Modelling of a linear proportional electromagnetic actuator and possibilities of

sensorless plunger position estimation

Ivor. Dülk; Tamás. Kovácsházy

Budapest University of Technology and Economics, Department of Measurement and Information Systems Budapest, Hungary

ivor@travex.hu; khazy@mit.bme.hu

Abstract— In flow control and in hydraulic systems used in

automotive industry, e.g. brakes or automatic transmission

units, one of the basic controlling problems is the precise

adjustment of oil (fluid) pressure. As an electromechanical

actuator, switching and linear variable solenoids are widely

used in such applications. In this paper, a magnetic valve is

analyzed in details. The main features of the system are

highlighted and a proper solenoid model is intended to be

established as well. Reliability and cost-efficiency are key

requirements for actuators in the field of automotive

engineering. For this purpose, sensorless position estimation

for linear translational valve actuators is also proposed in this

paper. According to this principle, the plunger position could

be predicted from the magnetic valve’s electrical impedance by

using its transducer characteristics during excitation. As a

great advantage, an external position or pressure transducer

becomes unnecessary thus saving additional costs and further

improving system robustness. This paper also presents and

studies two different methods for sensorless position prediction

which can be implemented under pulse width modulated drive

conditions.

Keywords: magnetic valve, modelling, sensorless estimation

I. INTRODUCTION

Cost efficiency and system reliability are key factors regarding engineering products and competitive edge in the market. In the field of flow control, solenoids are widely used because of their simple construction, ruggedness and low prime cost. These are translational, limited travel electromechanical devices consisting of two basic types, the switching (on-off) and linear ones respectively. The former is simpler in design and implemented in such applications as electrical contactors, valves etc., however, for either high precision force actuation or positioning, linear solenoids are used very often. Compared to the switching type, this family of devices has a linear transfer function but is more complex in design and might contain some sensing element, therefore resulting in additional expense. A simplified schematic of such a device is presented in Fig. 1.

This paper concentrates on the discussion and analyses of a linear solenoid used for pressure controlling purposes (hydraulic flow control) in automatic transmission units, though the analysis can be generalized to other fields of application as well. According to Fig. 1, such valve works as a variable hydraulic resistance by altering the outflow orifice, which is a direct function of the plunger position. In

order to achieve fast, precise and stable output pressure the output pressure or the position of plunger has to be accurately measured. The latter can be realized by either using an external transducer or a solenoid with linear characteristics, both solutions are rather expensive. As an alternate solution, sensorless position prediction [1], [6] might provide a much more cost efficient yet still robust choice since external or built in sensors become unnecessary. Nowadays, there is much research done on sensorless principle, a few, different solutions published in [1], [5], [8], [9]. Nevertheless, sensorless technique also makes it possible to use switching solenoids in applications where linear control is required [2], [3], thus further reducing the costs.

Figure 1. Cross section of a solenoid.

II. SYSTEM DECOMPOSITION

The working of a solenoid is generally as follows (shown in Fig. 1). Due to the excitation voltage applied to the electrical terminals an energizing current starts to build up in the solenoid winding. This current produces a magnetic flux that closes through the solenoid’s housing, plunger and air gaps which form a magnetic circuit. The magnetic field, through the main air gap, exerts an attractive force on the plunger intent to pull it inside the housing, which force is then counteracted by the compression of the valve return spring. In equilibrium position, these two forces are equal, thus the desired plunger position could be derived by adjusting the input voltage.

Based on the following reasoning, a solenoid actuator is actually an electromechanical converter, producing mechanical plunger displacement to input voltage, consisting of three unique subsystems [7]. The first one is the electrical stage of the input side that converts voltage to magnetizing current. The second one, which actually connects the input and output stages is the electromechanical or magnetic one converting the current to flux and magnetic force. It consists of a highly nonlinear magnetic circuit, which the housing,

89978-1-61284-361-2/11/$26.00 c©2011 IEEE

plunger etc. represent [4]. The third subsystem is the mechanical output side of the solenoid, such as the return spring and plunger (mass), producing displacement to force excitation. To be exact, the overall mechanical excitation is the sum of the magnetic force and external load, the latter is caused by the fluid pressure on the plunger’s sealing surface. Of course, ambient temperature also has a key role in the overall system behavior. According to the above explanation, a block diagram of an electromechanical converter/actuator can be drawn as shown in Fig. 2.

Figure 2. Block diagram of a magnetic valve.

The three major input quantities of the system are the temperature, input voltage and external load (either fluid pressure) respectively. Since magnetic properties (e.g. leakage) thus inductance are also determined by core geometry, both the impedance in the electrical side and current to force transfer in the electromechanical side are affected by the plunger position as indicated in Fig. 2. So as to establish a proper solenoid model all three subsystems have to be separately investigated and their approximate transfer functions obtained. References [4] and [7] provide some literature on modeling solenoids.

A. The Electrical Subsystem

The electrical input side mainly consists of the energizing coil that represents a resistive and inductive component [4], [7]. From Fig. 2, the impedance is also affected by temperature and plunger position. The voltage equation can be expressed as below:

( )dt

x,idRiV

ψ+⋅= (1)

In (1) the first term refers to the resistive voltage drop (coil resistance) and the second one is the induced voltage caused by the rate of change of flux linkage (term ) which is both current and position dependent. Current dependency can be explained by the highly nonlinear magnetizing curve which shows the effect of saturation and non constant permeability. By further expressing the flux linkage in (1) equation (2) can be obtained, which is equivalent to (3) with Li= substitution.

( ) ( )dt

dx

x

x,i

dt

di

i

x,iiRV ⋅

∂

ψ∂+⋅

∂

ψ∂+⋅= (2)

( ) ( )dt

x,idLi

dt

dix,iLiRV ⋅+⋅+⋅= (3)

The third term in (2) and (3) refers to the motional back e.m.f. caused by the change in air gap distance (core geometry) due to plunger movement. In case the plunger stands still it can be omitted. For compactness and reduced

costs the core is made of a high permeability material and operated also in the saturation region, therefore the flux linkage, thus the electrical impedance is expected to be greatly nonlinear with parasitic effects such as leakage, hysteresis and eddy current losses also present. A brief measurement process was carried out to get a glimpse of the electrical behavior. The test solenoid was connected to a precision component analyzer (LRC meter) and a series RL model was recorded versus the frequency at two fixed plunger positions. The results are presented in Fig. 3, 4.

Series inductances at 1 Vac, no bias, LRC meter

0

10

20

30

40

50

60

10 100 1000 10000 100000

Frequency [Hz]

Ind

uc

tan

ce

[m

H]

Plunger released Plunger pulled in

Figure 3. Measured inductance data versus frequency and position.

Series resistances at 1 Vac, no bias, LRC meter

1

10

100

1000

10 100 1000 10000 100000

Frequency [Hz]

Resis

tan

ce [

Oh

m]

Plunger released Plunger pulled in

Figure 4. Measured resistance data versus frequency and position.

The effect of plunger position (air gap distance) on the electrical impedance is significant. However, the series inductance diminishes with the frequency, most probably because of the more intense parasitic magnetic phenomena, such as eddy currents and hysteresis [5]. A change in plunger position also becomes less significant. On the other hand, the resistive part increases, which is most probably the contribution of increased core losses (eddy currents) [5]. According to Fig. 2, the temperature dependence also had to be counted with. It was briefly studied by putting the valve into a heat chamber and repeating the LRC measurement, however, measurement results are not discussed here. Further electrical analyses are presented in the next chapters.

90 2011 12th International Carpathian Control Conference (ICCC)

B. The Electromechanical Subsystem

An electromagnetic actuator incorporates a highly nonlinear magnetic circuit as it was already shown in Fig. 3. A key problem in modeling the core properties is the nonlinear relation between flux linkage and current; and the saturation region. Moreover, the temperature and plunger position dependence also have to be counted with. Papers [2], [4], [7] give examples of partial flux linkage approximation. The housing, plunger and air gaps form a magnetic circuit to which an approximate magnetic reluctance and magnetizing current to flux relation can be written in (4) and (5).

MRiN ⋅Φ=⋅ (4)

μ⋅μ⋅=

ri0i

iM

AR (5)

Since the relative permeability of air is at least three orders smaller than the core’s, the overall magnetic reluctance is mainly determined by this part given that the length of gap is not insignificant. As the air gap has a linear magnetizing curve, core nonlinearities are suppressed and the flux linkage function can be well approximated by a linear function as in [4]. Yet, flux linkage, besides plunger position, also depends on magnetizing current thus in the higher current operating region the core behavior will become dominant (saturation) despite any air gap. In [2], [4] this phenomena is approximated by a 1/x function. For a complete model, additional core nonlinearities like hysteresis, flux leakage and eddy currents ought to be modeled, which is an extremely difficult task. Regarding the electromechanical conversion, the attractive magnetic force on the plunger is the contribution of magnetic energy accumulated in the working air gap and core. From (6) the magnetic force can be derived from the overall co energy.

( ) ( )⋅

∂

Ψ∂=

∂

∂=

i

magnetic dix

x,i

x

x,iEF (6)

In order to establish the electromechanical transfer function the magnetizing current versus magnetic force relation has to be recorded. For this purpose and for further mechanical analyses, a designated test environment was built capable of measuring the plunger displacement and simulating an adjustable external load since fluid pressure might vary dynamically. With the test bench, the static current to plunger displacement behavior was recorded. The results are plotted in Fig. 5, from which a significant degree of nonlinearity and hysteresis is observable. The hysteresis is made up of both magnetic and mechanical, the ratio of them is yet unknown. As expected, the necessary coil current for the same plunger position also greatly depends on the external load, a parameter that has to be counted with in applying sensorless position prediction. In technical literature related to solenoids e.g. [1], [4], [5], [6], [7], the problem of magnetizing current (thus plunger position) dependence on a varying external load is not covered in details.

Input current vs. plunger position, load dependence

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

Current [A]

Plu

ng

er

po

sit

ion

[m

m]

2 N 3 N 4 N

Figure 5. Static electromechanical characteristics.

C. The Mechanical Subsystem

The mechanical side of the solenoid actuator basically consists of the plunger, return spring and some damping, thus a second order linear system approximation seems adequate. The equation of motion can be written as (7).

*

dampingspringfluidmagnetic FFFFFxm +−−+=⋅••

(7)

In (4) the term F* refers to additional forces that might be taken into account for a more accurate model, such as dry friction, mechanical hysteresis of the return spring and the gravitational force of the plunger if oriented in such direction. A simple way to model dry friction is a “signum” function. Since the magnetic valve is sunk into oil, this part is not so significant. The force Ffluid (external force) is produced by the fluid pressure integrated on the corresponding surface of plunger. By expanding (7) and omitting F* (8) can be derived.

•••

⋅−⋅−⋅+=⋅ xbxkdAPFxmA

fluidmagnetic (8)

The term x refers to plunger position, k to the spring constant, b to the viscous damping factor and m to the mass of the plunger. In real applications, the spring is initially pre compressed thus producing the necessary sealing force if no current is applied. In case the force caused by fluid pressure is not really comparable to the initial spring force, this term can be also neglected. The spring constant can be determined from the compression to return force characteristics (measured by the test bench) while the other parameters can be derived from the damped free oscillation of the mechanical system (plunger mass and damping factor).

III. METHODS FOR SENSORLESS PLUNGER POSITION

ESTIMATION

The sensorless principle utilizes the transducer characteristics of the actuator, namely there is information regarding the plunger position encoded in the electrical impedance. By measuring current and voltage signals, the actual plunger position can be determined without any external sensor [1], [6], [9].

2011 12th International Carpathian Control Conference (ICCC) 91

From Fig. 3 and Fig. 4 it has become visible that the plunger position has a significant effect on both series inductance and resistance, provided a series RL model is used. So as to measure for example the inductance, a non stationary test signal is required. Since solenoids are most commonly driven by pulse width modulation, the impedance measurement has to be superimposed on the PWM. In [1] the concept of incremental inductance is proposed that operates with rate of current change. Paper [5] uses the time constant of exponential response to PWM pulses, from which the position is calculated by determining the effective resistance as it also depends on the position (Fig. 4). In [6] a method similar to [1] is proposed based on the measuring of current slope. Next, two different techniques are discussed.

A. Estimation by Measuring the Flux Linkage

In the first case, the drive PWM itself is used as a test signal like in e.g. [1], [6]. During the ON period, the coil is being energized and flux starts to build up, the current increases. By taking an approximate, applied volt per seconds area on the magnetizing inductor during the PWM ON period and the corresponding change in coil current, an inductance value can be calculated. The difference compared to [1] and [6] is that a time integral of volt per second area (flux linkage swing) is measured instead of a i/ t current slope. If the PWM switching frequency is chosen such that mechanical vibration of the plunger is insignificant, the third terms in (2) and (3) (back e.m.f.) can be neglected. Further rearrangement of (3) yields (9).

( )( )

ON_PWM

ON_PWM

ON_PWM

ON_PWM*

ii

dtiRV

x,iLΔ

ΔΨ=

Δ

⋅⋅−

= (9)

The term R refers to the coil resistance, V to the terminal voltage and L* to the predicted, averaged inductance for the corresponding flux linkage and current swings. The inductance is actually the slope of the magnetizing (B/H)curve. From (9), parasitic effects (eddy current) were omitted. Since the denominator contains the current change during the period of interest, this method might be susceptible to noise and a considerable current ripple is essential. However, by choosing a relatively low switching frequency, thus higher current ripple, and taking averages of L* the signal to noise ratio can be enhanced. Equation (9) was tested to study the inductance characteristics under different PWM switching frequencies, terminal voltages and external load conditions. Such analyses were not discussed in the mentioned literature. The measurement setup and results are plotted in Fig. 6, 7 and 8.

The supply changes have little effect whereas reduction in switching frequency offset the inductance curve. This could be denoted to smaller core losses. However, the predicted inductance shows a saddle like waveform similar to [1], it begins to decrease over a certain position. Regarding the load dependence, the same phenomenon is observable but the curves are shifted with the magnitude of external load. Yet, despite the considerable plunger position dependence the flux linkage thus inductance is also current

dependent. For larger plunger displacement a higher magnetizing current is required and the core saturates [1], [7]; therefore the inductance decreases. Moreover, the working gap also becomes smaller so its linearizing effect wears off.

Figure 6. Measurement setup for inductance measurement.

Inductance vs. position

0,015

0,017

0,019

0,021

0,023

0,025

0,027

0,029

0,031

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

Displacement [mm]

Ind

ucta

nce

[H

]

12V_500Hz 9V_500Hz 12V_350Hz

Figure 7. Results for plunger position vs. inductance relation.

Inductance vs. position, load dependence, 500Hz, 12V

0,02

0,021

0,022

0,023

0,024

0,025

0,026

0,027

0,028

0 0,2 0,4 0,6 0,8 1 1,2 1,4

Position [mm]

Ind

uc

tan

ce

[H

]

3.35N 4.39N 5.75N

Figure 8. Load dependence of inductance.

Regarding further control algorithms, the saddle shape seems problematic as multiple inductance values belong to the same position. Since the slope of inductance curve changes sign the current working interval can be located by a small displacement “pulse”. As for the “breakdown” region, the dependence of external load might be compensated by monitoring the coil current. Performing a linear regression on both sides of the inductance curves gives a good approximation.

92 2011 12th International Carpathian Control Conference (ICCC)

B. Sinusoidal Duty Ratio Modulation

This method is different from the previous one but still applicable to PWM. The main concept is that by modulating the excitation PWM’s duty ratio by a low frequency sine a sinusoidal excitation voltage component is produced with the modulating frequency. By measuring the attenuation and phase shift between this test signal and the sinusoidal current response of the solenoid, an approximate impedance thus inductance can be determined at the test frequency. In [8] and [9] similar methods are introduced applying a sine as a test signal. However, [9] connects a parallel capacitor and utilizes the resonance frequency for position prediction.

Compared to the first technique, it has a considerably lower estimation bandwidth since the test frequency is much lower than the switching one, however, it is less susceptible to noise and phase shift can be measured very accurately. The measurement setup and an experimental result are plotted in Fig. 9 and 10. Unfortunately no more data is yet available.

Figure 9. Setup for sinusoidal duty ratio modulation method.

12 V Usupply, 1 Vpp 50Hz F_test, 3 kHz F_switch

0,015

0,017

0,019

0,021

0,023

0,025

0,027

0,029

0,031

0,033

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

Plunger position [mm]

Ind

ucta

nce

[H

]

Inductance

Figure 10. Inductance results for sinusoidal duty ratio modulation.

Compared to the previous method, a similar saddle like inductance curve was obtained, however, with larger inductance values. The difference is most probably caused by the various excitations. Most importantly, a considerable plunger position to inductance dependence has been highlighted.

IV. CONCLUSION

In this paper, a brief description of electromagnetic actuators was given with the most important features and modeling steps being highlighted. For more robust and cost efficient solenoid control, two methods were introduced for sensorless plunger position estimation which gave very similar results. Regarding future work, additional measurements are necessary to establish a complete model and evaluate the impedance to plunger position relation. Hardware implementation and testing in real applications are also of paramount importance in the future.

REFERENCES

[1] Muhammed Fazlur Rahman, Norbert Chow Cheung, Khiang Wee Lim, “Position Estimation in Solenoid Actuators”, IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 32, NO. 3, MAY/JUNE 1996.

[2] “Conversion of a switching solenoid to a proportional actuator,” in JlEE Proc. Int. Power Electron. Con$, IPEC’95, Yokohama, Japan, vol. 3, pp. 1628-1633, Apr., 1995.

[3] K. W. Lim, N. C. Cheung and M. F. Rahman, “Proportional control of a solenoid actuator,” in IEEE Proc. lnd. Electron. Soc. Annu. Meeting, IECON’94, Bologna, Italy, vol. 3, pp. 2045-2050, Sept., 1994.

[4] N. C. Cheung. K. W. Lim, and M. F. Rahman, “Modeling a linear and limited travel solenoid,” in IEEE Proc. Ind. Electron. Soc. Annu. Meeting, IECON’93, Hawaii, vol. 3, pp. 1555-1563, Nov., 1993.

[5] Dieter Pawelczak, Hans-Rolf Tränkler, “Sensorless Position Control of Electromagnetic Linear Actuator”, IMTC 2004 – Instrumentation and Measurement Technology Conference Como, Italy, 18-20 May 2004.

[6] Jyh-Chyang Renn, Yen-Sheung Chou, “Sensorless Position Control for a Switching Solenoid”, JSME International Journal, Series C., Vol. 47, No. 2, 2004.

[7] Y. Wang, T. Megli, and M. Haghgooie, K.S. Peterson and A.G. Stefanopoulou, “Modeling and Control of Electromechanical Valve Actuator”, SAE 2002-01-1106.

[8] Shang-Teh Wu, Wei-Nian Chen, „Self-sensing of a solenoid valve via phase detection”, Advanced Intelligent Mechatronics, 2009. AIM 2009. IEEE/ASME International Conference, 01 szeptember 2009, Print ISBN: 978-1-4244-2852-6.

[9] Maridor, J., Katic, N., Perriard, Y., Ladas, D., “Sensorless position detection of a linear actuator using the resonance frequency”, Electrical Machines and Systems, 2009. ICEMS 2009. International Conference, 15-18 Nov. 2009, Print ISBN: 978-1-4244-5177-7.

2011 12th International Carpathian Control Conference (ICCC) 93