11560 FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 36, No 4, December 2023, pp. 485 - 497 https://doi.org/10.2298/FUEE2304485S © 2023 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper ELECTRO-ACOUSTIC ANALOGIES BETWEEN THERMOELASTIC COMPONENT OF THE PHOTOACOUSTIC SIGNAL AND LOW-PASS RC FILTER Neda Lj. Stanojevic1, Dragana K. Markushev2, Sanja M. Aleksic1, Dragan S. Pantic1, Dragan V. Lukic2, Marica N. Popovic2, Dragan D. Markushev2 1Faculty of Electronic Engineering, University of Niš, Niš, Serbia 2Institute of Physics, University of Belgrade, Belgrade-Zemun, Serbia Abstract. This paper presents a new approach to the thermal characterization of aluminum, based on the electro-acoustic analogy between the thermoelastic component of the photoacoustic signal and the passive RC low-pass filter. The analogies were used to calculate the characteristic thermoelastic cut-off frequencies of the photoacoustic component and obtain their relationship with the thickness of the aluminum samples. Detailed numerical analysis showed that the required relationship is linear in the log- log scale and can serve as a reference curve for the given material. The results of the numerical analysis were also confirmed experimentally. Key words: electro-acoustic, thermoelastic component, photoacoustic signal, RC filter 1. INTRODUCTION Photoacoustics, as one of the very sensitive detection methods within photothermal sciences, is based on the photoacoustic effect [1,2]. The photoacoustic effect is the effect of the formation of sound waves after the interaction of light and the matter of the periodically illuminated sample in any aggregate state. In this article, we will deal with the analysis of solid samples. The sound generated in solids after light-matter interaction is usually called a photoacoustic signal. It is a complex combination of at least two components: thermo- diffusion and thermoelastic [3,4]. If a plate-shaped solid sample is illuminated by a modulated light source from one side, the thermo-diffusion component is generated by the periodic expansion and contraction of a thin layer of air adjacent to the unilluminated surface of the sample. On the other hand, periodic bending of the same sample occurs because of different temperatures at illuminated and nonilluminated sample surfaces. Such bending causes the Received February 13, 2023; revised April 10, 2023; accepted May 03, 2023 Corresponding author: Neda Stanojevic Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia E-mail: neda.stanojevic@elfak.ni.ac.rs 486 N. LJ. STANOJEVIC, D. K. MARKUSHEV, S. M. ALEKSIC, et al. compression of the gas in the sample vicinity, changing gas pressure, and thus the thermoelastic component is generated [5,6]. Both components bring physical characteristics of the illuminated sample material, usually given in the terms of different coefficients: thermal diffusion, DT, thermal conduction k, linear thermal expansion αT, etc. The thermoelastic component of the photoacoustic signal is specific in that its frequency response, both amplitude, and phase, is very similar to the frequency response of a low-pass RC filter [7-10]. Therefore, in this paper, we will try to establish an electro- acoustic analogy between the thermoelastic response and the response of a passive RC low-pass filter. We will show that the amplitude and phase graphs of the thermoelastic component can be understood as Bode plots and analyzed with the same transfer function as the aforementioned RC filter. Also, we will show that characteristic cut-off frequencies obtained from such an analysis can be used to obtain the thermal diffusion coefficient of the material from which the examined sample is made. 2. THEORETICAL BACKGROUND 2.1. Theory of thermoelastic bending A typical photoacoustic setup for material characterization (an open-cell setup) involves illuminating the sample with a modulated light source intensity (pure sinusoid I, Fig.1, with frequency ω=2πf, f is the modulation frequency) [5,11-13]. The sample is surrounded by air and placed on top of the microphone. Illumination leads to a change in the thermal state of the sample, which results in different temperatures on its illuminated and non-illuminated sides. Different temperatures cause the sample to bend (see Fig.1). The bending is periodic and elastic, following the rhythm of the light source modulation. The bending of the sample described in this way is called thermoelastic bending. Theory of thermoelastic bending [14,15] derived equations describing the thermoelastic bending of a uniform-thickness plate being heated. Fig. 1 The basic scheme of the open-cell setup shows the thermoelastic bending of a periodically illuminated sample of thickness l, radius R, and the thermoelastic sound component δpTE generation and propagation along the z - axis (axis of heat propagation) Electro-Acoustic Analogies Between Thermoelastic Component Of The Photoacoustic Signal And Low-Pass Rc Filter 487 Thermoelastic bending produces a back-side acoustic wave, so-called thermoelastic component of the photoacoustic signal δpTE (pure sinusoid, same frequency as I, different amplitude and phase), with a pressure detected by the microphone membrane defined as [16,17]: 4 0 TE 3 0 3 T T p R p M V l    = , (1) where αT is the coefficient of linear thermal expansion, γ, p0 and V0 are the adiabatic constant, pressure and volume of the air in microphone, respectively, and MT is the first moment of the plate temperature change along the z - axis, defined as [15-17]: ( ) /2 /2 , d l T s l M zT z z − =  . (2) Here Ts (z,ω) is the temperature distribution along the z - axis. When MT is calculated in the case of a surface absorber, it can be written as [15-17]: 0 3 tanh , 2 2σ i i T i I l l M k     =  −      (3) where I0 is the amplitude of excitation, k is the thermal conductivity of the sample material, and (1 ) / 2i Tσ j D= + is the complex thermal diffusion coefficient (j is the imaginary unit, DT is the diffusion coefficient). Being the complex number, δpTE has an amplitude ATE and a phase φTE (see Appendix I). Usually, their common modulation frequency f response plots are presented as in Fig. 2.a (ATE ) and Fig. 2.b (φTE ), where calculations are performed for the pure aluminum circular plate sample having the thickness, l = 100 μm, and a radius, R = 2.5 mm, with basic thermal properties given in Table I [17,18]. Table 1 Aluminum sample parameters used in Eq. (1-3) Thermal conductivity, k / (W/mK) 237 Heat capacity, C / (J/kgK) 900 Density, ρ / (kg/m3) 2700 Heat diffusion DT / (10-5 m2/s) 9.75 Linear thermal expansion, αT / (10-6 1/K) 23.1 Since δpTE represent sound, the ATE values can also be represented in decibels (dB), Figure 2a (AdB), using the equation: TE dB max TE 20log A A A = , (4) where ATE max is the maximal amplitude value in the given frequency domain. Corresponding phase φ values are represented in Figure 5b. Both quantities, AdB and φ, have the same shape in the frequency domain as ATE and φTE, but different numerical values. 488 N. LJ. STANOJEVIC, D. K. MARKUSHEV, S. M. ALEKSIC, et al. Fig. 2 a) Amplitudes ATE and b) phases φTE of the photoacoustic signal thermoelastic component δpTE (Eq. (1)) as a function of the modulation frequency f. Amplitude AdB in decibels (Eq.(4)) and corresponding phase φ values were depicted, also 2.2. Low-pass RC filter Analyzing δpTE response from Figure 2, one can see that our photoacoustic system can pass certain frequencies while attenuating others. In other words, our system acts as a filter, so the analogy can be made with a passive low-pass RC filter, presented in Figure 3 in (a) time and (b) frequency domain [7-10]. Here, Uin is an input voltage, Uout is an output voltage (voltage across the capacitor), ZR = Z is the resistor impedance, and ZC = 1/sC is the capacitor impedance, where s is a complex number s j = + (j is the imaginary unit, σ is the exponential decay constant, and ω is the sinusoidal angular frequency). By viewing the circuit (Figure 3b) as a voltage divider, and considering a special case of sinusoidal steady state in which the input (red line) and output (blue line) voltage consists of a pure sinusoid (no exponential decay, σ = 0, s = jω), the transfer function H(jω) from the input voltage to the voltage across the capacitor can be given in the form [7-10]: 0 ( ) 1 1 ( ) ( ) 1 1 out in U j H j U j j RC j      = = = + + (5) assuming zero initial conditions, where ω0=1/RC is the cut-off frequency, a boundary frequency at which energy flowing through the circuit begins to be attenuated. As a complex number, H(jω) has an amplitude |H(jω)| and the phase φ which can be presented in the forms: ( ) 2 0 1 1 H j   =   +     (6) and 0 arctan      = −     (7) Electro-Acoustic Analogies Between Thermoelastic Component Of The Photoacoustic Signal And Low-Pass Rc Filter 489 Fig. 3 A simple scheme of RC low - pass filter in a) time and b) frequency domain. Usually, the magnitude A is calculated in decibels (dB) using [7-10]: ( )20logA H j= (8) The standard representations of a given low-pass RC filter A and φ are Bode plots [19,20] (Figure 4), obtained using Eq.(6-8), taking into account that ω0 = 2πf0, and f0 = 1.5∙104 s-1. Fig. 4 Bode a) amplitude A and b) phase φ plots of the RC low-pass filter response in frequency domain. Dashed auxiliary lines define the position of f0 490 N. LJ. STANOJEVIC, D. K. MARKUSHEV, S. M. ALEKSIC, et al. 3. RESULTS AND DISCUSSION 3.1. Theoretical procedure Based on the introductory remarks, both the photoacoustic system (Figure 1) and the low - pass RC filter (Figure 3) can be considered as the linear time - invariant (LTI) systems [21,22] whose frequency responses after sinusoidal inputs (excitations) are shown in Figures 2 and 4. The cut - off frequency fTE of the thermoelastic component can be found, using the Bode diagrams from Figure 2 for l = 100 μm and R = 2.5 mm values, in two ways: 1) approximately, finding the intersection of the amplitude asymptotes (dashed lines in Figure 5a), and/or 2) explicitly, by fitting the amplitude (Figure 5.a) with a function yTE, based on Eqs. (6) and (8), given in the form: 2 2 TE 20log 1 n x y m −      = +          (9) where, x = f is the modulation frequency, and m and n are fitting parameters, fTE = m and the slope is n/2. This expression is commonly used in electronics to describe the cascade connection of the RC filters, but in our case, it is intended to cover a slope, different to the RC filter. The results of the δpTE amplitude fitting procedure (Figure 5a, blue line) are as follows: fTE = (15660±60) and n = (9786±6)∙10-4. The value of fTE is depicted in Figures 5.a.b, using dashed vertical line. Applying the obtained fTE value in A (Eq. (8)) and φ (Eq. (7)) one can calculate corresponding low - frequency bandpass RC filter responses (red lines in Figure 5.a, and Figure 5.b, respectively). Obvious discrepancies between blue and red lines can be observed at higher frequencies for both amplitudes and phases. This result is only a consequence of the fact that real systems (photoacoustics) are not the ideal analogies (RC filters). Fig. 5 Bode a) amplitude and b) phase plots of the photoacoustic signal thermoelastic component δpTE obtained theoretically (Eq. (1)) in the case of aluminum circular plate having the thickness l and radius R, with basic thermal parameters which are given in Table I. Cut-off frequency fTE is obtained using yTE fit (Eq. (9)). Red line A is obtained using Eq. (8) and fTE Electro-Acoustic Analogies Between Thermoelastic Component Of The Photoacoustic Signal And Low-Pass Rc Filter 491 Using the same procedure, we can fit thermoelastic components for aluminum samples of the same shape but different thicknesses: from lmin = 10 to 100 µm in steps of 10 µm, and from 100 to lmax = 1000 µm in steps of 100 µm (Figure 6). All presented thermoelastic components of the photoacoustic signals are obtained using Eq. (4) and sample parameters are given in Table 1. Fig. 6 Bode a) amplitude and b) phase plots of the photoacoustic signal thermoelastic component δpTE obtained theoretically (Eq. (1)) in the case of aluminum circular plate with different thicknesses l and constant radius R, with basic thermal parameters which are given in Table 1 The fitting results obtained by Eq. (9) are shown in Table II, based on which the dependence of fTE on the sample thickness l is drawn (red line, Figure 7). Table 2 Fitting results of an aluminum sample estimation of cut - off frequency as a function of sample thickness. Sample thickness l / (×10-6 m) Cut-off frequency fTE / (Hz) 10 1480460 20 376097 30 168860 40 95666.5 50 61560.5 60 42936.7 70 31658.7 80 24313.1 90 19261.3 100 15638.0 200 3965.95 300 1776.11 400 1004.17 500 645.126 600 449.404 700 331.063 800 256.923 900 204.806 1000 167.771 492 N. LJ. STANOJEVIC, D. K. MARKUSHEV, S. M. ALEKSIC, et al. Fig. 7 Log-log scale of fTE dependence on sample thickness l. Red line represent the fit line obtained using Eq. (11) Mathematically, the dependence of fTE on l in a logarithmic scale (Figure 7) can be obtained by finding the value of σi l/2 in the square bracket of Eq. (3), in the case of f=fTE, using: 2 i l b  = (10) or 2 TE 2 2 TD f b l =   (11) considering that /i TD = , and TE TE2 f  = = . Fitting the data from Figure 7 with Eq. (11) (red line), it is obtained that b = 1.545. Obtained data fit line represents the reference value of the aluminum samples considered as surface absorbers. It can be used to check the validity of measurements and deviations from theoretical model and/or literature values due to various sample impurities, physical damages, etc. 3.2. Experimental validation To validate the suggested δpTE analysis procedure based on electro-acoustic analogy with low - pass RC filter and Eq.(11), we measured and analyzed the photoacoustic response of aluminum sample in 20 Hz – 20 kHz modulation frequency f domain, using a typical open-cell photoacoustic experimental set-up (see Appendix II). The investigated sample was circular in shape, having a thickness, l = 155 μm, and a radius, R = 2.5 mm. The results of such analysis are shown in Figures 8 - 11. Using the well-known method of the signal "cleaning" from the instrumental influence [23-25], we obtained the "real" signal δp (blue line, Figure 8) from the measured experimental values Sexp (Figure 8, asterisks), within the experimental error of 5 %. Electro-Acoustic Analogies Between Thermoelastic Component Of The Photoacoustic Signal And Low-Pass Rc Filter 493 Fig. 8 a) Amplitude and b) phase of the aluminum A1 sample experimental signal Sexp (asterisks) and the "real" signal δp (blue line), signal freed from the instrumental influences Applying the composite piston model on the "real" signal δp [4,16,17], both components were obtained: thermo-diffusion, δpTD, and thermoelastic, δpTE, (Figure 9, green and red lines, respectively). One must take into account that model suggest simple relationship: TD TEp p p  = + . Obtained frequency response of these components correspond to the thermal characteristics of the investigated sample given in Table I. Fig. 9 a) Amplitude and b) phase of the "real" δp signal and its components (thermo- diffusion, δpTD, and thermoelastic, δpTE), obtained using the composite piston model [4,16,17] Analyzing obtained thermoelastic component with Eq. (12) (Figure 10), the value fTE155 = (6275±325) Hz is obtained for the measured sample. 494 N. LJ. STANOJEVIC, D. K. MARKUSHEV, S. M. ALEKSIC, et al. Fig. 10 Analysis of a) amplitude A and b) phase φ of δpTE using Eq. (9) (red line) and the analogue curve of the RC low - pass filter obtained by equations (7) and (8) and frequency f0 = fTE155 The value of fTE155 is plotted on the dependence graph fTE = f(l) (Figure 11, blue line) copied from Figure 7 (red line). It is obvious from Figure 11 that the full matching of fTE155 with the blue line confirms the correctness of Eq. (11). Fig. 11 Dependence of the characteristic thermoelastic cut-off frequency, fTE, on the thickness of the sample, l, with the value of fTE155 (red circle) of the experimental Al sample having a thickness of 155 microns 4. CONCLUSIONS In this paper, we have shown that it is possible to make an analogy between the response of the thermoelastic component of the photoacoustic signal in the frequency domain and the frequency response of the low - pass RC filter because both systems behave as linear time - invariant systems. It was established that the thermoelastic component could be numerically processed by a function created based on of the RC filter transfer function. As a result of such processing, the characteristic cut - off Electro-Acoustic Analogies Between Thermoelastic Component Of The Photoacoustic Signal And Low-Pass Rc Filter 495 frequency fTE of the thermoelastic component is obtained, the value of which changes by changing the thickness of the tested sample. It was observed that the dependence of fTE on sample thickness l in a logarithmic scale is linear, and that f is proportional to 1/l2. The proportionality coefficient contains the value of the thermal diffusion coefficient DT and the numerical value of the constant b that defines the value of temperature moment MT in the case when f = fTE. Since both DT and b constants are related to the tested sample, the dependence fTE ~ 1/l2 can be used to define a reference curve for the thermal characterization of the sample material. To confirm the slope of the reference curve, the photoacoustic response of a 155 µm thick aluminum sample was measured with the open-cell experimental set-up, and its thermoelastic component was processed. The obtained value of fTE entirely coincides with the established reference curve of aluminum obtained by theoretical analysis, thus confirming the correctness of the newly established methodology for the thermal characterization of the material. 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APPENDIX I Considering Eqs (1-3) the expression for δpTE can be written in the form 4 0 0 2 0 tanh 23 1 2 ( ) 2 i TE T ii l I p R p lV k l               =  −       , (12) as a complex number δpTE has its amplitude and phase, whose analytical expressions are complicated. But, for electro-acoustic analogies [26], with the series expansion of the expression in square brackets, one can obtain a simplified expression for δpTE of the form 4 0 0 0 1 3 16 1 TE T TE I p R p V k j           = −   +     , (13) https://web.archive.org/web/20200923154544/https:/www.ciaaw.org/publications.htm#P3 http://worldcat.org/isbn/0-387-25746-2 https://archive.org/details/analogdigitalsig00yarl_849 https://archive.org/details/analogdigitalsig00yarl_849/page/n271 Electro-Acoustic Analogies Between Thermoelastic Component Of The Photoacoustic Signal And Low-Pass Rc Filter 497 whose amplitudes and phases are given as 2 1 ( ) ( ) 1 ( / ) TE TE TE A j p j     =  + , and arctan TE      = −     . (14) Here 2 TE / ( / 2 )TD l b = , and 2 6b  , depending on number of terms in series expansion. APPENDIX II. EXPERIMENTAL SET-UP The experimental setup of the open-cell used in this work is presented in Figure 12. It is a homemade non-commercial setup, explained in detail somewhere else [23,24]. As a light source, a red laser diode with a wavelength of 650 nm, modulated by a frequency generator from the control unit, was used. The photodiode controls the operation of the laser diode and its signal is recorded by the signal processing unit. As a sample, a thin circular aluminum plate, 3 mm in diameter and 155 microns thick, was used, and placed on the microphone's opening. The sample and the microphone together form the so-called photoacoustic cell of minimal volume. photodiode Fig. 12 Simple scheme of the open-cell photoacoustics experimental setup used in our measurements. The sound created by the illumination of the sample spreads through the air in the cell to the microphone membrane, which records it as a photoacoustic signal, converts it into a voltage signal and forwards it to the signal processing unit for further processing. The PC is used simultaneously as a control (frequency generator) and a signal processing unit. (lock-in amplifier). As a replacement for the instrument, the computer's sound card emulates the lock-in amplifier's operation, with which we record the amplitude and phase of the measured signal (the signal from the photodiode is used as a reference signal).