With a clear idea of the basic electro-mechanical building blocks and their relationship to one another (Part 1), the use of simulation modelling tools makes it possible to derive the basic mathematical models of the hydraulic components, and more, importantly simulate and test them in a typical implementation and extract meaningful results that will be useful in coming up with a final design.

Tools such as the Matlab Simulink allow the developer to first derive a top level system design as shown in Figure A-1 below and then proceed to lower levels and derive models for each of the important elements in the design.

Figure A-1: Top level system diagram of a typical electro hydraulic system.

Modelling the Flow Control Servo Valve
The two stage nozzle-flapper servo-valve consists of three main parts: an electrical torque motor, hydraulic amplifier, and valve spool assembly and is shown in detail in Figure 8, below.

Figure 8. Valve Torque Motor Assembly (illustration courtesy of Moog)

The torque motor consists of an armature mounted on a thin-walled sleeve pivot and suspended in the air gap of a magnetic field produced by a pair of permanent magnets.

When current is made to flow in the two armature coils, the armature ends become polarized and are attracted to one magnet pole piece and repelled by the other. This sets up a torque on the flapper assembly, which rotates about the fixture sleeve and changes the flow balance through a pair of opposing nozzles, shown in Figure 9, below. The resulting change in throttle flow alters the differential pressure between the two ends of the spool, which begins to move inside the valve sleeve.

Figure 9. Valve Responding to Change in Electric Input (illustration courtesy of Moog)

Lateral movement of the spool forces the ball end of a feedback spring to one side and sets up a restoring torque on the armature/flapper assembly.

When the feedback torque on the flapper spring becomes equal to the magnetic forces on the armature the system reaches an equilibrium state, with the armature and flapper centred and the spool stationary but deflected to one side. The offset position of the spool opens flow paths between the pressure and tank ports (P_s and T), and the two control ports (A and B), allowing oil to flow to and from the actuator.

Modelling the Torque Motor
For simplicity, the electrical characteristics of the servo-valve torque motor may be modelled as a series L-R circuit, neglecting for the time being any back-EMF effects generated by the load. The transfer function of a series L-R circuit is:

where L_c is the inductance of the motor coil, and R_c the combined resistance of the motor coil and the current sense resistor of the servo amplifier. Values of inductance and resistance for series and parallel winding configurations of the motor are usually published in the manufacturer's data sheet.

The lateral force on the valve spool is proportional to torque motor current, but oil flow rate at the control ports also depends upon the pressure drop across the load.

Modelling the Valve Spool Dynamics
A servo-valve is a complex device which exhibits a high-order non-linear response, and knowledge of a large number of internal valve parameters is required to formulate an accurate mathematical model. Indeed, many parameters such as nozzle and orifice sizes, spring rates, spool geometry and so on, are adjusted by the manufacturer to tune the valve response and are not normally available to the user.

Practically all physical systems exhibit some non-linearity: in the simplest case this may be a physical limit of movement, or it may arise from the effects of friction, hysteresis, mechanical wear or backlash. When modelling complex servo-valves, it is sometimes possible to ignore any inherent non-linearities and employ a small perturbation analysis to derive a linear model which approximates the physical system.

Such models are often based on classical first or second order differential equations, the coefficients of which are chosen to match the response of the valve based on frequency plots taken from the data sheet.

A simple first or second order model yields only an approximation to actual behavior, however the servo-valve is not the primary dynamic element in a typical hydraulic servo system and is generally selected such that the frequency of the 90 degree phase point is a factor of at least three higher than that of the actuator.

 For this reason it is usually only necessary to accurately model valve response through a relatively low range of frequencies, and the servo-valve dynamics may be approximated by a second order transfer function without serious loss of accuracy.

Figure 10. Typical Servo-valve Frequency Response Curve (illustration courtesy of Moog)

A typical performance graph for a high-responsive servo-valve is shown in Figure 10, above. Assuming a second order approximation is to be used, suitable values for natural frequency and damping ratio will need to be determined from the graph.

Natural frequency (_v) can be read fairly accurately from the -3dB or 90 degree phase point of the 40% curve. Damping can be determined from an estimate of the magnitude of the peaking present. For an under-damped second order system, the damping factor (S_v) can be shown to be related to peak amplitude ratio (M_v) by the formula

In this example, a reasonable estimate of peaking based on the 40% response curve would be about 1.5 dB, which corresponds to an amplitude ratio of about 1.189. A suitable value of damping determined iteratively from Equation 2 is about 0.48. Armed with these values, a simplified model of the servo-valve spool dynamics may be constructed.

The input to the model will be the torque motor current derived from Equation 1 in Part 1 normalized to the saturation current obtained from the datasheet, and the output will be the normalized spool position. Shown in Figure A-2 below is the Simulink model of the servo valve.

Figure A-2. A Simulink model of the servo-valve

Modelling Valve Flow-Pressure
The servo-valve delivers a control flow proportional to the spool displacement for a constant load. For varying loads, fluid flow is also proportional to the square root of the pressure drop across the valve. Control flow, input current, and valve pressure drop are related by the following simplified equation:

In the above equation, Q_L, is the hydraulic flow delivered through the load actuator, Q _R the rated valve flow at a specified pressure drop P_R, and i*_v is normalized input current. P_R is the pressure drop across the valve given by P_V = P_S+ P_T + P_L, where P_S, P_T and P_L are system pressure, return line (tank) pressure, and load pressure respectively.

Maximum power is transferred to the load when P_L = 2/3 P_S, and since the most widely used supply pressure is 3,000 psi, it is common practice to specify rated valve flow at P = 1,000 psi (approximately 70 bar). The static relationship between valve pressure drop and load flow is often presented in manufacturer's datasheets as a family of curves of normalized control flow against normalized load pressure drop for different values of valve input current as shown in Figure 11, below.

Figure 11. Servo-valve Flow-pressure curves (illustration courtesy of Moog)

The horizontal axis is the load pressure drop across the valve, normalized to 2/3 of the supply pressure. The vertical axis is output flow expressed as a percentage of the rated flow, Q_R. The valve orifice equation is applied separately for the two control ports to obtain expressions for oil flow into each of the two actuator chambers. Since load flow is defined as the flow through the load: Q_L = Q_A = -Q_B

Figure A-3: Simulink model of the Hydraulic Actuator

As shown in the Simulink model in Figure A-3, above, the inputs are command voltage from the amplifier, supply and return oil pressures from the hydraulic power supply (P_S and P_T), and load pressures from the actuator chambers (P_A and P_B). Outputs are the flows to each side of the piston (Q_A and Q_B), and the load flow (Q_L).

Modelling the Linear Actuator
Cylinder Chamber Pressure. The relationship between valve control flow and actuator chamber pressure is important because the compressibility of the oil creates a "spring" effect in the cylinder chambers which interacts with the piston mass to give a low frequency resonance. This is present in all hydraulic systems and in many cases this abruptly limits the usable bandwidth. The effect can be modelled using the flow continuity equation from fluid mechanics which relates the net flow into a container to the internal fluid volume and pressure.

The left hand side of the equation is the net flow delivered to the chamber by the servo valve. The first term on the right hand side is the flow consumed by the changing volume caused by motion of the piston, and the second term accounts for any compliance present in the system. This is usually dominated by the compressibility of the hydraulic fluid and it is common to assume that the mechanical structure is perfectly rigid and use the bulk modulus of the oil as a value for .

Mineral oils used in hydraulic control systems have a bulk modulus in the region of 1.4 x 109 N/m. The above previous equation can be re-arranged to find the instantaneous pressure in chamber A as follows:

Piston Dynamics. Once the two chamber pressures are known, the net force acting on the piston (F_P) can be computed by multiplying by the area of the piston annulus (A_P) by the differential pressure across it.

F_P = (P_A - P_B) A_P

An equation of forces for piston motion can now be established by applying Newton's second law. For the purposes of this analysis, it will be assumed that the piston delivers a force to a linear spring load with stiffness K_L, which will allow us to investigate the load capacity of the actuator later. The effects of friction (F_f) between the piston and the oil seals at the annulus and end caps will also be included. The resulting force equation for the piston is shown below and may be modelled in Simulink using two integrator blocks.

The total frictional force depends on piston velocity, driving force (F_P), oil temperature and possibly piston position. One method of modelling friction is as a function of velocity, in which the total frictional force is divided into static friction (a transient term present as the actuator begins to move), Coulomb friction (a constant force dependent only on the direction of movement), and viscous friction (a term proportional to velocity). Assuming that viscous and Coulomb friction components dominate, frictional force (F_f) can be modelled as

where viscous and Coulomb friction coefficients are denoted by F_v0 and F_c0 respectively. Frictional effects are notoriously difficult to measure and accurate values of these coefficients are unlikely to be known, but order of magnitude estimates can sometimes be made from relatively simple empirical tests.

One test which can yield useful information is to subject the system to a low frequency, low amplitude sinusoidal input signal, and plot the output displacement over one or two cycles. A low friction system should reproduce the input signal, but the presence of friction will tend to flatten the tops of the sine wave as the velocity falls to a level blow that necessary to overcome any inherent Coulomb friction.

In actuators fitted with conventional PolyTetraFluoroEthylene(PTFE)-based bearings, friction is related fairly linearly to supply pressure and oil temperature and care should be taken to conduct testing under representative conditions.

In a first analysis, leakage effects in the actuator are sometimes neglected, however this is an important factor which can have a significant damping influence on actuator response. Leakage occurs at the oil seals across the annulus between the two chambers and at each end cap, and is roughly proportional to the pressure difference across of the seal. Including leakage effects, the flow continuity equation for chamber A

is similar with appropriate changes of sign. It is a relatively simple matter to modify the Simulink model to compute the instantaneous chamber leakages and subtract them from the total input flow.

Modelling the hydraulic power supply
The behavior of the hydraulic power supply described in Part 1 of this series may be modelled in the same way as the chamber volumes: by applying the flow continuity equation to the volume of trapped oil between the pump and servo-valve. In this case, the input flow is held constant by the steady speed of the pump motor, and the volume does not change. The transformed equation is

This equation takes into account the load flow (Q_L) drawn from the supply by the servo-valve, and accurately models the case of a high actuator slew rate resulting in a load flow which exceeds the flow capacity of the pump. In such cases the supply pressure (P_S) falls, leading to a corresponding reduction in control flow and loss of performance. The action of the pressure relief valve may be modelled using a limited integrator to clamp the system pressure to the nominal value.

Next in Part 3: Modelling a typical servo actuator
To  read Part 1 go to : "The basics of electro-hydraulic servo actuator systems."

Richard Poley is Field Application Engineer at Texas Instruments with focus on digital control systems.