Emi Filter Design Matlab

Filter Design

Sofiane Khelladi1, Khadidja Saci1,Abdelchafik Hadjadj1, and Achour Ales2 ABSTRACT In this paper, an optimization method of toroidal core-based Hybrid common mode chokes (HCMCs) for the design of an Electromagnetic interferences (EMI) filter is proposed. A dedicated algorithm is developed using MATLAB to characterize compact HCMCs that exhibit effective Common- Mode (CM) chokes with optimized leakage inductances by systematic variations in the winding patterns and geometric dimensions of their magnetic cores. It takes into consideration the physical limitations of these components as well as the constraints related to the design of EMI filters. The proposed algorithm allows through a small computational task to propose a variety of configurations for optimal HCMCs.

Finite element method (FEM) simulations are conducted on the HCMCs to extract their CM and leakage inductances. The results are compared with those calculated analytically and yield a good match. The performances of optimized HCMCs are evaluated through their implementation in the designed filter. All the cases of HCMCs including the smallest one allow the EMI filter to easily qualify a power converter to an electromagnetic compatibility (EMC) standard.

Index Terms

Electromagnetic interferences filter (EMI), Hybrid common mode choke (HCMC), Optimization algorithm.

He Proliferation Of Switch Mode Power Supplies

(SMPS) and their wide usage in all modern industrial and domestic energy applications are attributed to their high performances, compactness, and competitive prices .

However, Electromagnetic Compatibility (EMC) regulations require on these power devices to meet specific standards due to their high induced levels of electromagnetic interfer-

Ences (Emi) Mainly Conducted Common-Mode (Cm) And

Differential-Mode (DM) noises. This usually necessitates the implementation of passive EMI filters as they present the main countermeasure in the EMC arsenal against EMI due to their effective mitigation performance and their basic and low-cost conception process. On the other hand, EMI filters occupy a substantial portion of power converter volume and weight (up to 30%) . This comes back to the size of their CM and DM chokes in particular CM chokes (CMCs) which are usually the bulkiest filter components. Therefore, to achieve the design of compact and effective EMI fil- ters many optimization methods and integration techniques were vastly explored to reduce the size/number and weight of these components. In , the leakage inductance of a toroidal CMC is improved by inserting a smaller DM toroidal coil within. Although this structure reduces the vol- ume of the filter, it significantly heightens the total parasitic capacitance of the resulting choke. Similar to this approach, in the leakage inductance of a toroidal CMC is increased by inserting the latter into an EQ core made of MnZn ferrite, which enhances the DM filtering but still, the resulting structure will exhibit additional size and weight. Another approach described in suggests coating the outside of the CMC with a magnetic epoxy mixture which enhanced the leakage inductance. Yet, such component requires higher labor cost because of the fabrication and the coating process of the magnetic epoxy mixture. Moreover, in integration of a CM and DM inductor into a single magnetic core unit using EE-type and EIE-type cores was presented. This method helped a 69% reduction of the inductor size for single-stage EMI filter but still, this type of cores has a limited capacity of heat dissipation which affects their effi- ciency compared to discrete inductors. The work described in proposed the computation of the optimal volume of a toroidal CM inductor taking into account the problems related to the saturation of its core and the overlapping of the coil turns. This method achieved an optimal size for the CM inductor by providing a useful computation tool. The most frequently employed CMCs in commercial EMI filters are still constructed on toroidal cores owing to their low cost and wide availability according to . A toroidal CMC is usually realized with coupled inductors

56

whereas the DM inductance is achieved depending on the amount of required DM filtering either by actual discrete inductors or by the CMC parasitic leakage inductance.

This inductance depends on the CMC structure and has no physical size which limits the number of components and mounting area. Furthermore, since this inductance is an air-cored inductor, it never saturates at any level of current keeping sustainable filtering performance which favors the usage of a toroidal CMC for CM and DM noise suppression.

This inherent parasitical parameter has been addressed in several research papers, either as a constraint in the design of CMCs which is the case in – or to act as a formal DM

Inductor In Emi Filters Design As In –. References

, reported the designable parameter for a CMC.

This Parameter Defines The Cm And Dm Impedances Of

the CMC as well as its parasitic capacitances. The CMC

Dm Impedance Which Improves The Dm Noise Filtering

is defined by the CMC dimensions, the number of turns and the angle of its leakage inductances. However, due to their complex mechanical structures, toroidal CMCs provide leakage inductances with limited values ranging from 0.1 % to 2 % of their total values depending on the winding system. The CMCs with sectional winding exhibit a higher leakage inductance compared to CMCs with bifilar winding . Taking into account the features that this stray element offers, the concept of exploiting the leakage inductance of a standard toroidal CMC remains tempting compared to the aforementioned methods.

This paper presents an effective optimization approach intended for the design of EMI filters with compact and effective CMCs with optimized leakage inductances. The paper is structured as follows. Section II describes the design steps of the EMI filter. The HCMC optimization method is detailed in section III. Section IV is devoted to the application of the proposed optimization method in the design of an EMI filter dedicated to a power conversation system. Section V presents discussion. The conclusion of this work is given in section VI.

A. Sizing Of The Emi Filter

The EMI filter is constructed to fulfill the EMC standard requirements imposed on power converters by sizing its elements to provide the required mitigation of EMI. The design of the filter is based mainly on the levels of EMI disturbances and the EMC Standard that they are intended to qualify for. The proper sizing of the filter components is governed by several constraints related to power density, safety regulations, and the stability of the power electronic system. For the CM filter parameters, the value of its capacitors is limited by safety standards which define the maximum leakage current to earth thus, imposing the need to compensate by increasing the CM inductors value and thereby their sizes. The sizing of the DM filter parameters depends on the amount of DM noise to be attenuated taking into account the impact of its components impedance on the stability of the power conversion system. The design workflow of the EMI filter is summarized in the diagram shown in Fig. 1 including the optimization algorithm as an essential step to provide the recommendations for optimal HCMCs. The filter design procedure requires predefined

Parameters According To Emi Requirements I.E.:

• The EMC standards that EMI are intended to meet (e.g.

En 55032 Analog To Cispr 32);

• The conducted CM and DM EMI levels induced by the

• The Cm And Dm Filters Cutoff Frequencies (Fo_Cm,

fo_DM). The conducted CM and DM noises are evaluated separately to establish the key points in their frequencies spectra with respect to EMC limits and define the required CM and DM

(2)

Where (VLimit) is the limit defined by the standard and SM is a safety margin usually set to 6 dB to account for possible inaccuracies in conducted EMI estimation and stray inductive and capacitive components .

The filter topology is considered as a preliminary step in the filter design (e.g., 40 dB/dec for Γ type L-C single- stage, 60 dB/dec for Π or T type L-C single-stage... etc.).

Accordingly, fo_CM and fo_DM are delivered by (3):

(3)

where fo_CM/DM_h are the frequencies of the CM/DM har- monic components to be attenuated. ILfilter is the theoretical insertion loss of the filter depending on its topology .

With fo_CM/DM being defined, the required components forming the CM and DM sections of the filter are determined

Cdm = Cx1=Cx2 (5)

LCM_req and LDM_req are respectively the filter CM and DM required inductances, whereas CCM and CDM are the respective capacitances of its CM and DM sections. The design worflow of the filter is done with the following steps: • The pairs of components (LCM_req , CY) and (LDM_req, CX) that achieve the minimum required insertion losses profiles are determined according to EMI requirements

Where The Filter Cm And Dm Caps (Cy, Cx) Selec-

tion is done beforehand then their respective inductors

(Lcm_Req, Ldm_Req) Are Calculated Accordingly;

S. Khelladi et al.

57

• Based on the calculated pair of inductors (LCM_req, LDM_req) the proposed algorithm will define the best possible configurations of HCMCs with their feasible pairs of effective CM inductances and adequate leakage inductances taking into account their physical and

Geometric Limitations;

• The optimal HCMCs are chosen among the other cases

Used As Recommendations To Develop 3D Prototypes

then FEM simulations are carried out on each of them separately after including the material database of their

Magnetic Cores;

• The performances of optimized HCMCs are evaluated

The Designed Filter

model to qualify a power converter to an EMC stan- dard.

B. 3-D Fem Simulation Of The Hcmc

Three-dimensional (3-D) FEM simulations offer a powerful means to analyze electrostatic and electromagnetic problems linked to different power components and allow to model their behavior at high frequencies which is suitable for the characterization of CMCs. Usually, a proper adaptation of large or complex problems for a 3-D FEM simulation requires a certain degree of simplification to be made on the model’s structure to gain in simulation time and memory at the expense of accuracy. Considering this, the optimized HCMCs which reflect as close as possible the geometry of real choke (Fig. 2d) are modeled and treated separately.

The choice of the magnetic material is done beforehand by defining its main characteristics namely its real and complex magnetic permeability (µ′, µ′′) in the material model relative to the toroidal core. The 3-D FEM simulations performed on the HCMCs models took an average computational time of roughly 45 min for each case using a server comprising a CPU with 6 cores running each at 2.20 GHz and a 16 GB

A. The Hybrid Common Mode Choke

The CMC is the centerpiece in EMI filters and take a crucial part in their design. Its formed of two windings wound around a highly permeable core in the same direction for single-phase applications as shown in Fig. 2a. The magnetic field HCM induced by CM currents is cumulated within the toroidal core creating an impedance that acts against these currents. However, in a CMC, the leakage fluxes caused by the DM currents do not fully cancel out which creates magnetic leakage fields Hleak on each coil of the CMC (bleu curves) and defines its parasitic leakage inductance . Constructively, in the design of filters, this inherent parasitical parameter has been exploited by designers to act as a regular DM inductor which introduced the concept of HCMC. In Fig. 2b, d is the wire diameter estimated with respect to the power system rated current, e is the wire insulation thickness, and θ is the angular coverage of each winding. The CMC geometric parameters are shown in Fig. 2c where ID and OD are the inner and outer diameters of the core respectively whereas H is its height. Fig. 2d shows the 3D model of a CMC.

B. The Hcmc Cm Inductance

The sizing of a single-phase HCMC inductance to handle the CM and DM currents increase the design considerations due to the physical limits related to this device including saturation, high frequency operation, and others related to its geometric aspect. These considerations can be addressed by setting constraints on the design of this component to obtain its best performances. A single-phase CMC inductance is

(6)

where µ is the permeability of the core material. Ae and le are the effective cross-sectional area and the mean path length of the magnetic core respectively.

S. Khelladi et al.

(D)

FIGURE 2: (a) Sectional wound CMC, (b) Cross-sectional view of the CMC showing the toroidal core window area, (c) Dimensions of the CMC, (d) 3D model of the CMC CMCs are built on magnetic cores destined to exhibit a good dissipative behavior against the noise signals energy and the overall size of these magnetic cores is related to their capacity to handle this energy. Moreover, the core’s permeability is strongly dependent on frequency, and knowing that its typical curves decrease with frequency for ferrite and nanocrystalline core materials, the design of a specific inductance that maintains its value at high frequency imposes the increase of the core size and/or the number of turns N according to (6). The nonsaturation limit of the core is set by ensuring that the maximum magnetic field in the inductor Hmax is always beneath the saturation magnetic field Hsat of the used material as illustrated in (7):

(7)

where Imax is the maximum CM current and “r” is the inner radii of the toroidal core. For a single layer CMC, a high number of turns N will cause the saturation of the core according to (7) and at the same time raise the problem of the winding that must fit through the core window. Hence to avoid saturation and guarantee the effective usage of the core window area there are two design constraints to ensure when sizing the CM inductor.

The constraint on the saturation is verified with respect to ID taking into consideration (7) and it’s defined as in (8):

(8)

Secondly, when designing a CM inductor on a toroidal core the effective use of window area implies enough space left between windings of different phases. It also imposes to ensure that the required number of turns will fit on one- half of the core as it has to be enough angular spacing between the turns of the windings to decrease the effect of the parasitic capacitances . Thus, the number of turns included in (8) needs also to be included in the following relationship which defines the minimum inner diameter of the core that fulfills the condition for effective use of window area.

(9)

To acquire the proper CM filtering the theoretical value of LCM for the designed HCMC should attain an effective value always equal or above the required inductance defined according to EMI specifications. For an input line filter with a given CM attenuation, a higher value of LCM in a given (LCM , CY) pair will push the filter cutoff frequency towards low frequencies which improves furthermore the attenuation of the filter and in the same time diminish the CM currents at low frequencies. Hence, the designed HCMC CM inductance must achieve (10).

(10)

with LCM determined by (6) and LCM_req as the minimum required CM inductance defined from the CM filter design specifications.

C. The Hcmc Leakage Inductance

The unwounded section of the core evaluated in degrees is one of the parameters affecting directly the leakage induc- tance of a toroidal core-based CM choke according to .

Based on the methodology described in the calculation of the leakage inductance of a single-phase CM inductor is done by multiplying the effective mean path length of the leakage path of each phase leff by the inductance of an air- gap toroid. Presuming that the leakage magnetic field paths follow the blue lines shown in Fig. 2a, leff is empirically derived for windings which span a section of the core above

(11)

The leakage inductance of each phase can be therefore

(12)

Considering a specific core size and taking into account (12), an increased number of turns is one way to exploit a larger leakage inductance, but this will require a small wire size that limits RMS currents. Such a manner will also cause high winding losses, temperature rise and will expose the magnetic core to saturation as mentioned before . Conversely, a low number of turns N will lead to small leakage inductance. Another approach will be to enlarge the core, but this will result in large and high-cost filters.

Hence, the control of saturation and the proper amount of leakage inductance impose a compromise in the design of S. Khelladi et al.

59

the HCMC. Furthermore, the stability of power electronic systems invokes the Middlebrook criterion for input filters

(13)

This latter impose on the output impedance of the input DM filter Zo to be much lower than the power converter input impedance Zin thus, bounding the allowed values of the required DM inductor LDM_req to a maximum value denoted as LDM_max.

The feasible pairs of the DM filter components (LDM, CX) which verify (13) are defined in the allowed ranges of values determined with the intersection of Zo and fo_DM axes in a reactance graph. Given the substitution of the DM inductor with the CMC leakage inductance which doesn’t exhibit any physical size, the choice of a minimum value of CX is preferred to gain the highest value of leakage inductance.

Accordingly, the HCMC will be designed to harness the highest value of leakage inductance on each winding defined as in (14).

D. The Hcmc Volume

Improving power density imposes a high-frequency opera- tion with compact size in the design of any component and knowing that inductors are typically the largest components in electronic power systems, the key to small size will be the use of smaller inductors. Accordingly, the global volume V of the HCMC mainly contributed by the magnetic core size in addition to wire size is evaluated with (15) as follow:

E. Development Of The Optimization Prosses

The sizing of the HCMC for a given CM and DM filters im- pose on its pair of inductances (LCM,Lleak) the constraints predefined with (10) and (14) respectively. The constraints on the saturation and the proper use of the window area of the core are verified with respect to ID through (8) and (9) respectively. The geometrical aspect of the HCMC which is affected by all these constraints gets reflected on the total volume of the HCMC evaluated with (15). The optimization algorithm of the HCMC taking into account all the constraints mentioned before is described through the flow chart illustrated in Fig. 3. The development of this

Hsat;

• Select wire diameter with respect to Iphase_max; • Perform an iterative variation of ID, OD, H, and θ alongside N which is varied to exploit the increase in

The Number Of Turns For A Smaller Magnetic Core;

• Check every combination of the varied parameters against the constraints defined with (8) then (9) respec-

Tively;

• Evaluate LCM and Lleak respectively for every case approved by the previous step then check against the

Conditions Set With (10) Then (14) Respectively;

• Evaluate with (15) the volume V for every case ap-

Proved By The Previous Step;

• The output parameters ID, OD, H, θ, N, for all ap- proved cases are considered as recommendations for the best HCMCs in terms of their inductances (LCM, Lleak) and their volumes.

These geometric parameters are used to develop 3D pro- totypes of the HCMCs then FEM simulations are carried out on each of them separately. The simulation results of their respective inductances LCM and Lleak are then compared with those computed analytically.

Lcm_Req , Ldm_Req

FIGURE 3: Flow chart of the HCMC optimization algorithm S. Khelladi et al.

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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