This paper introduces the theoretical model for a commercial moving-coil loudspeaker as an electrical alternator. The audio loudspeaker comprises a paper cone or diaphragm, which is responsible for converting electrical signals into sound waves, and a magnet and coil assembly that drives the cone through electromagnetic interaction. Generally speaking, a loudspeaker that converts electrical energy into sound is an electro-mechanico-acoustical transducer (Beranek and Mellow, 2019). As shown in Figure 1, the loudspeaker attaches to a flange within a container which is composed of the electrical, mechanical, and acoustical parts. Here, we use a complex notation to describe physical quantities that oscillate harmonically at angular frequency ω, such as the acoustic pressure p(t) defined by p ( t ) = R e P e i ω t , where P is the complex amplitude of the pressure oscillation and i signifies the imaginary unit.

By linear and harmonic approximations, a simple linear model (Kleiner, 2013; Swift, 2017) for an electrodynamic device in the frequency domain can be given as follows: Δ P = B l A m I + Z m A m 2 U + (1a) and U + = A m B l Z e I . (1b)

In Eqs 1a, 1b, U ₊ stands for the complex amplitude of the volume velocity driven by the paper cone and A _m represents the effective surface area of the cone. Note that we suppose that the stiff paper cone is oscillated with a uniform velocity V ₊. Therefore, the volume velocity U ₊ driven by the cone is U ₊ = V ₊ A _m . Bl denotes the force factor, and I represents the complex electric current through the coil. Note that Bl is the product of the magnetic-flux density B and the total effective length of the coil l. Furthermore, ΔP represents the complex amplitude of the oscillation pressure difference between pressure oscillation P ₊ at the front and P _v at the rear of the cone, namely, ΔP = P ₊−P _v . In the control equations of the loudspeaker, the mechanical impedance Z _m and the electrical impedance Z _e are, respectively, given as Z m = R m + i ω m m − k m ω = R m + i X m (2a) and Z e = R e + R L + i ω L e − 1 ω C e = R e + R L + i X e , (2b) where R _m , m _m , and k _m , respectively, denote the mechanical resistance, mass, and stiffness and X _m represents the mechanical reactance; R _e and L _e , respectively, symbolize the electrical resistance and inductance of the coil; and X _e stands for the electrical reactance. In addition, R _L and C _e , respectively, stand for the external electrical resistance and capacitance connected to the terminals of the coil as an electrical load for extracting electrical power.

Because the loudspeaker is set within the cavity volume of the container, the paper cone functions as a reciprocating piston on the gas column. The sealed volume of the rear cavity therefore functions as a gas spring k _s , which relates the back pressure oscillation P _v of the loudspeaker as P v = k s U + i ω (Swift, 2017; Beranek and Mellow, 2019; Garrett, 2020). The mechanical stiffness of the volume of gas, k _s , is theoretically given by the adiabatic gas assumption with k s = γ P m A m 2 V v , (3) where γ is the ratio of the specific heats at constant pressure and constant volume, V _v is the volume of the rear cavity, and P _m is the mean pressure.

We replace P _v in terms of k _s in Eq. 1a and arrange Z _m in Eq. 2a by adding k _s in the stiffness term k _m in parallel, which gives total stiffness k _d of the loudspeaker system as k _d = k _m + k _s , which can be written as X m = ω m m − k d / ω . Eliminating I in Eqs 1a, 1b yields the acoustic impedance of the audio loudspeaker as Z a = P + U + = 1 A m 2 B l 2 Z e + Z m . (4)

We clearly express the real part and the imaginary part of Z _a with Z a = 1 A m 2 R m + R e + R L B l 2 R e + R L 2 + X e 2 + i X m − X e B l 2 R e + R L 2 + X e 2 . (5)

As an acoustical load of the impedance shown in Eq. 5, the real part of Z _a mainly represents the dissipative term in acoustics, while the imaginary part of Z _a denotes the acoustic reactance. It is worth noting that the imaginary part of Z _a is mainly dominated by X _m due to the much smaller value of X _e . Since the acoustical load receives acoustic power generated from the engine, the sign of the acoustic impedance of the loudspeaker should align with the thermoacoustic engine in the calculation model. We use the equation Z s p = − Z a (6) to couple the loudspeaker model with the thermoacoustic theory.

With Rott’s acoustic approximation (Tominaga, 1998; Swift, 2017) for small amplitude of gas oscillation with angular frequency ω, basic equations of energy, momentum, and continuity are linearized as the framework of thermoacoustic theory. For cylindrical pores with the gas-occupied cross-sectional area A, the thermoacoustic model in the frequency domain is expressed as follows: d P d x = − i ω ρ m A 1 1 − χ ν U = − Z U d U d x = − i ω A 1 + γ − 1 χ α γ P m P + χ α − χ ν 1 − χ ν 1 − σ 1 T m d T m d x U = − Y P + G U , (7) where P and U, respectively, stand for the complex amplitudes of the acoustic pressure and the complex amplitude of volumetric velocity. In the equations, Z stands for the fluid impedance per unit length, Y denotes the fluid admittance, and G represents the thermoacoustic source/damping term. Whether G is the source or damping term depends on the sign of the temperature gradient dT _m/dx. Note that the symbol σ in G represents the Prandtl number. Furthermore, χ _j (j = α or ν) is a thermoacoustic function, and subscripts ν and α, respectively, denote the viscous and the thermal effects (Swift, 2017; Biwa, 2021). If the geometries of flow channels are simple, χ _j is possible to be theoretically obtained (Swift, 2017; Biwa, 2021). It is worth noting that r/δ _j is a key non-dimensional parameter that dominates thermoacoustic functions, where δ _j ( = 2 j / ω ) represents the thermal or viscous penetration depths, using j = α or ν, respectively, for the thermal diffusivity or the kinematic viscosity of the working gas. For cylindrical tubes represented by radius r and parallel plates characterized by the half of spacing r _d, χ _j are calculated as χ j = 2 J 1 i − 1 r δ j i − 1 r δ j J 0 i − 1 r δ j  for circular pores tanh 1 + i r d δ j 1 + i r d δ j for parallel plates , (8) where δ _j ( = 2 j / ω ) represents the thermal or viscous penetration depths, using j = α or ν, respectively, for the thermal diffusivity or the kinematic viscosity of the working gas. Furthermore, J ₀ is the zeroth order Bessel function of the first kind, and J ₁ is the first-order Bessel function of the first kind. Drawing from the segments illustrated in Figure 2 and utilizing Eq. 8, we model the resonator tubes 1 and 2, branch resonator, thermal buffer tube, and stacked-screen regenerator using the circular cylindrical tube model, while the two heat exchangers are depicted through the theoretical expression for parallel plates. Importantly, even though the stacked-screen regenerator exhibits tortuous flow channels, we characterize it using cylindrical modeling with an effective radius, as reported by Ueda et al. (2009), when non-linear effects are negligible.

If the characteristic length or radius of cylindrical flow channels with simple and regular geometries are known, coefficients of Z , Y , and G can be considered constants for a short segment of length Δx. Combining Eq. 7 by eliminating U yields the Helmholtz equation for thermoacoustics, which is expressed as d 2 P d x 2 − G d P d x − Y Z P = 0 . (9)

By solving the aforementioned Helmholtz equation, we can obtain an acoustic network model for the whole system. With a transfer matrix M solved from Eq. 9, we therefore relate acoustic pressure P and volume velocity U at two ends of a short segment of length Δx as P x + Δ x U x + Δ x = e G Δ x 2 − G D sinh λ + cosh λ 2 Z D sinh λ 2 Y D sinh λ G D sinh λ + cosh λ P x U x , (10) with D = G 2 + 4 Y Z and λ = DΔx/2. Characterizing every element of the acoustic duct using Eq. 10 enables us to theoretically describe a matrix M _loop = M _tube,2 M _TBT M _H M _REG M _R M _tube,1 for the looped part of the thermoacoustic engine, where subscript tube, 1 or 2, R, REG, H, and TBT of matrices M denote the 30-mm-diameter waveguide tube, ambient-heat exchanger, stacked-screen regenerator, hot-heat exchanger, and thermal buffer tube, respectively. Note that we divide M _REG and M _TBT into 100 parts to describe axially linear temperature distributions imposed on these two regions, for accounting for the thermal properties of the working gas. By considering continuity at the T-junction, acoustic states at the origin are realized with U ₀ = U _L + U _R (referring to the inset of Figure 2) and P ₀ = P _L, where subscripts 0, L, and R, respectively, indicate acoustic states at positions of x = 0⁺ m, x = 1.635 m, and x = 0⁻ m, as illustrated in Figure 2. As a result, the transfer matrix M _loop can relate acoustic states at two ends of the looped part of the engine, through the continuity of P(x = 0) = P(x = L) and U(x = 0⁺) = U(x = 0⁻) + U(x = L), with P L U L = M loop P 0 U R + U L . (11)

Furthermore, acoustic states at two ends of the branch resonator are related by M _BR using Eq. 10 without any temperature gradients as M BR P en U en = P 0 U R , (12) where the subscript en represents the acoustic state at x = L _d = 1.494 m, which is the sum of the length of the branch resonator and the thickness of the front plate of the loudspeaker housing. We obtain the acoustic impedance Z _en at L _d by relating Eq. 11 and Eq. 12 with the continuity at the T-junction of x = 0, which is expressed as follows: Z en = P en U en . (13)

The acoustic impedance of the loudspeaker, Z _sp , represented by Eq. 6, should serve as the boundary conditions at the interface between the thermoacoustic engine and the loudspeaker. Replacing Z _sp of Eq. 6 with Z _en in Eq. 12 results in P 0 U R  = M BR Z s p 1 − U + = m 11 m 12 m 21 m 22 Z s p 1 − U + , (14) where m denotes elements of the matrix M _BR. Eliminating U ₊ in Eq. 14 gives the relationship between U _R and P ₀ as U R = m 21 Z s p + m 22 m 11 Z s p + m 12 P 0 . (15)

Equation 15 enables us to rewrite Eq. 11 as P L U L  = M loop 1 0 m 21 Z s p + m 22 m 11 Z s p + m 12 1 P L U L , = M All P L U L (16) where M _All stands for a representative transfer matrix of the combined system. Nonzero [ P L , A U L ] ⊺ in Eq. 16 results in a characteristic equation as det M All − E = 0 , (17) where E represents the identity matrix and det[ ] stands for the determinant of a matrix.

In this study, we used the numerical solver “fsolve” in MATLAB (MathWorks, Inc.) and set the tolerance at TolFun = 10^–25 to solve the aforementioned characteristic equation. Although the acoustical load of the loudspeaker strongly influences the fundamental frequency of the combined system, an initial guessing angular frequency ω ₀ = 2πa/(L + L _d ) × 1/4 = 174.6 rad/s, equivalent to 27.8 Hz, approximately estimated from the engine part is adopted for numerically solving Eq. 17, where a represents the adiabatic speed of sound for the working gas at the room temperature of 298 K. If a temperature difference across the regenerator is assigned, solving Eq. 17 yields a complex value of the angular frequency ω = ω _R + iω _I (Hsu and Li, 2023), where ω _R stands for the angular frequency of acoustic oscillation and ω _I represents the growth rate (ω _I < 0) or attenuation rate (ω _I > 0) over time, depending on the imposed temperature difference. In the light of |ω _I| < 10^–8, solutions of Eq. 17 provide the onset temperature difference and the frequency of the spontaneous oscillation for the thermoacoustic engine loaded with the loudspeaker.

A commercially available audio loudspeaker (Fostex FW168HS, Foster, Tokyo, Japan), having high stiffness diaphragm of the paper cone, is mounted in the circular center of a 15-mm-thick transparent flange of the container that is part of a stainless steel cylinder container with an inner diameter of 30 cm and an inner axial length of 34 cm. Table 1 lists detailed specifications and measured Thiele/Small (T/S) parameters of the commercial loudspeaker Fostex FW168HS used in the present study. The T/S parameters of the audio loudspeaker mounted in the container were measured according to the instructions provided in the acoustic textbook (Yu et al., 2011; Garrett, 2020). The moving mass m _m and stiffness k _m were measured by plotting the relationships between adding known masses on m _m and resonance frequencies of the loudspeaker. For the gas spring effect resulting from the volume of the rear cavity, we tested the audio speaker within and without the housing for experimentally determining the stiffness k _s , by monitoring the changes in resonance frequencies. The mechanical resistance R _m was determined using a simple free-decay experiment, while the internal electrical resistance R _e and inductance L _e of the coil were simply probed using a volt-ohm-milliammeter. Furthermore, the coil of the audio loudspeaker connects with resistances R _L and capacitance C _e in series as experimental variables for modulating the acoustic impedance. The influence both on the real and imaginary parts can be realized in Eq. 5.

In addition to examining the T/S parameters of the audio loudspeaker individually, we measured ΔP and U ₊ to experimentally determine Z _sp , supposing the uniform oscillation velocity V ₊ across the paper cone. Figure 1 illustrates the experimental setup used to ascertain the acoustic impedance of the loudspeaker. Another loudspeaker, acting as an acoustic driver, was attached at the end of a duct and positioned opposite the test loudspeaker. A pressure transducer (PMS-5M-2-1M, JTEKT, Nagoya, Japan) was mounted at the transparent front flange of the loudspeaker to measure P ₊. It is important to note that we determine P ₊ as ΔP since the loudspeaker was tested with the housing removed to simplify concerns related to the effects of the rear cavity. Through the pressure transducer, signals of the pressure oscillations were amplified 2,000 times by DC Amplifiers (AA6210, JTEKT, Nagoya, Japan) and then recorded using a fast Fourier transform (FFT) analyzer (OR35, OROS, Grenoble, French). To determine the oscillating velocity V ₊ of the paper cone, a laser vibrometer (PDV-100, Polytec, Waldbronn, Germany) was utilized. This device measured the oscillation velocity of the cone by passing a laser beam through the transparent flange to probe the paper cone. The signals generated by the vibrometer were recorded using the FFT analyzer along with the pressure signals. By reading peak values from the amplitude spectrum and phase spectrum via the fast Fourier transform algorithm, measured acoustic states were presented with the acoustic pressure p ( x ) = R e P ( x ) e i ω t and the oscillation volume velocity u ( x ) = R e U ( x ) e i ω t . We generated acoustic waves by driving the loudspeaker using a power amplifier (PX3, Yamaha, Shizuoka, Japan) and a function generator. Since we aimed to match the load impedance with the engine part at the initial onset states, a low amplitude of acoustic forcing was conducted for the experiments.

A schematic model has been illustrated in Figure 2 for our prototype of the thermoacoustic electric generator examined in this study. The prototype consists of the engine part and the load part, which were filled with ambient air at atmospheric pressure. The acoustic load part is the loudspeaker in the container mentioned in Section 3.1. The cone of the loudspeaker is set toward the engine part for receiving the acoustic power delivered from the engine part. The engine part is constructed using a 30-mm-inner-diameter tube of stainless steel, which can be divided by an averaged 1.635-m-long looped tube and a branch resonator tube of 1.479 m long. A pressure transducer (PMS-5M-2-1M) is mounted at x = −1.279 m to monitor pressure fluctuations of the gas column. The looped tube contains the thermoacoustic core consisting of a regenerator sandwiched between the hot- and ambient-heat exchangers. A thermal buffer tube is attached at one end of the hot-heat exchanger to moderate the thermal impact on the looped tubes. The heat exchangers consist of numerous brass fins with 0.5 mm in thickness and 1 mm spacing, which are inserted with three brass tubes, respectively, for installing electrical cartridge heaters and circulating cooling water. The geometrical parameters of the thermoacoustic system are shown in Table 2. Furthermore, the regenerator is made of a pile of stainless 304 wire woven mesh screens of #60 whose hydraulic diameter d _h and wire diameter d _w are, respectively, 0.4031 and 0.15 mm. The effective radius of empirical expressions (Ueda et al., 2009) for the stacked-screen regenerator, r 0 = d h d w / 2 , is used to model its flow channels for modeling into the linear theoretical framework. Two type-K thermocouples are inserted at both ends of the regenerator to observe the axial temperature difference ΔT = T _H−T _C across the regenerator, where T _H and T _C, respectively, represent the temperatures at the hot and ambient sides of the regenerator. To monitor these temperatures and record their time series data, we utilized a temperature recorder (LR8450, Hioki, Ueno, Japan) with a sampling interval set at 1 s.

In this study, we measured pressure, frequency, temperature, and velocity. We used a PMS-5M-2 pressure transducer to conduct pressure measurements, with a sensitivity of 7.49 Pa/mV and an accuracy within 0.15% FS for readings smaller than 2 kPa. The voltage-represented pressure signal is amplified 2,000 times using the DC amplifier (AA6210), which would introduce a peak-to-peak voltage noise of 15 μV, resulting in a maximum noise error of less than 0.1 Pa. The amplified voltage signals are subsequently analyzed using the OROS OR35 spectrum analyzer, which operates with 25,601 sampling points at a sampling frequency of 3276.8 Hz. The relative precision of frequency is within 1⁻⁵. It is worth noting that the sampling frequency is substantially higher than the maximum frequency (below 40 Hz) examined in this study. The OROS OR35 has a resolution of 24 bits with a dynamic range of 144 dB. When it is used to process voltage signals from the DC amplifier in the range of ±1 V, it is capable of discerning signal changes down to 0.1 μV. Temperatures are measured with type-K thermocouples (T35165H), which have an accuracy of ±2.5 K over the range of 273–923 K. We recorded the probed temperatures using the Hioki LR8450 temperature recorder, which maintains an accuracy of ±0.5 K within the range of 273–773 K and a resolution of 1 μV when used in the range of ±20 mV, corresponding to the type-K thermocouple. Furthermore, velocity is determined using the PDV-100 laser vibrometer, which has an accuracy of 1% FS within a range of ±100 mm/s.

The systematic measurement error σ _sys and random error σ _ran are, respectively, given as σ mea = σ sys 2 + σ ran 2 (18) and σ r a n = ∑ j = 1 n y j − y j ̄ 2 n − 1 n , (19) where y _j represents the jth measurement, y j ̄ is the arithmetic mean of measurements, and n is the number of repeated measurements. The systematic error σ _sys is due to the accuracy of the measurement devices, and the random error σ _ran is the standard deviation of repeated experiments. Note that not every physical quantity was subjected to repeated measurements in the present study. For these instances, we calculate the random error using σ ran = 1 2 ( resolution of device ) / 3 , where the term 3 originates from the assumption of a rectangular distribution of the measurement values’ probabilities ¹ .

Through these calculations, we estimated the maximum measurement errors for pressure, frequency, temperature, and velocity to be ±3.00 Pa, ±0.003 Hz, ±3.00 K, and ±1.00 mm/s, respectively.

For starting the thermoacoustic engine loaded with the loudspeaker, we introduce the external heat power through the power supply to electrical cartridge heaters at the hot-heat exchanger and circulate the cooling water through the ambient-heat exchanger to maintain a constant temperature of T _C = 298 K, to impose steep temperature gradients on the regenerator axially. In the experiments, we ensured thermally steady states to maintain a stable temperature difference ΔT. This stable ΔT was critical for reaching the instability threshold. Once this threshold was crossed, it triggered spontaneous pressure oscillations in the gas column, consistently coupled with the loudspeaker. The thermoacoustic engine loaded with the loudspeaker was thus excited in the natural resonant frequency of the combined system.

Figure 3 illustrates an example of our experimental approach, showcasing time-series data of both acoustic pressure and temperatures during the transition from a stable state to spontaneous oscillation states. During the startup phase, spontaneous oscillations within the gas column emerge at approximately 700 s, coinciding with a gradual increment of T _H at 0.85 K/min. In the spontaneous oscillation states, temperatures remain generally stable within specific ranges, specifically T _H within 3 K and T _C within 0.5 K. These stable conditions allowed us to experimentally determine the onset temperature difference based on these steady temperatures, thus demonstrating the practical application and validity of our experimental setup.

The measurements of Z _sp , obtained from forced oscillation experiments on the acoustic driver at low amplitudes of acoustic forcing (below 100 Pa, monitored by P ₊), are shown in Figure 4. Given the low amplitude of acoustic forcing, we assume that the input impedance of the loudspeaker remains unaffected by the amplitude of oscillation pressure. We conducted tests on the loudspeaker using forced steady oscillations at frequencies of 10, 20, 30, and 40 Hz. These test points are represented by black dots in the figure, while the real and the imaginary parts of Z _sp are shown on the separate vertical axes in two subfigures (a) and (b) in Figure 4. Note that every data point we plotted represents the mean of six repeated measurements. We used Eq. 19 to determine the random error, which is represented as the standard deviation shown in the attached error bars. Detailed information about the means of the measurements and corresponding standard deviations can be found in Table 3. The red curve illustrates the theoretical estimation from the linear model presented in Eq. 6. The T/S parameters used by the linear model depicted in Section 3.1 are each measured and detailed in Table 1.

Despite removing the loudspeaker housing to minimize the effects of the complex geometry in the loudspeaker’s rear cavity, the real part of the measured Z _sp deviated significantly from the theoretical prediction of the linear model. This deviation could be attributed to the geometry of the front cavity formed by the paper cone, highlighting the difficulties in accurately measuring mechanical resistance through the simple free amplitude decay experiments. However, the imaginary components of Z _sp exhibited qualitative consistency with the theoretical estimations. To mitigate these discrepancies, we calibrated the linear model based on the measurements of Z _sp .

Considering the linear model of the loudspeaker, as shown in Eq. 5, the real part of Z _a is primarily determined by the sum of the electrical resistances R _e + R _L and mechanical resistance R _m , while the imaginary part of Z _sp is influenced by the mechanical stiffness k _m of the loudspeaker since X _m ≫ X _e . Assuming constant values for electrical components, we can determine R _m and k _m from the measured Z _sp using the linear model shown in Eq. 5 while keeping the other T/S parameters constant. This calibration is achieved through the following equations: R m = − A m 2 R e Z s p − R e b l 2 Z e , (20) k m = ω A m 2 I m Z s p + ω 2 m m . (21)

The calibration values of R _m and k _m are, respectively, displayed in the subfigures (a) and (b) of Figure 5. As observed, R _m exhibits a cubic polynomial function of the frequency, while k _m demonstrates a nearly linear relationship with the frequency. Consequently, regression fitting is employed to obtain the following regression functions: R m = − 0.000682 ω 2 π 3 + 0.0665 ω 2 π 2 + − 2.2398 ω 2 π + 27.033 , k m = 22.0454 ω 2 π + 974.208 (22)

By incorporating the regression functions of R _m and k _m into Eq. 6, we obtained the calibrated Z _sp represented by the blue dotted curves in Figure 4. As evident from the figure, the calibrated Z _sp accurately reproduces the measured data. With Eq. 6 calibrated using R _m and k _m , we can effectively model the loudspeaker for practical scenarios within the tested frequency range of 10–40 Hz while considering the absence of significant amplitude effects.

Figure 6 illustrates the onset conditions of the thermoacoustic Stirling engine when loaded with the audio loudspeaker. The figure consists of the following two subfigures: (a) representing the onset temperature difference T _H−T _C and (b) representing the spontaneous oscillation frequency ω/(2π). Both subfigures are plotted on a logarithmic horizontal axis, with the external electrical resistance R _L parameterized accordingly. In Figure 6, experiments of the thermoacoustic engine loaded with the loudspeaker are denoted by markers (◦ for housing and * for no housing), while the theoretical estimation, obtained by solving the characteristic Eq. 17 using the calibrated loudspeaker model, is represented by curves (solid for housing and dotted for no housing). The spontaneous steady oscillation states are confirmed over a range of 20 Ω < R _L < 1,000 Ω. Note that for the analytical estimations involving the loudspeaker with housing, the calibrated values of R _m and k _m used in the model are obtained from the case without housing. To incorporate the housing effect, we include the measured gas spring k _s to account for the total mechanical stiffness k _d .

When examining the onset temperature difference ΔT of the combined system illustrated in Figure 6A, it can be observed that both the loaded loudspeaker with and without housing exhibit a hill-like trend as a function of the logarithmic horizontal axis R _L , which affects the onset temperature difference by approximately 20 K. The observation in Figure 6A suggests that ΔT reaches maximum values when R _L is approximately in the range of 40–80 Ω. Figure 6B demonstrates that the spontaneous oscillation frequency slightly increases with R _L by approximately 0.4 Hz, and this can be qualitatively explained by the expression of Im (Z _a ) in Eq. 5. It is realized that the acoustic reactance is minimally affected by R _L due to the dominance of X _e over X _m .

From Figure 6, it can be noted that the experimental measurements align reasonably well with the calculated results. By analyzing the onset temperature difference of the combined system, as depicted in Figure 6A, it becomes evident that the case of the audio loudspeaker without housing exhibits better consistency than that with housing. In Figure 6B, precise estimations for the onset frequency are displayed, with discrepancies of less than 0.5 Hz observed for both cases of the loudspeaker with and without housing. Notably, the case of the loudspeaker without housing demonstrates good agreement, with a maximum deviation of 1.7% observed at the data point of 80 Ω.

It is worth noting that in our previous research (Hsu and Li, 2023), we found a satisfactory level of consistency in the #60 stacked-screen regenerator, with discrepancies of approximately 18%. This was observed when using resonators with slightly varying lengths in the current thermoacoustic Stirling engine in the absence of any acoustical load. However, in the current study, we used the same regenerator in the loaded engine and found improved agreements, particularly when the loudspeaker was loaded on the engine in the case without housing. In this loaded thermoacoustic engine scenario, it is likely that the system experiences larger dissipation mechanisms. This infers that the sensitivity of the onset temperature estimation is intrinsically limited to a well-accepted fact, namely, that the onset ΔT is achieved when the thermoacoustic power generation slightly surpasses the dissipation at minimal oscillation amplitudes. Hence, once the load model is precisely theoretically estimated, implementing an appropriate acoustic load could help mitigate this sensitivity.

On the other hand, the larger discrepancies observed in the case of the loaded loudspeaker with housing could be attributed to the dissipation term in the rear cavity not being incorporated into the calculation model. In the calibration procedure for R _m and k _m for this study, we have only considered the front cavity of the paper cone. Furthermore, we simply account for the gas spring effect of the rear cavity in the loudspeaker housing. However, the complex geometries formed in the front and rear of the paper cone of the loudspeaker likely introduce non-negligible minor losses. This could be the primary reason for the larger discrepancies observed in our comparisons between calculations and experiments.

It should be noted that, in the range of 40–80 Ω, the comparisons agree relatively well. The higher onset temperature differences in the range of 40–80 Ω indicate that the system needs to overcome a greater load resistance, which our model predicts reasonably. However, for relatively lower onset temperature differences, the influence of the loudspeaker housing may affect accuracy, highlighting areas where our model’s predictive capability could be further improved.

Zorgnotti et al. (2018) highlighted this issue and carried out calibrations to experimentally determine the acoustic admittances of the two cavities by measuring the acoustic impedance through a two-load method. Their work not only is applicable to the engine onset but also has the capability of estimating larger limit cycle amplitudes of a loaded thermoacoustic engine through impedance matching. Our current research, which is based on solving the characteristic equation of the combined system, could benefit from the adoption of their calibration techniques.

In contrast, C _e directly impacts the imaginary part of Z _a . While maintaining R _L = 40 Ω, we also varied C _e to experimentally observe the applicability of the current calibration in describing the onset conditions of the combined system, as shown in Figure 7. The representations of markers and curves in Figure 7 are the same as in Figure 6, except for the horizontal axis which denotes C _e as the experimental parameter, ranging from 40 to 1,000 μF. In Figure 7A, it can be observed that measurements of the loaded loudspeaker, whether with or without housing, demonstrate a gradual increase with respect to the logarithmic C _e axis, indicating an approximate increase of 100 K in ΔT. The consistency between measurements and calculations for onset ΔT is only valid in the range from 40 to 200 μF. Because our calibration for k _m assumes X _m ≪ X _e , larger C _e values would introduce significant effects for X _e , leading to discrepancies. Meanwhile, as illustrated in Figure 7B, the current calculation model, based on the present loudspeaker calibration, well predicts spontaneous oscillation frequencies as C _e varies.