536 Proceedings of 20 th International Scientific Conference. Transport Means. 2016. Modeling of Current Limitation through the PWM Signal in LPG Injectors D. Szpica *Bialystok University of Technology, Faculty of Mechanical Engineering, 45C Wiejska Str., 15-351 Bialystok, Poland, E-mail: d.szpica@pb.edu.pl Abstract Computer calculations constitute a basis for modern design and analysis. Adopted mathematical models lead to the exploration of actual processes. This depends on the complexity of the model, availability of the input data or boundary conditions. The paper attempts to develop a mathematical model of current limitation used in LPG injectors. Based on the experimental data related to the operating parameters (i.e. inductiveness, resistance, weight of the working component etc.), the courses of current in the solenoid were determined and the results were validated experimentally. These types of models can be used in determining of the motion of the working components or checking whether a given modulating signal duty cycle is capable of holding the injector open. KEY WORDS: combustion engines, fuel supply, modeling, passenger car 1. Introduction At the turn of the XX century alternative fuel systems of spark ignition traction engines were dominated by the LPG systems. In the beginning LPG was fed in the vapor phase, through a mixer operating similarly to a carburetor. In subsequent solutions vapor injection - continuous and then periodic was introduced [1]. Currently, due to downsizing and the application of direct gasoline injection it is necessary to apply injection of a liquid phase LPG through additional indirect injectors [2] or utilizing the original gasoline direct injectors. Today, pulse vapor phase LPG injectors are the most widely applied subcomponents responsible for the supply of LPG to the engine. Systems equipped with these components are referred to as the IV generation systems, which, as a principle, are universal [3], though in some cases this is not confirmed [4]. Taking the external indexes into account, small differences in the maximum power output or torque were observed between the gasoline and the IV generation LPG systems. Important is the fact that the vapor LPG vapor occupies much larger volume of the cylinder compared to gasoline or the fact that LPG injectors have uneven fuel dosage (compared to gasoline, up to 10-times) [5]. Many factors influence the said unevenness such as workmanship, calibration or operational wear [6], or the diameter of the outlet nozzle [7]. A vapor phase LPG pulse injector is not a frequent object of scientific investigations. It is a simple electrical valve that remains closed under normal conditions. The unique feature of pulse injectors is the frequency of operation resulting from the fuel demand in the subsequent engine work cycles. The engine requires an abrupt opening and a possible quick closing, which would ideally remain in relation to the gasoline injectors, as it is the gasoline injector impulses that are used to control the LPG injectors [3]. In order to trigger an abrupt opening, the inertia force of the moving component, the friction force or the closing spring force have to be overcome. A current impulse is necessary to do that but only in the opening phase. Later, when the opening is upheld, the current value can be lower, which is why a current limitation by the PWM signal is applied. This allows keeping the injector open without excess coil (source of electromagnetic force) overheat. In the control system there is a transistor key with a Zener diode voltage regulator and the control is done through the injector short to ground. Eventually, the course of the current powering the injector coil assumes a shape presented in Fig. 1. In [8] Fig. 1 Phases of the course of current in the coil circuit of an LPG pulse injector [8] 2. Mathematical Modeling 4 phases describing the change of current throughout the course of the feed impulse have been presented. The empirical models proposed in [8] are applied in the calculations of the increment and holding of the current supplying the electromagnetic circuit, but they require a continuous adjustment of the boundary conditions. This is the ration containing a PWM current limitation.
537 Fig. 2 Design and schematics of the LPG pulse injector All mechanical devices that can move during operation (Fig. 2) are modeled using a conventional mass-springdamper system by mechanical equilibrium equations where the dynamic parameters are a function of the component position [9]., (1) where is the mass of the moving element, kg; x (t) is the acceleration, m/s 2 ; B is damping coefficient, Ns/m; x (t) is speed, m/s; k is the spring stiffness, N/m; x (t) is the displacement element, m; x 0 is initial tension the spring, m; F magn is the electromagnetic force, N. The description omits the force resulting from the pressure of fluid/gas that can be calculated from the relations presented in [9-11]. In the case of the injector, the damping may be the result of the friction between the piston and the body and the fluid. Then, the friction force can be described through static and viscous friction. F f N F s N if x 0; dx sign x w if x 0, dt (2) where s is coefficient of static friction; F N is the force, N; w is coefficient of viscous friction, Ns/m. The electromagnetic force being the result of the circuit operation can be obtained from the relation: 1 2 dl x 2 FE I N dx. (3) current supplying the electromagnetic circuit: di 1 dl x dx A I RI U dt L dx dt s. (4) By integrating an Eq. (4) and substituting it to (3) we can determine the force generated by the electromagnetic coil circuit. Then, substituting all components to (1) we will obtain an equation of equilibrium that, in order to be solved in the Matlab-Simulink environment, must be transformed according to the Kelvin principle.. (5) By twice integrating the relation (5) we will obtain a displacement of the piston. For the solution a calculation algorithm in the Simulink (Fig. 3) was developed. It allows conducting calculations of the LPG injector operation in the entire cycle, i.e. opening, holding the open position and closing. The parameters are entered via the Mask dialog windows. The PWM signal can be freely configured, i.e. the time after which it will occur and % of the duty cycle. The differential Eq. (5) was solved using the ode45 procedure, i.e. Runge-Kutta algorithms, one step method of iterative solution of differential equations by integration. It used the formulas of the fourth and fifth order at the same time to estimate the error and determine the time step [12].
Fig. 3 Structural diagram of the calculation algorithm Matlab-Simulink 538 3. Parameters, Functions and Boundary Conditions In order to initiate the simulation, selected parameters of the AC-W02 injector were used. For the measurement of the coil parameters, RLC CMR-417 Meter and a special test bed were used. The determined parameters, functions and boundary conditions have been presented in Table 1. 4. Analyzing the Results Parameters, functions and boundary conditions needed to initiate the simulation Upon introduction of the input data and the boundary conditions into the model (Fig. 3), a series of courses were obtained. The most important aspect discussed in the work is the modeling of the PWM signal, which is why in the analysis, attention was devoted to this problem. The PWM signal, its duty cycle mainly, aims at limiting the current flow that could overheat the coil. Table 1 Parameter Volume Parameter Volume Supply voltage 14 V - impulse Inductance function (100 Hz) -5.03x 3 +4.41x 2 +0.16x+3.02 e-3 H Injection time 10 ms Inductance function (2 khz) -3.03x 3 +2.59x 2 +0.06x+2.47 e-3 H Time to PWM 4 ms Spring stiffness 1000 N/m Filling PWM variables Initial tension the spring 1e-3 m Frequency PWM 1 khz Coefficient of static friction 0.03 Mass of the piston 2 gm Force 0.02 N Resistance (100 Hz) 2.1 Coefficient of viscous friction 0.009 Ns/m Resistance (1 khz) 2.8 x(t = 0) 0 m U(t = 0) 14 V short-circuit control 25 % PWM 50 % PWM 90 % PWM Fig. 4 The results of the simulation at variable PWM duty cycle On the other hand, it should be confirmed that the assumed limitation will not result in closing of the injector by the spring. Comparing the courses from (Fig. 4) one may observe that the duty cycles (25 and 50)% are not capable of keeping the injector in the fully open condition. In order to validate the simulation, flow research was conducted using the BRONKHORST F-113AC-M50- ABD-00-V flow meter. The values of the constant parameters were as follows: p supply = 1.2e5 Pa, n = 1000 rpm, t inj = 10 ms, modulation after 2.5 ms, U = 14 V, h piston = 0.6 mm, d nozzle = 3 mm. This was the way of confirming the simulation research that indicated an insufficient force generated by the coil during the Fig. 5 The relation between the volumetric flow current limitation with the PWM signal (Fig. 5). rate and the PWM signal duty cycle The next step was a direct comparison of the calculation results with the courses of the current for the actual object. To this end, technical data of the actual object were entered (due to the size of the data it has been omitted in the work) and
539 calculations were performed. The courses differed from one another (Fig. 6), which is mostly the result of the realization of the voltage impulse control via the transistor. This results in a change of the current fade during the modulation. Additionally, the reverse current reaching approx. 60 V, at the time of switching to the PWM signal, generates a current course of greater amplitude. The control key also causes a faster fading of the current as the control impulse seizes. At the moment, further improvements of the model are being made, particularly the simulation of the transistor key and the consideration of the flexibility of the piston displacement. 5. Conclusions Fig. 6 Comparison of the courses of the current The proposed calculation model of the operation of a LPG vapor phase pulse injector has confirmed its proper functioning. Obviously, this is a simplified model and some of the parameters are entered in the form of functional relations and rather than calculated from the input data. It should still be deemed sufficient for the preliminary evaluation of the injector operation. The sensitivity trial of the model to the PWM duty cycle resulted in various courses of the holding current. In this way, the variability of the electromagnetic force was obtained and, consequently, different injector openings. The main differences in the courses of the model and the experiment result from the application in the actual system of a specific type of control with the transistor key, which directly translates into the current courses in the fading phase. Acknowledgement The research has been carried out within work no. S/WM/2/2013 realized at Bialystok University of Technology and financed from the funding allocated for science by the Ministry of Science and Higher Education. References 1. Liquefied petroleum gas (LPG) as a mediumterm option in the transition to sustainable fuels and transport. Renewable & Sustainable Energy Reviews 2014. Vol. 32, pp. 513 525. http://dx.doi.org /10.1016/j.rser.2014.01.052. 2. Mitukiewicz G., Dychto R., Leyko J. Relationship between LPG fuel and gasoline injection duration for gasoline direct injection engines. Fuel 2015. Vol. 153, pp. 526 534. http://dx.doi.org/10.1016/j.fuel.2015.03.033. 3. Wendeker M, Jaklinski P, Czarnigowski J, Boulet P, Breaban F. Operational parameters of LPG fuelled SI engine comparison of simultaneous and sequential port injection. SAE Technical Paper 2007. 2007-01-2051. http://dx.doi.org/ 10.4271/2007-01-2051. 4. Borawski A. Modification of a fourth generation LPG installation improving the power supply to a spark ignition engine. Eksploatacja i Niezawodnosc - Maintenance and Reliability 2015. Vol. 17, No. 1, pp. 1-6. http://dx.doi.org/10.17531/ein.2015.1.1. 5. Szpica D., Czaban J. Operational assessment of selected gasoline and LPG vapour injector dosage regularity. Mechanika 2014. Vol. 20, No. 5, pp. 480-488. doi: http://dx.doi.org/10.5755/j01.mech.20.5.7780. 6. Szpica D. Fuel dosage irregularity of LPG pulse vapor injectors at different stages of wear. Mechanika 2016. Vol. 22, No. 1, pp. 44-50; http://dx.doi.org/10.5755/j01.mech.22.1.13190. 7. Szpica D. Research on the influence of LPG/CNG injector outlet nozzle diameter on uneven fuel dosage. Transport 2016. Vol. 3, No 3, pp. 1-11. http://dx.doi.org/10.3846/16484142.2016.1149884 8. Przeglad Elektrotechniczny 2014. Vol. 90, No. 3, pp. 195-198. http://dx.doi.org /10.12915/pe.2014.03.44. 9. Borawski A. Simulation studies of LPG injector used in 4 th generation installations. Combustion Engines 2015. Vol. 1, No. 160, pp. 49-55. ISSN 2300-9896. 10. Mathematical modelling of the pneumatic relay emergency valve for dual-line agricultural trailer braking systems. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 2012. Vol. 226, No. 5, pp. 603-612. http://dx.doi.org /10.1177/0954407011423133. 11. Mathematical modelling of the trailer brake control valve for simulation of the air brake system of farm tractors equipped with hydraulically actuated brakes. Eksploatacja i Niezawodnosc Maintenance and Reliability 2014. Vol. 16, No. 4, pp. 637 643. 12. Yang W.Y., Cao W., Chung T.S., Morris J. Applied Numerical Methods Using MATLAB; John Wiley & Sons Inc. 2005, Hoboken, New Jersey. http://dx.doi.org/10.1002/0471705195.