test-bench-force-sensor/index.org

Piezoelectric Force Sensor - Test Bench

• Introduction
• Change of Stiffness due to Sensors stack being open/closed circuit
  • Introduction
  • Load Data
  • Transfer Functions
• Generated Number of Charge / Voltage
  • Introduction
  • Steps
  • Add Parallel Resistor
  • Sinus

Introduction   ignore

In this document is studied how a piezoelectric stack can be used to measured the force.

• Section sec:open_closed_circuit: the effect of the input impedance of the electronics connected to the force sensor stack on the stiffness of the stack is studied
• Section sec:charge_voltage_estimation:

Change of Stiffness due to Sensors stack being open/closed circuit

<<sec:open_closed_circuit>>

Introduction   ignore

Load Data

  oc = load('identification_open_circuit.mat', 't', 'encoder', 'u');
  sc = load('identification_short_circuit.mat', 't', 'encoder', 'u');

Transfer Functions

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

  [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);

Zoom on the change of resonance

The change of resonance frequency / stiffness is very small and is not important here.

Generated Number of Charge / Voltage

<<sec:charge_voltage_estimation>>

Introduction   ignore

Two stacks are used as actuator (in parallel) and one stack is used as sensor.

The amplifier gain is 20V/V (Cedrat LA75B).

Steps

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

  figure;
  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
  nexttile;
  plot(t, v);
  xlabel('Time [s]'); ylabel('Measured voltage [V]');
  nexttile;
  plot(t, u);
  xlabel('Time [s]'); ylabel('Actuator Voltage [V]');

Time domain signal during the 3 actuator voltage steps

Three steps are performed at the following time intervals:

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

Fit function:

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

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

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

  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

$tau$ [s]	$V_0$ [V]
6.47	2.26
6.76	2.26
6.49	2.25

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

  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 $1.5\,M\Omega$ (specified at $1\,M\Omega$).

How can we explain the voltage offset?

As shown in Figure fig:force_sensor_model_electronics_without_R (taken from cite:reza06_piezoel_trans_vibrat_contr_dampin), an input voltage offset is due to the input bias current $i_n$.

Model of a piezoelectric transducer (left) and instrumentation amplifier (right)

The estimated input bias current is then:

  in = mean(V0)/Rin;

1.5119e-06

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:

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

79804

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

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

0.11454

Which is much more acceptable.

Add Parallel Resistor

A resistor $R_p \approx 100\,k\Omega$ is added in parallel with the force sensor as shown in Figure fig:force_sensor_model_electronics.

Model of a piezoelectric transducer (left) and instrumentation amplifier (right) with the additional resistor $R_p$

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

  figure;
  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
  nexttile;
  plot(t, v);
  xlabel('Time [s]'); ylabel('Measured voltage [V]');
  nexttile;
  plot(t, u);
  xlabel('Time [s]'); ylabel('Actuator Voltage [V]');

Time domain signal during the actuator voltage steps

Three steps are performed at the following time intervals:

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

Fit function:

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

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

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

  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, -0.2, 0.2];                        % Choose Appropriate Initial Estimates
      [B,rnrm] = fminsearch(nrmrsd, B0);     % Estimate Parameters ‘B’

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

And indeed, we obtain a much smaller offset voltage and a much faster time constant.

$tau$ [s]	$V_0$ [V]
0.43	0.15
0.45	0.16
0.43	0.15
0.43	0.15
0.45	0.15
0.46	0.16
0.48	0.16

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

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

101200.0

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

  in = mean(V0)/Rin;

1.5305e-06

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

Sinus

  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 fig:force_sensor_sin_u).

Driving Voltage

The full stroke as measured by the encoder is:

  max(encoder)-min(encoder)

5.005e-05

Its signal is shown in Figure fig:force_sensor_sin_encoder.

Encoder measurement

The generated voltage by the stack is shown in Figure

Voltage measured on the stack used as a sensor

The capacitance of the stack is

  Cp = 4.4e-6; % [F]

The corresponding generated charge is then shown in Figure fig:force_sensor_sin_charge.

Generated Charge

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

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

Almost linear relation between the relative displacement and the generated voltage

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

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

3.9838