A versatile Lock-In digital Amplifier (LIdA): the case of mechanical resonances

Resonances are of great importance regarding both the fundamental and functional properties of a system and occasionally have a large impact on everyday life. A characteristic example in this respect is the destructive effect of mechanical resonances on large-scale constructions such as bridges.

Resonant techniques are very well received by the scientific community and are relevant to electromagnetic resonances, e.g., electron paramagnetic resonance [1], nuclear magnetic resonance [2], and extensions up to high-energy physics, e.g., nuclear resonance scattering [3]. This is mainly because, on a resonance, the signal-to-noise ratio maximizes and peculiar effects appear which would otherwise be inaccessible.

The concept of a resonance is indeed covered several times in typical physics and engineering courses; however, the importance of experimental resonant techniques are barely highlighted in the curriculum, with possibly the only exception being resonances in electric circuits, e.g., resistor–inductor–capacitor (RLC) circuits [4].

Despite the importance of experimental resonant techniques in research, a gap between the experimental scientific and engineering community and undergraduate education is generally observed. Such a gap may be reasonably attributed to the significant effort and resources that need to be invested in order to quantitatively register resonances. Arguably, the detection of resonances in research and development usually requires sophisticated and expensive equipment or access to large-scale facilities such as high magnetic field laboratories, synchrotron radiation storage rings or high-energy particle colliders. As a result, access to experimental resonant techniques is usually limited to a few people at later stages of their careers.

Herein, we revive the case of mechanical resonances for a smooth transition between the undergraduate curriculum and research for prospective scientists and engineers. To this end, the assembly of a low-cost Lock-In digital Amplifier (LIdA) from accessible ready-made modules is described. Such a device may not only serve educational purposes but it could also complement the scientific laboratory. Details on excitation and detection of mechanical resonances are given and the importance of mechanical resonances in research is further discussed.

A simple mechanical oscillator such as a mass, m, suspended by a spring which is driven by an external alternating force with amplitude F₀ and frequency f has a mechanical resonance at a frequency ${f}_{{\rm{r}}}=\surd \overline{k/m}/2\pi$. It is already apparent that knowledge of the resonant frequency provides a direct measure of the spring constant, k, which is both of fundamental and practical interest.

The frequency profile of the oscillating motion, F(f), matches the Lorentzian profile $F(f)\sim {F}_{0}{f}_{{\rm{r}}}^{2}/(| {f}^{2}-{f}_{{\rm{r}}}^{2}| +{f}_{{\rm{r}}}^{2}{Q}^{2})$ , where $Q={f}_{{\rm{r}}}/\delta {f}_{{\rm{r}}}$ is the resonance quality factor which is indicative of the full width at half maximum of the Lorentzian profile, $\delta {f}_{{\rm{r}}}$. The well defined profile function of a resonance is an asset for resonant techniques. An additional asset is that on resonance, i.e., at $f={f}_{{\rm{r}}}$, the amplitude of the resonating system is maximum. It is hence straightforward to assume that the detection of mechanical resonances might be an easy task. This assumption is indeed valid for qualitative observations, i.e., when the amplitude of the excitation, F₀, is high enough and results in destructive phenomena, such as in the case of falling bridges. However, in the case of non-destructive techniques F₀ is kept low, and consequently the quantitative detection of mechanical resonances is reduced to discriminating a small-amplitude signal from the noise.

It is possible to distinguish a signal of small amplitude from noise both in the time and in the frequency domain. Here, the analysis will be done in the time domain since the transformation between time and frequency domains requires advanced mathematics [5], which might appear as a limitation for the inexperienced readership.

The main routes to distinguish a signal of small amplitude from noise in the time domain is either a reduction of the noise amplitude or so-called synchronous detection. Although reduction of the noise amplitude is always advisable, in most cases it is inherently difficult since some kind of noise, e.g. thermal noise, electronic noise, $1/f$ noise [6], are practically unavoidable. Synchronous detection, which is also known as lock-in amplification, introduced by Cosens [7] and further simplified by Michels and Curtis [8], is thus the method of choice.

The working principles of a lock-in amplifier are extensively discussed in the literature and are beyond our scope here. For a more detailed introduction into the principles of a lock-in amplifier the reader is encouraged to follow [9] and references therein. In short, a lock-in amplifier recovers a modulated signal with a well defined modulation frequency and phase buried in a noisy carrier signal. The trigger and the recorded signals are multiplied. The resulting signal is filtered out for high-frequency harmonics and all information relevant to the response of the studied system is encoded in the phase difference between the trigger and the recorded signal. The output of a lock-in amplifier comprises a signal, I, in-phase to the reference signal, and a quadrature signal, Q, which is out-of-phase to the reference signal. The magnitude signal, R, is defined as $R=\surd \overline{{X}^{2}+{Y}^{2}}$, where X and Y are given as $R{\rm{\cos }}\vartheta$ and $R{\rm{\sin }}\vartheta$, respectively, and ϑ is the phase.

In order to excite and detect mechanical resonances from a solid one needs a function generator, which produces modulated electrical signals of variable frequency, electro-mechanical transducers, which convert modulated electrical signals to analogous mechanical signals and vice versa, and a lock-in amplifier, which discriminates the actual signal from the noise.

Mechanical oscillations of solid bodies follow the classical wave mechanics equation, ${u}_{{\rm{s}}}=l\cdot {f}_{{\rm{r}}}$, with the typical speed for the propagation of the wave front equal to the speed of sound, u_s. The resonant frequencies depend on the dimensions of the body itself, i.e., the higher the dimensions, l, the lower the resonant frequencies. The volume of typical samples which are easy to obtain in the laboratory span between 1 mm³ and a few cm³. The typical speed of sound in solids ranges between 2 and 4 km s^–1. Thus, the typical resonant frequencies of such solids vary between 20 kHz and 2 MHz.

Commercially available lock-in amplifiers with high detection efficiency and accuracy do exist [10]; however, their wide use in the introductory laboratory is inhibited by their high cost. This is mainly due to the fact that synchronous detection is realized through analog electronics which, apart from cost, also adds volume, making such instruments non-portable. Recently, commercially available digital lock-in amplifiers have appeared [11]; however, the cost remains high. Several attempts to realize low-cost analog and digital lock-in amplifiers exist in the literature, e.g., [12–17]; however, most of them are dedicated to low frequencies, i.e., well below 1 MHz. Such lock-in amplifiers may indeed be used for registering mechanical resonances in relatively large samples. However, large-volume samples are not always available in the scientific and engineering laboratory.

The overall experimental scheme for the excitation and detection of mechanical resonances is shown in figure 1(a). A picture of the sample holder used for detection of mechanical resonances is depicted in figure 1(b). A picture of the LIdA assembly for registering mechanical resonances suggested in this study is shown in figure 1(c).

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The experimental setup comprises a set of cylindrical S-type piezoelectric transducers [18] with a thickness of 0.4 mm and a diameter of 10 mm. The first mechanical resonance of such a transducer is at 5 MHz, which is far from the mechanical resonances of a typical sample. The transducers are glued on plastic supports positioned on truncated cylindrical and hollow copper blocks. The lower copper block is a single piece, whereas the upper block is made of an inner and an outer part. The outer part is similar to the lower single part. Both parts are fixed using a primitive aluminum scaffold. The upper inner cylindrical piece is moveable. Such an elaborate design is not crucial for the operation of the spectrometer; it is only used in order to be able to accommodate samples of various dimensions.

The excitation transducer, which is selected to be the lower one, is fed through a thin coaxial cable with a modulated electrical signal that is produced by a separate direct digital synthesis GF-266 function generator [19]. The coaxial cable may either be glued using a conductive glue or soldered at one side of the transducer. The other side of the transducer is grounded. The operation of both transducers is acoustically inspected by scanning the modulation frequency in the hearing range, i.e., between 31 Hz and 21 kHz.

A reference signal with the same frequency and phase as the excitation signal is fed through a similar coaxial cable at one of the two channels, i.e., Vb, of a commercially available ready-made 14-bit AD7357 digitizer [20]. The role of the digitizer is to convert the analog signal produced by the transducer to a digital one. For this specific digitizer the sampling frequency is 4.2 MHz. According to the Nyquist–Shannon sampling theorem [21] reliable data acquisition up to 2.1 MHz can be achieved.

The detection transducer, which is selected to be the upper one, is connected similarly as the lower transducer to a ready-made commercially available PPA-100 preamplifier [22]. The role of the preamplifier is to match the impedance difference between the piezoelectric transducer, which is in the range of several MΩ, to the input impedance of the digitizer, which is 50 Ω, and to minimize potential electronic reflections. The output of the preamplifier is connected using a coaxial cable to the other channel, i.e., Va, of the digitizer. The digitizer is connected through a built-in 96-way connector to a commercially available Field-Programmable Gate Array (FPGA) CED1Z card [20] which has the same type of built-in connector. The acquired data are transferred through the FPGA card via a usb port using a usb cable to a personal computer for further processing and storage. Typical costs for all commercially available products in the assembly presented in this study are similar to the cost of a portable pc, i.e., around $\unicode{8364} 600$ ¹ . The cost of cabling is negligible. The mechanical work for the sample holder is done manually on small spare parts and takes only a few hours. Notably, the overall volume of the electronics and the mechanical parts of the spectrometer presented in figure 1 including the preamplifier does not exceed the volume of a portable pc.

The experimental setup is controlled via a freely available [23] Labview-based executable which includes the tedious FPGA programming. In this respect the user may use the program as it is without any knowledge of programming or need for further development. The source code of the Labview executable [24] is also available in case the user would like to further customize it, e.g., add a temperature or a magnetic field controller. The control panel of the Labview program is shown in figure 2. The graphical user interface is divided into sub-panels relevant to the function generator, data acquisition, signal analysis, and data storage. All white rectangularly shaped areas are inputs and can be modified by the user whereas the grey areas are either outputs or contain information relevant to the experimental setup.

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In the function generator sub-panel, the actual function can be selected; the initial/final frequency, the number of points, and the dwell time between the frequency points can be set. Moreover the amplitude or the attenuation of the output signal can be modified.

In the Data AcQuisition (DAQ) card info and the DAQ card sub-panel all information relevant to the digitizer and the FPGA card is given. The sampling frequency and the number of samples are set to their maximum values, 4.2 MHz and 4096 samples, respectively.

In the signal analysis sub-panel the amplitude of the input signals, both for the reference signal and the recorded waveform, are shown. A phase-locked loop [25] (PLL), which detects the frequency and the phase of the reference signal, is implemented. A variety of filter types, e.g., low-pass [26], together with the order of the filter and the cut-off frequencies, may be selected in the filter sub-panel and allows the user to filter out all the high-frequency components in the lock-in process.

The program applies synchronous detection in real time and the output of the lock-in process is provided in the lock-in amplifier sub-panel. Namely, the X- and Y-components of the lock-in process, together with the magnitude signal, R, and the phase, ϑ, are given both in the control panel as well as in the output data file. More importantly, the raw waveforms shown in the data generation and acquisition sub-panel may be saved for future reference or further processing, e.g., in the frequency domain.

Solid samples exhibit mechanical resonances at resonant frequencies, f_rⁱ. The detection of the frequency profile on resonance depends crucially on the quality factor, $Q={f}_{{\rm{r}}}^{i}/\delta {f}_{{\rm{r}}}^{i}$, which is in turn related to the micro-structure of the sample. A solid sample of any shape may be used for demonstration experiments. Herein, a ceramic–polycrystalline–alumina (Al₂O₃) composite is selected on a trial-and-error basis. The demonstration sample is cut using a saw from a widely available large ceramic alumina piece. No further treatment, such as polishing, or any special care about the parallelism of the opposite sample faces is applied. The dimensions of the sample were measured with a caliper and found to be $4.40(5)\,{\rm{mm}}\times 5.56(5)\,{\rm{mm}}\times 7.74(5)\,{\rm{mm}}$. The mass of the sample was 0.654(5) g. The selected modulation function was of sinusoidal type with 5 V amplitude. A frequency scan was carried out between 300 kHz and 1.1 MHz with a 50 Hz frequency step. The overall spectrum was collected in less than 10 minutes.

The X- and Y-components and the magnitude signal R of the lock-in amplifier for the empty spectrometer with an open gap between the transducers are shown in figure 3(a). The overall curves are relatively smooth. The inset to figure 3(a) shows the frequency regions where small disturbances are observed. The disturbance regions are reproducible between different runs and can be most likely attributed either to small instabilities in the amplitude of the frequency source, the cross-talk of the excitation and detection transducer through the metallic scaffold or, less likely, to environmental noise with the appropriate frequency and phase which successfully goes through the lock-in amplifier. Figure 3(b) shows the frequency response measured on the sample using the same sequence as in the background measurements. The sample was lightly held between two opposite edges by the transducers. More than 25 clearly resolved resonant lines can be identified between 300 kHz and 1.1 MHz. The first 14 resonances were fitted with a Lorentzian profile using the portable command-line driven graphing utility Gnuplot [27], and both the resonant frequencies, f_rⁱ, and the quality factor, Q, are extracted and shown in table 1.

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Mechanical resonances are an indispensable tool in characterizing the elastic properties of a material. Mechanical oscillations follow classical mechanics. The equation of motion in this case is Newton's law, which in its differential form is given by

$$\begin{eqnarray}&&\rho {\omega }_{{\rm{r}}}^{2}{\psi }_{i}+\sum _{i,j,k}{C}_{{ijkl}}\displaystyle \frac{{\partial }^{2}{\psi }_{k}}{\partial {x}_{j}\partial {x}_{l}}=0.\end{eqnarray} \tag{ 1 }$$

This is a typical eigenfunction, ψ, and eigenvalue, ${\omega }_{{\rm{r}}}$, problem, which can be solved numerically and provides the frequencies of the mechanical resonances ${f}_{{\rm{r}}}={\omega }_{{\rm{r}}}/2\pi$ if the elements of the elastic constants tensor, C_ijkl, and the mass density, ρ, are known. The inverse problem may also be solved numerically. If the frequencies of the mechanical resonances, f_rⁱ, are measured on a sample of mass density ρ and a shape which can be described by a set of basis functions ψ, then the elements of the elastic constant tensor C_ijkl can be extracted.

The elastic constant tensor in its general form has 36 elements. The number of independent elements, however, depends on the symmetry relation in the studied sample. For example, polycrystalline aggregates have only two independent elastic constants, namely, C₁₁ and C₄₄, whereas the maximum number of independent elastic constants in the most anisotropic, triclinic crystal structure is 21.

A program to refine the elastic constants of a rectangular parallelepiped solid is available online [28]. The output of the Los Alamos National Laboratory (LANL) Rectangular PaRallelepiped (RPR) code for refining the isotropic elastic constants (C₁₁ and C₄₄) of the ceramic alumina sample measured in this study is shown in figure 4. The resonant frequencies shown in table 1 together with the geometric dimensions and the sample mass are the input parameters to the program. The refined isotropic elastic constants at room temperature are C₁₁ = 467 GPa and C₄₄ = 139 GPa with an rms refinement error of 1.8%. Interested readers may find a more detailed manual on the use of this program in [29].

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The experimental setup described herein is assembled from commercially available components and programmed for synchronous detection. No need for advanced electronic design is required either for setting up or running the apparatus. It can thus be assembled and used both by undergraduate students and general physicists or inexperienced researchers. The control program is provided both as an executable for use as is, and as a source code for further research and development. The total cost of the equipment and its volume remains low. This makes the operation of multiple setups feasible and the setups themselves portable. The experimenter will be familiar with terms such as instrument control, data acquisition, digital signal processing and synchronous detection. The modular control program provides an  opportunity for understanding the operation of various analog electronics, e.g., frequency filters, without the actual presence of such analog devices.

The obtained data demonstrate the strength of resonant techniques. Although disturbances are recorded in an empty spectrometer, they do not follow a Lorentzian profile and thus are not of resonant nature. The sample oscillates at all frequencies; however, it is only at the resonant frequencies that the signal-to-noise ratio maximizes and a Lorentzian-type frequency response is detected. In this respect the experimenter will be introduced to experimental resonant techniques and the concept of signal-to-noise ratio. The high precision and accuracy of resonant techniques may be further highlighted by illustrating the sensitivity of the resonant frequencies to changes in temperature and the change of the frequency response due to macroscopic defects such as chips and cracks.

Such a device, apart from its use in demonstration experiments, may also be a useful tool for the advanced scientific research laboratory. Mechanical resonances are an indispensable tool for studying the elasticity of solids. A well established experimental technique known as resonant ultrasound spectroscopy [29], which focuses on the study of the coupling constants between stress and strain, i.e., elastic constants, may readily benefit from such a device. Elastic constants, apart from their obvious practical use in elasticity measurements, are also of fundamental interest. Elastic constants complement the knowledge obtained from the lattice parameters, the inter-atomic position at which the global minimum in potential energy is observed, with the curvature of the potential energy at the global minimum.

In this configuration the setup is optimized for registering mechanical resonances of specimens with small volumes without the requirement of advanced technical knowledge. The same electronic equipment, without hardware changes, may be used for general data acquisition. Once the electronic assembly is ready, further measurements in a modulated mode, i.e., resistivity/capacitance [30, 31], magnetic susceptibility [32], heat capacity [33], and thermal diffusivity/conductivity [34] are accessible. Modulated signals of any type may be registered and a frequency analysis [5] may be carried out either in real time or at a later time.

The assembly, control and operation of an open-source Lock-In digital Amplifier (LIdA) optimized for registering mechanical resonances is described. This  apparatus may introduce the experimenter to resonant techniques and bridge the gap between education and research. The elastic constants of a system, which are both of fundamental and practical interest, may be obtained using LIdA. Apart from mechanical spectroscopy, LIdA may also be used in other areas of science and technology.

The technical assistance of Mr J-P Celse and Mr A J E Lefering is acknowledged for the construction of the sample holder and the sample preparation, respectively. Mr M F J Boeije is acknowledged for careful proofreading. The authors acknowledge financial support from the Industrial Partnership Program (IPP-I28) of Foundation for Fundamental Research on Matter (FOM) (The Netherlands) and BASF New Business.