- Email: [email protected]
Dynamic Modeling of a Piezoelectric Actuated Fuel Injector
Proceedings of the 2009 IFAC Workshop on Engine and Powertrain Control, Simulation and Modeling IFP, Rueil-Malmaison, France, Nov 30 - Dec 2, 2009
Dynamic Modeling of a Piezoelectric Actuated Fuel Injector Chris A. Satkoski, Gregory M. Shaver, Ranjit More, Peter Meckl, Douglas Memering Energy Center, Herrick Labs, School of Mechanical Engineering, Purdue University, W. Lafayette, IN (e-mail: [email protected]). Abstract: As engine designers look for ways to improve efficiency and reduce emissions, piezoelectric actuated fuel injectors for common rail diesel engines have shown to have improved response characteristics over solenoid actuated injectors and may allow for enhanced control of combustion through multi-pulse profiles or rate shaping. This paper summarizes the development of a simulation model for a piezoelectric fuel injector and associated driver that can be used for injector design and control system verification. The model injection rate, piezo stack voltage, and piezo stack current are compared to experimental injection rig data for two different rail pressures. Keywords: Actuator Development; Combustion Control; Control Design; Controls; Engines; Modeling; Piezoelectricity; Simulation 1. INTRODUCTION Piezoelectric actuation of common rail type fuel injectors for diesel engines offers an enhanced ability to control injection over their solenoid counterparts. This can be accomplished due to their faster response time and an ability to generate larger forces. These characteristics can lead to a greater control of the injection event through multi-pulse profiles or rate shaping. With improved control of injection, engine designers will have better flexibility in managing efficiency and emissions at all operating points of an engine for both conventional and advanced mode combustion strategies. Previous studies have shown that piezoelectric injectors can improve air entrainment (Fettes and Leipertz, 2001), spray development, and boost injection velocity due to the fact that the needle lift process is noticeably faster than a solenoid actuated injector (Lee et al., 2006). Also, it has been shown that with advanced actuator driver techniques, such as pulse width modulation (PWM), better control of the piezo stack can be obtained and dwell times between pulses can be half that of a solenoid injector (Oh et al., 2007). This paper describes a dynamic simulation model of a prototype piezoelectric fuel injector. Initial testing has shown that pressure dynamics inside of the injector cause fueling from an injection event to influence a subsequent event, resulting in minimum separation times being achieved. A simulation model, such as the one outlined in this paper, can be used to develop design strategies to mitigate pulseto-pulse interactions and optimize performance. 2. THE PIEZOELECTRIC FUEL INJECTOR
Fig. 1. Operating Principle of a Prototype Piezoelectric Fuel Injector (Note: specific design details suppressed due to confidentiality)
The design of the injector used in this study incorporates a hydromechanical mechanism that has two essential functions - to convert the high downward expansion force of
the piezo stack into an upward needle lifting force and to convert the short stroke of the piezo stack into a longer stroke for the needle.
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2.1 General Operating Principle As shown in Fig. 1, when the piezo stack expands, the bottom link is forced down into the needle lower volume, raising the pressure, and forcing the needle upward. Upon upward motion of the needle, fluid is pressurized in the chamber above the needle, called the needle upper volume. Fluid is forced out of the needle upper volume into the body volume through spring-loaded check valve above the needle with an orifice in the center causing smooth, upward motion of the needle. Upon retraction of the piezo stack, the bottom link is forced upwards by the needle lower volume pressure. Upward bottom link movement lowers the pressure in the needle lower volume, closing the needle. At certain rail pressures, the check valve can be forced open by the body pressure above it, allowing the needle upper volume pressure to rapidly increase, quickly closing the needle. The actuator used in this injector is a 130 disc, 1000V piezoelectric stack. All of the discs are enclosed between electrodes and are wired in parallel to a single 1000V driver. This multi-disc, parallel actuator configuration is common for piezoelectric actuators, as it allows maximum deflection for a given amount of piezoelectric ceramic. Thinner discs can be used with a smaller voltage, but are not used currently in this application due to the high loads. The whole stack is encased in a protective housing. 3. MODELING
where Albot is the bottom link “ledge” area (critical for displacing liquid in the needle lower volume) and Anbot is the needle bottom area. The needle upper volume is much like the needle lower volume, except it has a flow term. dPuv dVuv = −ωcvf − (4) Cuv dt dt where ωcvf is the check valve flow of liquid out of the needle upper volume, Cuv = Vβuv is the needle upper volume liquid capacitance, and uv refers to the needle upper volume. dVuv dt is also a function of our states and is described by V˙ uv = Antop (y˙ − x). ˙ (5) where Antop is the area of the needle top. Flows are modeled with a standard orifice equation: 2 ω = Cd A0 ΔP (6) ρ where ω is the flow, Cd is the coefficient of discharge, A0 is the orifice area, and ρ is the liquid density. Terms in this equation can be grouped to fit the form of a flow resistance relationship: √ ρ Rω = ΔP where R = (7) 2Cd 2 A0 2 such that the check valve flow can be given as: ωcvf = Puv − Pbv /Rcv
(8)
The “lower body” includes all of the hydromechanical interaction from the bottom link to the injection of fuel. The “upper body” includes the electro-mechanical interaction from the voltage driver to the piezo stack, down to the bottom link. The upper and lower bodies are linked by the displacement of the piezo stack and the interaction force in between the bottom and top link. The coordinate systems for the injector can be seen in Fig. 1.
where Pbv is the pressure of the body volume and Rcv is the resistance of the check valve. It is important to note that the check valve has a different resistance when open and closed. The resistance is actually a function of its displacement relative to the plunger. It can be most accurately captured by a more complex parallel disc flow equation. For simplicity, this model will use an approximate constant flow resistance when open, and a different constant when closed.
3.1 Lower Body Modeling
After the needle opens, fuel will flow down the body, past the needle into the sac volume, which is the small volume enclosed between the bottom of the injector needle and the inlet to the spray holes, and then through the spray holes out of the injector. Two restrictions impede free flow from the body. One is the restriction of the spray holes themselves, Rsh . The other is the resistance past the needle, Rneed , which changes with needle lift such that the total restriction for fluid out of the injector is given by the following expression: Rtotal (x) = Rsh + Rneed (x) (9)
The injector control volumes are modeled using a coupled continuity and state equation for the operating liquid (Merritt, 1967). Temperature effects on properties are assumed negligible. V0 dP dV0 Σωin − Σωout = + (1) dt β dt where ωin is the volumetric flow into the control volume, ωout is the volumetric flow out of the control volume, V0 is the mean volume, β is the bulk modulus of liquid, and P is the pressure. In the case of the needle lower volume, we can assume that there are no flows in or out during needle opening and closing events. Then equation (1) becomes dPlv dVlv Clv =− (2) dt dt where the subscript lv refers to the needle lower volume and Clv = Vβlv is the liquid capacitance. dVdtlv is a function of our states and is described by V˙ lv = Albot y˙ + Anbot x. ˙ (3)
The flow out of the injector can be fully described by ωiof = Pbv − Pcyl /Rtotal (x) (10)
where ωiof is the flow out of the injector and Pcyl is the pressure of the combustion chamber. Rail-to-body flow can be described by: ωrtb = Prail − Pbv /Rrail
(11)
where Prail is the rail pressure, and Rrail is the sum of the restrictions between the rail and the body of the injector.
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Fig. 2. Injector Needle Free Body Diagram An equation can be written for the body, assuming no relevant change in volume: dPbv Cbv = ωrtb + ωcvf − ωiof (12) dt where Cbv is the effective liquid capacitance for the body of the injector. The forces on the needle, seen in the free body diagram of Fig. 2, can be used to write an equation of motion, assuming rigid behavior and neglecting friction: Fntip + Flv − Fuv1 − Fs1 = mneed x ¨
Fig. 3. Injector Needle Flow Resistance top surface of the valve or a momentum balance would give an approximation of the force, however a simpler modeling approach is to assume a certain effective area will see full body pressure and certain effective area will be at full needle upper volume pressure. The forces can then be approximated as follows: Fuv2 = Acvbot Puv Fcvtop =
(13)
Achannel Pbv
(19) if z = y
χAcvtop Pbv + (1 − χ)Acvtop Puv if z = y
(20)
The force terms are defined as follows: Flv = Anbot Plv
(14)
Fuv1 = Antop Puv
(15)
Fs1 = P Ls1 + ks1 (x − z)
(16)
where mneed is the mass of the needle, ks1 is the spring stiffness of the needle upper volume spring, and P Ls1 is the spring’s preload. Fntip represents the intermediate pressure between the body and the spray holes: 2 Fntip =
Pbv − (Rneed (x) · wiof ) · Antip
Fuv2 + Fs1 − Fcvtop = mcv z¨
(21)
if x > 0
(17)
Pcyl Asac + Pbv (Antip − Asac ) + Fseatreaction if x=0
where Asac is the area of the needle tip exposed to the sac volume upon closing, Antip is the area of the needle tip, and Fseatreaction is the reaction force from the needle seat. Notice that when the needle is open, the equation is representative of the pressure drop across the needle due to the flow out of the injector. This can be seen graphically in Fig. 3. Using the free body diagram in Fig. 2, the check valve motion can similarly be characterized by the following equation of motion: Fuv2 + Fs1 − Fcvs − Fcvtop = mcv z¨
where Achannel is the small area on top of the check valve exposed to body pressure when the valve is closed, Acvtop is the area of the top of the check valve, χ is a constant between 0 and 1, and Acvbot is the area of the bottom of the check valve. It is important to note that as long as Fcvs > 0, then there is contact between the check valve and its seat, and z = y. However, once contact is lost, the EOM changes to be independent of the reaction force.
(18)
where mcv is the mass of the check valve, Fcvs is the reaction force from the check valve seat and Fuv2 is the pressure force from the needle upper volume below the check valve. The force Fcvtop is the pressure force from the body pressure above the check valve. This force is somewhat straightforward when the valve is closed, but becomes noticeably more complex when the valve is open. One observation is that the check valve sees body pressure at the center of the valve and needle upper volume pressure around the edge. An integral across the
During most of the simulation, logic is used to calculate the reaction force between the check valve and its seat to verify that it remains positive. If Fcvs ≤ 0, then the model states of z and z˙ are calculated independently. Once Fcvs ≥ 0 and z ≥ y, then again z = y and z˙ = y˙ until contact is lost again. The states of the lower body can currently be defined as x, x, ˙ z, z, ˙ Puv , Plv , and Pbv . 3.2 Upper Body Modeling The upper body of the injector includes the piezo stack and the linkage between the bottom link and the piezo stack and associated driver. The driver used in experimentation is the Kinetic Ceramics Model PDI-10 Piezoelectric Actuator Driver. It utilizes a TTL signal to trigger 1000V pulses to the actuator. The driver circuit can be modeled as a supply voltage, Vin , in series with a resistance, R, an inductor, L, and current, I, such that: Vin = LI˙ + RI + Vs (22) Piezoelectric materials show an electric polarization when they are strained. These materials also show a converse effect, where they strain in response to an applied electric
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field. These properties result from unit cell distortions of the crystal structure which causes a net polarization. Piezoelectrics also exhibit dielectric properties like any insulator. The polarization dynamics that occur in a piezoelectric material are complex, involving multiple modes of polarization and domain wall oscillation that can lead to non-linear effects including hysteresis. Also, all of the properties that describe these dielectric phenomenons are frequency dependent (Moulson and Herbert, 1990). However, in many cases linear, quasi-static, constant property, and frequency independent piezoelectricity relations can be assumed (Standards Committee of the IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society, 1988). D = dX + X E and e = sE X + dE
1 X
The reaction force with the check valve is assumed negligibly small and the other reaction forces are calculated as follows:
(23)
where D is the charge density (also called electric displacement) of the material (charge/area), d is the piezoelectric coefficient, X is the material stress, X is the permittivity of the material at constant stress, E is the electric field, e is the material strain, and sE is the material compliance under constant electric field. Manipulation of these equations gives the field and strain as a function of the stress and the charge density: 1 E = −gX + X D and e = sD X + gD (24) where g =
Notice that there are three springs acting on the mass, as well as body pressure, and an additional pressure force from the needle lower volume.
· d.
The complete, three-dimensional equations for a piezoelectric material are specified in tensor form to account for all directions, but can be simplified for piezoelectric discs which are polarized in a single direction and experience uniaxial stress. To correlate these equations to the axis of interest, they can be rewritten in the following form: D3 = d33 X3 + 33 X E3
(25)
e3 = s33 E X3 + d33 E3 1 D3 E3 = −g33 X3 + 33 X
(26)
e3 = s33 D X3 + g33 D3
(28)
(29)
Fs3 = P Ls3 − k3 · y
(30)
Fs4 = P Ls4 + k4 · y
(31)
Fbv = Alink Pbv
(32)
Flv2 = Apbot (Plv − Pbv )
(33)
Fdamp = by˙
(34)
where P Lsj signifies a spring preload for the j th numbered spring, ksj is a spring stiffness for the j th numbered spring, Fbv is the body force pressure on the top link, Flv2 is the differential pressure force across the bottom link, and Atlink is the area of the top link that is sealed from body pressure. Fdamp is the damping force such that b is the effective dampening coefficient of the upper body including the stack. From these equations, we can develop an equation of motion for the upper body. Fs2 + Fs3 + Flv2 + Fbv − Fdamp − Fs4 − Fpiezo = mub y¨
(35)
Fpiezo = Fs2 + Fs3 + Flv2 + Fbv − Fdamp − Fs4 − mub y¨
(36)
Or
where mub is the total mass of the upper body components. The piezo stack is encased in a protective spring housing and the force from this component can be defined as follows: Fhousing = P Lhousing − khousing · y
(27)
Fig. 4 shows a free body diagram of the mass relevant to the upper body of the injector. For simplification, the top and bottom links will be lumped as a single mass, mub . This assumes that no disconnection occurs between the pieces, and that they remain in contact during operation.
Fs2 = P Ls2 − k2 · y
(37)
Because the housing force and the reaction force with the upper body are the only two significant external forces on the stack, the stress in the stack can fully be described. The constitutive equations are considered valid here for a single piezoelectric element. In the case of an actuator stack, there are N identical elements, each of area Adisc and thickness t and it is assumed the stress in all discs is the same. With that assumption, the constitutive equation for strain (equation (28)) can be transformed to represent the stack motion, y. y = s33 D tN
(Fhousing + Fpiezo ) − g33 tN D Adisc
(38)
Where Fhousing + Fpiezo are in equations (36) and (37). The signs used in the above equation reflect the coordinate system for this injector. As this is uniaxial motion, references to D, E will no longer carry the tensor subscript.
Fig. 4. Injector Upper Body Free Body Diagram
The free charge density can be related to the current in the electric circuit by the equations N AIdisc = D˙ and I˙ ¨ = D. N Adisc
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Plugging these into equation (22): ¨ + RAdisc N D˙ + Vs Vin = LAdisc N D
(39)
Vs can be derived from equation (27), knowing that a good approximation of the voltage potential across a disc is the electric field multiplied by the distance. (Fhousing + Fpiezo ) t Vs = g33 t + D (40) Adisc 33 X Note that this equation assumes that all of the discs are experiencing equal stress and charge density. At this point in the analysis, all of the equations necessary to describe the upper body have been enumerated. The ¨ and equations (29) two highest derivatives are y¨ and D, through (40) can be coupled to give the driving differential equations for the upper body. y¨ =
Fig. 5. Bosch Type Flow Measurement Rig The rig is capable of measuring injection rate, piezo stack voltage, and injector current. The comparisons of those quantities with model results can be seen in Fig. 6, Fig. 7, and Fig. 8, respectively.
(P Ltot +Apbot (Plv −Pbv )+Alink Pbv ) mub
−
b Adisc kef f Adisc g y˙ − ( D + )y − D mub s33 tN mub mub s33 D mub
(41)
This equation was simplified by lumping the preloads and stiffnesses into “effective” terms. Similarly, a differential equation can be derived for charge density: ¨ = D
Vin LN Adisc
−
R ˙ LD
2
− ( LN Agdiscd s33 D + g y − 2 LN Adisc s33 D
d )D 33 X LN Adisc
(42)
With these two differential equations the upper body states can be defined as y, y, ˙ D, and D˙
Fig. 6. Comparison of Injector Flow Rate
3.3 Model Summary The non-linear equations described for the lower body and linear equations listed for the upper body are used to simulate the injection rate in MATLAB. The inputs to the model are rail pressure and the commanded voltage input profile to the driver. Numerical computation is done using a basic Euler iteration method. The simulation time is 3.5 ms and the time step utilized is 1 μs. Again, the states of the entire model are: x, x, ˙ z, z, ˙ Puv , Plv , Pbv , y, y, ˙ D, and ˙ D. A complete listing of the differential equations can be seen in equations (43) through (49). 4. EXPERIMENTAL SETUP
Fig. 7. Comparison of Piezo Stack Voltage
The injector is fed from a common rail and high pressure fuel pump. A Kinetic Ceramics PDI-10 driver actuates the stack with a TTL signal from a PC. The flow is measured by a Bosch type measurement device seen in Fig. 5. Rate shape data is collected on a 12-bit ultra highspeed data acquisition system (UHSDA) collected at a sampling frequency of 1 MHz. 5. RESULTS AND DISCUSSION The model was compared to the injection rig data at two rail pressures where the second is twice the first. The input to the model and experimental injector was a square voltage wave of 1000 volts for an on-time of 2.505 ms.
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−Albot Anbot P˙lv = y˙ − x˙ Clv Clv ˙ =− Puv
(Puv − Pbv )1/2 Antop Antop − y˙ + x˙ Rcv Cuv Cuv Cuv
(Prail − Pbv )1/2 (Puv − Pbv )1/2 (Pbv − Pcyl )1/2 P˙bv = + − Rrail Cbv Rcv Cbv Rtotal (x)Cbv Fntip Anbot Antop P Ls1 ks1 ks1 x ¨= + Plv − Puv − − x+ z mneed mneed mneed mneed mneed mneed ⎧ ⎨ Pbv − (Rneed (x) · wiof )2 · Antip if x > 0 where Fntip = ⎩ P A + P (A cyl sac bv ntip − Asac ) + Fseatreaction if x=0 Acvbot P Ls1 ks1 ks1 Puv + + x− z − Fcvtop mcv mcv mcv mcv if z = y Achannel Pbv where Fcvtop = χAcvtop Pbv + (1 − χ)Acvtop Puv if z = y z¨ =
y¨ =
(P Ltot + Apbot (Plv − Pbv ) + Alink Pbv ) b Adisc kef f Adisc gN − y˙ − ( D + )y − D mub mub s33 tN mub mul s33 D mub Vin g2 d R d g ¨= D − D˙ − ( + )D − y LN Adisc L LN Adisc s33 D 33 X LN Adisc LN 2 Adisc s33 D
Fig. 8. Comparison of Piezo Current The model captures the start of injection, end of injection, steady-state injection rate, and ramp rates with reasonable accuracy. Driver and piezoelectric dynamics such as the stack voltage profile and current are predicted well by the model. 6. CONCLUSION A simulation model of a piezoelectric injector has been shown to accurately predict the injection rate, piezo stack voltage, and piezo stack current of the prototype injector at two different rail pressures. Simplified driver circuits, linear piezo response, and rigid body assumptions were utilized. Future work will explore model simplification for synthesis of flow rate estimation and closed-loop injection rate control strategies. REFERENCES Fettes, C. and Leipertz, A. (2001). Potentials of a piezodriven passenger car common rail system to meet future emission legislations - an evaluation by means of incylinder analysis of injection and combustion. SAE 2001-01-3499. Lee, J., Min, K., Kang, K., and Bae, C. (2006). Hydraulic simulation and experimentatal analysis of needle response and controlled injection rate shape characteristics in a piezo-driven diesel injector. SAE 2006-01-1119. Merritt, H.E. (1967). Hydraulic Control Systems. John Wiley and Sons, Inc., New York. Moulson, A. and Herbert, J. (1990). Electroceramics. Chapman and Hall, New York. Oh, B., Oh, S., Lee, K., and Sunwoo, M. (2007). Development of an injector driver for piezo actuated common rail injectors. SAE 2007-01-3537. Standards Committee of the IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society (1988). An
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(43) (44) (45) (46)
(47)
(48) (49)
American National Standard: IEEE Standard on Piezoelectricity. The Institute of Electrical and Electronics Engineers, Inc., New York.
Dynamic Modeling of a Piezoelectric Actuated Fuel Injector Chris A. Satkoski, Gregory M. Shaver, Ranjit More, Peter Meckl, Douglas Memering Energy Center, Herrick Labs, School of Mechanical Engineering, Purdue University, W. Lafayette, IN (e-mail: [email protected]). Abstract: As engine designers look for ways to improve efficiency and reduce emissions, piezoelectric actuated fuel injectors for common rail diesel engines have shown to have improved response characteristics over solenoid actuated injectors and may allow for enhanced control of combustion through multi-pulse profiles or rate shaping. This paper summarizes the development of a simulation model for a piezoelectric fuel injector and associated driver that can be used for injector design and control system verification. The model injection rate, piezo stack voltage, and piezo stack current are compared to experimental injection rig data for two different rail pressures. Keywords: Actuator Development; Combustion Control; Control Design; Controls; Engines; Modeling; Piezoelectricity; Simulation 1. INTRODUCTION Piezoelectric actuation of common rail type fuel injectors for diesel engines offers an enhanced ability to control injection over their solenoid counterparts. This can be accomplished due to their faster response time and an ability to generate larger forces. These characteristics can lead to a greater control of the injection event through multi-pulse profiles or rate shaping. With improved control of injection, engine designers will have better flexibility in managing efficiency and emissions at all operating points of an engine for both conventional and advanced mode combustion strategies. Previous studies have shown that piezoelectric injectors can improve air entrainment (Fettes and Leipertz, 2001), spray development, and boost injection velocity due to the fact that the needle lift process is noticeably faster than a solenoid actuated injector (Lee et al., 2006). Also, it has been shown that with advanced actuator driver techniques, such as pulse width modulation (PWM), better control of the piezo stack can be obtained and dwell times between pulses can be half that of a solenoid injector (Oh et al., 2007). This paper describes a dynamic simulation model of a prototype piezoelectric fuel injector. Initial testing has shown that pressure dynamics inside of the injector cause fueling from an injection event to influence a subsequent event, resulting in minimum separation times being achieved. A simulation model, such as the one outlined in this paper, can be used to develop design strategies to mitigate pulseto-pulse interactions and optimize performance. 2. THE PIEZOELECTRIC FUEL INJECTOR
Fig. 1. Operating Principle of a Prototype Piezoelectric Fuel Injector (Note: specific design details suppressed due to confidentiality)
The design of the injector used in this study incorporates a hydromechanical mechanism that has two essential functions - to convert the high downward expansion force of
the piezo stack into an upward needle lifting force and to convert the short stroke of the piezo stack into a longer stroke for the needle.
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2.1 General Operating Principle As shown in Fig. 1, when the piezo stack expands, the bottom link is forced down into the needle lower volume, raising the pressure, and forcing the needle upward. Upon upward motion of the needle, fluid is pressurized in the chamber above the needle, called the needle upper volume. Fluid is forced out of the needle upper volume into the body volume through spring-loaded check valve above the needle with an orifice in the center causing smooth, upward motion of the needle. Upon retraction of the piezo stack, the bottom link is forced upwards by the needle lower volume pressure. Upward bottom link movement lowers the pressure in the needle lower volume, closing the needle. At certain rail pressures, the check valve can be forced open by the body pressure above it, allowing the needle upper volume pressure to rapidly increase, quickly closing the needle. The actuator used in this injector is a 130 disc, 1000V piezoelectric stack. All of the discs are enclosed between electrodes and are wired in parallel to a single 1000V driver. This multi-disc, parallel actuator configuration is common for piezoelectric actuators, as it allows maximum deflection for a given amount of piezoelectric ceramic. Thinner discs can be used with a smaller voltage, but are not used currently in this application due to the high loads. The whole stack is encased in a protective housing. 3. MODELING
where Albot is the bottom link “ledge” area (critical for displacing liquid in the needle lower volume) and Anbot is the needle bottom area. The needle upper volume is much like the needle lower volume, except it has a flow term. dPuv dVuv = −ωcvf − (4) Cuv dt dt where ωcvf is the check valve flow of liquid out of the needle upper volume, Cuv = Vβuv is the needle upper volume liquid capacitance, and uv refers to the needle upper volume. dVuv dt is also a function of our states and is described by V˙ uv = Antop (y˙ − x). ˙ (5) where Antop is the area of the needle top. Flows are modeled with a standard orifice equation: 2 ω = Cd A0 ΔP (6) ρ where ω is the flow, Cd is the coefficient of discharge, A0 is the orifice area, and ρ is the liquid density. Terms in this equation can be grouped to fit the form of a flow resistance relationship: √ ρ Rω = ΔP where R = (7) 2Cd 2 A0 2 such that the check valve flow can be given as: ωcvf = Puv − Pbv /Rcv
(8)
The “lower body” includes all of the hydromechanical interaction from the bottom link to the injection of fuel. The “upper body” includes the electro-mechanical interaction from the voltage driver to the piezo stack, down to the bottom link. The upper and lower bodies are linked by the displacement of the piezo stack and the interaction force in between the bottom and top link. The coordinate systems for the injector can be seen in Fig. 1.
where Pbv is the pressure of the body volume and Rcv is the resistance of the check valve. It is important to note that the check valve has a different resistance when open and closed. The resistance is actually a function of its displacement relative to the plunger. It can be most accurately captured by a more complex parallel disc flow equation. For simplicity, this model will use an approximate constant flow resistance when open, and a different constant when closed.
3.1 Lower Body Modeling
After the needle opens, fuel will flow down the body, past the needle into the sac volume, which is the small volume enclosed between the bottom of the injector needle and the inlet to the spray holes, and then through the spray holes out of the injector. Two restrictions impede free flow from the body. One is the restriction of the spray holes themselves, Rsh . The other is the resistance past the needle, Rneed , which changes with needle lift such that the total restriction for fluid out of the injector is given by the following expression: Rtotal (x) = Rsh + Rneed (x) (9)
The injector control volumes are modeled using a coupled continuity and state equation for the operating liquid (Merritt, 1967). Temperature effects on properties are assumed negligible. V0 dP dV0 Σωin − Σωout = + (1) dt β dt where ωin is the volumetric flow into the control volume, ωout is the volumetric flow out of the control volume, V0 is the mean volume, β is the bulk modulus of liquid, and P is the pressure. In the case of the needle lower volume, we can assume that there are no flows in or out during needle opening and closing events. Then equation (1) becomes dPlv dVlv Clv =− (2) dt dt where the subscript lv refers to the needle lower volume and Clv = Vβlv is the liquid capacitance. dVdtlv is a function of our states and is described by V˙ lv = Albot y˙ + Anbot x. ˙ (3)
The flow out of the injector can be fully described by ωiof = Pbv − Pcyl /Rtotal (x) (10)
where ωiof is the flow out of the injector and Pcyl is the pressure of the combustion chamber. Rail-to-body flow can be described by: ωrtb = Prail − Pbv /Rrail
(11)
where Prail is the rail pressure, and Rrail is the sum of the restrictions between the rail and the body of the injector.
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Fig. 2. Injector Needle Free Body Diagram An equation can be written for the body, assuming no relevant change in volume: dPbv Cbv = ωrtb + ωcvf − ωiof (12) dt where Cbv is the effective liquid capacitance for the body of the injector. The forces on the needle, seen in the free body diagram of Fig. 2, can be used to write an equation of motion, assuming rigid behavior and neglecting friction: Fntip + Flv − Fuv1 − Fs1 = mneed x ¨
Fig. 3. Injector Needle Flow Resistance top surface of the valve or a momentum balance would give an approximation of the force, however a simpler modeling approach is to assume a certain effective area will see full body pressure and certain effective area will be at full needle upper volume pressure. The forces can then be approximated as follows: Fuv2 = Acvbot Puv Fcvtop =
(13)
Achannel Pbv
(19) if z = y
χAcvtop Pbv + (1 − χ)Acvtop Puv if z = y
(20)
The force terms are defined as follows: Flv = Anbot Plv
(14)
Fuv1 = Antop Puv
(15)
Fs1 = P Ls1 + ks1 (x − z)
(16)
where mneed is the mass of the needle, ks1 is the spring stiffness of the needle upper volume spring, and P Ls1 is the spring’s preload. Fntip represents the intermediate pressure between the body and the spray holes: 2 Fntip =
Pbv − (Rneed (x) · wiof ) · Antip
Fuv2 + Fs1 − Fcvtop = mcv z¨
(21)
if x > 0
(17)
Pcyl Asac + Pbv (Antip − Asac ) + Fseatreaction if x=0
where Asac is the area of the needle tip exposed to the sac volume upon closing, Antip is the area of the needle tip, and Fseatreaction is the reaction force from the needle seat. Notice that when the needle is open, the equation is representative of the pressure drop across the needle due to the flow out of the injector. This can be seen graphically in Fig. 3. Using the free body diagram in Fig. 2, the check valve motion can similarly be characterized by the following equation of motion: Fuv2 + Fs1 − Fcvs − Fcvtop = mcv z¨
where Achannel is the small area on top of the check valve exposed to body pressure when the valve is closed, Acvtop is the area of the top of the check valve, χ is a constant between 0 and 1, and Acvbot is the area of the bottom of the check valve. It is important to note that as long as Fcvs > 0, then there is contact between the check valve and its seat, and z = y. However, once contact is lost, the EOM changes to be independent of the reaction force.
(18)
where mcv is the mass of the check valve, Fcvs is the reaction force from the check valve seat and Fuv2 is the pressure force from the needle upper volume below the check valve. The force Fcvtop is the pressure force from the body pressure above the check valve. This force is somewhat straightforward when the valve is closed, but becomes noticeably more complex when the valve is open. One observation is that the check valve sees body pressure at the center of the valve and needle upper volume pressure around the edge. An integral across the
During most of the simulation, logic is used to calculate the reaction force between the check valve and its seat to verify that it remains positive. If Fcvs ≤ 0, then the model states of z and z˙ are calculated independently. Once Fcvs ≥ 0 and z ≥ y, then again z = y and z˙ = y˙ until contact is lost again. The states of the lower body can currently be defined as x, x, ˙ z, z, ˙ Puv , Plv , and Pbv . 3.2 Upper Body Modeling The upper body of the injector includes the piezo stack and the linkage between the bottom link and the piezo stack and associated driver. The driver used in experimentation is the Kinetic Ceramics Model PDI-10 Piezoelectric Actuator Driver. It utilizes a TTL signal to trigger 1000V pulses to the actuator. The driver circuit can be modeled as a supply voltage, Vin , in series with a resistance, R, an inductor, L, and current, I, such that: Vin = LI˙ + RI + Vs (22) Piezoelectric materials show an electric polarization when they are strained. These materials also show a converse effect, where they strain in response to an applied electric
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field. These properties result from unit cell distortions of the crystal structure which causes a net polarization. Piezoelectrics also exhibit dielectric properties like any insulator. The polarization dynamics that occur in a piezoelectric material are complex, involving multiple modes of polarization and domain wall oscillation that can lead to non-linear effects including hysteresis. Also, all of the properties that describe these dielectric phenomenons are frequency dependent (Moulson and Herbert, 1990). However, in many cases linear, quasi-static, constant property, and frequency independent piezoelectricity relations can be assumed (Standards Committee of the IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society, 1988). D = dX + X E and e = sE X + dE
1 X
The reaction force with the check valve is assumed negligibly small and the other reaction forces are calculated as follows:
(23)
where D is the charge density (also called electric displacement) of the material (charge/area), d is the piezoelectric coefficient, X is the material stress, X is the permittivity of the material at constant stress, E is the electric field, e is the material strain, and sE is the material compliance under constant electric field. Manipulation of these equations gives the field and strain as a function of the stress and the charge density: 1 E = −gX + X D and e = sD X + gD (24) where g =
Notice that there are three springs acting on the mass, as well as body pressure, and an additional pressure force from the needle lower volume.
· d.
The complete, three-dimensional equations for a piezoelectric material are specified in tensor form to account for all directions, but can be simplified for piezoelectric discs which are polarized in a single direction and experience uniaxial stress. To correlate these equations to the axis of interest, they can be rewritten in the following form: D3 = d33 X3 + 33 X E3
(25)
e3 = s33 E X3 + d33 E3 1 D3 E3 = −g33 X3 + 33 X
(26)
e3 = s33 D X3 + g33 D3
(28)
(29)
Fs3 = P Ls3 − k3 · y
(30)
Fs4 = P Ls4 + k4 · y
(31)
Fbv = Alink Pbv
(32)
Flv2 = Apbot (Plv − Pbv )
(33)
Fdamp = by˙
(34)
where P Lsj signifies a spring preload for the j th numbered spring, ksj is a spring stiffness for the j th numbered spring, Fbv is the body force pressure on the top link, Flv2 is the differential pressure force across the bottom link, and Atlink is the area of the top link that is sealed from body pressure. Fdamp is the damping force such that b is the effective dampening coefficient of the upper body including the stack. From these equations, we can develop an equation of motion for the upper body. Fs2 + Fs3 + Flv2 + Fbv − Fdamp − Fs4 − Fpiezo = mub y¨
(35)
Fpiezo = Fs2 + Fs3 + Flv2 + Fbv − Fdamp − Fs4 − mub y¨
(36)
Or
where mub is the total mass of the upper body components. The piezo stack is encased in a protective spring housing and the force from this component can be defined as follows: Fhousing = P Lhousing − khousing · y
(27)
Fig. 4 shows a free body diagram of the mass relevant to the upper body of the injector. For simplification, the top and bottom links will be lumped as a single mass, mub . This assumes that no disconnection occurs between the pieces, and that they remain in contact during operation.
Fs2 = P Ls2 − k2 · y
(37)
Because the housing force and the reaction force with the upper body are the only two significant external forces on the stack, the stress in the stack can fully be described. The constitutive equations are considered valid here for a single piezoelectric element. In the case of an actuator stack, there are N identical elements, each of area Adisc and thickness t and it is assumed the stress in all discs is the same. With that assumption, the constitutive equation for strain (equation (28)) can be transformed to represent the stack motion, y. y = s33 D tN
(Fhousing + Fpiezo ) − g33 tN D Adisc
(38)
Where Fhousing + Fpiezo are in equations (36) and (37). The signs used in the above equation reflect the coordinate system for this injector. As this is uniaxial motion, references to D, E will no longer carry the tensor subscript.
Fig. 4. Injector Upper Body Free Body Diagram
The free charge density can be related to the current in the electric circuit by the equations N AIdisc = D˙ and I˙ ¨ = D. N Adisc
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2009 IFAC E-CoSM (E-CoSM'09) Nov 30 - Dec 2, 2009, IFP, Rueil-Malmaison, France.
Plugging these into equation (22): ¨ + RAdisc N D˙ + Vs Vin = LAdisc N D
(39)
Vs can be derived from equation (27), knowing that a good approximation of the voltage potential across a disc is the electric field multiplied by the distance. (Fhousing + Fpiezo ) t Vs = g33 t + D (40) Adisc 33 X Note that this equation assumes that all of the discs are experiencing equal stress and charge density. At this point in the analysis, all of the equations necessary to describe the upper body have been enumerated. The ¨ and equations (29) two highest derivatives are y¨ and D, through (40) can be coupled to give the driving differential equations for the upper body. y¨ =
Fig. 5. Bosch Type Flow Measurement Rig The rig is capable of measuring injection rate, piezo stack voltage, and injector current. The comparisons of those quantities with model results can be seen in Fig. 6, Fig. 7, and Fig. 8, respectively.
(P Ltot +Apbot (Plv −Pbv )+Alink Pbv ) mub
−
b Adisc kef f Adisc g y˙ − ( D + )y − D mub s33 tN mub mub s33 D mub
(41)
This equation was simplified by lumping the preloads and stiffnesses into “effective” terms. Similarly, a differential equation can be derived for charge density: ¨ = D
Vin LN Adisc
−
R ˙ LD
2
− ( LN Agdiscd s33 D + g y − 2 LN Adisc s33 D
d )D 33 X LN Adisc
(42)
With these two differential equations the upper body states can be defined as y, y, ˙ D, and D˙
Fig. 6. Comparison of Injector Flow Rate
3.3 Model Summary The non-linear equations described for the lower body and linear equations listed for the upper body are used to simulate the injection rate in MATLAB. The inputs to the model are rail pressure and the commanded voltage input profile to the driver. Numerical computation is done using a basic Euler iteration method. The simulation time is 3.5 ms and the time step utilized is 1 μs. Again, the states of the entire model are: x, x, ˙ z, z, ˙ Puv , Plv , Pbv , y, y, ˙ D, and ˙ D. A complete listing of the differential equations can be seen in equations (43) through (49). 4. EXPERIMENTAL SETUP
Fig. 7. Comparison of Piezo Stack Voltage
The injector is fed from a common rail and high pressure fuel pump. A Kinetic Ceramics PDI-10 driver actuates the stack with a TTL signal from a PC. The flow is measured by a Bosch type measurement device seen in Fig. 5. Rate shape data is collected on a 12-bit ultra highspeed data acquisition system (UHSDA) collected at a sampling frequency of 1 MHz. 5. RESULTS AND DISCUSSION The model was compared to the injection rig data at two rail pressures where the second is twice the first. The input to the model and experimental injector was a square voltage wave of 1000 volts for an on-time of 2.505 ms.
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2009 IFAC E-CoSM (E-CoSM'09) Nov 30 - Dec 2, 2009, IFP, Rueil-Malmaison, France.
−Albot Anbot P˙lv = y˙ − x˙ Clv Clv ˙ =− Puv
(Puv − Pbv )1/2 Antop Antop − y˙ + x˙ Rcv Cuv Cuv Cuv
(Prail − Pbv )1/2 (Puv − Pbv )1/2 (Pbv − Pcyl )1/2 P˙bv = + − Rrail Cbv Rcv Cbv Rtotal (x)Cbv Fntip Anbot Antop P Ls1 ks1 ks1 x ¨= + Plv − Puv − − x+ z mneed mneed mneed mneed mneed mneed ⎧ ⎨ Pbv − (Rneed (x) · wiof )2 · Antip if x > 0 where Fntip = ⎩ P A + P (A cyl sac bv ntip − Asac ) + Fseatreaction if x=0 Acvbot P Ls1 ks1 ks1 Puv + + x− z − Fcvtop mcv mcv mcv mcv if z = y Achannel Pbv where Fcvtop = χAcvtop Pbv + (1 − χ)Acvtop Puv if z = y z¨ =
y¨ =
(P Ltot + Apbot (Plv − Pbv ) + Alink Pbv ) b Adisc kef f Adisc gN − y˙ − ( D + )y − D mub mub s33 tN mub mul s33 D mub Vin g2 d R d g ¨= D − D˙ − ( + )D − y LN Adisc L LN Adisc s33 D 33 X LN Adisc LN 2 Adisc s33 D
Fig. 8. Comparison of Piezo Current The model captures the start of injection, end of injection, steady-state injection rate, and ramp rates with reasonable accuracy. Driver and piezoelectric dynamics such as the stack voltage profile and current are predicted well by the model. 6. CONCLUSION A simulation model of a piezoelectric injector has been shown to accurately predict the injection rate, piezo stack voltage, and piezo stack current of the prototype injector at two different rail pressures. Simplified driver circuits, linear piezo response, and rigid body assumptions were utilized. Future work will explore model simplification for synthesis of flow rate estimation and closed-loop injection rate control strategies. REFERENCES Fettes, C. and Leipertz, A. (2001). Potentials of a piezodriven passenger car common rail system to meet future emission legislations - an evaluation by means of incylinder analysis of injection and combustion. SAE 2001-01-3499. Lee, J., Min, K., Kang, K., and Bae, C. (2006). Hydraulic simulation and experimentatal analysis of needle response and controlled injection rate shape characteristics in a piezo-driven diesel injector. SAE 2006-01-1119. Merritt, H.E. (1967). Hydraulic Control Systems. John Wiley and Sons, Inc., New York. Moulson, A. and Herbert, J. (1990). Electroceramics. Chapman and Hall, New York. Oh, B., Oh, S., Lee, K., and Sunwoo, M. (2007). Development of an injector driver for piezo actuated common rail injectors. SAE 2007-01-3537. Standards Committee of the IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society (1988). An
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(43) (44) (45) (46)
(47)
(48) (49)
American National Standard: IEEE Standard on Piezoelectricity. The Institute of Electrical and Electronics Engineers, Inc., New York.