Hysteresis Current Control Matlab

1. Introduction

Hysteresis inverters are used in many low and medium voltage utility applications when the inverter line current is required to track a sinusoidal reference within a specified error margin. Line harmonic generation from those inverters depends principally on the particular switching pattern applied to the valves. The switching pattern of hysteresis inverters is produced through line current feedback and it is not pre-determined unlike the case, for instance, of Sinusoidal Pulse-Width Modulation (SPWM) where the inverter switching function is independent of the instantaneous line current and the inverter harmonics can be obtained from the switching function harmonics.

This chapter derives closed-form analytical approximations of the harmonic output of single-phase half-bridge inverter employing fixed or variable band hysteresis current control. The chapter is organized as follows: the harmonic output of the fixed-band hysteresis current control is derived in Section 2, followed by similar derivations of the harmonic output of the variable-band hysteresis controller in Section 3. The developed models are validated in Section 4 through performing different simulations studies and comparing results obtained from the models to those computed from MATLAB/Simulink.

The chapter is summarized and concluded in section 5.

2.1 System Description

Fig.1 shows a single-phase neutral-point inverter. For simplicity, we assume that the dc voltage supplied by the DG source is divided into two constant and balanced dc sources, as in

C

V . The RL element on the ac side represents the combined line and



through the action of the relay band and the error current



. In Fig.2, the fundamental frequency voltage at the inverter ac terminals when the line current equals the reference current is the reference voltage,

. Fig.2

compares the reference voltage to the instantaneous inverter voltage resulting from the action of the hysteresis loop.

Di

Fig. 1. Single-phase half-bridge inverter with fixed-band hysteresis control. Referring to Fig.2, when valve Q is turned on, the inverter voltage is

; This

forces the line current ai to slope upward until the lower limit of the relay band is reached



. At that moment, the relay switches on Q and the inverter voltage becomes



, forcing the line current to reverse downward until the upper limit of the relay



. Fig. 2. Reference voltage calculation and the instantaneous outputs. The bang-bang action delivered by the hysteresis-controlled inverter, therefore, drives the instantaneous line current to track the reference within the relay band 

. With Reference To

Fig.3 and Fig.4, the action of the hysteresis inverter described above produces an error current

Ae

t close to a triangular pulse-train with modulating duty cycle and frequency.

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Modeling & Simulation of Hysteresis Current Controlled Inverters Using MATLAB

2.2 Error Current Mathematical Description

The approach described in this section closely approximates the error current produced by the fixed-band hysteresis action, by a frequency-modulated triangular signal whose time- varying characteristics are computed from the system and controller parameters.

Subsequently, the harmonic spectrum of the error current is derived by calculating the Fourier transform of the complex envelope of frequency modulated signal. Results in the literature derived the instantaneous frequency of the triangular error current

(3)

and M is the amplitude modulation index of the inverter expressed in terms of the peak



.

Ae

t .

A

v on the error current duty cycle.

Carrier Frequency

cf and a modulating part that explicitly determines the bandwidth of the error current spectrum, as it will be shown later in this chapter. Notice that the modulating frequency is twice the fundamental frequency, that is,

2 F

. Now, with the help of Fig.3, we define the instantaneous duty cycle of the error current



, we obtain after using (1), (2) and manipulating,



.

V . The Relation Between The Instantaneous Duty

cycle and the reference voltage can be demonstrated in Fig.4: the duty cycle reaches its

A

v ; it becomes 0.5 (symmetric form) at the zero of

V ;

and it reaches its minimum value (tilt in the opposite direction) at the crest of

Ae

t by the Fourier series of a triangular pulse-train having an instantaneous



.

(6)

As the Fourier series of the triangular signal converges rapidly, the error current spectrum is approximated using the first term of the series in (6). Therefore truncating (6) to

(9)

that contains 98% of the spectral energy of the modulated sinusoid in (7). To simplify (7) further, we use the following convenient approximation (see Appendix-A for the



.



.



D t from (5) into (11) and manipulating, we obtain

(12)

Next, the cosine term in (12) is simplified by using the infinite product identity and

(13)

Substituting (13) into (12) and manipulating, the error current approximation becomes:

(15)

where  denotes convolution. In order to calculate





.

Nj Is The Bessel Function Of The First Kind And

order n . Substituting (17) into (15), and convoluting, we obtain:



.

(18)

Using the recurrence relation of the Bessel functions,

(19)

the positive half of the error current spectrum takes the final form:



.



Fig. 5. Effect of changing  on the harmonic spectrum. The calculation of the non-characteristic harmonic currents using (20) is easily executed numerically as it only manipulates a single array of Bessel functions. The spectral energy is distributed symmetrically around the carrier frequency

1

2 f . Fig.5 shows the harmonic spectrum of the error current as a function of the frequency modulation index . If the operating conditions of the inverter forces  to increase to , then the spectral energy shifts to higher carrier frequency

Cf . Additionally,

as the average spectral energy is independent of  and depends on the error bandwidth ,

103

the spectral energy spreads over wider range of frequencies,

, With An Overall

decrease in the band magnitudes to attain the average spectral energy at a constant level as shown in Fig.5. The Total Harmonic Distortion (THD) of the line current is independent of  and is directly proportional to the relay bandwidth .

2.3 Model Approximation

The harmonic model derived in the previous section describes the exact spectral characteristics of the error current by including the duty cycle

A

v on the error current amplitude and tilting. Moreover, the



D t in (6) predicts the amplitude of the error current precisely, which in turn, would result in accurate computation of the spectrum bands magnitudes according to (20). The model can be further simplified to serve the same functionality in without significant loss of numerical accuracy. As the instantaneous frequency of the error current,

D T To Its Average Value 0.5 Will Slightly

affect the magnitude of the spectrum bands according to (7). Subsequently, the error current

Where The Carrier (Average) Frequency

cf is given by (3), the frequency modulation index  is given by (8). The 3 dB frequency bandwidth BW that contains 98% of the spectral energy is given by (9).









Fig. 6. AC harmonics transfer to the inverter dc side.

2.4 Dc Current Harmonics

The hysteresis switching action transfers the ac harmonic currents into the inverter dc side through the demodulation process of the inverter. As the switching function is not defined

104

for hysteresis inverters, the harmonic currents transfer can be modeled through balancing the instantaneous input dc and output ac power equations. With reference to Fig.1, and assuming a small relay bandwidth (i.e.



.

(23)

The power balance equation over the switching period when Q is on is given by:





.

(26)

where x is the derivative of x with respect to time. Using the product-to-sum



.

(27)

The positive half of the dc current spectrum is thus computed from the application of the Fourier transform and convolution properties on (27), resulting in

E

f is the error current spectrum given by (22). The average, fundamental, and harmonic components of the dc current spectrum are respectively given by

2

.

(29)

Each spectrum band of the ac harmonic current creates two spectrum bands in the dc side due to the convolution process implicitly applied in (28). For instance, the magnitude of the

As Shown In Fig.6. Consequently, Every Two

successive bands in the ac spectrum create one corresponding dc spectrum band that is located half the frequency distance between the two ac bands. 2.5 Harmonic generation under distorted system voltages The harmonic performance of the hysteresis inverter in Fig.7 under distorted dc and ac system voltages is analyzed. The presence of background harmonics in the ac and dc voltages will affect the instantaneous frequency of the inverter according to (30) as



.



.

A

Fig. 7. Hysteresis inverter operating with distorted system voltages. Notice that in (31), k and h need not be integers. Substituting (31) in (30) and assuming small distortion magnitudes, the instantaneous frequency of the error current

(33)

are the frequency noise terms due to the system background distortions. The amplitude modulation indices of the ac and dc harmonic distortions are given by :





.



.

In (35): The Carrier Frequency

cf is given by (3); the frequency modulation index  is given

1

.





.

The

corresponding ac and dc frequency modulation indices are given by

;

.

(37)

Applying the Fourier transform and convolution properties on (35), the positive half of the

(39)

are the ac and dc modulating spectra. Generally, for any H number of ac voltage distortions and K number of dc distortions, (40) is applied first to calculate the total ac and dc modulating spectra, then (38) is used to compute the error current harmonic spectrum.

,

.

3.1 Error Current Mathematical Description

The harmonic line generation of the half-bridge inverter of Fig.1 under the variable-band hysteresis current control is derived. The constant switching frequency of the error current in (2), i.e.



, is achieved by limiting the amplitude of the error current to stay

(41)

where the maximum value of the modulating relay bandwidth is

And

of is the target switching frequency. Subsequently, the error current is approximated

(43)

Substituting (41) in (43) and then applying the Fourier transform, the positive half of the



.

(44)

The error current spectrum in (44) consists of a center band at the switching frequency



. The frequency bandwidth that contains the spectral

1

4 f .

3.2 Dc Current Harmonics

The approach developed in 2.2.4 also applies to compute the dc current harmonic spectrum when the variable-band hysteresis control. The positive half of the dc current harmonic spectrum is computed by substituting (44) in (28).

3.3 Harmonic generation under distorted system voltages The presence of background harmonics in the ac and dc voltages, given in (31) will affect the instantaneous frequency of the inverter according to (30). Subsequently, to achieve the

Constant Switching Frequency

of , the modulating error band in (41) will also contain the



is the error under zero background distortion given by (41), and

M

M define the modulation index of the ac and dc background distortion terms respectively as (34). The new terms introduced by the background distortion appear as amplitude modulations



.

E

f is the zero-background-distortion error as in (44), and the new terms due to

Δ

Δ

Δ

Δ

.

(49)

Examining (49), the presence of the harmonic distortions in the system tends to scatter the spectrum over lower frequencies, more specifically, to



.

4. Simulation

The harmonic performance of the half-bridge inverter under the fixed- and variable-band hysteresis control is analyzed. Results computed from the developed models are compared to those obtained from time-domain simulations using MATLAB/Simulink. Multiple simulation studies are conducted to study the harmonic response of the inverter under line and control parameter variations. The grid-connected inverter of Fig.1 is simulated in

. In Order

to limit the THD of the line current to 10%, the line current tracks the sinusoidal reference





.

4.1 Fixed-Band Hysteresis Current Control

The ac outputs of the half-bridge inverter under the fixed-band hysteresis current control

109

peak value of 263.7 V. the inverter line current ai tracks the sinusoidal reference within an absolute error margin . The error current resulting from the fixed-band hysteresis action resembles a frequency-modulate triangular signal of constant amplitude. The implicit relation between the error current duty cycle and the reference voltage

V Is Clearly Seen In

Fig.8. The symmetric duty cycle, i.e.

, Happens Whenever The Reference Voltage

approaches a zero crossing.

Time(Sec)

Fig. 8. Inverter ac outputs under fixed-band hysteresis control. 9. Simulation results obtained from the developed model and Simulink.

110

The harmonic parameters of the model are computed the system and controller parameters as follows: substituting the reference voltage in (4) results in an amplitude modulation index

E

f computed from (20) to that obtained from the Fourier analysis of the time-domain simulation results using Simulink. The figure shows a good agreement between the two spectra in terms of frequency order, magnitude and angle.

The spectrum bands are concentrated around the order of the carrier frequency and are

2 F As Shown In Fig.9. With Reference

to (9) and Fig.9, it is shown that 98% of the spectrum power is laying in the bandwidth



. Therefore, the spectrum bands outside this range contribute insignificantly to the total spectrum power and thus can be truncated from the spectrum for easier numerical applications.

To study the effect of line parameter variations on the harmonic performance of the inverter, the DG source voltage is decreased to have the dc voltage

, Then The Harmonic

spectrum is recomputed using the model and compared to the results obtained from

C

V will increase M and  according to (4) and (8) respectively, but

Will Decrease

cf according to (3). Fig. 10. Ea(f)| when Vc is decreased to 350V. With reference to the results shown in Fig.10, the harmonic spectrum

F Will Shift To The

lower frequency order of, approximately, 18, and will span a wider range, as  is greater.



. The total spectral energy of the error current depends on the relay bandwidth  and it is independent of . As  increases the spectrum energy redistributes such that the bands

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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