test-bench-piezo-amplifiers/index.org

Measurement of Piezoelectric Amplifiers

• Introduction
• Effect of a change of capacitance
  • Cedrat Technology
  • PI
• Effect of a change in Voltage level
  • Cedrat Technology
  • PI
• Comparison PI / Cedrat
  • Results
• Impedance Measurement
  • Introduction
  • Cedrat Technology
    • Compute Impedance
    • Effect of Impedance on the phase drop
  • PI
• Effect of filters configuration on the PI-E505 dynamics
  • PI
  • Transfer function of the Voltage Amplifier
  • With Load

Introduction   ignore

The following two voltage amplifiers are tested:

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

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

This document is divided into the following sections:

• Section sec:effect_change_capacitance: The effect of a change in load capacitance on the amplifier dynamics is studied
• Section sec:effect_change_voltage: The effect on the voltage level on the amplifier dynamics is studied
• Section sec:comp_pi_cedrat: The dynamics of the E-505 and LA75B are compared
• Section sec:impedance_meas: The output impedance of both amplifiers are measured
• Section sec:pi_e505_filters: The effect of the internal filters of the E-505 on its dynamics is studied

Effect of a change of capacitance

<<sec:effect_change_capacitance>>

Cedrat Technology

Load Data

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

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

PI

  piezo1 = load('pi_505_high.mat', 't', 'V_in', 'V_out');
  piezo2 = load('pi_505_high_2_stacks.mat', 't', 'V_in', 'V_out');
  piezo3 = load('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')));

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

Effect of a change in Voltage level

<<sec:effect_change_voltage>>

Cedrat Technology

  hi = load('cedrat_la75b_high_1_stack.mat', 't', 'V_in', 'V_out');
  me = load('cedrat_la75b_med_1_stack.mat', 't', 'V_in', 'V_out');
  lo = load('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')));

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

PI

  hi = load('pi_505_high.mat', 't', 'V_in', 'V_out');
  lo = load('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);

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

Comparison PI / Cedrat

<<sec:comp_pi_cedrat>>

Results

  ce_results = load('cedrat_la75b_high_1_stack.mat', 't', 'V_in', 'V_out');
  pi_results = load('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')));

Comparison of the two Amplifier transfer functions

Impedance Measurement

<<sec:impedance_meas>>

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);

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

Effect of filters configuration on the PI-E505 dynamics

<<sec:pi_e505_filters>>

PI

Three measurements are done:

• Slew Rate limitation at maximum
• Slew Rate limitation at minimum
• Notch Filter at maximum frequency

  pi_sr_min       = load('pi_slew_rate_min.mat');
  pi_sr_max       = load('pi_slew_rate_max.mat');
  pi_sr_max_notch = load('pi_slew_rate_max_notch_high.mat');
  pi_sr_load      = load('pi_slew_rate_max_notch_high_2stacks.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);
  [tf_sr_load, ~] = tfestimate(pi_sr_load.V_in, pi_sr_load.V_out, win, [], [], 1/Ts);

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

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);

With Load

  R = 2.78; % Output Impedance [Ohm]
  C = 9e-6; % Load capacitance [F]

  G_amp = 10/(1 + s*R*C);