Dynamic Signals¶

Preparation¶

In this tutorial we generate a test signal of 2.5 MHz with an amplitude modulated at a frequency of 1 Hz. Then we measure the test signal using two different filter settings.

Figure 1: Tutorial dynamic signals setup

Note

This tutorial can be performed both on the HF2LI Lock-in Amplifier and
on the HF2IS Impedance Spectroscope and will use the Input
connector. The generation of multi-frequency signals is simple on the
HF2LI with the HF2-MF option or on the HF2IS, where there is no need to
make use of the ADD connector.

Connect the cables as described above. Make sure the HF2 unit is powered on, and then connect the HF2 to your computer with a USB 2.0 cable. Finally launch LabOne (Start Menu/Programs/Zurich Instruments/LabOne User Interface).

Generate the Test Signal¶

In this section you generate a 2.5 MHz sinusoidal signal whose amplitude oscillates at 1 Hz. This is also called the beat signal. In order to obtain this test signal you add two sinusoids of the same amplitude but with a 1 Hz difference in the frequency.

Table 1: Settings: generate the test signal

Output 1 range	1 V / ON
Output 2 range	1 V / ON
Oscillator 1 frequency	2'500'000 Hz
Demodulator 7 Amp 1	0.3 V / ON
Oscillator 2 frequency	2'500'001 Hz
Demodulator 8 Amp 2	0.3 V / ON
Signal Output 1 Add	ON

When connecting an oscilloscope to the Output 1 connector, you should be able to observe the superposition of the 2 sinusoids. To see the acquired signal inside the LabOne User Interface switch to the Scope tab. The Scope view looks like this with the following settings.

Table 2: Settings: acquire the test signal

Scope Signal input	Signal Input 1
Scope Trigger signal	Signal Input 1
Scope sampling rate	6.4 kSa, 320 ms
Run/Stop	Run
Signal Input 1 range	1 V
Signal Input 1 AC / 50 / Diff	ON / OFF / OFF

Figure 2: LabOne UI displaying the acquired signal

The beat signal has a maximum amplitude of 0.6 V, thus it falls within the set range of 1 V. The range setting will prevent any higher voltage than what is set - even if 2 sinusoidals of 0.7 V amplitude each would be added like done in this section, the output would be clipping at 1 V which is the set range. Try to change the output range to 0.1 V, and see how the output voltage is changed to prevent inconsistent settings.

Measure the Test Signal¶

First you change to the Plotter tab, set the scale in order to view an interesting set of data, and set the demodulator filters to a low time constant to measure the amplitude of the 2.5 MHz signal (Hull curve).

Table 3: Settings: filter with a low time constant

Time constant (TC)	10 ms (approximated to 10.2 ms)
Filter order	2
Resulting measurement bandwidth (BW 3dB)	\~10 Hz
Data transfer rate	100 Hz (approximated to 112 Hz)
Data transfer enable (En)	ON

These settings set the demodulation low-pass filter to a 10 ms time constant (the corresponding bandwidth is around 10 Hz) and the filter slope to second order. The output of the filter is sampled at a rate of 100 Hz, and the samples are sent to the host computer.

Figure 3: LabOne UI displaying the measured signal

If you stop the acquisition by pressing the button "Run/Stop" you can conveniently measure the amplitude of the 1 Hz signal by using the 2 cursors C1 and C2: 394.4 mV_RMS, half period 498.1 ms. You can achieve higher measurement precision by using a even lower time constant (e.g. 1 ms), increasing the readout rate (e.g. 1.8 kHz), and zooming into the Plotter view.

Next you use a high time constant to separate the 2 original sinusoids even though they are superposed in one signal. In the Lock-in tab apply the following settings.

Table 4: Settings: filter with a high time constant

Time constant (TC)	2 s
Filter order	2
Resulting measurement bandwidth (BW 3dB)	\~35 mHz
Data transfer rate	100 Hz (approximated to 112 Hz)
Data transfer enable (En)	ON

These settings set the demodulation low-pass filter to a time constant of 2 s, with a resulting measurement bandwidth of 35 mHz. With these settings the HF2 is able to distinguish between the signal component at 2'500'000 Hz and the signal component at 2'500'001 Hz as the measurement bandwidth is considerably less than the frequency spacing of the 2 signal components. The output of the demodulator is stable after a settling time.

Figure 4: LabOne UI displaying the measured signal

The output of the demodulator does not show any oscillations like before: the numerical value is 214 mV_RMS. If you switch to the oscilloscope view, you see that the signal at Input 1 is still beating as before, while the demodulator filter is set such to ignore the interferer at 2'500'001 Hz. Try to switch off the interferer.

Table 5: Settings: remove the interferer

Signal output 2 ON/OFF	OFF

The time it takes to settle depends on various parameters like filter setting and switch-off timing. The difference in amplitude of the measurement at 2'500'000 Hz with or without interferer is in the range of 50 μV. With different filter settings it is possible to do better than that.

Consider this: you have 2 signals with relevant amplitude (0.3 V) interfering with each other as their frequency is very close (1 Hz at 2.5 MHz). The power of lock-in amplification consists of extracting the relevant signal energy at exactly one frequency. The "immunity" from nearby interferer is the capability to ignore them. This a simple definition of the dynamic reserve.

Filter Setting Discussion¶

This section aims to summarize a few basic concepts of filtering in connection with lock-in amplification. In this tutorial, you have used different filter settings to measure different signal properties.

Table 6: Settings: filter with a high time constant
Time constant	Measurement bandwidth	Measurement noise	Changes upon steady state change	Example
Low setting	High, capability to detect fast events	More noise in measurement result	Fast adaptation of result	BW = 10 kHz, capability to detect events at 2-5 kHz, prone to pick-up noise
High setting	Low, capability to determine the steady state	Less noise in measurement result	Slow adaptation of result	BW = 50 mHz, exact determination of steady state - events more frequent than 0.1 Hz are filtered

Filtering constitutes a trade-off between measurement speed and measurement accuracy. In order to measure fast events, it is necessary to open up the filters allowing also more noise in the measurement result. The opposite is to measure with narrow filters which increases the signal-to-noise ratio, but limits the capability to detect the changes in the signal of interest. This trade-off is in common with any lock-in amplifier. The power of the HF2 is that it allows to do both at the same time thanks to the multiple demodulators per input channel.