Thyristor Converter Matlab

Stability Analysis Of Converter-Based Systems

Abstract—Thyristor rectifiers are a well-established and cost- effective solution for controlled high-power rectification, com- monly used for hydrogen electrolysis and HVDC transmission.

However, small-signal modeling and analysis of thyristor rectifiers remain challenging due to their line-commutated operation and nonlinear switching dynamics. This paper first revisits conven- tional RMS-based modeling of thyristor rectifiers and subse- quently proposes a novel nonlinear state-space EMT model in the dq domain that can be linearized for small-signal analysis.

The proposed model accurately captures all the relevant dynamic phenomena, including PLL dynamics, the commutation process, and switching delays. It is derived in polar coordinates, offering novel insights into the impact of the PLL and commutation angle on the thyristor rectifier dynamics. We verify the RMS and EMT models against a detailed switching model and demonstrate their applicability through small-signal stability analysis of a modified interfaced hydrogen electrolyzers, synchronous generators, and grid-forming converters.

Index Terms—thyristor rectifiers, small-signal stability, EMT

I. Introduction

Thyristor rectifiers are the most mature and prominent technology for high-power rectification, offering robustness, cost-effectiveness, and DC voltage controllability. These char- acteristics make thyristor rectifiers the preferred choice in applications such as HVDC transmission, large motor drives (e.g., in mining applications), and controlled rectification for electrochemical processes (e.g., hydrogen electrolysis). Never- theless, analytical modeling and stability analysis of thyristor rectifiers remain challenging due to their line-commutated operation and the nonlinearities of the switching process.

The design and analysis of electrical systems that incor- porate thyristor rectifiers often rely on computer-based time- domain simulation tools capable of accurately reproducing instantaneous voltage and current waveforms, including thyris- tor switching patterns. These simulations are suitable for the analysis of power quality and large-signal events such as commutation failure , short-circuit faults , and operation under current limits . Although these simulations offer valu- able insights, they are computationally intensive and cannot provide deeper system-theoretic insights. Therefore, analytical modeling for small-signal stability analysis is required to uncover the properties of operating points, identify critical operating conditions, and provide tuning guidelines .

Small-signal stability analysis in power systems is com- monly performed separately for Root Mean Square (RMS) and Electromagnetic Transient (EMT) models to address different aspects of system behavior and to meet the different O. Stanojev, M. Schweizer are with ABB Corporate Research, Baden- P. J. Soneira, G. Stomberg are with ABB Corporate Research, Mannheim, demands of grid design and grid operation tasks. RMS models, also known as quasi-static phasor models, simplify the system by representing voltages and currents as quasi-steady-state phasor quantities. This allows for computationally efficient analysis of slower dynamic phenomena such as interactions of outer control loops, frequency stability, etc. EMT models ex- tend the validity of RMS models to also consider fast dynamic phenomena, such as network resonances, PLL transients, and interactions of device-level converter controls .

The RMS modeling of thyristor rectifiers is well estab- lished, with its formulation and applications extensively docu- mented in classical references –. These models assume a constant, balanced three-phase supply and use fundamental frequency relationships derived from the converter switching waveforms, thereby enabling tractable analysis and efficient simulation. However, RMS models often neglect commutation angle variations or rely on their quasi-steady-state approxima- tions , . As a result, the dynamics of the commutation process are not precisely characterized. Several studies have sought to extend the applicability of the RMS framework

–. For Example, Augments The Formulation With

filter dynamics, an AC-system representation, and PLL syn- chronization, while , introduce additional control loops to broaden the modeling envelope. Nevertheless, the key assumptions remain, thus limiting the accuracy of the model.

To improve the model quality while retaining analytical tractability, recent work has pivoted toward impedance-based models that capture multi-harmonic and interaction dynam- ics. Within this line of work, small-signal impedance and admittance models of thyristor rectifiers (and inverters) have been developed primarily in the context of Line Commutated Converter-based HVDC (LCC-HVDC) systems. In , a gen- eral sequence-frame impedance formulation was proposed that incorporates the overlap angle via switching functions. Sim- ilarly, frequency-domain admittance models based on single- side modulated state functions were derived in to enable multi-harmonic linearization and provide a clearer view of har- monic interactions. Further advances include harmonic state- space formulations that employ multiple switching functions to model the variable commutation angle dynamics accurately . For hybrid HVDC systems, impedance-based stability analysis has been extended to multi-terminal cascaded config- urations, where sensitivity analysis and impedance reshaping help mitigate oscillatory interactions . In parallel, the influence of control-link time delays on small-signal dynamics has been quantified in , showing that even modest delays can significantly affect stability margins. Although these con- tributions enhance the understanding of harmonic interactions, they remain predominantly frequency-domain-oriented, which limits readiness for time-domain simulation, analysis of nonlin- ear effects, and generalization beyond HVDC-specific contexts.

The need for high-fidelity EMT modeling of thyristor recti-

2

fiers for eigenvalue analysis and transient studies has driven the development of state-space formulations based on state

Averaging –. In , A Linearized State-Space Repre-

sentation of thyristor rectifiers is derived using an infinite- series converter approach and subsequently transformed into the frequency domain to enable stability assessment through the generalized Nyquist criterion. The concept of motion equations has been employed in to develop an EMT state- space model for multi-infeed LCC-HVDC systems. Building on these foundations, proposes a continuous-time state- space model that incorporates firing-angle control, DC-side dynamics, and commutation overlap for small-signal analysis.

The work in focused on accurately modeling the commu- tation dynamics. More recently, introduced a Linear Time- Periodic (LTP) framework to embed switching dynamics and address the nondifferentiable nature of thyristor commutation.

Complementing these analytical approaches, presents an enhanced RMS model utilizing neural networks to capture EMT transients and mitigate commutation-failure risks.

Despite these advancements, several limitations persist in existing EMT modeling approaches for thyristor rectifiers. First, a systematic treatment of delays introduced by switching and PLL mechanisms is often absent, even though these delays can significantly shape dynamic performance during

Fast Transients , , , . This Gap Also Appears

in earlier dq-frame treatments of thyristor-controlled devices such as thyristor controlled rectifiers and thyristor controlled static compensators, which prioritize fundamental-frequency dynamics over switching-induced timing effects , .

Second, LTP-based and neural-network-enhanced formulations introduce substantial complexity and computational burden, limiting practicality in large-scale system studies , .

Third, most models are tailored to LCC-HVDC and do not generalize readily to other high-power rectifier-based systems (e.g., hydrogen electrolyzers). In addition, many formulations assume balanced network variables and periodic operation, which constrains accuracy under asymmetrical operation or nonstationary grid conditions. Finally, most existing studies on thyristor rectifier dynamics have focused on relatively simple system configurations, such as HVDC systems fed by synchronous generators and voltage sources , elec- tem . While such system configurations provide valuable insights into local control interactions and rotor-angle stability, they do not capture the complexity of large interconnected grids where multiple thyristor-based loads interact with diverse generation sources. The absence of studies addressing large- scale power systems containing thyristor rectifiers is largely due to the reliance on impedance models, which are well-suited for frequency-domain analysis but impractical for large-scale networks with several generation and load types.

In this paper, we revisit the RMS-based modeling of thyristor rectifiers and propose a novel EMT formulation suitable for large-scale stability studies. Both RMS and EMT models are presented to provide a complete perspective on the relevant characteristics of a small-signal model. We begin by examining existing RMS models, tracing their derivation, and highlighting the assumptions that limit their accuracy under fast transients.

Building on this foundation, we extend the state-of-the-art in several directions. Firstly, we derive a continuous-time nonlinear state-space model in the dq domain that incorporates PLL dynamics, accurately represents the commutation process, and accounts for switching-related delays. The model is derived in polar coordinates, which offers novel insights into the impact of the PLL and commutation angle on the thyristor rectifier dynamics. Unlike previous approaches that relied on analytical linearization during model development, which introduced additional assumptions and potential inaccuracies, our method derives a nonlinear model to ensure higher fidelity.

When required, linearization can be performed numerically, while the nonlinear formulation remains directly applicable for EMT simulations in time-domain studies. Secondly, although the proposed model employs a state-averaging technique, we demonstrate that its validity extends well beyond the switching frequency of the rectifier. This is confirmed through time- domain simulations and extensive frequency-domain analysis, which demonstrate close agreement with the behavior of a switching thyristor rectifier model. Finally, we conduct a large- scale system-level analysis using eigenvalue-based stability hydrogen electrolyzer loads, synchronous generators, and grid- forming converters. This configuration has been overlooked in the literature, despite its critical importance for future grids.

The remainder of the paper is structured as follows. Firstly, in Sec. II, we briefly review the basics of small-signal analysis and present the RMS model. Subsequently, in Sec. III, we rigorously derive a novel EMT model of thyristor rectifiers.

The RMS and EMT models are validated both in frequency and time-domain in Sec. IV. Finally, small-signal stability analysis loads interfaced via thyristor rectifiers is performed in Sec. V.

Ii. Modeling And Stability Analysis Preliminaries

This section provides an overview of power system model- ing using differential-algebraic equations for EMT and RMS simulations and outlines the adopted approach to small-signal analysis. We then introduce two conventional RMS models of thyristor rectifiers which conform to this framework.

A. Rms And Emt Power System Modeling

A power system model represents a network of intercon- nected generation and load components in the form of explicit differential-algebraic equations. By grouping the system’s dif- ferential variables x, algebraic variables z, and inputs u, the

(1B)

where xini denotes the initial state of the differential variables. The structure of the above DAE system is influenced by the modeling approach adopted. RMS models focus on system behavior at the fundamental frequency, representing network variables as phasors under the assumption of quasi-steady- state conditions. This simplifies the grid model to algebraic equations and is typically valid for analyzing phenomena up to around 20 Hz, such as electro-mechanical interactions of machines. In contrast, EMT models retain full time-domain resolution (with averaged converter switching to enable lin- earization) and capture fast dynamics, including device-level converter control, machine flux behavior, and line transients.

These models extend the validity in the frequency range up to 1 kHz or higher, resulting in detailed and higher-order DAEs.

B. Small-Signal Stability Analysis

The small-signal model of a power system can be ob- tained by linearizing (1) around a desired equilibrium ξo =

(2C)

where the respective deviations from the desired operating point are represented by ∆x = x−xo, ∆u = u−uo, ∆z = z−zo. The matrices Axx, Bxu, and Axz denote the Jacobians of f with respect to x, u, z, respectively, and the matrices Azx, Bzu, and Azz denote the Jacobians of g with respect to x, u, z, respectively. The Jacobians are evaluated at ξo. We assume that DAE (1) is of index-1. Then, Azz is regular and

Zz Bzu

are the state-space matrices. To determine whether the system is small-signal stable, we

A) Of The Reduced State-Space Ma-

trix. If all eigenvalues have negative real parts, then the power system is said to be small-signal stable at the equilibrium ξo.

C. Thyristor Rectifier Rms Modeling

The RMS modeling of thyristor rectifiers is well established in the power systems literature and is readily available in classical references , . In these works, the thyristor rectifier is modeled as a voltage source on the DC side and a current source on the AC side. The RMS behavior of a six-pulse thyristor rectifier, with input phase-to-ground voltage magnitude Vm, firing angle α, and output DC current Idc is

(4B)

with Rdc = 3ωgLc/π modeling the commutation-induced voltage drop, and Lc denoting the commutation inductance. Thus, Vdc defines the voltage of the DC-side votlage source and Im and φ define the magnitude and phase displacement angle of the AC-side current source, respectively. This model captures the basic power conversion principles of a six-pulse thyristor rectifier. To more accurately capture the commutation- related effects, this model has been extended in , ,

(5D)

where µ is the commutation angle and kic is defined as

. Although (5) provides greater accuracy, it does not introduce additional dynamics compared to (4); both models therefore exhibit identical dynamic behavior. Multipulse thyristor recti- fier models, e.g., 12 pulse or 18 pulse, adopt the same structure, differing only in the voltage and current coefficient values.

D. Rms Model Assumptions And Limitations

The previously presented model captures the dominant rectifier behavior at the fundamental frequency, making it suitable for RMS simulations. Its derivation is based on several simplifying assumptions, which limit its applicability to EMT simulations. Firstly, the input is assumed to be a balanced three- phase voltage of constant magnitude and frequency, whereas more comprehensive models would treat it as time-varying and adopt a dq domain representation. Secondly, the model omits the dynamics of the PLL, which is typically used to syn- chronize thyristor firing; such dynamics can introduce delays that lead to delayed or premature thyristor firing. Furthermore, a constant DC output current is assumed, although practical systems exhibit time-varying currents as a result of finite output inductance. Moreover, commutation inductance is considered only in terms of its low-frequency impact on the average DC voltage, neglecting its dynamic behavior at higher frequencies.

Finally, AC-side currents are derived via Fourier series, based on assumed waveforms, rather than through direct, physics- based dynamic modeling, that would yield increased accuracy.

Iii. Emt Modeling Of Thyristor Rectifiers

In this section, we develop a more detailed nonlinear thyris- tor rectifier model suitable for EMT small-signal analysis by considering the thyristor differential equations during commu- tation and during conduction and averaging them over a switch- ing period (2π/p interval, where p is the number of pulses of the thyristor rectifier). This model aims to overcome the limitations of the previously discussed RMS models. Without loss of generality, we consider the six-pulse rectifier topology depicted in Fig. 1 for the subsequent derivations.

The DC voltage of a six-pulse thyristor rectifier consists of six identical segments within each fundamental cycle. That is, the DC voltage trajectory over each fundamental cycle is given by the concatenation of six equivalent segment trajectories.

Thus, it is sufficient to model the DC voltage behavior over

G2

Fig. 1: Configuration of a six-pulse thyristor rectifier supplied from a three-phase voltage source and feeding a constant current load.

4

a single segment, i.e., by studying a single π/3 interval. Throughout the commutation interval, as the current is redi- rected from T6 and T5 to T6 and T1, the behavior of the system

(6C)

where vdc,p and vdc,n denote the voltages of the positive and negative DC rails. Similarly, during the conduction interval, when T6 and T1 are conducting, the circuit model becomes:

Vc = Vc,

ic = 0.

(7C)

The equations outlined above are the basis for subsequent development of the EMT model.

A. Dc Voltage Dynamics

In order to capture both the commutation and conduction effects in a single continuous time expression, we average the conduction and commutation dynamics over a switching interval. To that end, the DC voltage vdc = vdc,p −vdc,n can firstly be calculated for both the commutation (superscript com) and the conduction (superscript con) intervals, as:

= Va −Vb −2Lc D

dtIdc.

(8B)

Let us now consider a dq frame defined by an angle θr. A three-phase voltage vector v is described in the dq frame by a vector v = vd + jvq or in the polar form v = ∥v∥exp (jθe).

The phase voltages in (6) and (7) supplying the rectifier can be converted to the dq domain by using the inverse Park transform:



.

(9C)

Expressing the phase quantities in their dq forms, the DC voltage equations in the two intervals (8) become:

−2Lc D

dtIdc. Unlike in the RMS model derivation, no assumptions are made here about the form of the three-phase AC source voltages.

These expressions are hence exact, as no simplifications have been adopted thus far. Furthermore, it is interesting to observe that the commutation inductance appears on the DC side, i.e., multiplied by the derivative of the DC current. The DC dynamics can now be averaged over a π/3 segment to obtain

,

which, after evaluating the integrals and using vd + jvq = ∥v∥exp (jθe), yields the following DC voltage model

3/Π And ˜Lc(Μ) = (2 −3Μ/2Π)Lc. Averaging

the system dynamics inevitably introduces modeling errors, as it captures only the mean behavior rather than the full range of dynamic responses. Nevertheless, this approximation is essential for transforming the hybrid switched differential- algebraic equation system into a standard DAE formulation that is amenable to small-signal analysis. The extent of the modeling error introduced by this step is examined in the results section. Figure 2 illustrates the equation (10) in the form of a control block diagram. The variable inductance is included in the circuit representation, derived later in this section.

B. Dynamics Of The Commutation Process

The DC voltage dynamics, previously derived in (10), are a function of the commutation angle, denoted by µ. As in the derivation of the RMS model, the commutation angle can be determined by analyzing the commutation process, which is

≈0

.

(11)

The differential equation is derived from the circuit in Fig. 1 assuming commutation from thyristor T5 to thyristor T1, where phases a and c are shorted during the commutation interval until the current in T1, i.e., ia, reaches Idc. The last term in (11) considers the effect of DC current variation during the commutation period, which is typically insignificant and can thus be neglected . It is important to note that there are no inherent mathematical challenges in solving the differential equation in its original form. The introduced simplification is rather adopted since the resulting expression depends on the difference between the DC current at the beginning and end of the commutation period. These quantities are not captured by the EMT model, as our approach relies on averaging.

The phase voltages va and vc can be expressed in the dq- domain as previously introduced in (9) and their difference can



.

Dtθ = Ω D

dθia.

Vdc

Fig. 2: Control block diagram of the output DC voltage calculation.

3

Fig. 3: Control block diagram of the commutation angle calculation. Hence, the differential equation can be written as



= Idc.

(15)

The differential equation can be solved by integrating the



.

(16)

In order to determine the commutation angle µ, let us impose

2Ωlc

(cos(α + θe) −cos (α + µ + θe)) .



−α −θe.

(18)

This equation is illustrated in Fig. 3 in the form of a control diagram. An absolute-value operator is introduced to enforce strictly unidirectional conduction within the thyristor rectifier.

C. Input Current Dynamics

The average input current dynamics are derived following the same approach that is used to derive the DC voltage dynamics. As established at the beginning of this section, the phase currents during the commutation interval are given by

(19)

which we transform to the dq domain through the Park transform, resulting in the following d- and q-components:



.

(21)

On the other hand, the phase currents during the conduction

(22)

which we also transform to the dq domain, resulting in

3 Idc Cos (Θ −Π

3 ).

(24)

The input current dynamics can now be averaged over a

!

. The final expressions depend on how the phase current ia(θ) during commutation is modeled. The expression in (16) is accurate but complex. For this reason, we first employ a simplified form assuming that the current increases linearly during commutation. In this case, the phase current becomes

Μidc(Θ + Π

3 −α).

(25)

Under this assumption, the average current dynamics take the

3 Idc Sin (Α + Μ

2 ).

(27)

Notice the resemblance between the equations above and the current approximation obtained in the RMS model (5). However, if the exact expression in (16) is adopted, we obtain:

!

. The proposed input current model is depicted in Fig. 4 as a control block diagram for clarity. With the behaviors on both the AC and DC sides now defined, an equivalent circuit can be constructed. The representation shown in Fig. 5 illustrates that the thyristor rectifier EMT model is represented as a current sink on the AC side and as a voltage source on the DC side.

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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