Behavioural Prediction of HYDRAULIC Step-Up Switching Converters
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BEHAVIOURAL PREDICTION OF HYDRAULIC STEP-UP SWITCHING CONVERTERS
Victor J. De Negri1, Pengfei Wang2, Andrew Plummer2, and D. Nigel Johnston2 1Federal University of Santa Catarina, Department of Mechanical Engineering, LASHIP, Trindade, Florianópolis, SC, 88040-900, Brazil, e-mail: [email protected] 2University of Bath, Department of Mechanical Engineering, Claverton Down, Bath, BA27AY, United Kingdom, e-mail: [email protected]; [email protected]; [email protected]
Abstract
In this paper the fundamental principles of energy-conservative hydraulic control based on the fluid inertance principle are discussed and a detailed analysis of step-up switched-inertance control is presented. A non-loss system comprising an inertance tube and switching valve is modelled and its operational curves are presented as a reference for an ideal behaviour. Considering the load loss at both the tube and the PWM switched valve, a linear mathematical model for the step-up switched-inertance hydraulic system is presented which describes the pressure response as a function of the PWM duty cycle. Mathematical expressions of the flow rates through the tube and the supply and return ports as well as the system efficiency are also presented. A system prototype is evaluated on a test rig and the experimental data compared with the theoretical results, demonstrating the model accuracy. The proposed model simplifies the analysis process for step-up switching converters and thus their restrictions and potential can be investigated more quickly.
Keywords: Digital hydraulics, Hydraulic switching converter, Hydraulic valve, PWM switched valve
1. Introduction As discussed by Scheidl et al. (2008), the hydraulic switching control was used in the late 18th century. Last decade this subject became focus of research In most power hydraulic systems the speed and/or again as an alternative for energy efficiency increase as force of the load are controlled using valves to throttle can be seen in Manhartsgruber et al. (2005), Kogler the flow and limit or reduce the hydraulic pressure or and Scheidl (2008), Hettrich et al. (2009), and Johnston reduce the flow rate. Since this is an energy dissipative (2009), among others. process, the hydraulic circuit efficiency is often lower than 50%. Furthermore, a valve can only serve to Using the nomenclature from the electrical area, basic reduce, and not increase, the flow or pressure. Other types of switching devices have been studied. These alternatives using variable displacement pump or are the step-up or boost circuit, the step-down or buck motor, fixed pump driven by variable speed electrical circuit, and the Cuk-converter which can step up or motor, and hydro-mechanical transformer are available step down (Brown, 1987, Kogler and Scheidl, 2008). technologies and applicable depending on the working Brown (1987) also proposed another concept named cycle and equipment cost constraints. the switching gyrator which was not normally associated with electrical design. An alternative design, named switched-reactance hydraulics, was presented by F. Brown in the 1980s as Several aspects can influence the switching fluid being an energy-conservative control (Brown, 1984, inertance including pressure wave propagation, control Brown, 1987, and Brown et al., 1988). The hydraulic orifice switching time, non-linear load losses, leakages, control principle is based on the cyclical acceleration and fluid capacitances. Despite of that, it is shown in and deceleration of fluid using pulse-width modulation. this paper the well known linear modelling of buck and As mentioned by Brown (1987) the switched-reactance boost electrical circuits is applicable for hydraulic hydraulic system was developed independently of circuit prediction. electrical switched power converters, but the concepts are the same and a direct analogy exists between them. This paper is organized as follows. In section 2, the step-up and step-down converters are described and in section 3 the hydraulic step-up converter is modelled. Fig. 1: Step-up circuit: a) Hydraulic circuit; b) An ideal device is analysed as well as a realistic one Electrical circuit; c) PWM signal. including the resistance associated with tube and switching orifices. Section 4 presents the experimental setup and the system parameters. In section 5 experimental and theoretical results are compared confirming the model validity and section 6 exemplify the model use predicting the system performance according to the switching frequency. Finally, section 7 provides the main conclusions.
2. Step-Up and Step-Down PWM Valves
From the electrical point of view the function of switching-converter circuits is to convert an unregulated DC input into a regulated DC output. The efficiency is usually up to 98% and these devices can provide an output that is greater than the input. Fig. 2: Step-down circuit: a) Hydraulic circuit; b) The step-down or buck converter is used to convert a Electrical circuit; c) PWM signal. DC voltage to a lower DC voltage of the same polarity. In the step-up principle, when the flow path P-T is In the step-up or boost regulator the output voltage is p higher than the input voltage. In an electric-hydraulic active the internal pressure ( Sin ) is reduced and the analogy, an equivalent hydraulic system will be fluid is accelerated through the tube. As the valve is working as a pressure control valve. A constant supply switched to the other position (P-A and T blocked), the pressure is presupposed to be available. On the other fluid momentum in the tube causes the internal hand, in the switching gyrator proposed by Brown pressure to increase and, consequently, the load (1987) the output current is proportional to the input pressure ( pL ) to increase. In the next time period the voltage, and vice-versa. Therefore, the corresponding load port will be blocked again while the fluid is hydraulic system operates as a flow control valve. accelerated. Under a theoretical point of view, with a duty cycle ( ) of 100% (P-T and A blocked) the load The fundamental circuit of step-up and step-down pressure tends to infinity and with a duty cycle of 0% switching converters and their electrical counterparts (P-A and T blocked) pL and pSin are equal to pS . are shown in Fig. 1 and Fig. 2, respectively. In both Therefore, ideally the load pressure ( pL ) can be systems, the directional valve is driven by a pulse- width modulated (PWM) signal such that it switches modulated from the supply pressure value to infinity cyclically and rapidly, modulating the time in which achieving twice the supply pressure for 50% . each flow path remains active. The other major element On the other hand, the step-down principle consists of is the hydraulic tube which ideally has only inertance a tube installed between the switching valve and the effect (L), but hydraulic resistance (R) and capacitance external load port. As can be seen in Fig. 2, when the (C) effects are also present in practical systems, as flow path P-A is active the internal pressure ( p Ain ) represented in these figures. tends to increase and, consequently, the fluid accelerates through the tube. When the valve switches to the other position the internal chamber is connected to the return port (T) but the fluid momentum causes the fluid to continue to move through the tube, drawing the fluid from the T port despite the adverse (low to high) pressure gradient. With the duty cycle ( ) equal to 100% (P-A and T blocked) the load pressure ( pL ) is ideally equal to p Ain and pS . With 0%, the T port is connected to A and P is blocked such that pL and p Ain are equal to the return pressure ( pT ). Duty cycle of 0% or 100% are never applied in practice such that the load pressure ( pL ) can be modulated from higher than the return pressure value to lower than the supply pressure value. The basic assumption for the operation of both systems successive pressure steps is as shown in Fig. 4. The is that there is a capacitance ( CL ) associated with the tube flow rate can be expressed by load circuit such that it absorbs the flow variation pS pT produced by the switching process and the load qV (t) qV1(0) t for 0 t Tsw (3) 1 L pressure remains constant for a specific duty cycle and value. Therefore, a system comprised of the inertance and capacitance, whose natural frequency is given by p p q (t) q (T ) S L (t T ) Eq. (1), must filter the switching frequency. V 2 V 2 sw L sw (4) for T t T 1 sw sw n (1) LC L A good design rule is to choose a load capacitance value such that the natural frequency is less than a tenth of the switching frequency (Brown, 1987). Since the step-up and step-down circuit are pressure regulators they must deliver an average flow rate equal to the flow rate consumed by the main load. Consequently, the average load pressure will remain constant and theoretically proportional to the duty cycle.
3. Step-Up PWM Valve Modelling
3.1. Ideal Valve Model Fig. 4: Inertance tube response for a square wave (ideal system): a) General response; b) Response for = Considering the step-up circuit presented in Fig. 1 and 0.5. assuming that the load capacitance is high enough for the load pressure to be considered constant, the Substituting t Tsw in Eq. (3) and t Tsw in Eq. (4) switching circuit can be analysed separately from the the tube flow amplitude can be writen as loading circuit. Therefore, one can perform the pS pT pS pL behavioural analysis based on the fundamental circuit qVI Tsw (1 )Tsw (5) L L p shown in Fig.3a where the pressure difference ( ) Consequently, through the inertance tube is considered as the input and the inertance tube flow rate ( qVI ) as the output. p S pT pL (6) In the step-up circuit the pressure at tube upstream end 1 Observing Fig. 1 and Fig. 4a one concludes that the is constant and equal to pS while the pressure at the average load flow rate ( qVL ) can be calculated by downstream end is being switched between the return integration of Eq. (4) through the interval Tsw t Tsw pressure ( pT ) and the load pressure ( pL ). and dividing by Tsw . Consequently, being
qV 1( 0 ) qV 2 (Tsw ) ,
qVL g 1 f qV (t) Tsw (1 ) t for 0 t Tsw (7) 1 1 L 2 L and Fig.3: Step-up fundamental hydraulic circuit: a) Ideal q g 1 g q (t) VL T (1 ) (t T ) system; b) System with resistances V 2 1 L 2 sw L sw (8) The ideal inertance tube behaviour can be expressed by for Tsw t Tsw where f pS pT and g pS pL , dq p L VI (2) dt the average supply flow rate ( qVS ) is equal to the where p pS pT for 0 t Tsw , average flow rate through the inertance tube ( qVI ) and
p pS pL for Tsw t Tsw , in [0,1] defines can be obtained by integration of the sum of equations (7) and (8) through the entire interval and dividing by the duty cycle, and Tsw is the PWM signal period. Tsw , resulting in Therefore, the inertance tube corresponds to an integrator such that the flow rate behaviour for qVL qV qV (9) S I 1
The average return flow rate ( qVT ) can be obtained by integration of Eq. (7) through the interval 0 t T and diving by Tsw , resulting in
qVL qV (10) T 1 The step-up efficiency can be determined through the following equation where the use of equations (6), (9), and (10) provides an efficiency of 100% as expected for an ideal system. p q L VL (11) p q p q S VS T VT Fig. 5: Inertance tube response for a square wave (system with resistance): a) General response; b) 3.2. Model Including Linear Resistance Response for = 0.5.
Combining Eq. (13) for t Tsw and Eq. (14) for In this section the influence of the valve and tube t Tsw one can obtain the maximum flow rate through resistances is included, as shown in Fig.3b. As q previously stated, in the step-up circuit there are two the inertance tube ( VImax ), that is, different valve flow paths that are switched alternately, 1 1 Tsw qV1(Tsw ) f (1 e ) consequently connecting the tube downstream end to 1 eTsw R the load or the reservoir. Assuming that both flow paths have the same resistance and adding the tube T / T (15) 1 g(e sw e sw ) resistance, the circuit model can be described by R q (T ) q (T ) L dqVI 1 and V 2 sw V 1 sw qVI p (12) R dt R and the minimum flow rate through the inertance tube ( where p pS pT for 0 t Tsw , qVImin ), that is,
p pS pL for Tsw t Tsw , and R Rv Rtb . 1 1 (1)Tsw qV 2 (Tsw ) g(1 e ) Aiming to achieve the time response of this system for 1 eTsw R a square wave input the approach presented by (16) Millman & Taub (1965) for electric circuits is used. 1 f (e (1)Tsw e Tsw ) Therefore, since the steady-state output corresponds to R the steady-state gain multiplied by the step magnitude, and qV 1(0) qV 2 (Tsw ) the flow rate through the hydraulic step-up circuit can The amplitude of the flow rate wave can be calculated be expressed by by subtracting Eq. (16) from Eq. (15) such that 1 1 t / qV (t) f (qV1(0) f )e (1)Tsw Tsw Tsw 1 R R (13) (1 e e e ) 1 qV I T ( f g) (17) for 0 t Tsw (1 e sw ) R and The average load flow rate ( qVL ) corresponds to the
(tT ) / integral of Eq. (14) through the interval Tsw t Tsw q (t) 1 g (q (T ) 1 g)e sw V 2 R V 2 sw R (14) divided by Tsw . The result using Eq. (15) is for Tsw t Tsw f p p g p p L R 1 where S T , S L , and . qVL gy fx Tsw (18) (1 e )RTsw Fig. 5a shows the graphic representation for these where: functions as well as identifies their specific values at T T instants zero, sw , and sw . As demonstrated in x (1 e(1)Tsw )(1 eTsw ) (19) Millman & Taub (1965), the average output value ( qVI and ) is equal to the average input value (multiplied by the (1)Tsw Tsw steady-state gain) for an entire period. Fig. 5b presents y Tsw (1 ) (e 1)(1 e ) a specific condition where the duty cycle is equal to (20) 50%. (1 e(1)Tsw )2 eTsw
The average supply flow rate ( qVS ) is equal to the
average flow rate through the inertance tube ( qVI ) and can be obtained by integration of the sum of equations (13) and (14) through the entire interval and dividing corresponds to the average of the linear coefficients by Tsw , resulting in obtained from the experimental curves shown in Fig. 8.
1 qVS qVI g(x y) f (x z) Tsw (21) (1 e )RTsw where
Tsw Tsw z Tsw (e 1)(1 e ) (22) (1 eTsw )2 e(1)Tsw
The average return flow rate ( qVT ) is obtained by integration of Eq. (13) through the interval 0 t Tsw and dividing by Tsw , resulting in:
1 qVT gx fz Tsw (23) (1 e )RTsw
Substituting f pS pT and g pS pL in Eq. (18) Fig. 6: Test rig at the Centre for Power Transmission the load pressure can be written as a function of the and Motion Control. supply pressure, return pressure, average load flow rate, duty cycle, and switching period as Table 1: Proportional valve data. p (x y) p x q (1 eTsw / )RT p S T VL sw (24) L y Nominal flow rate 0.67 L/s (40L/min) @ pp 3.5 MPa The step-up valve efficiency is determined by Eq. (11) ( qVn ) presented above. Equivalent 9 3 1 3.8810 Pa.s/m @ U c = ± U cn The equations presented above are used resistance ( Re ) 2 straightforwardly in sections 5 and 6 to calculate the 3.5 ms @ Uc = 0 → 100% t average values of flow rates and load pressure as Settling time ( s ) 1 6.25 ms @ U c = -100 → 100% function of both load flow rate and duty cycle. These theoretical results are validated by comparison with Natural frequency ( 120 Hz@90(Uc = ± 90%) experimental results as shown in section 5. n ) 1Experimental data; 2Catalogue data
The inertance tube has an internal diameter ( dt ) of
4. Experimental Setup 7.1 mm and a length ( lt ) of 1.7 m . The hydraulic fluid has a density ( ) of 870 kg/m3 and is assumed to have 9 Fig. 6 shows the test rig where the hydraulic circuit an effective bulk modulus ( e ) of 1.6 10 Pa . shown in Fig. 7 was implemented. A directional Considering that proportional valve (Parker D1FPE50MA9NB01) driven by a PWM signal was used to accomplish the 4lt Lt 2 (25) function of the directional valve shown in Fig. 1a. dt Error: Reference source not found presents the the tube inertance is 3.75107 kg/m 4 . experimental values for the valve used in this paper where the equivalent resistance of the valve
Fig. 7: Test hydraulic circuit diagram. The tube hydraulic resistance ( Rt ) was estimated by and thus average values related to each load flow rate linear approximation based on experimental data and were calculated based on experiments for all duty the value obtained was 1.67 109 Pa.s/m3 and thus the cycles (Table 2). total resistance ( R ) (valve and tube) is The average values for the experimental results were 5.55109 Pa.s/m3 . compared with the theoretical results as presented in next section.
Fig. 8: Characteristic curves for the proportional valve a)
5. Theoretical and Experimental Results
5.1. Introduction
The step-up valve behaviour in steady-state is discussed in this section using the model with linear resistance presented in Section 3.3 and experimental results. The valve settling time presented in Error: Reference source not found limited the minimum switching period of the PWM signal. Therefore, switching periods of 62.5 ms ( f sw 16 Hz) and 125 ms
( f sw 8 Hz) were adopted such that the spool achieved the total displacement even under limiting conditions b) of 10% and 90% duty cycles. Fig. 9: Experimental time response for fsw = 16 Hz, The experiments were carried out for different duty qVL = 0.2 L/s (12 L/min), and = 0.25: a) Pressure cycles while the average load flow rate remained curves; b) Flow curves. constant, adjusted by valve V2 (Fig. 7). The average Table 2: Average return pressures. supply pressure ( pS ) was 2.4 MPa . Average return pressure As an example, Fig. 9 presents the experimental time Load flow rate response for a load flow rate of 0.2 L/s, 16 Hz For Tsw 62.5ms For Tsw 125ms switching frequency, and 25% duty cycle ( 0.25 ) 0 L/s 0.33 MPa 0.20 MPa which is observable through the proportional valve spool position voltage (Us). The supply and load 0.1L/s 0.37 MPa 0.35 MPa pressures vary around their steady-state values 0.2 L/s 0.35 MPa 0.36 MPa although the spikes are reduced using accumulators. 0.3 L/s 0.28 MPa 0.36 MPa The turbine flow meters did not capture the effective dynamic behaviour; however the average value is recorded. 5.2. Case 1: Switching Period of 62.5 ms Similar experimental results were obtained for a Considering the system parameters presented in switching frequency of 8 Hz. Based on these results the Section 4 and the operational conditions described average values were calculated from the last 16 cycles above the step-up valve analysis was carried out for a when the system was in steady-state. The return f 16 Hz T pressure was a consequence of the return line load loss switching frequency ( sw ) of ( sw = 62.5 ms). Fig. 10 presents the load pressure versus duty cycle for different load flow rates. As one can see the summing of the tube and valve load losses has a great influence on the system performance and thus the expected pressure booster effect (ideal model) is not reached as the load flow rate increases. In this figure, and also in those which follow, the lines correspond to theoretical results according to the equations presented previously. The points relate to average experimental values.
Fig. 12: Return flow rate versus duty cycle for 16 Hz
Fig. 10: Load pressure versus duty cycle for 16 Hz Fig. 11 andFig. 12 present the average supply flow rate and the average return flow rate, respectively, where the experimental points confirm the model prediction. However, one can observe in Fig. 10 that the effective operational range of the step-up valve is limited for load pressure values greater than zero. Therefore, only Fig. 13: Efficiency versus duty cycle for 16 Hz a range of the flow theoretical values is reached under real conditions and these limits are defined by the duty cycle values at pL 0 Pa . 5.3. Case 2: Switching Period of 125 ms
Experiments and calculations were carried out under the same conditions described in the previous section
but using a switching frequency ( f sw ) of 8 Hz ( Tsw = 125 ms). As observed in Fig. 14, the model gives a good prediction of the valve performance despite the valve load loss being described by a linear coefficient (
Re ). Equivalent results to figures 11 to 13 were also obtained operating at 8 Hz.
Fig. 11: Supply flow rate versus duty cycle for 16 Hz The system efficiency calculated according to Eq. (11) is shown in Fig. 13. The difference between the experimental values and the theoretical prediction is intensified as a consequence of the small errors observed in the load pressure and supply and return flow rates.
Fig. 14: Load pressure versus duty cycle for 8 Hz 6. Performance of Step-Up Converters
As presented on previous sections, the model including linear resistance gives a very good description of the average response of a hydraulic step-up switching converter. Therefore, it is expected that this model can be used to predict the valve performance under both different operational conditions and with different values for parameters such as inertance, resistance, switching frequency, and supply pressure. So the model can be used to help design a system which has maximum efficiency. As an example, the influence of the switching frequency is analysed in Fig. 15 and Fig. 16. In this case the same inertance and supply pressure as used in Fig. 16: Predicted influence of the switching frequency the previous section are being considered however the on the valve efficiency return pressure is assumed to be zero. The valve load loss is neglected such that the total resistance The responses at 16 Hz shown in Fig. 15 and Fig. 16 corresponds to the tube resistance presented before. As can be compared with Fig. 10 and Fig. 13, respectively, can be observed, the efficiency increases as the where the significant influence of the load loss can be switching frequency increases but small influence is observed closely. Switching valves as reported in observed for frequencies higher than 100 Hz. Winkler et al. (2010) are potential solutions for Therefore, for this system a fast switching valve with 1 achieving reduced load loss. Furthermore, the tube load ms of settling time will be enough to carry out duty loss can be reduced by increasing the tube length and cycles between 10% and 90 %. New valve designs as diameter or working with lower viscosity fluids as, for presented by Murrenhoff (2003), Winkler et al. (2008), example, on water hydraulic systems. Uusitalo et al. (2010), and Winkler et al. (2010) are potential solutions to reach this requirement. 7. Conclusions
Two detailed models for step-up PWM valves which are of interest for the analysis and design of new systems using the switched inertance principle were presented. Firstly, an ideal valve was studied as a reference for the analysis of real valves. Using the model including linear resistance the theoretical and experimental results presented show that it is possible to predict the average value of the controlled pressure and the flow rates in several parts of the system. The modelling also gives the minimal and maximum values of flow rate through the inertance tube and thus an effective idea of the valve operation can be achieved including a prediction of reverse flow rates. Therefore, despite both flow-pressure non-linearity and Fig. 15: Predicted influence of the switching frequency limited time response of the switching valve as well as on the load pressure the pressure wave propagation in the inertance tube, the presented linear model describes the global behaviour It is important to observe that for fsw = 100 Hz and qVL = 0.1 L/s (6 L/min) the efficiency is higher than 75 % of step-up switching converters very well. for duty cycle lower than 0.7, controlling the pressure The time responses presented are complex. Several up to approximately 60 bar (2.5 times greater than the dynamic phenomena occur, such as fluid supply pressure). compressibility in the internal chambers and wave Based on this example, it can be seen that the definition propagation through the tube. Determining the system of the maximum flow rate for a specific step-up performance and the effectiveness of designs based converter and the consequent tube and switching only on this kind of information is difficult. In this context, the model presented in this paper can be used device sizing is imperative for achieving an efficient for the preliminary design of switching converters and device. a time or frequency analysis can be used for system Acknowledgments optimization. According to the presented equations, the step-up This work was supported by the Engineering and converter performance depends on the PWM signal Physical Sciences Research Council (EPSRC-UK) period, resistance, inertance, as well as the average [grant number EP/H024190] and the Coordenação de load flow rate. Therefore, using this model it is Aperfeiçoamento de Pessoal de Nível Superior possible find the ideal parameter combination for (CAPES-Brazil) [grant number BEX 3659/09-7]. maximum efficiency. The same modelling approach was applied by the authors for step-down converters and validated on a References test rig. As in the case of step-up converters, the results are very promising and the model represents a very good approach for the analysis and design of inertance Brown, F. T. 1984. On switching circuits for fluid servosystems. Proceedings of National Conference on switching valves. Fluid Power, Illinois, USA, pp. 209-218. Brown, F. T. 1987. Switched reactance hydraulics: a new way to control fluid power. Proc. National Nomenclature Conference on Fluid Power. Chicago, USA, pp. 25-34. Brown, F.T. Tentarelli, S. C., Ramachandran, S. A. C Hydraulic capacitance [m3/Pa] 3 1988. Hydraulic rotary switched-inertance servo- CL Load capacitance [m /Pa] transformer, Transactions of ASME: Journal of fsw Switching frequency [Hz] Dynamic Systems, Measurement, and Control. Vol. f, g Auxiliary variables [Pa] 110, pp.144-150. L Tube inertance [kg/m4] R Hydraulic resistance [Pa.s/m3] Hettrich, H., Bauer, F., Fuchshumer, F. 2009. Speed pAin Internal working pressure [Pa] controlled, energy efficient fan drive within a constant pL Load pressure [Pa] pressure system. Proceedings of the Second Workshop pS Supply pressure [Pa] on Digital Fluid Power. Linz, Austria, pp. 62-71. pSin Internal supply pressure [Pa] Johnston, D. N. 2009. A switched inertance device for pT Return pressure [Pa] 3 efficient control of pressure and flow. Proceedings of qVI Tube flow rate [m /s] 3 the ASME 2009 Dynamic Systems and Control qVImax Maximum tube flow rate [m /s] 3 Conference - DSCC2009. Hollywood, USA. qVImin Minimum tube flow rate [m /s] 3 qVL Load flow rate [m /s] Kogler, H., Scheidl, R. 2008. Two basic concepts of 3 qVn Nominal flow rate [m /s] hydraulic switching converters. Proceedings of the 3 qVML Main load flow rate [m /s] First Workshop on Digital Fluid Power. Tampere, 3 qVS Supply flow rate [m /s] Finland, pp. 113-128. q Return flow rate [m3/s] VT Manhartsgruber, B., Mikota, G., Scheidl, R. 2005. R Hydraulic resistance [Pa.s/m3] Modelling of a switching control hydraulic system. Re Equivalent resistance [Pa.s/m3] Mathematical and Computer Modelling of Dynamical T Switching period [s] sw Systems, Vol. 11, N. 3, pp. 329-344. t Time [s] ts Settling time [s] Millmann, J., Taub, H. 1965. Pulse, digital, and Uc Command signal [1] switching waveforms. McGraw-Hill, New York. U Spool position voltage [V] s Murrenhoff, H. 2003. Trends in valve development. x, y, z Auxiliary variables [s] O+P – Ölhydraulik und Pneumatik. V. 46, N. 4. p Pressure difference [Pa] 3 qVI Tube flow amplitude [m /s] Scheidl, R., Manhartsgruber, B., Winkler, B. 2008. λ Duty cycle [1] Hydraulic Switching control – Principles and state of Efficiency [1] the art. Proceedings of the First Workshop on Digital
n Natural frequency [rad/s] Fluid Power. Tampere, Finland, pp. 31-49. τ Time constant [s] Uusitalo,J.-P., Ahola, V., Soederlund, L., Linjama, M., Juhola, M., Kettunen, L., , 2010. Novel bistable hammer valve for digital hydraulics. International Journal of Fluid Power. N. 3, pp. 35-44. Winkler, B., Plöckinger, A., Scheidl, R. 2008. Components for digital and switching hydraulics. Proceedings of the First Workshop on Digital Fluid Power. Tampere, Finland. pp. 53-76. Winkler, B., Plöckinger, A., Scheidl, R. 2010. A novel piloted fast switching multi popet valve. International Journal of Fluid Power. N. 3, pp. 7-14.
Andrew Plummer He received his PhD degree from the University of Bath in 1991, for research into adaptive control of electrohydraulic systems. He has worked for Rediffusion Simulation on flight simulator control systems, as a lecturer at the University of Leeds, and as R&D manager for Instron. Now Professor of Machine Victor Juliano De Negri Systems at the University of Bath, and Director of the Centre for Power Transmission and Motion Control. He received his D. Eng. degree in 1996, from the Federal University of Santa Catarina (UFSC). In 2010 he took a 7-month sabbatical at PTMC, University of Bath, UK. He has been a Professor at the Mechanical Engineering Department at UFSC since 1995. He is currently the Head of Department and the Head of the Laboratory of Hydraulic and Pneumatic Systems (LASHIP). His interest areas include hydraulic components, power generating plants, mobile hydraulics, pneumatic systems and positioning systems. Nigel Johnston He is a Senior Lecturer in the Department of Mechanical Engineering at the University of Bath, and teaches computer programming, simulation, numerical analysis and fluid power. His PhD was for research into fluid-borne noise characteristics in hydraulic systems. This work resulted in a new ISO Standard for pump fluid-borne noise testing (ISO 10767-1: 1996). His current research interests include: modelling of the dynamic behaviour of pumps, pipelines and valves, Pengfei Wang noise in fluid power systems, valve stability, vehicle hydraulics including power-assisted steering systems, He was a Research Officer in the Department of flow and pressure transients in aircraft fuel systems. Mechanical Engineering at the University of Bath. His PhD was also awarded at the same university on hardware in the loop testing of continuously variable transmission (CVT). His research covered fluid power systems like power-assisted steering systems, CVTs, among others.