Mosfet Buck Converter Matlab

Arxiv:1211.4415V1 [Cs.Sy] 19 Nov 2012

Discrete-Time Poles and Dynamics of Discontinuous Mode Boost

Chung-Chieh Fang∗

Manuscript: July 2010, revised on November 27, 2024

Abstract

Nonlinear systems, such as switching DC-DC boost or buck converters, have rich dynamics. A simple one-dimensional discrete-time model is used to analyze the boost or buck converter in discontinuous conduction mode. Seven different control schemes (open-loop power stage, voltage mode control, current mode control, constant power load, constant current load, constant- on-time control, and boundary conduction mode) are analyzed systematically. The linearized dynamics is obtained simply by taking partial derivatives with respect to dynamic variables. In the discrete-time model, there is only a single pole and no zero. The single closed-loop pole is a linear combination of three terms: the open-loop pole, a term due to the control scheme, and a term due to the non-resistive load. Even with a single pole, the phase response of the discrete-time model can go beyond -90 degrees as in the two-pole average models. In the boost converter with a resistive load under current mode control, adding the compensating ramp has no effect on the pole location. Increasing the ramp slope decreases the DC gain of control-to- output transfer function and increases the audio-susceptibility. Similar analysis is applied to the buck converter with a non-resistive load or variable switching frequency. The derived dynamics agrees closely with the exact switching model and the past research results.

KEY WORDS: Nonlinear system, DC-DC power conversion, discrete-time model, non-resistive load, discontinuous conduction mode, small-signal analysis

2.1

Nonlinear Discrete-Time Model . .

Steady-State (Fixed-Point) Analysis

.

Limitation On The Input Space

.

2.4

Linearized Open-Loop Dynamics . .

2.5

Agreement with the Exact Switching Model . .

2.6

Open-Loop Frequency Responses . .

3

General Closed-Loop Dynamics with Non-Resistive Load

The Pole Is A Linear Combination Of Three Terms

.

Closed-Loop Frequency Responses

.

4

Scheme Two (S2): Voltage Mode Control (VMC) with Resistive Load

5

Scheme Three (S3): Current Mode Control (CMC) with Resistive Load

Linearized Dynamics

.

5.2

DC Gains and the Effects of the Compensating Ramp . .

6

Buck Converter with a Resistive Load: A Short Review

7.1

Scheme Four (S4): Boost Converter with a Resistive Load in Parallel with CPL . .

7.2

Scheme Five (S5): Boost Converter with a Resistive Load in Parallel with CCL . .

7.2.1

General Case . .

7.2.2

Special Case: Pure CCL . .

8.1

Scheme Four (S4): Buck Converter with a Pure CPL . .

8.1.1

Open-Loop Power Stage . .

Current Mode Control (Cmc)

.

8.2

Scheme Five (S5): Buck Converter with a Pure CCL . .

9

Scheme Six (S6): Valley Voltage Constant-On-Time Control (V-COTC)

21

10 Scheme Seven (S7): Boundary Conduction Mode (BCM)

10.1 Boost Converter

.

25

10.1.1 BCM with Variable On-Time . .

25

10.1.2 BCM with Constant On-Time (BCM-COT) . .

25

10.2 Buck Converter . .

27

10.2.1 BCM with Variable On-Time . .

27

10.2.2 BCM with Constant On-Time (BCM-COT) . .

2

Table 1: Summary of seven schemes analyzed.

Scheme

Part I. Resistive Load (Fixed Switching Frequency) (summarized in Table 2)

Current Mode Control (Cmc)

Part II. Non-Resistive Load (Fixed Switching Frequency) (summarized in Table 3)

S4

Power stage/VMC/CMC with constant power load (CPL)

S5

Power stage/VMC/CMC with constant current load (CCL) Part III. Part III. Variable Switching Frequency Control (summarized in Table 4)

Introduction

Nonlinear systems, such as switching DC-DC boost or buck converters, have rich dynamics. Many efforts have been made in the past three decades to analyze the boost converter power stage in discontinuous conduction mode (DCM) based on average models . Fewer efforts have been made to model the boost converter under current mode control (CMC) in DCM. The analysis of the DCM is generally believed to be complex because DCM has three stages in a switching cycle.

Combination of CMC and DCM further increases the complexity. Adding a non-resistive load also increases the complexity . This paper presents an alternative and accurate modeling in addition to the average models.

Similar modeling approach has been applied to the fixed-switching-frequency buck converter with a resistive load . Compared with , this paper makes three additional extensions. First, it extends to the boost converter.

Second, it extends to the non-resistive load case.

Third, It

extends to variable switching frequency control. This paper focuses on the boost converter. In , a discrete-time model for the boost converter power stage in DCM is proposed. The model accurately predicts subharmonic oscillation in a boost converter with proportional voltage feedback . However, its potential advantage has not been fully appreciated. In this paper, the discrete-time model is applied to analyze the boost and buck converters in seven different schemes in a unified way (summarized in Table 1): open-loop power stage, voltage mode control (VMC), current mode control, constant power load (CPL), constant current load (CCL), constant-on-time control (COTC), and boundary conduction mode (BCM).

In the past, the analysis of these different schemes were reported in separate references [1, 3– 7, 9–12, 16–18], instead of in a single reference as this paper.

Here, The Linearized Dynamics Of

the discrete-time model is obtained simply by taking partial derivatives with respect to dynamic variables. In the discrete-time model, the pole will be shown to have a simpler expression and is a linear combination of three terms: the open-loop pole, a term due to the control scheme, and a term due to the non-resistive load. The discrete-time model provides a simpler alternative to design or analyze the converter different from the circuit-averaging approach . The same methodology developed here can be extended to analyze other types of converters , or applications, such as power factor correction and digital control of DC-DC converters directly based on discrete-time dynamics. For example, as shown in this paper, similar analysis can be readily extended to analyze the converter with a non-resistive load.

This paper presents theoretical analysis of already experimentally observed phenomena in [5,9, 12,15–17,20,21]. Seven simulation examples based on the exact switching model have been made.

R

Figure 1: A boost converter power stage. All the obtained results agree coherently with the past observations, and they are verified by the exact switching model. The analysis of the seven different schemes is presented next. For each scheme, the flow of analysis is as follows.

1. Identify the dynamic variables. 2. Identify the feedback variables. 3. Derive the switching constraint (when the switch is turned on/off).

4. Derive the large-signal dynamics. 5. Using partial derivatives, derive the small-signal (linearized) dynamics. 6. Determine the pole, control-to-output and audio-susceptibility frequency responses.

This paper has three parts. In Part I, the boost converter with resistive load is analyzed, and similar results for the buck converter are reviewed. In Part II, pole shifting due to non-resistive load is analyzed. In Part III, variable frequency control is analyzed. At the end of each part, the key results are summarized in a table.

Scheme One (S1): Open-Loop Power Stage

Consider a boost converter power stage (Fig. 1) with a switching frequency fs and a switching period T = 1/fs. Let ωs = 2πfs. Denote the source voltage as vs, the capacitor voltage as v, the inductance as L, the capacitance as C, and the equivalent series resistance (ESR) as Rc. The load, either resistive or non-resistive, has a steady-state effective resistance R. If the load is non-resistive, the dynamic resistance in the n-th cycle is denoted as Rn.

Nonlinear Discrete-Time Model

In DCM, there are three stages in the switching period. Let the durations of the first and the second stages be DT and D2T, respectively. The inductor current iL is zero in the third stage, and one discrete-time pole is zero . The discrete-time dynamics is thus one-dimensional.

Throughout the paper, to simplify the dynamics, all continuous-time variables are sampled at the beginning of each cycle. A subscript n is used for a dynamic variable of the n-th cycle. For

❅

❅

❅

❅

Figure 2: An illustrative signal plot of iL in each cycle for DCM.

✲Vn

Figure 3: Open-loop power stage large-signal dynamics. example, Dn denotes the duty cycle in the n-th cycle. Also, trailing-edge modulation (where the switch is turned on at the beginning of each cycle) is assumed. An illustrative signal plot of iL is shown in Fig. 2. The nonlinear large-signal discrete-time model (mapping) reported in is

(1)

where Kn = 2L/RnT, βn = ρnT 2/2LC, and ρn = Rn/(Rn + Rc). The model dynamics is shown in Fig. 3. Note that Kn, βn, and ρn are dimensionless variables. Also note that Rn, Kn, βn, and ρn are dynamic variables (varying in each cycle) if the load is non-resistive. If the load is resistive, they are constant and denoted as R, K, β, and ρ, respectively. The short notation vsn, instead of vs,n, is used for brevity. This applies to other variables. For Rc = 0, ρn = 1. Here, under fixed switching frequency, T is constant. In Sec. III, under variable switching frequency, and the switching period is a dynamic variable, denoted as Tn.

Steady-State (Fixed-Point) Analysis

In steady state, let (fixed-points) Dn = D, Rn = R, Kn = K = 2L/RT, ρn = ρ = R/(R + Rc), βn = β = ρT 2/2LC, vsn = vs and vn+1 = vn = v = Mvs, where M is the conversion ratio .

(2)

which is a quadratic equation of M and has two solutions. Ignoring the negative solution, one has

5

Assume ESR is small. By simple algebra based on the steady-state inductor current slopes, one

(4)

Using (3) and (4), one has D2 = KM/D and D = (M −1)D2 =

Km(M −1). These Equations

greatly simplify the linearized dynamics and are used throughout the paper.

Limitation On The Input Space

The nonlinear dynamics (1) is derived under the assumption that the converter operates in DCM, not all inputs (vn, vsn, Dn, Rn) = (v, vs, D, R) are legitimate, unless an additional CCM model is included. An input (v, vs, D, R) which makes the converter leave DCM is illegitimate. Assume that the variation of the capacitor voltage v is small within a switching period T, then D2/D = vs/(v −vs) = 1/(M −1). The converter operating within DCM requires D +D2 = Dv/(v −vs) < 1, which leads to the following limitation on the input space (v, vs, D, R) for the mapping (1) to be

Linearized Open-Loop Dynamics

A hat ˆ is used to denote small perturbations (e.g., ˆvn = vn −v and ˆDn = Dn −D). For a resistive load, ˆRn = R −R = 0. The effect of a non-resistive load will be discussed later. The linearized

(9)

The converter is stable if |p0| < 1. Saddle-node bifurcation occurs when p0 = 1, and subhar- monic oscillation (period-doubling bifurcation) occurs when p0 = −1. In [10, p. 427], the negative continuous-time pole is ωp = (2M −1)/RC(1 −M), equivalent (through a mapping) to the discrete-time pole p0 ≈eωpT if T ≪RC and Rc ≪R. Thus, given a discrete-time pole (9), one can easily obtains its corresponding continuous-time pole.

Agreement With The Exact Switching Model

In , based on the exact switching model, the exact value of the discrete-time pole p0 is obtained,

6

For a small θ, eθ ≈1 + θ, sin(θ) ≈θ and cos(θ) ≈1 −θ2/2, then the exact discrete-time pole (10) becomes (7). This shows that the derived linearized model (8), although based on the approximate nonlinear model (1), is close to the exact switching model.

Open-Loop Frequency Responses

The output voltage is close to ρv. In the power stage, Dn is the control variable to control the output voltage. Given the dynamics (8), the open-loop control-to-output transfer function is

P

KM(M −1). The DC gain, agreed with [10, p. 427], is

(13)

Similarly, the open-loop audio-susceptibility (source-to-output transfer function) is

(15)

agreed with . Given a transfer function in the z-domain, say T(z), its DC gain is T(1), and its effective frequency response is T(ejωT ), which is valid in the frequency range |ω| < ωs/2 (half the switching frequency).

Different from a single-pole continuous-time system, in which the phase response cannot go beyond -90 degrees, a single-pole discrete-time system has phase response beyond -90 degrees , giving similar results as in two-pole average models as shown in the next example.

Example 1. (The frequency response of the discrete-time model agrees with the experimental results reported in .) Consider a boost converter power stage from with parameters fs = 100 kHz, vs = 5 V, R = 20 Ω, L = 5 µH, C = 40 µF, Rc = 0, and D = 0.7.

The pole from (7) is 0.9703. The exact pole from (10) based on the exact switching model is 0.9707. Both agree closely. Throughout the paper, the exact switching model means the circuit as in Fig. 1 with the ideal switch, where the exact switching instants depend on the particular control scheme. Simulation based on the exact switching model is expected to be accurate as other circuit simulators such SIMPLIS, PSIM and SABER.

The control-to-output frequency response of the discrete-time model (12) is shown in Fig. 4, compared with that of the average model in . The frequency response of the discrete-time model matches well with the experimental data (reproduced and marked as * in Fig. 4) reported in based on SABER simulation. This example shows that the discrete-time model, even though with only one pole and no zero, still gives accurate frequency responses.

Here, both the discrete-time model and the average model in agrees closely with the exact switching model. In this example, the average model has good agreement because the discrete-time pole here is real positive. With additional feedback as discussed later, the closed-loop discrete-time

Phase (Deg)

Figure 4: Control-to-output frequency responses of discrete-time model (solid line), average model (dashed line) and experimental data marked as *. pole may be real negative, and the converter is oscillatory . In that case, the discrete-time model would give more accurate results than the average model (as shown in Example 2 with voltage feedback). In the average model of the boost converter, if the ESR zero is located between the two poles in the complex plane, the root loci of poles would remain on the real axis, and the converter with any feedback gain is not oscillatory. However, if the ESR zero is located to the left of the high-frequency pole in the complex plane, based on the root locus, the converter may have complex poles, but the oscillation frequency is not subharmonic (contradicting to simulations or real circuit experiments shown in Example 2 or ). Therefore, the average models are accurate only in some conditions (when the discrete-time pole is real positive) as reported in , whereas the discrete model does not have such a limitation.

The Pole Is A Linear Combination Of Three Terms

For a closed-loop converter, another switching constraint associated with the duty cycle is placed on the power stage dynamics (1). Generally, the constraint can be represented directly in terms of

(16)

where, in VMC, the control variable vcn controls the output voltage; while in CMC, the control variable vcn controls the peak inductor current. The closed-loop dynamics is shown in Fig. 5. Generally, the dynamic load Rn can be represented as a function of the capacitor voltage vn, Rn = R(vn). For general cases about other switching constraints or other load representations, similar dynamics can be derived and are omitted to save space.

From (1) and (16), the linearized closed-loop dynamics is

✲

Figure 5: Closed-loop large-signal dynamics. where p is the closed-loop pole and can be expressed as a combination of three terms:

(20)

where ∆pc = (∂f/∂Dn)(∂D/∂vn) denotes a pole shifting due to the closed-loop control scheme, and ∆pl = (∂f/∂Rn)(∂R/∂vn) denotes a pole shifting due to the non-resistive load. Note that the term ∆pc depends on the control scheme, and may be converter-dependent, whereas the ∆pl is generally converter-independent. If the load is purely resistive, one has ∂R/∂vn = 0 and ∆pl = 0.

Note that a resistive load affects the pole location through R, as shown in (9), not through ∆pl.

Closed-Loop Frequency Responses

Given the dynamics (18), the closed-loop control-to-output transfer function is

(21)

Similarly, the closed-loop audio-susceptibility (source-to-output transfer function) is

(22)

Compared with other modeling approaches, the discrete-time modeling is simpler.

Given A

converter with a particular load under a particular control scheme, the discrete-time pole is just a linear combination of different terms. The dynamics for the boost converter under fixed-switching- frequency VMC or CMC with a resistive load is presented next, followed by the non-resistive load case in Sec. 7, and the variable-switching-frequency case in Sec. III.

Given a particular control scheme, the first step of analysis is to determine the switching con- straint (16). Once the constraint is obtained, the closed-loop dynamics and pole can be easily obtained from (18) and (20).

4

Scheme Two (S2): Voltage Mode Control (VMC) with Resistive

Load

Consider a VMC boost converter shown in Fig. 6. Assume that the output voltage variation is small within a cycle and Rc is small. Consider a voltage feedback with a gain g. Let the ramp amplitude in

To Switch

Figure 6: A boost converter under voltage-mode control. the PWM module be Vh and the reference voltage be vc. The duty cycle is determined by equating the voltage loop output g(vc −vn) to the ramp DnVh at the switching instant, g(vc −vn) = DnVh It is rearranged in terms of the duty cycle Dn as a function of vn,

(23)

It is a simple state feedback and the linearized closed-loop dynamics is ˆvn+1 = pˆvn +Γs0ˆvsn, where the closed-loop pole is p = p0 + ∆pc = p0 −gΓc0/Vh. The converter is stable if −1 < p < 1. Since ∆pc here is generally negative, which may shift the pole to the left and make the pole p = p0 +∆pc to be negative. Subharmonic oscillation occurs when p = p0 −gΓc0/Vh < −1, rearranged as

(24)

Example 2. (The discrete-time model gives better prediction of gain margin than the average

Model.)

Consider a VMC boost converter from with parameters fs = 3 kHz, vs = 16 V, R = 12.5 Ω, L = 208 µH, C = 222 µF, Rc = 0, and output voltage v = 25 V. It is shown in that subharmonic oscillation occurs when g > 0.08 by simulation based on the exact switching model.

From (24), the critical gain (when the subharmonic oscillation occurs) is 0.076. Thus, a feedback gain greater than 0.076 is expected to be destabilizing, which agrees with the simulation result in noted above. With g = 0.076, the closed-loop discrete-time pole is -1.08, indicating that the converter is oscillatory at the half switching frequency (9424.9 rad/s).

Next, the frequency responses of the discrete-time model and the average model are compared. It will be shown that the discrete-time model gives more accurate results. The control-to-output frequency response of the discrete-time model (21) is shown in Fig. 7, compared with that of the average model based on . The gain margin of the discrete-time model is -22.4 dB (corresponding to g = 10−22.4/20 = 0.076) at the half switching frequency. The gain margin agrees with the exact switching model reported in .

In contrast, the gain margin based on the average model is -9.28 dB (corresponding to g = 10−9.28/20 = 0.3436) at frequency 15900 rad/s, which does not accurately predict the critical gain for the subharmonic oscillation. With g = 0.076, the closed-loop poles of the average model are −4508.3 ± 7115.2i, which are stable with a transient oscillation frequency at 7115.2 rad/s. This contradicts with the simulation that, with g = 0.076, the converter is unstable with a (subharmonic) oscillation frequency at 9424.9 rad/s. In this example, the discrete-time model gives more accurate

Frequency (Rad/Sec)

Figure 7: Control-to-output frequency responses of discrete-time model (solid line) and average model (dashed line). The gain margin -22.4 dB predicted by the discrete-time model is more accurate than the gain margin -9.28 dB predicted by the average model.

results than the average model both qualitatively (about the stability) and quantitatively (about the oscillation frequency).

5

Scheme Three (S3): Current Mode Control (CMC) with Resis-

Tive Load

In CMC shown in Fig. 8, let the compensating ramp slope be ma and the inductor current slope in the first stage of each cycle be m1 = vs/L. The duty cycle is determined by these two slopes and the control variable vcn (which controls the peak inductor current):

(25)

This feedback control law adds a nonlinear constraint to the discrete-time dynamics (1). As noted in for the boost converter in DCM, CMC adds feedforward from vs but adds no voltage feedback. Since Dn is not a function of vn, one has ∆pc = (∂f/∂Dn)(∂D/∂vn) = 0 and p = p0. The pole for CMC is the same as the open-loop power stage pole, agreed with . The results for CCM and DCM are different. In CCM, CMC does add state feedback. In contrast, in DCM, the CMC control law (25) does not add any state feedback because the initial current at the beginning of each cycle is zero.

Linearized Dynamics

Let mc = 1 + ma/m1 as in . Taking partial derivative of (25) with respect to vcn and vsn, the

(26)

The closed-current-loop linearized dynamics, (7) with (26), can be simplified as

To Switch

Figure 8: A boost converter under current-mode control.

Dc Gains And The Effects Of The Compensating Ramp

From (21) and (27), the DC gain of control-to-output transfer function, agreed with , is

(28)

Compared with (13), for no compensating ramp added (mc = 1), the DC gain for CMC is larger than that for the power stage if L > Tvs. However, since mc ≥1, adding the ramp or increasing the ramp slope decreases the DC gain.

(29)

The audio-susceptibility for CMC is smaller than that for the open-loop power stage (see (15)). However, based on (29), one has ∂Tos(1)/∂mc > 0, and increasing the ramp slope increases the audio-susceptibility. From (29), the DC gain of audio-susceptibility is nulled if mc = 1−1/(2M −1).

Since M > 1 for the boost converter, a negative ramp (with mc < 1) is required to null the audio- susceptibility. Without the ramp compensation (mc = 1), (29) becomes

(30)

which is close to 1/2 for a large M. Different from the CCM case, the effects of the compensating ramp for DCM are summarized. First, since the pole location in DCM is not shifted by CMC, adding the ramp also does not shift the pole and does not affect the stability. Second, increasing the ramp slope decreases the DC gain of control-to-output transfer function. Third, increasing the ramp slope increases the DC gain of audio-susceptibility. These three effects raise the question whether the ramp is needed for the boost converter in DCM. The ramp, beneficial in CCM to stabilize the current loop, may be unnecessary in DCM since the current loop itself is not oscillatory. The need of the ramp is also questioned for the buck converter in DCM .

Example 3. Consider a CMC boost converter with parameters fs = 700 kHz, vs = 12 V, R = 24 Ω, L = 1 µH, C = 125 µF, Rc = 0, and output voltage v = 24 V.

Frequency (Rad/Sec)

Figure 9: Control-to-output frequency responses of discrete-time model (solid line) and average model (dashed line). The control-to-output frequency response (21) is shown in Fig. 9, compared with that of the average model reported in which was shown in agreement with the experimental results. Both have similar magnitude frequency responses. The discrete-time model generally has a larger phase lag close to the half switching frequency. This can be explained by the fact, in the discrete-time model, the output is measured at the start of the period (t = nT), while the control is exerted at t = nT + DnT (with a delay). The discrepancy can be mitigated if the output is also measured at t = nT + DnT .

6

Buck Converter with a Resistive Load: A Short Review Similar results for the buck converter based on are summarized here for completeness and also for comparison. From , the nonlinear large-signal discrete-time dynamic for the buck converter

(31)

The open-loop power stage pole for the buck converter, agreed with [10, p. 427], is

(32)

From , the pole for the CMC buck converter, agreed with , is

(33)

With no compensating ramp, mc = 1, and the CMC pole is

(34)

The pole (34) is greater than 1 (unstable) for M > 2/3, implying occurrence of saddle-node bifur- cation . The possibility of instability for M > 2/3 was also reported in .

13

Table 2: Summary for the power stage and CMC for boost and buck converters, some agreed with past research results .

3M−2

A summary for the power stage and CMC for boost and buck converters is given in Table 2.

Boost Converter

With a non-resistive load, either under open loop, VMC or CMC, the pole is shifted by ∆pl as discussed in Sec. 3. Two different loads, CPL and CCL, are considered.

7.1

Scheme Four (S4): Boost Converter with a Resistive Load in Parallel with

Cpl

Let the load be a resistive load R0 in parallel with a CPL (with a constant power P) as shown in Fig. 10. Assume that the output voltage variation within a cycle is small and ESR is also small, then the effective resistance of the CPL is close to v2

N/P And

the load is a pure CPL. For P = 0, Rn = R0 and the load is a pure resistor.

R0

Figure 10: A boost converter power stage with CPL and resistive load R0. In steady state, the effective resistance is R = R0v2/(R0P + v2), which leads to

(36)

which is independent of R0. Note that, as discussed above, a resistive load such as R0 affects the pole location through the effective resistance R as shown in (9), not through ∆pl. Also note that, with a small R or R0, the converter may not operate in DCM, and the DCM analysis does not apply. For P = 0, one has R = R0 and ∆pl = 0 because with a resistive load, the pole is not shifted by an additional term.

As noted above, for the CMC boost converter, ∆pc = 0. With CPL, the poles for the power stage and CMC are the same. From (9) and (36), for either the power stage or CMC,

(37)

When the load is a pure CPL (R0 = ∞and the total effective resistance R = v2/P), for example, the converter is generally stable (with p < 1) and agreed with . Subharmonic oscillation (with p < −1) may occur if M < 1 + ρT/2RC (close to 1), which is rare because another condition with M > 1/(1 −D) for the boost converter is required in DCM.

7.2

Scheme Five (S5): Boost Converter with a Resistive Load in Parallel with

General Case

Let the load be a resistive load R0 in parallel with a CCL (with a constant current Io) as shown in Fig. 11. This load can model a light emitting diode (LED), and the boost converter is an LED driver. Assume that the output voltage variation within a cycle is small and ESR is also small, then the effective resistance of the CCL is close to vn/Io. The total load resistance (as a function of vn) is Rn = R(vn) ≈R0 ∥(vn/Io) = R0vn/(R0Io + vn). For R0 = ∞, one has Rn = vn/Io and the load is a pure CCL. For Io = 0, one has Rn = R0 and the load is a pure resistor.

(38)

For Io = 0, one has R = R0 and ∆pl = 0 because with a resistive load, the pole is not shifted by an additional term. If Io > 0, then ∆pl > 0 and the pole is shifted to the right. If Io < 0, then ∆pl < 0 and the pole is shifted to the left. The effect of CCL on the power stage or CMC is discussed next.

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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