Influence of eddy current on transient characteristics of common rail injector solenoid valve

LI Pi-mao(李丕茂), ZHANG You-tong(張幽彤)

(Low Emission Vehicle Research Laboratory, Beijing Institute of Technology, Beijing 100081, China)

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Influence of eddy current on transient characteristics of common rail injector solenoid valve

LI Pi-mao(李丕茂), ZHANG You-tong(張幽彤)

(Low Emission Vehicle Research Laboratory, Beijing Institute of Technology, Beijing 100081, China)

Influence of eddy current on transient characteristics of common rail injector solenoid valve was studied in this paper. Experimental investigations of drive current and power source voltage of both drive current ascending and descending process were conducted on a common rail injector solenoid valve. A new discretizing calculation method of solenoid valve flux linkage was put forward for the first time based on the experimental results and drive circuit principle, and flux linkage of both drive current ascending and descending process were evaluated. New inductance calculation methods for drive current ascending and descending process respectively were also presented. Influence of parasitic inductance was evaluated. Results indicate that the air gap, under which the transient flux linkage of the solenoid valve is the biggest, varies with drive current due to eddy current. Flux linkage of drive current descending process is bigger than that of drive current ascending process under the same drive current and the same air gap width. Eddy current can reduce the delay between the time that drive current begins to descend and the time that armature begins to move downward. Inductance of drive current descending process is bigger than that of drive current ascending process over larger scope of drive current, but the difference becomes smaller with the increasing of air gap width. The differences of both flux linkage and inductance between drive current ascending and descending process are caused by the eddy current in core and armature materials.

common rail injector; solenoid valve; transient characteristic; flux linkage; inductance; eddy current

Economy and emission performance of diesel engine are closely related to the accuracy of fuel injection rate control. Opening and closing of electromagnetic common rail injector are controlled by a solenoid valve, transient characteristics of which are the key of accurate fuel injection rate control. The transient characteristics are affected by properties of core and armature materials, especially eddy current.

An estimation method of solenoid actuators’ position based on experimental results was presented[1]. Finite element method was widely applied to the investigation of solenoid valve.B-Hcurve was used to characterize the property of core and armature materials[2-5]. The nonlinear relationship between flux linkage and magnetization current was used in the computer-aided design method for solenoid actuators[6-7]. Permeability of magnetic materials was thought to be constant and do not vary with magnetization current[8-13]. Constant permeability of core and armature material was used in the modeling of common rail injector[14-15]. Researches above focus mainly on modeling of solenoid valve, special attentions have not been paid to influence of eddy current on transient characteristics of solenoid valve.

In this paper, experimental investigations of drive current and power source voltage of both drive current ascending and descending processes were conducted on a common rail injector solenoid valve. A new discretizing calculation method of solenoid valve flux linkage was put forward for the first time based on the experimental results and the principle of drive circuit, and flux linkage of both drive current ascending and descending process were assessed. New inductance calculation methods for the drive current ascending and descending process respectively were also presented. Influence of parasitic inductance was evaluated.

1 Experimental investigations

1.1 Structure of solenoid valve

Fig. 1 shows structure of common rail injector solenoid valve. Core and stroke limiter are fixed. Direction of electromagnetic force is upward. Spring 1 and spring 2 are compressed all the time. Stiffness of spring 1 is far bigger than that of spring 2. If electromagnetic force is bigger than the force acted on armature by spring 1, the armature will move upward and the solenoid valve will open. Otherwise, the solenoid valve keeps closed.

Fig.1 Structure of common rail injector solenoid valve (δ is the width of air gap)

1.2 Experimental scheme

Fig.2 presents drive current of the solenoid valve. The drive current consists of three stages including current ascending stage A, high current stage B and current descending stage E. The drive current and power source voltage of stage A under different air gaps were recorded and shown in Fig.3, and the drive current and power source voltage of stage E under the same condition were recorded and shown in Fig.4. In order to compare the characteristics of stage E with stage A conveniently, drive current and power source voltage of stage E are plotted in the form of stage A. In fact, the drive current of stage E decreases with the increasing of time, and the power source voltage of stage E increases with the increasing of time. The minimum displacement of armature is 0.05 mm, so 0.06 mm, 0.1 mm, 0.16 mm, 0.31 mm and 0.6 mm wide air gaps are used.

Fig.2 Drive current of common rail injector

1.3 Experimental results

Fig.3 Drive current and power source voltage of stage A under different air gaps

Fig.4 Drive current and power source voltage of stage E under different air gaps

The increasing rate of drive current decreases first and then increases with the rising of drive current whenδis less than 0.16 mm, but decreases with the rising of drive current whenδis more than 0.31 mm (Fig.3a). Besides, the time taken by drive current changing from 0 to 18 A increases first and then decreases with the increasing ofδ. The maximum time is needed whenδis about 0.1-0.16 mm.

The drive current changing of stage E withδshows a similar law to that of stage A. The time taken by drive current changing from 0 to 16 A, is maximum whenδis about 0.16-0.31 mm. The time needed whenδis 0.1 mm is obviously less than whenδis 0.16 mm.

2 Analysis of electromagnetic force

2.1 Relationship between electromagnetic force and flux linkage

Electromagnetic force can be calculated by[10]

(1)

whereμ0is vacuum permeability,Sis equivalent cross-sectional area of air gap in magnetic circuit,φis magnetic flux. According to the definition of flux linkage,φcan be obtained as

φ=ψ/N

(2)

whereψis flux linkage andNis the number of coil turns. Replacingφin Eq.(1) with Eq.(2), electromagnetic force can be expressed as

(3)

whereμ0,SandNare all constants. Electromagnetic force is proportional to the square of flux linkage, so flux linkage of both stages were analyzed instead of electromagnetic force.

2.2 Flux linkage under stage A

Fig.5 Equivalent drive circuit of the solenoid valve under stage A

Fig.5 shows the equivalent drive circuit of the solenoid valve under stage A.Rinjis resistance of the solenoid valve,Linjis inductance of the solenoid valve,Rcis the resistance including parasitic resistance and the resistance of other components in the drive circuit,iis drive current andUis power source voltage. The drive circuit in Fig.5 meets the following equation

dψ/dt+i(Rinj+Rc)=U

(4)

Forward difference is used to discretize the derivative in Eq.(4), and yielded as

(5)

whereMis sampling number. The sampling rate is constant, let Δt=tn+1-tn, and the flux linkage attn+1can be expressed as

ψn+1=Δt[Un-in(Rinj+Rc)]+ψn

n=0,1,…,M-1

(6)

Core and armature materials are all soft magnetic materials, and the residual magnetization of these materials at the start of stage A is very low, so the residual magnetization is neglected and the initial flux linkageψ0is 0 Wb. Flux linkage of stage A under different air gaps were calculated by Eq.(6).inandUnwere replaced with the experimental data shown in Fig.3a and Fig.3b respectively.

The square of flux linkage decreases with the widening of air gap under the same drive current when the drive current is low, but increases first and then decreases with the widening of air gap when the drive current is high in Fig.6. According to the ohm law of magnetic circuit[16], the flux linkage of solenoid valve decreases with the widening of air gap, which is different from the results shown in Fig.6. The change of flux can lead to the generation of eddy current inside core and armature materials. The flux linkage changes quickly when transient drive current is applied. The smaller the width of air gap, the faster the changing of flux. So eddy current is more serious when the width of air gap is small. It can be concluded that the eddy current inside core and armature materials lead to the results shown in Fig.6. Besides, it can be seen from Fig.6 that magnetic saturation of core and armature materials happened.

Fig.6 Relationships between the square of flux linkage of stage A and drive current under different air gaps

The square of flux linkage under different air gapsis almost the same at the beginning in Fig.7. Then the square of flux linkage under 0.1 mm wide air gap becomes the biggest with the rising of drive current. Therefore, in order to acquire the maximum electromagnetic force of this solenoid valve, the 0.1 mm wide air gap rather than the 0.06 mm wide air gap should be chosen.

Fig.7 Relationships between the square of flux linkage of stage A and time under different air gaps

2.3 Flux linkage under stage E

The equivalent drive circuit of stage E is similar to that of stage A shown in Fig.5. The difference of equivalent drive circuit between stage A and E is that the power source is charged in stage E and the drive current direction of stage E is opposite to the direction ofishown in Fig.5. Besides, the value ofRcunder stage E is different from the value ofRcunder stage A. In order to guarantee that the flux linkage is continuous, the flux linkage of stage A and E under the maximum current should be the same on condition that the maximum drive current under the same air gap are equal. The flux linkage of stage A under the same current is used as the initial flux linkage of stage E approximately. Drive current of stage E decreases with the increasing of n. The square of flux linkage of stage E under different air gaps were calculated by Eq.(7) and shown in Fig.9.

ψn+1=ψn-Δt[Un+in(Rinj+Rc)]

n=0,1,…,M-1

(7)

Fig.8 indicates that flux linkage of the solenoid valve under stage E increases first and then decreases with the widening of air gap under the same drive current. Besides, flux linkage of the solenoid valve under stage E are different and are not 0 when the drive current is 0 under different air gaps.

Fig.8 Relationships between the square of flux linkage under stage E and drive current under different air gaps

Fig.9 Relationships between residual flux linkage under stage E and air gap

The residual flux linkage shown in Fig.9 increases first and then decreases with the widening of air gap, and it approaches 0 when the air gap is wide enough. Residual flux linkage can be caused by either hysteresis or eddy current. The residual flux linkage caused by hysteresis can not disappear unless coercive force is applied, but the residual flux linkage caused by eddy current can disappear automatically after the transient drive current decreases to 0. Therefore, the residual flux linkage corresponding to 0 drive current is caused by the eddy current inside the core and armature materials.

2.4 Comparison of the flux linkage between stage A and E

Flux linkages of both stages under two different air gaps are shown in Fig.10. Besides, Fig.10 shows that the difference of flux linkage under the same air gap becomes smaller with the widening of air gap. The flux linkage difference between stage A and stage E is also due to eddy current.

Fig.10 Comparison of the flux linkage between stage A and E

2.5 Effect of eddy current on the closing process of the solenoid valve

Assuming that the armature begins to move downward when the flux linkage of the solenoid valve is less than 2.5×10-3Wb, it can be seen from Fig.11a that the drive current of stage A and E is about 9.2 A and 6.5 A respectively when the flux linkage is 2.5×10-3Wb. If the maximum drive current of stage E is 16 A, the interval of stage A between 16 A and 9.2 A is Δt2, and the interval of stage E between 16 A and 6.5 A is Δt1in Fig.11b. Due to the eddy current inside core and armature materials, the delay between the time that drive current begins to decrease from 16 A and the time that armature begins to move downward is Δt1. If eddy current doesn’t exist like static state, drive current of stage E will fall along the drive current curve of stage A and the delay is Δt2. It is obvious that Δt1is smaller than Δt2, which means that eddy current can reduce the delay. Although it is shown in Fig.11a that flux linkage of stage E is bigger than that of stage A under the same drive current and the same air gap, eddy current can still reduce the delay because drive current of stage E decreases faster than that of stage A. The eddy current becomes less apparent with the widening of air gap, so influence of eddy current on the delay will become smaller with the widening of air gap.

Fig.11 Effect of eddy current on the closing process of solenoid valve

3 Inductance of the solenoid valve

3.1 Inductance of the solenoid valve

According to the definition of flux linkage

ψ=Linji

(8)

If remanence exists, flux linkageψis not 0 when the drive currentiis 0. The only way that can make Eq.(8) reasonable when remanence exists, is thatLinjapproaches infinite when the drive current approaches 0. Infinite inductance means that the drive current is impossible to increase, which is inconsistent with the experimental results. So Eq.(8) is only fit for the case that remanence does not exist and when remanence exists should be replaced with

ψ-NBrS=Linji

(9)

whereBris remanence intensity, andNBrSis the residual flux linkage of solenoid valve when the drive current decreases to 0. When remanence exists inductance of solenoid valve should be calculated by

(10)

When remanence does not exist inductance of solenoid valve should be calculated as

Linj=ψ/i,i≠0

(11)

As presented above, flux linkage of stage A is 0, but flux linkage of stage E is not 0 when drive current is 0. The residual flux linkage of stage E under different air gaps is shown in Fig.9. Inductance of solenoid valve under stage A and E should be calculated with Eq.(11) and Eq.(10) respectively. Fig.12a and Fig.12b demonstrate the inductance of stage A and E respectively.

Comparing Fig.6 with Fig.12, it can be concluded that the flux linkage under stage A is the maximum, and the inductance of solenoid valve is the biggest under the same air gap. The flux linkage under stage E is also biggest under the same air gap as stage A, but inductance of stage E under the same air gap is not the biggest due to the eddy current inside core and armature materials.

Fig.12 Calculated inductance of the solenoid valve under stage A and E

Now good explanations of the interesting phenomenon presented at the end of section 2.3 can be acquired. Inductance of stage E is obviously smaller than that of stage A over large scope of drive current whenδis 0.06 mm, and this is why the time needed in stage A is more than that in stage E under the same air gap on condition that drive current changes from 0 to 16 A. Besides, it is the difference of the law of inductance changing with drive current between stage A and stage E that leads to the difference of drive current changing between these two stages. It is also shown in Fig.13 that the difference of inductance between stage A and stage E under the same air gap becomes smaller with the widening of air gap.

Fig.13 Comparisons of the inductance between stage A and stage E

3.2 Evaluation of parasitic inductance

It is well-known that the permeability of soft magnetic materials increases first and then decreases with the increasing of magnetic field intensity, and inductance of solenoid valve is proportional to the permeability of core and armature materials[17]. Therefore, it can be inferred that the inductance of solenoid valve increases first and then decreases with the rising of drive current, which is different from the results shown in Fig.14.

Fig.14 Calculated inductance of the air core solenoid

In order to further investigate the reason for this difference, another experiment was conducted on an air core solenoid. The relative permeability of air is 1 and does not vary with drive current, so inductance of the solenoid is considered to be constant when it is driven by current with certain frequency. Inductance of the air core solenoid is 343 μH measured by precise inductance test equipment.

Drive current of the air core solenoid is similar to the drive current shown in Fig.3b. The drive current and power source voltage of stage A were recorded. Flux linkage and inductance of the solenoid were calculated with Eq.(6) and Eq.(11) respectively. The calculated inductance of the solenoid is shown in Fig.14. The calculated inductance of the solenoid decreases with the rising of drive current in Fig.14. The inductance decreases quickly when the drive current is less than 2 A, but decreases slowly and approaches a certain valve gradually when the current is more than 2 A. The inductance of the solenoid is 343 μH and keeps constant, so the parasitic inductance was calculated by subtracting 343 μH from the calculated inductance of the air core solenoid. The parasitic inductance is also shown in Fig.14. The parasitic inductance of the drive circuit may be caused by properties of some electronic devices like diode when the drive current is low, and it will not be further discussed here.

From Fig.15, the flux linkage under 0.6 mm wide air gap is the minimum. The flux linkage due to parasitic inductance is shown in Fig.16 and far less than the flux linkage of the solenoid valve. The difference becomes larger with the rising of drive current.

Fig.15 Inductance of the solenoid valve

Fig.16 Flux linkage due to the parasitic inductance

The parasitic inductance of drive circuit is subtracted from the inductance of the solenoid valve of stage A and E under 0.06 mm wide air gap in Fig.14 respectively, and the results are shown in Fig.15. The inductance of both stages increases first and then decreases with the rising of drive current, but the inductance of stage E is smaller than that of stage A apparently over large scope of drive current.

4 Conclusions

A new discretizing calculation method of the solenoid valve flux linkage was proposed for the first time based on experimental results and the principle of the drive circuit were evaluated. The results indicate that the air gap, under which the transient flux linkage of the solenoid valve is the biggest, varies with drive current due to eddy current. The flux linkage of drive current descending process is bigger than that of drive current ascending process under the same drive current. The flux linkage is not 0 when the drive current descends to 0. Eddy current can reduce the delay between the time the drive current begins to descend and the time the armature begins to move downward.

New inductance calculation methods for the drive current ascending and descending process respectively were also presented. Influence of parasitic inductance was evaluated through a new experiment conducted on an air core solenoid. The inductance of drive current descending process is bigger than that of drive current ascending process over larger scope of drive current, but the difference becomes smaller with the increasing of air gap width. The differences of both flux linkage and inductance between drive current ascending and descending process are caused by eddy current inside the core and armature materials.

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(Edited by Cai Jianying)

10.15918/j.jbit1004-0579.201524.0105

TK 42 Document code: A Article ID: 1004- 0579(2015)01- 0026- 09

Received 2013- 10- 10

Supported by the National Natural Science Foundation of China(51076014); the Research Fund for the Doctoral Program of Higher Education(20101101110011)

E-mail: youtong@bit.edu.cn