A novel formula easily applied with high precision is proposed in this paper to fit the
The fuel system has a vital impact on the overall performance of a diesel engine, and a highpressure common rail system enables the cyclic injection quantity, injection timing, and fuel injection law to be controlled exactly and flexibly. Therefore, a diesel engine that is equipped with a highpressure common rail system has the potential to achieve the optimized design goals of highefficiency combustion and ultralow emission [
The dynamic response characteristic is an important evaluation index of the HSV in the fuel injector. This is because a fast response speed of the HSV is beneficial to achieve multiple injections and more precise control of the fuel injection timing of the highpressure common rail system. The dynamic response of the HSV is determined by both its electromagnetic force characteristic and moving part quality, but the former has a more significant impact. Only when the maximum static electromagnetic force of the HSV meets its requirements can the response time requirements of the opening and closing phases of the HSV be met by using the drive circuit. Hence, research on the electromagnetic force is of great importance for the design of the HSV.
In research on the static electromagnetic force of the HSV, most scholars used the finite element method (FEM). Liu et al. [
Elmer and Gentle [
Most of the literature on the HSV tends to ignore the influence of the soft magnetic material reluctance in developing the electromagnetic model of the solenoid valve. Topçu et al. [
Coppo et al. [
It can be found from the above references that a certain research insufficiency on the electromagnetic mathematical model of the HSV still exists. Some electromagnetic mathematical models use an electromagnetic force fitting formula based on experimental data, some models do not consider the reluctance of the soft magnetic materials, and some only took segmental magnetic properties of the magnetic materials into account. Therefore, to provide more valuable information on the electromagnetic energy conversion of the HSV and more deeply understand the solenoid valve’s electromagnetic conversion process, more detailed research on the electromagnetic mathematical model of the HSV has been carried out in this paper. The resulting model considers the nonlinear magnetization characteristics and magnetic saturation characteristics of the HSV. Based on this mathematical model, research on the electromagnetic force characteristics of the common rail injector HSV and its key influence factors can follow. It provides a certain theoretical basis and design tools for the design of HSVs.
Figure
The structure schematic of HSV.
For the HSV, the functional relationship between the total magnetic flux and the total reluctance of the magnetic circuit is described as
To accurately describe the nonlinear magnetization and magnetic saturation of the HSV, the mathematical model should take the reluctance of the iron core and armature soft magnetic materials into consideration. Due to the different equivalent crosssectional areas of the main and side poles under normal circumstances, the gap reluctances between the main pole, side pole, and armature are described separately in the mathematical model, and the fringe effects of the air gap and the leakage flux are ignored. Based on the equivalent magnetic circuit shown in Figure
Equivalent magnetic circuit of HSV.
Finally, the electromagnetic force is computed by
The permeability of the soft magnetic material shows its influence by two aspects in the electromagnetic mathematical model of the HSV. On the one hand, the reluctance of the soft magnetic material has an effect on the electromagnetic conversion, and, on the other hand, the drive current has an effect on the reluctance. The electromagnetic coupling of the HSV is exactly implemented by the permeability of the soft magnetic material in the electromagnetic force mathematical model. Therefore, the accuracy of the
Jiles and Atherton [
Chan et al. [
The onedimensional commercial software AMESim [
As described above, previous studies used different fitting formulas for the
Comparison between simulated results by formula proposed by Chan and experimental results of
As shown in Figure
Comparison between simulated results by formula proposed by AMESim and experimental results of
Figure
Schematic diagram of
In numerous functional forms, the curve of the logarithmic function
Every magnetic material has a different saturation induction density
Influence of
Equation (
Influence of
It can be observed from the above analysis that the two coefficients
As shown in Figure
Comparison between simulated results by (
As shown in Figure
Comparison between simulated results by (
To experimentally validate the accuracy of the electromagnetic model for the HSV, the test bench of the static electromagnetic force depicted in Figure
Measurement accuracy of main equipment.
Equipment  Force sensor  Current probe 

Type  CZLYB3  1146A 
Producer  Chengdu Xingpu Transducer Co., Ltd.  Agilent Technology 
Measurement range  0 
1 
Measurement accuracy 


Test bench of electromagnetic force for HSV.
The schematic block diagram of the test bench
Test apparatus
Table
Structural parameters of HSV.
Parameters  Reference value 

Iron core  
Height 
13.7 
Diameter 
20.4 
Diameter of hole 
7 

From Figure 
Coil  
Number of turns 
52 
Height 
7.6 
Inside diameter 
12 
Outside diameter 
17.7 
Armature  
Thickness 
1.8 
Diameter 
20 

From Figure 
Assembly  
Working air gap 
0.1; 0.12 
Control  
Driving current 
1 
A comparison between the simulated and experimental results of the electromagnetic force at various drive currents and working air gaps is shown in Figures
Comparison between simulated and experimental results of electromagnetic force at
Comparison between simulated and experimental results of electromagnetic force at
As shown in Figure
Experimental results of electromagnetic force at different working air gaps.
The electromagnetic force increment at different working air gaps.
It can be observed from Figure
It can be observed from Figure
The total magnetic reluctance as a function of drive current at different working air gap.
It can be observed from Figure
Based on the electromagnetic transient coupling principle, an electromagnetic mathematical model of the HSV is developed in Fortran language that takes the magnetic saturation characteristics into account, and it is validated by experimental data of the static electromagnetic force.
Experimental research on the variation relationship between the static electromagnetic force of the HSV and its drive current at different working air gaps is carried out. It is concluded that, with the increasing drive current, the increment of the electromagnetic force first increases rapidly and then decreases at different working air gaps, and there exists a drive current value that makes the increment of the electromagnetic force achieve its maximum value.
Based on the established electromagnetic mathematical model of the HSV, it is revealed that the electromagnetic energy conversion characteristics of the HSV are affected by the drive current and the total reluctance and that these two influence factors within the scope of the different drive currents have different contribution rates to the electromagnetic energy conversion efficiency.
The authors declare that there are no conflicts of interest regarding the publication of this article.
This work is supported by the Natural Science Foundation of Heilongjiang Province of China (LC201422), the National Natural Science Foundation of China (NSFC 51279037, 51379041, 51475100, and 51679048), the Fundamental Research Funds for the Central Universities (HEUCF160304), and Recruitment Program of HighEnd Foreign Experts of the State Administration of Foreign Experts Affairs (GDW20162300256).