Voltage Amplifier PD200 - Test Bench

In order to measure the transfer function from the input voltage \(V_{in}\) to the output voltage \(V_{out}\), the test bench shown in Figure 6 is used.

For this measurement, the sampling frequency of the Speedgoat ADC should be as high as possible.

Then the maximum output current of the amplifier is reached, the amplifier automatically shuts down itself. We should then make sure that the output current does not reach this maximum specified current.

The maximum current is 1A [rms] which corresponds to 0.7A in amplitude of the sin wave.

The impedance of the capacitance is: \[ Z_C(\omega) = \frac{1}{jC\omega} \]

Therefore the relation between the output current amplitude and the output voltage amplitude for sinusoidal waves of frequency \(\omega\): \[ V_{out} = \frac{1}{C\omega} I_{out} \]

Moreover, there is a gain of 20 between the input voltage and the output voltage: \[ 20 V_{in} = \frac{1}{C\omega} I_{out} \]

For a specified voltage input amplitude \(V_{in}\), the maximum frequency at which the output current reaches its maximum value is:

with:

• \(\omega_{\text{max}}\) the maximum input sinusoidal frequency in Radians per seconds
• \(C\) the load capacitance in Farads
• \(V_{in}\) the input voltage sinusoidal amplitude in Volts
• \(I_{out,\text{max}}\) the specified maximum output current in Amperes

\(\omega_{\text{max}}/2\pi\) as a function of \(V_{in}\) is shown in Figure 7.

When doing sweep sine excitation, we make sure not to reach this maximum excitation frequency.

Here the small signal dynamics of all the 7 PD200 amplifiers are identified.

A (logarithmic) sweep sine excitation voltage is generated by the Speedgoat DAC with an amplitude of 0.1V and a frequency going from 1Hz up to 5kHz.

The output voltage of the PD200 amplifier is measured thanks to the monitor voltage of the PD200 amplifier. The input voltage of the PD200 amplifier (the generated voltage by the DAC) is measured with another ADC of the Speedgoat. This way, the time delay related to the ADC will not be apparent in the results.

The obtained transfer functions from \(V_{in}\) to \(V_{out}\) are shown in Figure 8.

We can see the very well matching between all the 7 amplifiers. The amplitude is constant over a wide frequency band and the phase drop is limited to less than 1 degree up to 500Hz.

The identified dynamics in Figure 8 can very well be modeled this dynamics with a first order low pass filter (even a constant could work fine).

Below is the defined transfer function \(G_p(s)\).

Gp = 20/(1 + s/2/pi/25e3);

Comparison of the model with the identified dynamics is shown in Figure 9.

And finally this model is saved.

save('mat/pd200_model.mat', 'Gp');

The PD200 amplifiers will most likely not be used for large signals, but it is still nice to see how the amplifier dynamics is changing with the input voltage amplitude.

Several identifications using sweep sin were performed with input voltage amplitude ranging from 0.1V to 4V. The maximum excitation frequency for each amplitude was limited from the estimation in Section 3.2.

The obtained transfer functions for the different excitation amplitudes are shown in Figure 10. It is shown that the input voltage amplitude does not affect that much the amplifier dynamics.

As the output noise of the PD200 voltage amplifier is foreseen to be around 1mV rms in a bandwidth from DC to 1MHz, it is not possible to directly measure it with an ADC. We need to amplify the noise before digitizing the signal. To do so, we need to use a low noise voltage amplifier with a noise density much smaller than the measured noise of the PD200 amplifier.

Let’s first estimate the noise density of the PD200 amplifier. If we suppose a white noise, this correspond to an amplitude spectral density:

The input noise of the instrumentation amplifier should be then much smaller than the output noise of the PD200. We will use either the amplifier EG&G 5113 that has a noise of \(\approx 4 nV/\sqrt{Hz}\) referred to its input or the Femto DLPVA amplifier with an input noise of \(\approx 3nV/\sqrt{Hz}\).

The gain of the low-noise amplifier is then increased until the full range of the ADC is used. This gain should be around 1000 (60dB).

A representation of the measurement bench is shown in Figure 11.

Note that it is quite important to load the amplifier with the “Load Box” including a \(10\,\mu F\) capacitor as the (high frequency) noise of the amplifier depends on the actual load being used.

As shown in Figure 12, there are 4 elements involved in the measurement:

• a Digital to Analog Convert (DAC)
• the Voltage amplifier to be measured with a gain of 20 (PD200)
• a low noise voltage amplifier with a variable gain and integrated low pass filters and high pass filters
• an Analog to Digital Converter (ADC)

Each of these equipment has some noise:

• \(q_{da}\): quantization noise of the DAC
• \(n_{da}\): output noise of the DAC
• \(n_p\): input noise of the PD200 (what we wish to characterize)
• \(n_a\): input noise of the pre-amplifier
• \(q_{ad}\): quantization noise of the ADC

In the next sections, we wish to measure all these sources of noise and make sure that we can effectively characterize the noise \(n_p\) of the PD200 amplifier.

The quantization noise is something that can be predicted from the sampling frequency and the quantization of the ADC. Indeed, the Amplitude Spectral Density of the quantization noise of an ADC/DAC is equal to:

with:

• \(q = \frac{\Delta V}{2^n}\) the quantization in [V], which is the corresponding value in [V] of the least significant bit
• \(\Delta V\) is the full range of the ADC in [V]
• \(n\) is the number of bits
• \(f_s\) is the sample frequency in [Hz]

Let’s estimate that with the ADC used for the measurements:

%% ADC Quantization noise
adc = struct();
adc.Delta_V = 20; % [V]
adc.n = 16; % number of bits
adc.Fs = 20e3; % [Hz]
adc.Gamma_q = adc.Delta_V/2^adc.n/sqrt(12*adc.Fs); % [V/sqrt(Hz)]

The obtained Amplitude Spectral Density is 6.2294e-07 \(V/\sqrt{Hz}\).

We now wish to measure the noise of the pre-amplifier. To do so, the input of the pre-amplifier is shunted with a 50Ohms resistor such that the pre-amplifier input voltage is just its input noise. Then, the gain of the amplifier is increased until the measured signal on the ADC is much larger than the quantization noise.

The Amplitude Spectral Density \(\Gamma_n(\omega)\) of the measured signal \(n\) is computed. Finally, the Amplitude Spectral Density of \(n_a\) can be computed taking into account the gain of the pre-amplifier:

The gain of the low noise amplifier is set to 50000 for the measurement.

The obtained Amplitude Spectral Density of the Low Noise Voltage Amplifier is shown in Figure 14. The obtained noise amplitude is very closed to the one specified in the documentation of \(4nV/\sqrt{Hz}\) at 1kHZ.

It is also verified that the quantization noise of the ADC is much smaller and what we are measuring is indeed the noise of the pre-amplifier.

Similarly to Section 4.4, the noise of the Femto amplifier is identified.

The obtained Amplitude spectral density is shown in Figure 15.

The measurement setup is shown in Figure 16. The input of the PD200 amplifier is shunted with a 50 Ohm resistor such that there in no voltage input expected the PD200 input voltage noise. The gain of the pre-amplifier is increased in order to measure a signal much larger than the quantization noise of the ADC.

The measured low frequency (<20Hz) output noise of one of the PD200 amplifiers is shown in Figure 17. It is very similar to the one specified in the datasheet in Figure 3.

The obtained RMS and peak to peak values of the measured output noise are shown in Table 2 and found to be very similar to the specified ones.

The measurement setup is the same as in Figure 16.

The Amplitude Spectral Density \(\Gamma_n(\omega)\) of the measured signal by the ADC is computed. The Amplitude Spectral Density of the input voltage noise of the PD200 amplifier \(n_p\) is then computed taking into account the gain of the pre-amplifier and the gain of the PD200 amplifier:

And we verify that we are indeed measuring the noise of the PD200 and not the noise of the pre-amplifier by checking that:

The Amplitude Spectral Density of the measured input noise is computed and shown in Figure 18.

It is verified that the contribution of the PD200 noise is much larger than the contribution of the pre-amplifier noise of the quantization noise.

In order not to have any quantization noise and only measure the output voltage noise of the DAC, we “ask” the DAC to output a zero voltage.

The measurement setup is schematically represented in Figure 19. The gain of the pre-amplifier is adjusted such that the measured amplified noise is much larger than the quantization noise of the ADC.

The Amplitude Spectral Density \(\Gamma_n(\omega)\) of the measured signal is computed. The Amplitude Spectral Density of the DAC output voltage noise \(n_{da}\) can be computed taking into account the gain of the pre-amplifier:

And it is verified that the Amplitude Spectral Density of \(n_{da}\) is much larger than the one of \(n_a\):

The obtained Amplitude Spectral Density of the DAC’s output voltage is shown in Figure 20.

Let’s now measure the noise of the full setup in Figure 21 and analyze the results.

The Amplitude Spectral Density of the measured noise is computed and the shown in Figure 22.

We can very well see that to total measured noise is the sum of the DAC noise and the PD200 noise.

Let’s now measure the output voltage noise of another DAC called the “SSI2V” (doc). It is a 20bits DAC with an output voltage range of +/-10.48 V and a very low output voltage noise.

The measurement setup is the same as the one in Figure 19.

The obtained Amplitude Spectral Density of the output voltage noise of the SSI2V DAC is shown in Figure 23 and compared with the output voltage noise of the 16bits DAC. It is shown to be much smaller (~1 order of magnitude).

Let’s design a transfer function \(G_n(s)\) whose norm represent the Amplitude Spectral Density of the input voltage noise of the PD200 amplifier as shown in Figure 24.

A simple transfer function that allows to obtain a good fit is defined below.

%% Model of the PD200 Input Voltage Noise
Gn = 1e-5 * ((1 + s/2/pi/20)/(1 + s/2/pi/2))^2 /(1 + s/2/pi/5e3);

The comparison between the measured ASD of the modeled ASD is done in Figure 25.

Let’s now compute the Cumulative Amplitude Spectrum corresponding to the measurement and the model and compare them.

The integration from low to high frequency and from high to low frequency are both shown in Figure 26.

The fit between the model and the measurements is rather good considering the complex shape of the measured ASD and the simple model used.

The obtained RMS noise of the model is 286.74 uV RMS which is not that far from the specifications.

Finally the model of the amplifier noise is saved.

save('mat/pd200_model.mat', 'Gn', '-append');

In this section, three similar voltage amplifiers are compared:

• the PD200 from PiezoDrive
• the LA75B from CedratTechnologies
• the E-505.00 from PI

These are compared in term of dynamic from input voltage to output voltage for a load of \(10\,\mu F\) in Section 5.2 and then in term of input voltage noise in Section 5.3.

The characteristics that I could find for the three amplifiers are summarized in Table 3.