Cuk Converter Design Matlab

International Journal of Advanced and Applied Sciences, 4(6) 2017, Pages: 175-180

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175

Designing a PI controller for Cuk converter using converter dynamics

Accepted 2 May 2017

This paper dedicated to study indirect control of Cuk converter. Cuk converter has a 4th order non minimum phase transfer function. Extraction of converter’s dynamical equation is not an easy task with pencil-and-paper analysis. Converter’s dynamical equations are obtained using “Kocaeli (RHP) zeros force us to use two feedback loops. Controllers are designed for this two loops using MATLAB’s control system toolbox. Close loop system is

Simulink®

environment.

The

performance of designed controller.

State Space Averaging

© 2017 The Authors. Published by IASE. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Current For The Load. Obtaining A Stable Output

voltage or current in presence of disturbances like:

Changes Seems Impossible Without Some Form Of

control.

Considerable Progress Over Recent Decades, Most

applications use PID controllers, because of their low

Electronics Converters Control. Usually A P Or Pi

controller is all that is required. Designing a classical P or PI controller for a power electronics converter is started by obtaining the model of converter.

Mathematical Description Of The System. Obtaining

the mathematical model of system is the first step

Controller Design Techniques. Switching Power

converters are nonlinear variable structure systems.

Methods Are: Current Injected Approach, Circuit

* Corresponding Author.

Https://Doi.Org/10.21833/Ijaas.2017.06.025

2313-626X/© 2017 The Authors. Published by IASE. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Small Signal Linearization Is Key Steps Of These

methods.

Describe

converters that work in CCM while is less suitable for

Method Kislovski Et Al. (1991) And Mohan And

Undeland (2007) can do the job of modeling in either

Ccm Or Dcm. Circuit Averaging Gained A Lot Of

attention recently due to its generality (Hren and Slibar, 2005). At the time of this writing, there is no software to

Power

electronics converters. Using available commercial

Software Only A Frequency Response Plot Can Be

obtained. No information about location of poles and

Zeros Are Given By Software So A Pencil-And-Paper

analysis is required to obtain the algebraic transfer

Function Of Converter. Obtaining The Dynamical

equation of converter in presence of circuit’s non idealities such as Equivalent Series Resistance (ESR) of capacitors and inductors, voltage drop of diodes,

Consuming And Error Prone Task For Pencil-And-

paper analysis. If a parameter is changed, i.e. load, all the calculation must be done from scratch. Developed software can be used to extract dynamical equation of buck, boost, buck-boost, Cuk, SEPIC, fly back, forward and full bridge converters in presence of mentioned non-idealities. Developed software can Asadi et al/ International Journal of Advanced and Applied Sciences, 4(6) 2017, Pages: 175-180

176

give both algebraic transfer function and frequency

Paper Shows How A Control System Can Be Designed

for a Cuk converter. Cuk converter has a 4th order, i.e.

Theorem That Presence Of Rhp Zeros Degrades The

achievable close loop performance. Presence of RHP

Used To Design Controller For Other Non-Minimum

phase converters like SEPIC.

Foundation Of State Space Averaging (Ssa) Was

laid down in Middlebrook and Cuk (1977). The first

Attempt To Model Discontinuous Conduction Mode

(DCM) is presented in Cuk and Middlebrook (1977).

Accurate Small Signal Models For Dcm Operation

were developed by Sun et al. (2001). A unified SSA

Survey Of The Modeling Issues Can Be Found In

Maksimovic et al. (2001). Application of different control methods to power electronics converters has been studied in many papers. For example, feedback linearization (Sanders et al., 1986), sliding mode

(Venkatanarayanan And Saravanan, 2014) And 𝐻∞

design (Rodriguez et al., 2005) has been applied to Cuk converter, Linear Matrix Inequality(LMI) control has been applied to conventional boost by Reddy et

Al. (2015). Discrete Time Controller Has Been

designed for a boost converter in Alkrunz and Yazıcı (2016). A cascade state space controller is designed

Ocilka And Béreš (2010). Pid Control Of Sepic

converter is studied in Veenalakshmi et al. (2014).

3. Working Principle Of Cuk Converter

Topology of Cuk converter is shown in Fig. 1. Output voltage can be either smaller or larger than that of input. There is a polarity reversal on the

Output. Energy Transfer From Input Source To Load

depends on the capacitor C1. Analysis of this circuit is

Based On The Following Assumptions:

1. Both inductors (capacitors) are large enough so current in (voltage across) them are constant. 2. Circuit operating in steady state, i.e. transients has been passed.

3. Switch and diode are ideal (i.e. no resistance and voltage drop). 4. Switch is closed for time DT and open for (1-D) T.

Corresponding Circuit For Switch Closed And

opened is shown in Figs. 2 and 3, respectively.

Fig. 1: Topology Of Cuk Converter

Fig. 2: Cuk converter’s equivalent circuit for closed switch Fig. 3: Cuk converter’s equivalent circuit for opened

Assuming An Ideal Capacitor, I.E. No Equivalent

Series Resistance (ESR), output voltage ripple can be

Cuk Converter, The Inductors Average Current Must

be greater than one half the changes in current. This

(5)

converter works in CCM if L1>L1,min and L2>L2,min.

(6)

Asadi et al/ International Journal of Advanced and Applied Sciences, 4(6) 2017, Pages: 175-180

Eqs. 6 And 7 Are Written Under Switch Close And

switch open condition, respectively. 𝑥 is state vector, i.e. capacitors’ voltage and inductors’ current, 𝑢 is control input and 𝑣𝑜 is output voltage of converter.

𝑇)𝑋]𝑑̃ (9)

Tilde (𝑥,̃ 𝑑̃, 𝑣̃𝑜) shows small signal variables and D

Switch Open’S Equations For Cuk Converter In

presence of non-idealities such as capacitor’s and inductor’s ESR, MOSFET’s on resistance and voltage

𝑅𝐿×𝐶2 𝑣𝐶2

for switch opened.

Obtain

dynamical equation of this 4th order converter is a

Rl1=10Mω, Rl2=21Mω, C1= 20Μf,C2=4Μf, Rc1= 10Mω,

rC2=5mΩ, VDiode_on=0.7, rDiode_on=0.05 Ω, rMOSFET=40 mΩ, RLoad=8.1Ω.

−9.09×109×𝑠2+3.618×1013×𝑠−2.992×1017

𝑠4+3.228×104×𝑠3+4.428×108×𝑠2+1.8×1012×𝑠+6.903×1015

𝑑̃(𝑠) Is Minimum Phase. Suggested

control structure is shown in Fig. 5. Fig. 4: Entering Cuk converter’s parameters to developed software

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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