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.

It should be mentioned that the transduction coefficients, T1 and T2, are usually known to be the same for piezoelectric actuators [30]. In the case, for the 1_3 tube array actuator from MSI, they are not the same. T2 is 5dB bigger than T1. A first possible explanation could be the fact that the required blocked condition of the actuator while measuring T1 was not totally respected (as mentioned previously in Section 6.2). Another eventual possible explanation could come from the particular tube array shape of the actuator where the tubes, inside the urethane matrix, may not all exert a force output with the

83 same effectiveness (if they are not all perfectly in contact with the extremities of the

actuator) . Therefore, when calculating T1 (Cf: equation (6.1)), the force measured may not be as high as expected giving this lower value for T1. T2 is not comparatively as effected as T1 by this particular configuration of the 1_3 tube array actuator since its ratio (Cf: equation (6.2)) does not depend on the force. However, this unexpected difference between T1 and T2 for a piezoelectric actuator may also have another explanation. It could be very interesting to deeper investigate this problem, but that would be the subject of another research .

Figure 6.4: Magnitude of different transfer function giving the internal parameters of the 1_3 tube array actuator at 500Hz.

Once the complex values for these four coefficients are obtained, the force and velocity can be calculated for any desired impedance, only as a function of the measured current signal (Cf: equations (6.10) and (6.11)).

To verify the accuracy of the model, the adaptive feedforward controller was used to simulate various desired impedances. During these various control tasks, force, velocity and current time signals have been measured to be compared at the same frequency with

84 values provided by the model. For every case of different impedance, the corresponding measured current signal has been used to perform all the calculations.

Figures 6.5 and 6.6 show the velocity and force outputs of the 1_3 tube array actuator at 500Hz versus a real load impedance whose magnitude varies from 1 to 10e6 N.s/m. The plots show the magnitude of the velocity and the force expressed in dB normalized with respect to input voltage. For those load impedances, corresponding to pure dampers, the predicted performances using the two-port network model (dashed line) matches very well the measured performances (solid line). At low impedance levels, the velocity output, u0, is relatively constant whereas the force, F0, increases linearly with the load impedance. This can be explained by the fact that for low external load impedances, Zext, the actuator exerts most of its effort to overcome its own internal impedance, Zint, (related to its stiffness) which is relatively high in magnitude compare to the external load impedance. As a consequence, the motion of the actuator does not really increases with the magnitude of the load impedance. Therefore, since the velocity does not change in this low impedance region, as the magnitude of the load impedance increases, so does the force output (Cf: Figure 6.7). After it has reached about 10 thousands N/m/s, the load impedance becomes higher than the internal impedance of the actuator. As the actuator approaches a blocked condition, the velocity falls off and the force tends to stabilize to the blocked force (Fo=Fint).

85 Figure 6.5: Velocity of the 1_3 tube array actuator for different real impedances at 500Hz

Figure 6.6: Force for the 1_3 tube array actuator for different real impedances at 500Hz

86 F u 0 0 F0 = Z ext u0 Zext F Fint u = int i.e. for small Z : u = . 0 ext 0 Z Sample Zint (Z ext + Z int ) int Fint Actuator Z ext Z ext F0 = Fint » Fint for small Zext (Z ext + Z int ) Z int

Figure 6.7: Behavior of the force and velocity for small external load impedance

Figures 6.8 and 6.9 represent the same thing as Figures 6.5 and 6.6, except that the load impedances are now purely imaginary (equivalent to the impedance of a mass) but with the same magnitude. Therefore, instead of adding damping to the system (previous case), if the sample actuator is seen as a mass-spring-damper system, increasing the load impedance is equivalent to adding mass. In that case, when the load impedance gets close to Z=j10e4, where j=Ö(-1) it matches the internal impedance of the actuator which is mainly imaginary if there is not too much damping:

Zint= f/v » kint/(jw) (6.12)

where Zint is the internal impedance, kint the internal stiffness, j=Ö(-1), and w is the circular frequency. Because the added “mass” has an impedance term that is of opposite sign to the internal impedance created by the actuator stiffness, these two terms cancel at this point and the actuator begins to resonate and both velocity and force become very large. For this resonance condition, the controller had to drive the control actuator extremely hard (see Section 3.3: Limitation of Control), and therefore could not perform very well for this corresponding desired impedance. This explains the mismatch between the predicted and measured performances.

87 Figure 6.8: Velocity of the 1_3 tube array actuator for different pure imaginary impedances at 500Hz

Figure 6.9: Force from the 1_3 tube array actuator for different pure imaginary impedances at 500Hz

88 In an effort to further characterize the performance of an actuator, the mechanical power output was calculated using:

1 W = Â{F *u} (6.13) 2 where  denotes the real part and * denotes the complex conjugate.

Figures 6.10 and 6.11 show the mechanical power output of the 1_3 tube array actuator for both cases with real and purely imaginary load impedances. As expected, in both cases, there is a peak when the load impedance reaches the magnitude of 10e4 N/m/s. The peak is even higher in the less damped case where the impedance is purely imaginary.

What is emphasized by these two graphs is the importance for actuators to match the application in which they will be used. Indeed, on both cases presented here, there are very huge differences in power output for a same actuator used under different load impedance conditions. Considering the case of a real impedance condition at 500 Hz (Cf: Figure 6.10), the mechanical power output of the actuator for a load impedance Z=10e4 N/m/s, is –75dB. On the same graph, but for the load impedance condition Z=10 N/m/s the mechanical power output is then –105dB. There is a difference of almost 30dB in the mechanical power output of the actuator between these two different cases of load impedance condition. The mechanical output power for Z=10e4 N/m/s is then one thousand times bigger than for Z=10 N/m/s. Therefore, to be more efficient, it seems obvious that this actuator should be used under a load impedance condition Z=10e4 N/m/s rather than a load impedance condition Z=10 N/m/s.

Since it can give good insight into actuator behavior, the two-port network model can be a very useful tool for actuator characterization, and can be used to match actuators to applications.

89 Figure 6.10: Mechanical power from the 1_3 tube array actuator for different real impedances at 500Hz

Figure 6.11: Mechanical power from the 1_3 tube array actuator for different pure imaginary impedances at 500Hz

90 6.5 Discussion

The two-port network model is a model that relates the electrical properties of a transducer to its mechanical properties. With this two-port network model that is studied in this Chapter, the actuator's force and displacement are only considered in one direction and on only one side (the other side is assumed to be blocked). If the force and the displacement where considered on both sides of the actuator, the model would be a little more complicated and probably called a three-port network model. Indeed, with two force outputs and two velocity outputs, the characteristic matrix of the model would be of size three by three.

However, in our case, the two-port network model is easy to calculate since it only requires a single set of measurements per frequency in order to get the four characteristic coefficients (Zint, Zele, T1 and T2) that provide a complete frequency domain description of an actuator’s linear behavior.

Force and velocity outputs of an actuator are linked together by the applied load impedance condition. Therefore, once the four characteristic coefficients of an actuator are measured at a given frequency, it is very simple to obtain its different force and voltage outputs, as a function of current for a wide range of desired load impedance conditions without extra measurements. Furthermore, the tests that have been proceeded for different cases of load impedance conditions prove that this model is fairly accurate compared to the results provided with the test set-up using the controller.

In addition, with this model, the performance of an actuator can be predicted even for load impedance conditions where the controller does not work very well (e.g.: lack of authority due to the control actuator or resonance of the system). That was the case for the load impedance condition, Z=j10e4 N/m/s, of opposite sign of the internal stiffness of the actuator. The actuator begun to resonate and the controller had to drive the control

91 actuator very hard, and therefore could not perform very well for this corresponding desired impedance.

Even if this method appears to be very convenient, it cannot substituted the active control of impedance test set-up. Indeed, in order to develop and test the two-port network model, the active control of impedance test set-up is necessary. However, it can be a good complement.

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