ieee lumped-parameter model of the delay solenoid … · imovo~~~~~v im 0 (c) fig. 2. equivalent...

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. IE-29, NO. 3, AUGUST 1982

Lumped-Parameter Model of the Delay Solenoid Valvewith Integral Thermistor

WILLIAM G. HURLEY, MEMBER, IEEE

Abstract-In the past, many applications have been found for delaysolenoid valves using an integral thermistor for delay action. Theseapplications include oil and gas flow control in hydraulic andpneumatic systems used in a wide variety of situations, ranging fromthe residential heating furnace to large industrial controllers. Becauseof the large number of parameters involved in the design of thesevalves, no known analytical method exists to accurately predict timedelay and power consumption. This has resulted in a trial-and-errorapproach to design and thermistor selection involving expensiveprototypes for testing purposes. In this paper the author has de-veloped an accurate lumped-parameter model. This model clearlyillustrates those parameters which govern the terminal variables suchas power consumption and also allows thermistor selection for a givendelay based on manufacturers' specifications.The scope of applications of delay valves should be increased as a

result of greater predictability in valve performance and greaterflexibility and reduced cost in thermistor selection. The principle ofgeometric similarity is also established, which leads to scaled-downprototypes of larger valves for testing purposes.

I. INTRODUCTIONTHE BASIC electromechanical system employed in a typical

delay solenoid valve is shown in Fig. 1. It consists of athermistor in series with a coil, wound on a ferromagnetictube. The tube constrains the movement of the plunger. Cur-rent flowing in the coil sets up a mechanical force of electricorigin which attracts the plunger into the tube to allow fluidflow. The magnetic force on the plunger is opposed by a springwhich closes the valve when the system is de-energized. A cer-tain value of current is required to counteract the spring forceand open the valve. As soon as the system is energized, a cur-rent flows which is insufficient to open the valve. This causesthe negative temperature coefficient thermistor to heat upwhich in tum reduces its resistance, allowing more current toflow. Eventually, the current will increase to a value whichopens the valve and allows fluid to flow through it. This situa-tion is complicated by the fact that heat is also generated inthe coil resistance and in the ferromagnetic tube and plunger.In order to set up a consistent model, the electromechanicalsystem must be modeled to predict the current transient andthe thermal system must be modeled to predict thermistorresistance as a function of time. The two models are combinedto predict time delay. The steady-state power consumptioncan also be predicted using these models.

The key to the modeling process is the heat balance be-tween the thermistor and its surroundings. The thermistortemperature controls its resistance which in turn controls thecurrent in the system that sets up the electric force which

Manuscript received August 5, 1980; revised November 30, 1981.The author is with Ontario Hydro, Toronto, Ont. M5G 1X6, Canada.

Fig. 1. Electromechanical system of the delay valve.

opens or closes the valve. Because of the complicated relation-ships involved, a finite-difference method is used to solve thegoverning differential equations.

II. LUMPED-PARAMETER MODEL DEVELOPMENTTwo equivalent circuits of the delay valve are shown in

Fig. 2: Fig. 2(a) shows a parallel combination to represent thecore, while Fig. 2(b) shows the equivalent series combination.Rt is the thermistor resistance which is a function of tempera-ture. R, is the coil resistance and R,0 represents the corelosses equivalent series resistance. L, is the coil self-inductancewhich is a function of plunger position. The internal coilvoltage Eo will depend somewhat on the variations in R, andRt during operation, but these changes are insignificant.Magnetic nonlinearities are neglected since the terminalvoltage is constant in any given application. In the followingsections, each parameter will be examined and relationshipsestablished.

Coil Self-Inductance LcBy considering the magnetic field system of Fig. 1, it can

be easily shown that the inductance varies inversely withplunger position [11

x1 -

g

Lo= APo2g

where Lo is the value of L.(x) for x = 0, i.e., with the valveopen (Fig. 1). g is a constant which depends on the gap be-tween the plunger and tube; x is the plunger position; , is themagnetic permeability of the plunger material; Ap is theplunger cross-sectional area; andN is the number of coil turns.

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225

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+Ivo R L gco bM

-((a)

EoRt+ RC)O

IC

Imovo~~~~~~vIm 0

(c)

Fig. 2. Equivalent circuits and phasor diagram of the delay value. (a)Parallel circuit. (b) Series circuit. (c) Phasor diagram.

In practice, Lo and g are found from measurements of coilself-inductance in the valve open and closed positions.

Equivalent Core Resistance R,oThe core losses consist of hysteresis and eddy current

losses, and so Rco would be a function of plunger positionwith a relationship similar to that for L(x). Before the valveopens, the thermistor resistance is very large compared to Rco.We require the value of Rco for the closed valve to determinedelay, we require the open-valve value to determine steady-state core losses. These two values are sufficient for modelingpurposes.

Thermistor Resistance RtFor negative temperature coefficient thermistors, the re-

sistance R is related to the absolute temperature T by [2]

RR-oefl[(I1T)-( 1 TO) l

where Ro is the resistance at temperature To(K), and ,(K) isa constant of the thermistor. Normally, Ro is specified at250C. Ro and j3 are given in the manufacturer's specifications.

For a voltage E and current I applied to the thermistor, thedissipation constant is [3]

6 = EI MX/OC.T- Ta

EI is the thennistor internal heat generation, and T is thethermistor temperature with ambient temperature Ta. Be-sides being a function of ambient temperature, the dissipationconstant also depends on the mounting method.

The heat balance for a thermistor cooling in air is [31

-MCpdT = 6o(T -Ta) dt

where M is the mass and Cp the specific heat of the thermis-

tor. 60 is used to indicate free air cooling. Integrating theabove equation for thermistor initial temperature T1 gives

T- Ta = (T1 -T)e-"T°, MCpTo =

o0

so

r060 = MCp

where T0 is the time constant for the above cooling process.Thus the time constant is also a function of mounting of thethermistor, but the product r6 is a constant for any giventhermistor. Usually, the manufacturer supplies r0 and 60 sothat for any given application, r or 6 can be measured and theother found from the above relationship since

1T6 1 = T00.

Coil Resistance RCThe coil resistance is a function of coil temperature given

by

RC =R25 [1 + 025(T- 25)]

where R2 5 is the resistance at 250C and a!2 5 is the correspond-ing temperature coefficient of resistance. For copper a2 5 =0.00385/0C. Note that the coil has a positive temperaturecoefficient of resistance in opposition to the negative coeffi-cient of the thermistor. In general, for a wire of length L andcross section a we have

LR =p -

a

where p is the conductor resistivity. Thus the resistance isproportional to the number of coil turns. If we maintain thesame volume of conductor in the coil, and increase the turnsby a factor x, then the conductor cross section is reduced by xso that the coil resistance will vary with x2 or the square ofthe turns ratio

R2 N2 2

RI N,

III. STEADY-STATE ANALYSISAt any instant during valve operation for an applied voltage

VO and current Io, the following relationships hold (refer toFig. 2 for phasor diagram):

Power factorConsumed power

Induced voltageCore loss

Core loss componentof current

= cos oPo = VolO cos 0

Eo = Vo-Io(Rc +Rt)Pco =Po-(Rc+R yo)2

I =PcoICQ=CO

226


227HURLEY: LUMPED-PARAMETER MODEL OF DELAY SOLENOID VALVE

Core conductance

Magnetizing compon-ent of current

Magnetizingsusceptance

Core resistance

Core reactance

Coil inductance

Pcog0o =E2

IV. TRANSIENT ANALYSIS

Dynamic Model-Current to Open I, p

The magnetic co-energy of the system in Fig. 1 is

,MO -1o 'cO

b ImoR o =

Eo

bmo=gco2 + bo2

L --2rrf

So by measuring PO, VO, Io, and R, and Rt, all otherparameters can be found. Rt should be measured immediatelyafter power is removed since its value changes quickly. f is thefrequency of the applied voltage (in units of hertz).

After the valve opens, the thermistor continues to heat upuntil an equilibrium is established between the heat generatedin the valve coil and thermistor and the heat dissipated toambient.

If Po, VO, Io are measured in the steady state after the valveopens, then Lo and R,o can be found since x = 0. If the valveis forced closed, and again PO, VO, Io measured, then R,o =RC0C in the valve closed position is found (the thermistor canbe removed for this test to facilitate readings). Also, gis foundfrom

LoXc I + XC/9

where Xc is the value of x, the plunger position when the valveis closed.

Finally, the steady-state losses are

Coil copper losses P_=RcJ 2

Wm' = f X(i' x) di'.

SinceX=LiandL,(x)=Lo/(I +x/g) we obtain [1I

WIn L0i2Wm'- o2(1 + x/g)

The force of electric origin is then

aWm'fe =ax

Loi2

2g(1 +±Xlg)2

Applying Newton's Law to the plunger, neglecting damping wehave for a current Io

d2x L0102M- +K(x-i)= _toodt2 2g(1 +x/g)2fS(x) = -K(x -1).

fS(x) is the spring force acting on the plunger, K is the springconstant, and I is the value of x for which the spring force iszero. For stable equilibrium

d2xM- =0

dt2

so

K(x-l2)g+ ± / 0.2g(1 + Xlg)2

Thermistor Pt =R to2dissipation

Core loss Pco = R oIo2Total dissipation PV = PC Pt + PcO.In these equations, Rt and Rc are steady-state values. Just

after the valve opens, the coil is still close to its ambient tem-perature. Since the thermistor has a negative temperaturecoefficient of resistance, and the coil has a positive coefficient,their sum tends to remain constant during the transient. Thusthe steady-state total power dissipation may be estimated bymeasuring Rc when the coil is cold, and measuring Rt justafter the valve opens. This was found to give good results inpractice. For this calculation, the value of current is thatwhich is required to open the value Ip, which will be calcu-lated in the next section.

The graphic solution of this cubic equation is shown inFig. 3. At X1 and X2 the plunger is statically balanced. Thethird root has a negative value for x. It is obvious from thegraph that X1 is dynamically unstable and that X2 is dynam-ically stable. Perturbations at XT to reduce x cause it to befurther reduced by the electric force; on the other hand, aperturbation to increase x at X2 results in the plunger return-ing to X2 under the influence of the electric force. This can beshown mathematically by comparing the derivatives of theforce functions given above at X1 and X2. In Fig. 3, curvesA and B illustrate situations where no statically stable pointsexist and where one statically stable point exists, respectively.In a delay solenoid valve, when the system is first energized,the current is limited by the thermistor, and curve B describesthe system. As the thennistor heats up, the current increasesand the dynamically unstable equilibrium point at X1 movestowards Xc: the value of x corresponding to the valve closed



Ope n ClosedXc

Fig. 3. Equilibrium points for the delay valve.

position. When X1 = Xc, the valve opens and curve C applies.The current continues to increase, and eventually reaches asteady-state value represented by curve A and the valve re-mains open.

When the valve opens, the current is reduced due to theincreased inductance at x = 0 (Lo). As long as the electricforce at x = 0 corresponding to this reduced current is greaterthan the spring force, the valve remains open. Solving thecubic equation for x = Xc gives the current required to openthe valve

K(l XC)2g(1 +Xc/g)2 1/2Iop =Lo

ThermalModel-Time to Open TdHaving found Iop, we now wish to determine the time

taken to reach that value. As stated in Section II, the dissipa-tion constant and the time constant for the thermistor aredependent on the method of mounting. As an integral thermis-tor, there is some electrical insulating material between thethermistor and the coil. Since the time constant of the coppercoil is much greater than that of the thermistor, we treat thecoil as a heat sink at ambient temperature for transient analysis.

The dissipation constant 6, for the thermistor mounted inthe valve may be found as follows;

Apply a voltage E (ac or dc) to the thernistor in the valveand measure the steady-state current I, the resistance R is

R = E/I

so the temperature T can be found from

R =ROefN(lT)-(/To)Jand

EIT- Ta

where Ta is the ambient temperature during the test.From Section II, the thermistor time constant is

0oroTV =

6 o, ro, Ro, and ,B, are given by the manufacturer.

From Fig. 2 and our equivalent circuit in Section III, thecurrent in the coil just before the valve opens is

VORC + RC0C ±Rt(t) +j2TfLc(XJ)

where Xc is the value of x in the valve closed position as be-fore.

The heat generated in the thermistor is

Pt = RtIo(t)2

so at time t, the heat balance for the thermistor is

Pt(t)dt =MCpdT+ v(T- Ta)dt. (1)

Combining the above expressions for R(T), Pt, and IO(t), weobtain

Roelf(l/T)-(l/TO) V02[RC ± Rcoc +RoeP[(1IT)- (1ITO) I] 2 + [ 27rfLC(Xc)] 2 d

==MCpdT + 6v(T-Ta) dt.

This is a very complicated differential equation for thermistortemperature as a function of time. To simplify the solution,we introduce a finite-difference type solution.

We assume Pt is constant in the time interval (ti-t1) andintegrate (1) to obtain

±Pt(t)1)t"iTa = (T-Ta)e tITu + t(1 _eAt/r)5u

where At = t1 - ti.

By taking sufficiently small time intervals, acceptable accuracyis obtainable.

In the interval ti-tj, we have

Rti = Roe fl I Ti3-(i ITO) I

I- ~~~V0oi = RC + RcOc + Rti + !2iTf c(Xc)

Pti = Rtijoi2

(Tj-a= (Ti-T)e-T Jr (i(-e-lv

t= ti + At.

The initial conditions are T1 = Ta, t1 = 0.When Ioi becomes equal to Iop calculated in the last

section, the elapsed time is equal to Td, the valve delay. Theabove method is amenable to solution on a programmablehand-held calculator.

V. EXPERIMENTAL RESULTS AND DESIGN EXAMPLEThe above model was applied to two valves which had

completely different physical characteristics and also haddifferent thermistors. The results are shown in Table I.

228


HURLEY: LUMPED-PARAMETER MODEL OF DELAY SOLENOID VALVE

TABLE IEXPERIMENTAL RESULTS

Valve 1 Valve 2

Calcu- Calcu-lated Measured lated Measured

Power at open Pop (W) 8.31 _ 4.0Steady state power Pv (W) - 8.3 3. 7

In general, there is excellent agreement between theoreticalpredictions and practical measurements. It is interesting tonote that the valve power when it is opened tumed out to be avery good estimate of the steady-state power as discussed inSection III.

Let us redesign Valve 1 to reduce its power consumptionwhile maintaining the same time delay. We assume that thesame volume of copper is to be maintained. From our lumpedparameter model in Section II

Lo OcN2

Rco ocN2

Rc o N2

Thus by increasing the number of turns we can find thenew values of the above parameters. The model is used to findthe new value of time delay and power consumption. Increas-ing the number of turns with smaller wire reduces the overallpower consumption but increases the time delay. Having fixedthe turns for the correct dissipation, we must choose a newthemistor to give the original delay. In Section IV, we foundthat the delay was proportional to the product r8 = MCp, thethermistor heat capacity. We stipulate that the delay is alsoproportional to Ro since it limits the initial current in thecoil. In general, the delay is

TdccROMCp or TdocRor.

These values are supplied by the manufacturer. If we want a

thermistor to give half the time delay then we must choose

one with a product Ro0r reduced by one half. This techniquproved very successful. This procedure eliminates the tediouand costly task of building a new sample for each thermistoibeing considered. Note that the power consumption will no,change very much if the thermistor is changed since it makerthe least contribution to overall power consumption comparecwith the coil and core losses.

VI. CONCLUSIONSA model has been developed which simulates the operatior

of the delay solenoid valve. Practical measurements haveshown that the model is very accurate in predicting the twcmost important parameters: power consumption and timedelay. The real strength of the model lies in its ability to checkvarious design modifications without resorting to costlyprototype models.

Another area where the model is of immense value is in thequality and process control for the valve. From the model, thetwo values most likely to vary are the thermistor resistan-ceand the core equivalent resistance. As stated in Section II,the equivalent core resistance is negligible when the valve isclosed so that fluctuations in time delay would probably bedue to variations in the thermistor resistance. On the otherhand, core losses make a significant contribution to overallpower losses, therefore, loss variations are most likely due tochanges in the valve magnetic material.

In all, over 15 parameters are used in the model, whichmeans that every component in the electromechanical systemof the valve can be studied and fine tuned for optimumdesign.

ACKNOWLEDGMENTAppreciation is extended to K. R. Cribb and members of

the Evaluation Laboratory at Honeywell Limited, Canada.

REFERENCES[11 H. H. Woodson and J. R. Melcher, Electromechanical Dynam-ics. New York: Wiley, chs. 1, 2, 3, and 5.[21 F. J. Hyde, Thermistors. London: Iliffe, 1971, ch. 2.[31 Electronic Industries Association, Standard RS-275-A, ThermistorDefinitions and Test Methods, Washington, DC, 1971.

22(


ieee lumped-parameter model of the delay solenoid … · imovo~~~~~v im 0 (c) fig. 2. equivalent...

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