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Measurement of Piezoelectric Amplifiers

Introduction   ignore

Two voltage amplifiers are tested:

  • PI E-505.00 (link)
  • Cedrat Technology LA75B (link)

The piezoelectric actuator under test is an APA95ML from Cedrat technology. It contains three stacks with a capacitance of $5 \mu F$ each that can be connected independently to the amplifier.

Effect of a change of capacitance

Cedrat Technology

Load Data

  piezo1 = load('mat/cedrat_la75b_med_1_stack.mat', 't', 'V_in', 'V_out');
  piezo2 = load('mat/cedrat_la75b_med_2_stack.mat', 't', 'V_in', 'V_out');
  piezo3 = load('mat/cedrat_la75b_med_3_stack.mat', 't', 'V_in', 'V_out');

Compute Coherence and Transfer functions

  Ts = 1e-4;
  win = hann(ceil(0.1/Ts));

  [tf_1, f] = tfestimate(piezo1.V_in, piezo1.V_out, win, [], [], 1/Ts);
  [co_1, ~] = mscohere(piezo1.V_in, piezo1.V_out, win, [], [], 1/Ts);

  [tf_2, ~] = tfestimate(piezo2.V_in, piezo2.V_out, win, [], [], 1/Ts);
  [co_2, ~] = mscohere(piezo2.V_in, piezo2.V_out, win, [], [], 1/Ts);

  [tf_3, ~] = tfestimate(piezo3.V_in, piezo3.V_out, win, [], [], 1/Ts);
  [co_3, ~] = mscohere(piezo3.V_in, piezo3.V_out, win, [], [], 1/Ts);

We remove the phase delay due to the time delay of the ADC/DAC:

  angle_delay = 180/pi*angle(squeeze(freqresp(exp(-s*Ts), f, 'Hz')));

/tdehaeze/test-bench-piezo-amplifiers/media/commit/447ea1b6fb8e3718d5de9d988544673ebe91f356/figs/change_capa_cedrat.png

Effect of a change of the piezo capacitance on the Amplifier transfer function

PI 

  piezo1 = load('mat/pi_505_high.mat', 't', 'V_in', 'V_out');
  piezo2 = load('mat/pi_505_high_2_stacks.mat', 't', 'V_in', 'V_out');
  piezo3 = load('mat/pi_505_high_3_stacks.mat', 't', 'V_in', 'V_out');
  Ts = 1e-4;
  win = hann(ceil(0.1/Ts));

  [tf_1, f] = tfestimate(piezo1.V_in, piezo1.V_out, win, [], [], 1/Ts);
  [co_1, ~] = mscohere(piezo1.V_in, piezo1.V_out, win, [], [], 1/Ts);

  [tf_2, ~] = tfestimate(piezo2.V_in, piezo2.V_out, win, [], [], 1/Ts);
  [co_2, ~] = mscohere(piezo2.V_in, piezo2.V_out, win, [], [], 1/Ts);

  [tf_3, ~] = tfestimate(piezo3.V_in, piezo3.V_out, win, [], [], 1/Ts);
  [co_3, ~] = mscohere(piezo3.V_in, piezo3.V_out, win, [], [], 1/Ts);

We remove the phase delay due to the time delay of the ADC/DAC:

  angle_delay = 180/pi*angle(squeeze(freqresp(exp(-s*Ts), f, 'Hz')));

/tdehaeze/test-bench-piezo-amplifiers/media/commit/447ea1b6fb8e3718d5de9d988544673ebe91f356/figs/change_capa_pi.png

Effect of a change of the piezo capacitance on the Amplifier transfer function

Effect of a change in Voltage level

Cedrat Technology 

  hi = load('mat/cedrat_la75b_high_1_stack.mat', 't', 'V_in', 'V_out');
  me = load('mat/cedrat_la75b_med_1_stack.mat', 't', 'V_in', 'V_out');
  lo = load('mat/cedrat_la75b_low_1_stack.mat', 't', 'V_in', 'V_out');
  Ts = 1e-4;
  win = hann(ceil(0.1/Ts));

  [tf_hi, f] = tfestimate(hi.V_in, hi.V_out, win, [], [], 1/Ts);
  [co_hi, ~] = mscohere(hi.V_in, hi.V_out, win, [], [], 1/Ts);

  [tf_me, ~] = tfestimate(me.V_in, me.V_out, win, [], [], 1/Ts);
  [co_me, ~] = mscohere(me.V_in, me.V_out, win, [], [], 1/Ts);

  [tf_lo, ~] = tfestimate(lo.V_in, lo.V_out, win, [], [], 1/Ts);
  [co_lo, ~] = mscohere(lo.V_in, lo.V_out, win, [], [], 1/Ts);

We remove the phase delay due to the time delay of the ADC/DAC:

  angle_delay = 180/pi*angle(squeeze(freqresp(exp(-s*Ts), f, 'Hz')));

/tdehaeze/test-bench-piezo-amplifiers/media/commit/447ea1b6fb8e3718d5de9d988544673ebe91f356/figs/change_level_cedrat.png

Effect of a change of voltage level on the Amplifier transfer function

PI 

  hi = load('mat/pi_505_high.mat', 't', 'V_in', 'V_out');
  lo = load('mat/pi_505_low.mat', 't', 'V_in', 'V_out');
  Ts = 1e-4;
  win = hann(ceil(0.1/Ts));

  [tf_hi, f] = tfestimate(hi.V_in, hi.V_out, win, [], [], 1/Ts);
  [co_hi, ~] = mscohere(hi.V_in, hi.V_out, win, [], [], 1/Ts);

  [tf_lo, ~] = tfestimate(lo.V_in, lo.V_out, win, [], [], 1/Ts);
  [co_lo, ~] = mscohere(lo.V_in, lo.V_out, win, [], [], 1/Ts);

/tdehaeze/test-bench-piezo-amplifiers/media/commit/447ea1b6fb8e3718d5de9d988544673ebe91f356/figs/change_level_pi.png

Effect of a change of voltage level on the Amplifier transfer function

Comparison PI / Cedrat

Results 

  ce_results = load('mat/cedrat_la75b_high_1_stack.mat', 't', 'V_in', 'V_out');
  pi_results = load('mat/pi_505_high.mat', 't', 'V_in', 'V_out');
  Ts = 1e-4;
  win = hann(ceil(0.1/Ts));

  [tf_ce, f] = tfestimate(ce_results.V_in, ce_results.V_out, win, [], [], 1/Ts);
  [tf_pi, ~] = tfestimate(pi_results.V_in, pi_results.V_out, win, [], [], 1/Ts);

We remove the phase delay due to the time delay of the ADC/DAC:

  angle_delay = 180/pi*angle(squeeze(freqresp(exp(-s*Ts), f, 'Hz')));

/tdehaeze/test-bench-piezo-amplifiers/media/commit/447ea1b6fb8e3718d5de9d988544673ebe91f356/figs/tf_amplifiers_comp.png

Comparison of the two Amplifier transfer functions

Impedance Measurement

Introduction   ignore

The goal is to experimentally measure the output impedance of the voltage amplifiers.

To do so, the output voltage is first measure without any load ($V$). It is then measure when a 10Ohm load is used ($V^\prime$).

The load ($R = 10\Omega$) and the internal resistor ($R_i$) form a voltage divider, and thus: \[ V^\prime = \frac{R}{R + R_i} V \]

From the two values of voltage, the internal resistor value can be computed: \[ R_i = R \frac{V - V^\prime}{V^\prime} \]

Cedrat Technology

Compute Impedance 

  R = 10;    % Resistive Load used [Ohm]
  V = 0.998; % Output Voltage without any load [V]
  Vp = 0.912; % Output Voltage with resistice load [V]
  R * (V - Vp)/Vp;
0.94298

  R = 47;    % Resistive Load used [Ohm]
  V = 4.960; % Output Voltage without any load [V]
  Vp = 4.874; % Output Voltage with resistice load [V]
  R * (V - Vp)/Vp;
0.8293

Effect of Impedance on the phase drop 

  C_1 = 5e-6; % Capacitance in [F]
  C_2 = 10e-6; % Capacitance in [F]
  C_3 = 15e-6; % Capacitance in [F]

  Ri = R * (V - Vp)/Vp; % Internal resistance [Ohm]
  G0 = 20;

  G_1 = G0/(1+Ri*C_1*s);
  G_2 = G0/(1+Ri*C_2*s);
  G_3 = G0/(1+Ri*C_3*s);

/tdehaeze/test-bench-piezo-amplifiers/media/commit/447ea1b6fb8e3718d5de9d988544673ebe91f356/figs/change_capa_cedrat.png

Effect of a change of the piezo capacitance on the Amplifier transfer function

PI 

  R = 10;    % Resistive Load used [Ohm]
  V = 1.059; % Output Voltage without any load [V]
  Vp = 0.828; % Output Voltage with resistice load [V]
  R * (V - Vp)/Vp
2.7899

  R = 10;    % Resistive Load used [Ohm]
  V = 2.092; % Output Voltage without any load [V]
  Vp = 1.637; % Output Voltage with resistice load [V]
  R * (V - Vp)/Vp
2.7795

New PI amplifier measurements

PI

Three measurements are done:

  • Slew Rate limitation at maximum
  • Slew Rate limitation at minimum
  • Notch Filter at maximum frequency
  pi_sr_min = load('mat/pi_slew_rate_min.mat');
  pi_sr_max = load('mat/pi_slew_rate_max.mat');
  pi_sr_max_notch = load('mat/pi_slew_rate_max_notch_high.mat');
  Ts = 1e-4;
  win = hann(ceil(0.1/Ts));

  [tf_sr_min, f] = tfestimate(pi_sr_min.V_in, pi_sr_min.V_out, win, [], [], 1/Ts);
  [tf_sr_max, ~] = tfestimate(pi_sr_max.V_in, pi_sr_max.V_out, win, [], [], 1/Ts);
  [tf_sr_max_notch, ~] = tfestimate(pi_sr_max_notch.V_in, pi_sr_max_notch.V_out, win, [], [], 1/Ts);
  angle_delay = 180/pi*angle(squeeze(freqresp(exp(-s*Ts), f, 'Hz')));

/tdehaeze/test-bench-piezo-amplifiers/media/commit/447ea1b6fb8e3718d5de9d988544673ebe91f356/figs/pi_slew_rate_notch.png

Effect of a change in the slew rate limitation and notch filter

Transfer function of the Voltage Amplifier

The identified transfer function still seems to match the one of a notch filter at 5kHz.

  w_nf = 2*pi*5e3; % Notch Filter Frequency [rad/s]
  G = 10.5*(s^2 + 2*w_nf*0.12*s + w_nf^2)/(s^2 + 2*w_nf*s + w_nf^2);