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Chapter 6 Two-Port Network Model Chapter 6 Two-Port Network Model 6.1 Introduction In this chapter a two-port network model of an actuator will be briefly described. In Chapter 5, it was shown that an automated test setup using an active system can re- create various load impedances over a limited range of frequencies. This test set-up can therefore be used to automatically reproduce any load impedance condition (related to a possible application) and apply it to a test or sample actuator. It is then possible to collect characteristic data from the test actuator such as force, velocity, current and voltage. Those characteristics can then be used to help to determine whether the tested actuator is appropriate or not for the case simulated. However versatile and easy to use this test set-up may be, because of its limitations, there is some characteristic data it will not be able to provide. For this reason and the fact that it can save a lot of measurements, having a good linear actuator model can be of great use. Developed for transduction theory [29], the linear model presented in this chapter is 77 called a Two-Port Network model. The automated test set-up remains an essential complement for this model, as it will allow the development and verification of accuracy. This chapter will focus on the two-port network model of the 1_3 tube array actuator provided by MSI (Cf: Figure 5.5). 6.2 Theory of the Two–Port Network Model As a transducer converts energy from electrical to mechanical forms, and vice- versa, it can be modelled as a Two-Port Network that relates the electrical properties at one port to the mechanical properties at the other port. On the two-port network, Figure 6.1, we can see that the voltage, V, and the current, I, (electrical properties) are both linked to the force, F, and the velocity, u, (mechanical properties). V is the voltage across the electrical inputs to the actuator. I is the current supplied to it. If it is assumed that one side of the actuator is fixed (i.e.: only one side of the actuator moves), F is the force output and u is its velocity. I u ACTUATOR F V Electrical Side Mechanical Side Figure 6.1: Two-port network model of an actuator 78 Associated with those four different variables, four different ratios that correspond to different properties of the system under different boundary conditions can be measured. First if the electrical side of the circuit is open, it is easy to measure what is called the internal mechanical impedance directly linked to the internal dynamic stiffness of the actuator and defined as the ratio of force over velocity for the frequency w: Zint(w)=F(w)/u(w) for I(w)=0, (6.1) and a transduction coefficient defined as the ratio of voltage over velocity: T2(w)=V(w)/u(w), for I(w)=0. (6.2) Then, if the mechanical side of the actuator is blocked (the actuator is clamped on both sides), the electrical impedance of the actuator defined as the ratio of voltage over current can be measured as: Zele(w)=V(w)/I(w), for u(w)=0, (6.3) and another transduction coefficient defined as the ratio of force over current can also be measured: T1(w)=F(w)/I(w), for u(w)=0. (6.4) In the model shown in Figure 6.1, voltage has been chosen to be the analog of force, and current to be the analog of velocity. Therefore, using the different mechanical and electrical actuator characteristics described above (equations (6.1) through (6.4), both force and voltage can be expressed as a function of current and velocity for the frequency w: F(w)=Zint(w)u(w) +T1(w)I(w), (6.5) 79 And V(w)=T2(w)u(w) + Zele(w)I(w) (6.6) Rearranging this into a matrix form, the following expression is obtained: ìF(w)ü éZint (w) T1(w) ù ìu(w)ü í ý = ê ú * í ý (6.7) îV (w)þ ë T2 (w) Zele (w)û îI(w)þ This relation describes completely the linear electro-mechanical behavior of an actuator since it links all its electrical and mechanical characteristics together. When the actuator drives a load, or in our case the impedance Zd, set by the control system, the following relation for the actuator output force and velocity can be made: Zd(w)=F(w)/u(w) (6.8) Substituting equation (6.8) into equation (6.5), a new expression for the velocity as a function of current is: -1 u(w)={(Zd(w)-Zint(w)) T1(w)}I(w), (6.9) By substituting this relationship into equation (6.7), the force and the voltage now become a function of current and desired load impedance: -1 F(w)=Zd(w){(Zd(w)-Zint(w)) T1(w)}I(w) (6.10) -1 V(w)=[T2(Zd(w)-Zint(w)) T1(w)+Zele(w)]I(w) (6.11) Therefore, once the four matrix coefficients (equation (6.7)) are measured for a frequency, the actuator performance (output force and velocity) can then be predicted for any given value of the current over its range of linear behavior. The quality of the prediction will mainly depend on the quality of the measurements of these four 80 characteristic coefficients and this is due in part to the accuracy of the boundary conditions during measurements. The condition, I=0, necessary to get Zint and T2 , is easy to satisfy but the other one, u=0, to get Zele and T1 is more critical. Using the controller to block the actuator gives a very low level of the velocity on top of the sample actuator. However, because the limits of the sensitivity of the impedance head are reached during this operation, it cannot be totally ensured that the sample actuator is perfectly blocked. This can then lead to slight errors in the calculation of Zele and T1. 6.3 Obtaining Data for the Two–Port Network Model To measure all the characteristic coefficients, the test set-up (shown in Figure 5.7) has been used. A stack actuator (E100P-2) from EDO has been used as the control actuator instead of the 1_3 tube array actuator from MSI. This modification was made because the stack actuator has larger blocked-force and free displacement capabilities than the 1_3 tube array actuator, providing more control authority and the ability to re- create difficult impedance conditions. The first two coefficients, Zint and T2, were calculated using time signals of force, voltage and velocity measured when the sample actuator had no input current. The only excitation was due to the control actuator driven with a harmonic signal. Force and velocity time signals have been measured with the impedance head. The voltage time signal has been measured directly out of the actuator itself since its level was low enough (Cf: Figure 6.2). 81 Control actuator driven F u V Sample actuator passive Figure 6.2: Set-up to measure V(t), F(t) and u(t) when the sample actuator has no input current. The second two coefficients, Zele and T1, have then been calculated using time signals of force, voltage and current measured when the sample actuator was blocked (u»0). To do so, the adaptive feedforward controller was used with a very high desired impedance (Zd= 10e5) so that the control system practically cancelled the velocity measured with the impedance head. The high blocked force capability of the EDO stack actuator was necessary for this part of the control providing the boundary condition, u=0. Therefore, to reach this very high desired impedance, the controller had to increase the force, canceling the velocity. The force time signal has been measured using the impedance head (I-H). The voltage, V, to the sample actuator, was too high if measured directly out of the power amplifier (±200V) to be recorded on the time data acquisition system. Therefore, it was aquired by measuring the output voltage divided by 20, that is provided by the power amplifier. For the current time signal, a resistor of resistance R=176W was placed in series with the control actuator and the voltage across it, Vi=RI, was used to calculate the current, I (as the ratio Vi/R) (Cf: Figure 6.3). In the case where the actuator was blocked, the assumption that Vi was relatively small compare to V was made and it was assumed that the voltage, V, out of the power amplifier was the voltage input to the sample actuator. This assumption was valid since V was 30 times bigger than Vi. 82 Control signal Voltage VI=RI Sample signal Voltage V/20 Control actuator I Resistor, R Force, F I-H OUT/20 V Power amplifier Sample actuator IN OUT Figure 6.3: Set-up to measure I(t), V(t) and F(t) when the sample actuator is blocked 6.4 Application for the 1_3 Tube Array Actuator from MSI Using the two different configurations described above, and setting the excitation signal as an harmonic signal of frequency 500Hz, the four matrix coefficients were first calculated using the corresponding measured time signals (Cf: equations (6.1) through (6.4)). Figure 6.4 shows the magnitude of all those coefficients on the same plot. These plots are from 0 to 2000Hz, but, as the signal was harmonic, the values are only consistent at 500Hz. Elsewhere, as expected, the signals are mainly noise.
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