on target response. Bedard et al. (2002) found that while left-side (driver-side) impacts accounted for only 13.5% of all crashes, the fatality rate among these

Crashes Was 68.3% In Comparison To Front Impact

(48.3%), right-side impact (31.3%), and rear impact (38.4%). These studies underscore the importance of oc- cupant safety during side-impact collisions. In the last years, the current market requested to reduce the time and cost airbag development. In order to achieve this result, virtual simulations play an important role since they allow to minimize the number of experimental tests (Pei et al. 2013; Cao et al. 2014). Several simulation models of airbag were established (Wang et al. 2007). It is feasible to optimize the parameters of airbag deploy- ment using simulation technology. Experimental and numerical studies have quantified injury risks to close- proximity occupants from deploying side airbags. These studies have focused on the prevention of the most ad- verse effects of airbag deployment (Duma et al. 2003).

Other studies have proposed airbag characteristics to minimize particular biomechanical responses (Haland and Pipkorn 1996). In a more recent study, Marklund and Nilsson (2003) compared deformation patterns with experimental data as well as the computational costs associated with three different airbag deployment simu- lation methods; they concluded that the SPH method is relatively inexpensive and produces incremental deform- ation patterns that compare most closely to the experi- mental results. The process of inflation of an airbag is one of the determining factors in saving lives. The duration from the initial impact of the crash to the full inflation of an airbag is about 40 ms, and during this time, the airbag goes from being in a folded state to a fully inflated state, with a high internal pressure. After achieving this state, the airbag begins to deflate, thus providing a nice cushion for the body impacting it.

Ideally, the person in the crash should come into contact with the airbag at this time. In the present study, a large volume passenger side airbag model is developed to handle different collision scenarios. The main aim is evaluate the performance of deploying of passenger side airbag using finite element methods (FEM).

Materials

The tensile specimens were made in different airbags (P: Peugeot, R: Renault, and VW: Volkswagen) with a length of 200 mm long and a width of 40 mm. Table 1 shows the mechanical properties of the airbag.

Tensile Tests

To determine the mechanical properties of the material of airbag used in the test pieces, tensile tests were performed on Lloyd EZ20 universal testing machine in Constantine. These tests were conducted using rect- angular samples. The axial force and axial displacement acquired during a test are converted into stress and the strain in order to be used for the fabric material model.

The continuous recording of the stress-strain data was performed during both the load and unload phases. A minimum of five samples were made in order to check the repeatability of the measurements. All the data was collected by using a PC-based data acquisition system and analyzed by commercial software. The picture frame test device that is made for this study is shown in Fig. 2.

Fig. 1 a Frontal and side airbags. b Oblique view of facet occupant model in sitting posture following airbag deployment (Lim et al. 2014)

0.150

Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

Page 2 Of 9

Figure 3 shows the stress-strain relationship of the airbag sample under axial tensile loads. The results are showing a linear increase in extension with the increas- ing stresses. This is an expected output and it confirms with the theoretical behavior of a sample subjected to tensile stress. The rupture strain values for different airbags (R/P/VW) were 0.322, 0.441, and 0.472, respect- ively. The measured elastic parameters (i.e., Young’s modulus E and initial yield strength) and Poisson’s ratio are summarized in Table 2. The tensile tests of the woven fabrics can show differences on mechanical prop- erties because woven fabrics can resist in-plane shear loads once the yarn lock-up angle has been reached. The differences of material property on material direction can affect the shape of fully deployed bag (see Fig. 3b).

Theoretical Background

Numerical simulations of airbags use very complex and techniques such as an orthotropic model to identify the mechanical behaviors during the airbag inflation and the fluid mechanics (gas flow) to describe the inflator gas flow (pressure gradient) and improve the representation of the pressures within the airbag. To model the airbag as an orthotropic model, three material constants have to be provided. Assuming a plane stress condition, the

Ð1Þ

where σ is the normal stress and τ is the shear stress, the subscript refers to the principal material directions, i.e., the fill and warp directions. Also, ε and γ are the strain components. The material elastic constants Qij are

Ð2Þ

where E1 and E2 are the Young’s modulus in the fill and wrap directions and G12 is the shear modulus of the fabric material. νij is the Poisson ratio of the material.

The gas exerts a pressure load on the airbag causing it to expand. This expansion puts the airbag under tensile stress lowering the expansion rate. In this study, heat conduction and heat transfer is not taken into account.

Fig. 2 A photograph of Lloyd EZ20 universal testing Fig. 3 Stress versus strain using Lloyd EZ20 machine for a three different airbags at 0° and 90° and b VW airbag test specimens at

Different Angles

Table 2 Physical and mechanical properties of the airbag

Page 3 Of 9

In the deployment of an airbag, an inflator supplies high velocity gas into an airbag causing it to expand rapidly. The gas inside the airbag is assumed to be ideal, to be of constant entropy, and to satisfy the equation of state:

Ð3Þ

Here p, ρ, and e are respectively the pressure, density, and specific internal energy, and γ is the ratio of the heat capacities of the gas. The gas flow is described by the conservation laws for mass, momentum, and energy that

Ð4Þ

here, V is a volume, A is the boundary of this volume,

N Is The Normal Vector Along The Surface A, And U

denotes the velocity vector in the volume. Applying Bernoulli’s equation in the case of an ideal gas with

Ð5Þ

Here, the subscript ex denotes quantities at the throat of the tube. Furthermore u, p, and ρ denote the quan- tities inside that part of the tube that is supplying mass.

Materials And Boundary Conditions

The airbag system mainly consists of three parts: the airbag itself, the inflator unit, and the crash sensor or diagnostic unit. Thus, to study the behavior of the airbag using FE simulations, we need to have an FE model of the airbag in the folded position. A FE model of the airbag was used to simulate the test condition as shown in Fig. 5. LS-DYNA® material model FABRIC (MAT_34) is used to simulate the airbag material. It is a variation of the layered orthotropic material model. Additionally, in the LS-DYNA® material model, fabric leakage can be accounted for. However, for this CAB material, the leak- age is almost negligible and therefore no leakage is specified. The mechanical properties can be determined from the physical test. Typical material properties for airbag fabrics are taken as given in Chawla et al. (2004a) (Table 3). These properties are used to simulate inflation process of airbag (see Table 1). The car dashboard is modeled as the rectangular thin plate using a MAT_RI-

Gid Material, And The Degrees Of Freedom Are Con-

strained in all the directions. The similar properties of thermoplastic polymer are assigned for contact purposes. The porosity of the fabric is assumed zero. The nitro- gen gas is taken for inflating the airbag. Properties of nitrogen gas and initial bag conditions are shown in Table 4. The example on which we perform the study is a typical passenger side airbag. The geometric de- tails have been measured from a commercially avail- able airbag. The initial state of the airbag is a closed rectangular whose sides are to be finished to 482 × 635 mm2 and is shown in Fig. 4.

Table 3 Material properties of airbag and rigid plate used in FE

–

Table 4 Initial values used for FE simulation of the swelling of

3.33 × 10−4

Fig. 4 The initial airbag geometry in the form of a rectangular Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

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