Piezoelectric Force Sensor - Test Bench

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Ts = 1e-4; % Sampling Time [s]
win = hann(ceil(10/Ts));

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[tf_oc_est, f] = tfestimate(oc.u, oc.encoder, win, [], [], 1/Ts);
[co_oc_est, ~] = mscohere(  oc.u, oc.encoder, win, [], [], 1/Ts);

[tf_sc_est, ~] = tfestimate(sc.u, sc.encoder, win, [], [], 1/Ts);
[co_sc_est, ~] = mscohere(  sc.u, sc.encoder, win, [], [], 1/Ts);

Important

The measured data is loaded.

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load('force_sensor_steps.mat', 't', 'encoder', 'u', 'v');

The excitation signal (steps) and measured voltage across the sensor stack are shown in Figure 6.

The measured voltage shows an exponential decay which indicates that the charge across the capacitor formed by the stack is discharging into a resistor. This corresponds to an RC circuit with a time constant $\tau =RC$.

In order to estimate the time domain, we fit the data with an exponential. The fit function is:

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f = @(b,x) b(1).*exp(b(2).*x) + b(3);

Three steps are performed at the following time intervals:

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t_s = [ 2.5, 23;
       23.8, 35;
       35.8, 50];

We are interested by the b(2) term, which is the time constant of the exponential.

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tau = zeros(size(t_s, 1),1);
V0  = zeros(size(t_s, 1),1);

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for t_i = 1:size(t_s, 1)
    t_cur = t(t_s(t_i, 1) < t & t < t_s(t_i, 2));
    t_cur = t_cur - t_cur(1);
    y_cur = v(t_s(t_i, 1) < t & t < t_s(t_i, 2));

    nrmrsd = @(b) norm(y_cur - f(b,t_cur)); % Residual Norm Cost Function
    B0 = [0.5, -0.15, 2.2];                        % Choose Appropriate Initial Estimates
    [B,rnrm] = fminsearch(nrmrsd, B0);     % Estimate Parameters ‘B’

    tau(t_i) = 1/B(2);
    V0(t_i)  = B(3);
end

The obtained values are shown below.

With the capacitance being $C=4.4\mu F$, the internal impedance of the Speedgoat ADC can be computed as follows:

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Cp = 4.4e-6; % [F]
Rin = abs(mean(tau))/Cp;

1494100.0

The input impedance of the Speedgoat’s ADC should then be close to $1.5M\mathrm{\Omega }$ (specified at $1M\mathrm{\Omega }$).

As shown in Figure 6, the voltage across the Piezoelectric sensor stack shows a constant voltage offset.

We can explain this offset by looking at the electrical model shown in Figure 7 (taken from (Reza and Andrew 2006)).

The differential amplifier in the Speedgoat has some input bias current $i_n$ that produces a voltage offset across its own internal resistance. Note that the impedance of the piezoelectric stack is much larger that that at DC.

The estimated input bias current is then:

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in = mean(V0)/Rin;

1.5119e-06

Be looking at Figure 7, we can see that an additional resistor in parallel with $R_{in}$ would have two effects:

• reduce the input voltage offset $$V_{off}=\frac{R_a{R}_{in}}{R_a+R_{in}}{i}_n$$
• increase the high pass corner frequency $f_c$ $$C_p\frac{R_{in}{R}_a}{R_{in}+R_a}={\tau }_c=\frac{1}{f_c}$$ $$R_a=\frac{R_i}{f_c{C}_p{R}_i-1}$$

If we allow the high pass corner frequency to be equals to 3Hz:

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fc = 3;
Ra = Rin/(fc*Cp*Rin - 1);

79804

With this parallel resistance value, the voltage offset would be:

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V_offset = Ra*Rin/(Ra + Rin) * in;

0.11454

Which is much more acceptable.

A resistor $R_p\approx 100k\mathrm{\Omega }$ is then added in parallel with the force sensor as shown in Figure 8.

After the resistor is added, the same steps response is performed.

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load('force_sensor_steps_R_82k7.mat', 't', 'encoder', 'u', 'v');

The results are shown in Figure 9.

Three steps are performed at the following time intervals:

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t_s = [1.9,  6;
       8.5, 13;
      15.5, 21;
      22.6, 26;
      30.0, 36;
      37.5, 41;
      46.2, 49.5]

The time constant and voltage offset are again estimated using a fit function. And indeed, we obtain a much smaller offset voltage and a much faster time constant.

Knowing the capacitance value, we can estimate the value of the added resistor (neglecting the input impedance of $\approx 1M\mathrm{\Omega }$):

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Cp = 4.4e-6; % [F]
Rin = abs(mean(tau))/Cp;

101200.0

And we can verify that the bias current estimation stays the same:

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in = mean(V0)/Rin;

1.5305e-06

This validates the model of the ADC and the effectiveness of the added resistor.

The measured data is loaded and the first 25 seconds of data corresponding to transient data are removed.

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load('force_sensor_sin.mat', 't', 'encoder', 'u', 'v');

u       = u(t>25);
v       = v(t>25);
encoder = encoder(t>25) - mean(encoder(t>25));
t       = t(t>25);

The driving voltage is a sinus at 0.5Hz centered on 3V and with an amplitude of 3V (Figure 10).

The corresponding displacement as measured by the encoder is shown in Figure 11.

The full stroke is:

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max(encoder)-min(encoder)

5.005e-05

The generated voltage by the stack is shown in Figure 12.

The capacitance of the stack is

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Cp = 4.4e-6; % [F]

The voltage and charge across a capacitor are related through the following equation:

where $U_C$ is the voltage in Volts, $Q$ the charge in Coulombs and $C$ the capacitance in Farads.

The corresponding generated charge is then shown in Figure 13.

The relation between the generated voltage and the measured displacement is almost linear as shown in Figure 14.

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b1 = encoder\(v-mean(v));

With a 16bits ADC, the resolution will then be equals to (in [nm]):

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abs((20/2^16)/(b1/1e9))

3.9838

Bibliography