A method of characterising performance of audio loudspeakers for linear alternator applications in low-cost thermoacoustic electricity generators

Applied Acoustics 72 (2011) 260–267

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A method of characterising performance of audio loudspeakers for linear alternator applications in low-cost thermoacoustic electricity generators Zhibin Yu, Patcharin Saechan, Artur J. Jaworski ⇑ School of Mechanical, Aerospace and Civil Engineering, University of Manchester, Sackville Street, Manchester M13 9PL, UK

a r t i c l e

i n f o

Article history: Received 7 September 2009 Received in revised form 10 August 2010 Accepted 22 November 2010 Available online 28 December 2010 Keywords: Linear alternator Acoustic–electric efficiency Measurement method Thermoacoustic generator

a b s t r a c t This paper investigates the feasibility of using commercially available loudspeakers as low-cost linear alternators for thermoacoustic applications, to convert acoustic power to electricity. The design of a purpose built experimental apparatus, in which a high intensity acoustic wave is induced by using a high power woofer, is described. The rig is used to excite loudspeakers (referred here as ‘‘alternators’’) under test, while a pair of microphones and a laser displacement sensor are used to enable acoustic power measurements. The paper presents a case study in which characteristics of acoustic-to-electric energy conversion of a candidate loudspeaker (alternator) – selected from the viewpoint of general performance, as well as parameters such as: high force factor, low electrical resistance and low mechanical loss – are measured. The measurements of acoustic power absorbed by the alternator and the electric power extracted from it by the load resistor, which allow estimating acoustic-to-electric efficiencies, are presented. The alternator has been tested at different operating frequencies, cone displacements and load resistance values. The measurement results are discussed and compared in detail with the calculations based on the linear acoustics model. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Thermoacoustic heat engines directly convert heat input into acoustic power, by utilising the thermoacoustic effect: a spontaneous generation of sound waves in a compressible fluid due to a temperature gradient imposed along the solid material in contact with the fluid. Thermoacoustic engines are thought to be particularly attractive because their only moving component is the gas undergoing acoustic motion [1,2]. The absence of mechanical moving parts provides a potential for high reliability and low cost. The working fluid in thermoacoustic engines is usually a noble/inert gas, making this technology environmentally friendly. Furthermore, the required operating temperature difference could be relatively small. For example, de Blok’s [3] travelling-wave thermoacoustic engine starts the acoustic oscillation at a temperature difference of only 65 K. Therefore the technology shows a lot of potential for utilising industrial waste heat or renewable solar power. The thermoacoustic engine’s acoustic power derived from the heat input can be utilized in different ways for different applications. However, generally it can be used for two main purposes: one is to use the obtained acoustic power to drive coolers or heat pumps [4], which can be either thermoacoustic or pulsed-tube type. The other is to directly convert the acoustic power to ⇑ Corresponding author. Tel.: +44 (0) 161 275 4352; fax: +44 (0) 161 306 3755. E-mail address: [email protected] (A.J. Jaworski). 0003-682X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2010.11.011

electricity through the electro-dynamic transduction mechanism. Usually, flexure-bearing-supported linear alternators are an excellent solution due to their high reliability and efficiency [5]. However, high costs of commercially available linear alternators limit the advantages of the thermoacoustic heat engines for low-cost electricity generators. Usually, ordinary audio loudspeakers are excluded as prospective candidates for linear alternators due to their relatively low power transduction efficiency, a fragile paper cone, and a limited stroke, especially when the researchers aim at obtaining generators with a high power, high efficiency and high pressure difference. However, it is possible to consider niche applications where the main driver is cost of the device, not the power transduction (or even the overall) efficiency. This is particularly true for the above mentioned waste heat and solar energy utilisation applications, where a low grade thermal energy is abundant and could be considered a limitless source [6]. Then the actual efficiency figures may become a secondary issue, as long as the electric power could be extracted at very low cost per kW he. Similar reasoning may be true for designing cheap electricity generators for the third world countries, as exemplified by SCORE Project [7], which is the immediate background of current work. SCORE Project aims at designing a cheap thermoacoustic electricity generator for rural areas of developing countries, where heat typically obtained from biomass burning for cooking applications can be used for generating small amounts of electricity to improve

Z. Yu et al. / Applied Acoustics 72 (2011) 260–267

the quality of life of rural people. The electricity generated (and stored in a battery) would be used for domestic lighting, charging of mobile phones or powering radios. Given that the target cost of a 100–150 W system is below 100 US dollars, one has to consider cheap linear alternators. These are likely to be commercially available audio loudspeakers, or purpose designed units based on the components of audio loudspeakers. For the research in this paper, as a first step, a measurement methodology and an experimental set-up were developed to enable characterisation of the performance of candidate loudspeakers as linear alternators. Secondly, a commercially available 6.5-in. diameter audio loudspeaker was selected and tested. In this case study, the alternator’s acoustic-to-electric power transduction efficiency is measured at various operating frequencies, cone displacements and load resistances. The experimental results are compared with the calculated ones using an analytical linear model of an alternator based on the measured Thiele/Small parameters.

261

Assuming all parameters are linear and independent from frequency, ignoring hysteresis losses, the model in Fig. 1b can be described approximately by the following linear equations:

  BlI1 Rm þ j xMm  Kxm Dp ¼ U1 ; þ S S2 BlU 1 ¼ ðRe þ RL þ jxLe ÞI1 : S

ð1Þ ð2Þ

The input acoustic power to the alternator Pa,in is defined as

Pa;in ¼

1 e 1  ¼ 1 jp jjU 1 jcosðh1 Þ: Re½p1 U 2 2 1

ð3Þ

In Eq. (3), h1 is the phase angle between p1 and U1. Accordingly, the acoustic power that runs out from behind of the alternator is defined as

Pa;out ¼

1 e 1  ¼ 1 jp jjU 1 jcosðh2 Þ: Re½p2 U 2 2 2

ð4Þ

In Eq. (4), h2 is the phase angle between p2 and U1. Therefore the acoustic power absorbed by the alternator Pa is defined as

2. Theoretical analysis Fig. 1 shows schematically a simple linear model [8] describing the loudspeaker as a linear alternator. Referring to Fig. 1a, the acoustic wave exerts an oscillatory pressure on the diaphragm, which has an effective area S. The total mass of the diaphragm and the coil is Mm. The alternator has a mechanical stiffness, Km, and a mechanical resistance, Rm. The coil has an inductance, Le, and a resistance, Re. The force factor is Bl. A load resistor RL is connected to the terminals of the coil to extract the electrical power converted from the acoustic power by the alternator. Fig. 1b shows the equivalent impedance circuit of the physical model shown in Fig. 1a. The acoustic circuit is connected to mechanical circuit through a transformer and the mechanical circuit is connected to the electric circuit through a gyrator. The pressure difference (pressure drop) between the front and the back of the diaphragm is Dp, the volumetric velocity due to the diaphragm displacement is U1. The force exerted on the diaphragm due to the pressure drop is F, and velocity of the diaphragm is u1. The voltage on the load resistor is VL, and the current is I1.

Pa ¼

1 e 1  ¼ Pa;in  Pa;out : Re½Dp U 2

ð5Þ

In Eq. (5), Dp is defined as Dp = p1  p2 in a vector sense. The extracted electric power by the load resistor (assuming it is purely resistive), Pe, is defined as

Pe ¼

1 1 jV L j2 RL jI1 j2 ¼ : 2 2 RL

ð6Þ

In Eq. (6), |I1| is the amplitude of current I1. Accordingly, the acoustic–electric efficiency can be defined as

g¼

Pe : Pa

ð7Þ

According to the analysis above, to measure the acoustic–electric transduction efficiency of the alternator, one needs to measure p1, p2, U1, h1, h2 and |VL| (or |I1|).

Fig. 1. Schematic of the alternator’s physical model (a); the equivalent impedance circuit (b).

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Furthermore, substituting Eqs. (1) and (2) into (5) leads to

1 e 1 Re½Dp U 2 " # 1 jU 1 j2 ðBlÞ2 Re ðBlÞ2 RL ¼ R þ þ : m 2 S2 ðRe þ RL Þ2 þ x2 L2e ðRe þ RL Þ2 þ x2 L2e

3. Experimental set-up

Pa ¼

ð8Þ

From Eq. (8), it can be found that the input acoustic power, Pa, is dissipated by the alternator by three different mechanisms: the mechanical resistance, the resistance of the coil and the load resistor. According to Eq. (8) and neglecting the inductance of the coil, the electrical power extracted by the load resistor can be written as

Pe ¼

1 jU 1 j2 ðBlÞ2 RL : 2 S2 ðRe þ RL Þ2

ð9Þ

The maximum extracted power is therefore

Pe;max ¼

1 jU 1 j2 ðBlÞ2 ; 8 S2 Re

ð10Þ

when

RL ¼ Re :

ð11Þ

For the loudspeaker tested in this paper, xLe is much less than Re or RL. Therefore, substituting Eqs. (8) and (9) into (7) and neglecting the inductance of the coil, the efficiency can be approximately obtained as

g¼

ðBlÞ2 RL Rm ðRe þ RL Þ2 þ ðBlÞ2 ðRe þ RL Þ

:

ð12Þ

When

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðBlÞ2 RL ¼ Re 1 þ ; Rm Re

ð13Þ

the maximum efficiency can be obtained as

gmax

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi ðBlÞ2 1 þ RðBlÞ m Re ¼   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi : ðBlÞ Rm Re 1 þ 1 þ Rm Re þ ðBlÞ2 1 þ 1 þ RðBlÞ m Re

ð14Þ

Substituting Eq. (2) into (1) to cancel I1 also leads to the total acoustic impedance of the alternator as

Z total

"  # Dp 1 ðBlÞ2 Km : ¼ ¼ þ Rm þ j xM m  U 1 S2 ðRe þ RL þ jxLe Þ x

ð15Þ

Because xLe is much less than Re or RL, the term involving the inductance of the coil can be neglected. Thus, the phase angle between the pressure drop, Dp and volumetric velocity, U1, can be approximately expressed as:

2

Km

3

xMm  x 5 : h ¼ arctan4 ðBlÞ2 þ Rm ðRe þRL Þ

A dedicated test rig has been designed and built in order to perform the necessary experiments to characterise the candidate loudspeakers as alternators. It is shown schematically in Fig. 2. It is essentially a 50 cm long acoustic duct with 7.6 cm  7.6 cm internal cross section made out of aluminium. On one end of the duct (see the left of Fig. 2) there is a 300 mm long conical section, which connects it to an 18-in. diameter ‘‘driving’’ loudspeaker (PD 1850 loudspeaker manufactured by Precision Devices, with a rated power of 800 W). The other end of the duct (see the right of Fig. 2) is used to mount the alternator under test. The tested alternator is connected to the duct using a short (50 mm long) conical section. To obtain a desired relative pressure amplitude in the rig during tests, the alternator is enclosed by a metal cover, which has an inside diameter of 22 cm and a length of 13 cm, for the case considered in this paper. The driving loudspeaker is excited by a power amplifier, driven by a sinusoidal wave generated by a TTI TG 1010A digital function generator, which has a frequency resolution of 0.1 mHz. Four pressure transducers (model 112A22, made by PCB) are used to measure the dynamic pressures in front and behind the diaphragm of the alternator. These have a resolution of 7 Pa, and are frequently calibrated to ensure high data quality. They have also been interchanged to double check the measurement reliability. They are denoted as P1, P2, P3 and P4 in Fig. 2. P1 and P3 are flush mounted just in front of the diaphragm. They both measure the dynamic pressure in front of the diaphragm in this rig. As they are installed symmetrically, the pressure measurements from sensors P1 and P3 are expected to be identical assuming a planar wave in the rig (in effect P1 and P3 were a backup for each other in case one of them failed). P2 is flush mounted on the side wall of the metal back cover, just behind the diaphragm. P4 is flush mounted on back wall of the metal cover. They both measure the pressure oscillations behind the diaphragm. The original intention was to use P4 as a backup of P2. However, in the experiments, it was found that the results from P2 and P4 are slightly different when the operating frequency approaches the maximum testing frequency of 120 Hz. The difference between the results from P2 and P4 indicates that the back volume does not behave as an ideal acoustic compliance, especially at relatively high operating frequency (e.g. above 100 Hz). However from the measurement methodology point of view P2 is the correct signal to be used as it is just behind the diaphragm. For practical calculations, the readings from P1 and P2 were used in the analysis. The displacement of the cone is measured by a high speed laser displacement sensor, MICROTRACK II (manufactured by MTI Instruments Inc.), which has a 50 mm standoff and ±5 mm measuring range, with a resolution of 1.25 lm. Its sampling frequency is 40 kHz. The laser displacement sensor is installed in the centre of the cavity in front of the alternator. It is around 50 mm away from the diaphragm. It has a cross-sectional area of 74.6 mm by

ð16Þ

In Eq. (16), when x2 = Km/Mm, i.e. the alternator is at mechanical resonance, the phase angle between Dp and U1 is

h ¼ np; ðn ¼ 0; 1; 2; 3:::Þ:

ð17Þ

Therefore, by monitoring the phase angle h, one can find out whether the alternator is at resonance. However, it should be noted that the above analysis is based on the ideal linear conditions and neglects the hysteresis losses. In a real linear alternator, the mechanical resistance depends on frequency because the windage and spring losses depend on the frequency. Furthermore, at high pressure amplitudes, the harmonics will also be induced.

Fig. 2. Schematic diagram of the test rig.

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24.6 mm. The blockage caused by the sensor is around 10% of the local cross-sectional area. However, this does not cause any pronounced constraints on the acoustic movement of air because the local air passage (where the displacement sensor is located) is still 1.5–2 times bigger than that in the long acoustic duct. Furthermore, the pressure signal is measured by P1 which is installed between the displacement sensor and the diaphragm. Therefore, the effect of the displacement sensor on the acoustic field can be neglected. The output of the measurement devices, together with the voltage drop on the load resistor are recorded directly using a computer acquisition card (OMB-DAQTEM 14), with the maximum sampling frequency of 200 kHz. The sampling frequency used in the experiments is 10 kHz. The phase angles between the signals are measured by SR830 DSP lock-in amplifier, with an accuracy of 0.01°. A variable high-power resistor box is built to work as a load resistor. The voltage difference and the current flowing through the load resistor are measured using a standard voltmeter (with a resolution of 0.001 V) and ammeter (with a resolution of 0.01 A). As a result, the electrical power extracted by the load resistor can be deduced. The loudspeaker (alternator) investigated in this paper is B&C 6PS38 woofer manufactured by B&C Speakers. Its specifications and the Thiele/Small parameters are summarized in Table 1. As the measuring range of the laser displacement sensor is ±5 mm, the tested displacement amplitudes were chosen not to exceed 4.5 mm in all experiments. This is below the maximum excursion of the selected alternator (±6 mm). For each case, fine adjustment of the output voltage of the function generator was performed to reach the required displacement amplitude of the alternator’s diaphragm. The signals from the sensors were recorded via the data acquisition card. For cross-referencing purposes, RMS amplitudes of these signals have also been read using a digital multi-meter (TTi 1604; 40,000 count digital multi-meter). Following the linear model described in Section 2, according to Eq. (12) the transduction efficiency is independent of frequency. Furthermore, Eq. (9) shows that the extracted electric power only depends on the volumetric velocity |U1| (=x|n1|) for a given alternator. Therefore, once |U1| is given, the extracted electric power is also independent from frequency. Furthermore, the intended application of the alternator studied in this work was a thermoacoustic generator with the nominal working frequency around 70 Hz (background information can be found in Ref. [10]). Therefore, the experimental evaluation focussed in the range between 40 and 120 Hz with particular attention paid to frequencies around 70 Hz. Furthermore, the objective of the engine design is to extract

as much electrical power as possible. According to Eq. (11), the highest electrical power can be extracted from the alternator when the load resistance is equal to the coil resistance. Therefore, most of the tests focussed on relatively low load resistances.

4. Experimental results and discussion 4.1. Measurement of alternator’s parameters Standard methods [11,12] were followed to measure the parameters of the alternator under test. The first step was to test the resonance frequency of the alternator through measuring the total electrical impedance of the alternator while varying the frequency. The highest impedance appears at the resonance frequency. Two cases were tested: (i) both sides of the alternator exposed to free air; (ii) the back of the alternator is covered with the metal cap as shown in Fig. 2, but the front of the diaphragm is exposed to free air. Fig. 3 shows the measured impedance curves for the two cases. The squares show the impedance of the alternator when it was tested in free air; the resonance frequency is around 62.0 Hz, which is different from its nominal frequency of 75 Hz provided by the manufacturer (see Table 1). The triangles represent the impedance curve when the alternator has the back volume attached, as shown in Fig. 2, whilst free air is only in front of the diaphragm. In this case, the measured resonance frequency is around 103.0 Hz. Evidently the gas spring effect of the back volume increases the resonance frequency. The resistance and inductance of the coil were measured using an RLC meter. They are 5.41 X and 0.479 mH, respectively. The alternator’s moving mass and suspension stiffness were determined by measuring the mechanical resonance frequency as a function of mass. A known mass (12.31 g) was added to the diaphragm, which reduced the measured resonance frequency to 47.6 Hz when both sides of the alternator were exposed to free air. The inferred moving mass is 17 g, and the suspension stiffness is 2621 N/m. The mechanical resistance is determined by measuring the mechanical quality factor, Qm, of the alternator using the flowing equation:

Rm ¼

90

Unit

Nominal [9]

Measured

Deviationa

Resonance frequency Force factor Electric inductance Electric resistance Moving mass Stiffness Mechanical resistance Maximum excursion Effective area

Fs

Hz

75

62.0

Bl Le

N/A mH

10.8 0.6

9.6 0.479

Re

X

5.4

5.4

Mm Km Rm

g mm/N kg/s

14 2778 0.64

17.0 2621 0.96

±0.1 (±0.2%) ±0.3 (±3%) ±0.002 (±4.1%) ±0.1 (±1.8%) ±0.3 (±2%) ±10 (±4%) ±0.02 (±2%)

Xmax

mm

±6

Free air Cover

80

Impedance (Ω)

Symbol

cm

ð18Þ

:

100

Parameter

S

Qm

The force factor Bl was measured in a static experiment. The tested alternator was mounted upright. The laser displacement sensor was used to monitor the position of the diaphragm. A

Table 1 Specifications of the loudspeaker B&C 6PS38 [9].

2

xM m

70 60 50 40 30 20 10 0 10

132

a The estimated uncertainties are based on the differences in the results obtained in different runs.

100

1000

Frequency (Hz) Fig. 3. Impedance curves of the alternator with and without the back cover.

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known mass, m, was added to the diaphragm, and a dc current, I, was passed through the coil to bring the diaphragm back to the equilibrium position. The force factor was inferred as

Bl ¼

mg ; I

ð19Þ

where g is the gravity constant. All experimental runs were repeated several times. The averaged values of measured parameters and the standard deviation are shown in Table 1 as ‘‘measured’’ and ‘‘deviation’’ columns, respectively. As is well known, the value of Bl strongly depends on the location of the coil in the magnet gap [13,14]. It drops dramatically as the coil reaches the maximum excursion (e.g. 50% drop was observed in Ref [13]). The force factor Bl shown in Table 1 is measured according to Eq. (19) when the coil is close to the equilibrium position where the magnet field intensity is the strongest. Therefore, it is the maximum value of Bl which this alternator can achieve. The dependency of Bl on the location in the magnet gap can be measured by different means [13,14]. However, this is not the objective of the current study. Here, the loudspeaker functions as an alternator. Therefore, it is more crucial to study the time-averaged transduction effect of the alternator over an acoustic cycle. In this respect, based on the rig shown in Fig. 2, a measurement was conducted to study the impact of the displacement amplitude on the effective value of Bl in the working range of the displacement, defined as:

Bleff ¼

jV open j xjn1 j

ð20Þ

Here, |Vopen| is the amplitude of the induced voltage measured when the terminals are open, |n1| is the amplitude of the coil displacement. Assuming the coil undergoes a simple harmonic oscillation, the effective value of Bl can be determined by measuring |Vopen| and |n1|. Fig. 4 shows the results obtained using this method. It was measured at three different frequencies. It can be found that there is no dependence of Bleff on the frequency, as expected according to the linear model. It can also be seen that, as the displacement amplitude increases, the effective value of Bl drops only slightly: around 5% when the displacement amplitude increases from 0.25 to 4 mm. This is due to the fact that although Bl decreases significantly as the displacement increases, its effect on the voltage amplitude that the coil can produce is relatively small since the coil undergoes a nearly simple harmonic motion with the coil/diaphragm velocity being the largest at the equilibrium position where the Bl is the largest [14].

4.2. Measurement of the alternator’s acoustic–electric transduction efficiency The second step was to test the acoustic–electric power transduction efficiencies under various conditions using the rig shown in Fig. 2. The acoustic input power to the alternator was controlled by varying the input power to the driving loudspeaker. The frequency was controlled by the function generator. The acoustic pressure amplitudes behind and in the front of the alternator diaphragm were measured by the two pressure sensors. In this way, the time dependent pressure drop was obtained. Simultaneously, the displacement sensor recorded the displacement of the diaphragm, which was converted to the velocity of the diaphragm. Using the signal of the displacement sensor as a reference, the lock-in amplifier measured the phase angles of the p1, p2 and the voltage drop on the load resistor. Combining the pressure drop, velocity of the diaphragm and the phase angle between them, the acoustic power absorbed by the alternator could be calculated using Eq. (5). The electric power extracted by the load resistor could be calculated using Eq. (6). Subsequently, the acoustic–electric transduction efficiency was calculated using Eq. (7). In the experiments discussed below, the alternator is tested with the metal back cover. Figs. 5 and 6 show one sample case of the measurements described above. The load resistance is 5.57 X. Fig. 4 shows the simultaneous pressure oscillations p1 and p2, the displacement of the alternator diaphragm, n1, and the voltage drop on the load resistor. The volumetric velocity (which can be easily converted from the displacement values) is also shown for the convenience of further discussion. In Fig. 5, it can be seen that p2 is almost in phase with the displacement. The measured leading phase angle is 2.4°. This indicates that the back volume does not work as an ideal lumped acoustic compliance. The amplitudes of p1 and p2 are 1164 and 1216 Pa, respectively. However, p1 leads n1 with a phase angle around 48.5°. The voltage drop VL on the load resistor has the amplitude of around 5.72 V, which leads the displacement by almost 90°, and is almost in phase with the volumetric velocity U1. This indicates that the inductance of the coil is very small compared with the resistance. The results shown in Fig. 5 can also be converted into a phasor diagram shown in Fig. 6. It can be seen that, the pressure drop lags volumetric velocity around 29°. According to Eq. (16), in this case, the alternator is off resonance. Using Eqs. (3)–(5), the calculated input acoustic power is 7.25 W, and the electric power extracted by

4 P1 (kPa)

3

10

VL (2V)

2

9 8

Effective Bl (N/A)

P2 (kPa)

f=60 Hz f=75 Hz f=100 Hz

7 6

U1 (0.01 m3/s) Displacement (mm)

1 0

5

-1

4

-2

3

-3

2 -4

1

0

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Displacement amplitude (mm) Fig. 4. Effective force factor Bl measured when the amplitude of displacement varies.

1

2

3

4

5

6

7

8

9

10 11 12 13 14

Time (ms) Fig. 5. A sample measurement for excitation frequency of 70 Hz. Five curves are shown: pressure oscillations behind (P2) and in front (P1) of the alternator diaphragm, displacement of the diaphragm, voltage drop on the load resistor, and calculated volumetric velocity.

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the load resistor is around 2.94 W. Accordingly, the obtained acoustic–electric efficiency is around 40.6%. This case is also shown in Fig. 7. Fig. 7 shows the relationship between the measured acousticto-electric efficiency versus the displacement amplitude for three selected frequencies. The load resistance is set to 5.57 X. The tested displacement amplitude is between 0.25 mm and 4.5 mm with a step of 0.25 mm for the frequencies of 55, 62 and 70 Hz. However, for other frequencies, only two typical displacements: 1 and 2 mm have been tested. It can be found that the displacement amplitude does not affect the conversion efficiency significantly in the tested range. The measured efficiency decreases only slightly when the displacement amplitude increases from 0.25 to 4.5 mm. This is because of the fact that, although the value of Bl drops significantly as the displacement increases, the effective value of Bl in the working range of the displacement amplitude drops only slightly as shown in Fig. 4. The time-averaged electric power that the alternator produces depends on the effective value of Bl. Therefore, the results obtained in Fig. 7 in essence agree with those shown in Fig. 4. This is quite a useful observation from the viewpoint of using audio loudspeakers as alternators in low-cost thermoacoustic electricity generators. This is because a higher displacement and a lower pressure drop are preferred due to the fragile paper cone; unlike for the flexure-bearing-supported linear alternators where a higher pressure drop and a lower displacement condition is preferred. According to the specifications of the sensors and instrumentation described in Section 3, the uncertainty introduced by their resolution is very low. However, it should be noted that the

50

Acoustic-Electric Efficiency (%)

Fig. 6. Phasor diagram related to the oscillations shown in Fig. 5.

experimental realisations have their own ‘‘fluctuations’’ due to the complex coupling between the acoustic, mechanical and electrical systems. To further analyze the overall uncertainty of measurements, all of the cases were measured five times. The mean values are shown as points in Figs. 7–10, while the corresponding standard deviation is in the range of ±1–2%. In Fig. 7, for the three different frequencies, the measured results are relatively close to one another. However, the differences between different frequencies are in the range of uncertainty. It is difficult to conclude weather the frequency affects the measured transduction efficiencies. To further investigate the possible impact of frequency on the alternator’s transduction efficiency, a series of experiments were conducted. The measured results are shown in Fig. 8. Two displacement amplitudes, 1 mm and 2 mm, are shown. The load resistance is kept as 5.57 X. It can be found that as the operating frequency varies, the measured efficiencies fluctuate in the range of 42–45%. It seems as though the measured efficiencies at the high frequencies are slightly higher than those at the low frequencies. Furthermore, the measured efficiencies are slightly higher when the operating frequency is close to the two resonance frequencies of the alternator with and without the cover. It should be noted that the nonlinear effects in the alternator are neglected by the linear model described in Section 2, which shows that the alternator’s

40 35 30 25 20 15 10

Displacement: 2mm

5

Displacement: 1mm

0 40

50

60

70

80

90

100

110

120

Frequency (Hz) Fig. 8. Acoustic-to-electric conversion efficiencies versus frequency for two selected displacement amplitudes; load resistance kept constant as 5.57 X.

50

10

45 10

1

0

40 10

35

55 Hz 62 Hz 66 Hz 70 Hz 80 Hz 95 Hz 100 Hz

30 25 20 15

Power as MSA

Acoustic-Electric efficiency (%)

45

10 10 10 10

-1

-2

-3

-4

-5

10 10

5

10

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Displacement amplitude (mm) Fig. 7. Acoustic-to-electric conversion efficiencies versus displacement amplitude plotted for three values of frequency.

-6

-7

0

100

200

300

400

500

Frequency (Hz) Fig. 9. Power spectrum of the pressure signal (P1) when the operating frequency is 100 Hz.

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Acoustic-Electric efficiency (%)

70 60 50 40 30 Measured: 70 Hz

20

Measured: 62 Hz Measured: 55 Hz

10

Calculation using measured parameters

0 0

5

10

15

20

25

30

35

40

45

50

Load Resistance (Ω) Fig. 10. Acoustic-to-electric conversion efficiencies versus load resistance for three selected frequencies, the displacement kept constant at 3 mm.

efficiency is independent of frequency. However some nonlinear effects are inevitable. For example, the mechanical resistance depends on frequency, because the mechanical windage and spring losses depend on frequency. The particular alternator (loudspeaker) tested here, does not have a hole on the poleplate (or yoke). This means that a volume of air constrained by the dust cap has to be forced out (or sucked in) through the very narrow gaps between the coil and wall of the magnet gap when the cone moves towards (or away from) the magnet. The windage losses caused by this mechanism depend on both the displacement and frequency. As the displacement increases, this windage loss will increase because more air has to be moved in and out. However, it decreases as the frequency increases because the small gap between the coil and wall of magnet will work as a ‘‘clearance seal’’ at a high frequency which prevents the air passing through these gaps. However, it should also be noted that although the measured points fall into a narrow range there are some fluctuations. It is difficult to tell whether these fluctuations are some sort of frequency dependency or just caused by the uncertainty of the measurement itself (which is around ±1–2% as described above). Furthermore, it is still unclear why there might be two possible ‘‘peaks’’ as shown in Fig. 8. Furthermore, a potential impact of harmonics on the tendency shown in Fig. 8 has also been studied. It was found that the power level of the harmonics is trivial compared with the fundamental mode. Fig. 9 shows a typical power spectrum of the measured pressure oscillation obtained from P1 when the operating frequency is 100 Hz. In Fig. 9, the coordinate ‘‘Power as MSA’’ means that the ‘‘Mean Square of Amplitude’’ is used as an indicator of power in the Fast Fourier Transform process. It can be seen that the power level of the second and third harmonics are several orders of magnitude smaller than that of the fundamental mode. Therefore, the impact of harmonics can be neglected. Fig. 10 shows the relationship between the load resistance and measured acoustic–electric efficiencies. Here, the displacement amplitude is kept constant at 3 mm. Three different frequencies: 55, 62 and 70 Hz were tested. Furthermore, based on the measured parameters of the alternator, the transduction efficiency of the alternator can be estimated using Eq. (12). The calculated results are also shown in Fig. 10. It can be seen that as the load resistance increases, the measured efficiencies increase and reach a maximum value; after which they slightly decrease. This tendency agrees well with the calculated results shown in Fig. 10. Furthermore, in terms of absolute values, the experimental results are very close to the calculated ones using the measured parameters. Both the calculation

and experiments show there is an optimal load resistance region which leads to a high transduction efficiency. However, the optimal region of the load resistance is relatively broad, e.g. there is only a 2% change in efficiency when the load resistance changes in the range of 15–40 X. This is an advantage when applying a linear alternator in a practical thermoacoustic generator where the load is usually not fixed all the time. The measured optimal region of load resistance agrees with the calculated one to a reasonable extent. It can also be found from Fig. 10 that the measured efficiencies are very close to one another for the three tested frequencies at a low load resistance, while the discrepancy reaches up to 5–6% when the load resistance is 46.9 X. This dependence of efficiency on frequency is similar with that shown in Fig. 8. However, it is difficult to conduct further quantitative comparisons between the measured results and the calculation based on the results obtained in Fig. 10, because the Thiele/Small parameters used for the calculation were also measured with an appreciable uncertainty as shown in Table 1.

5. Conclusion The work presented in this paper addressed two important issues: firstly, an experimental method was developed to characterise the performance of commercially available loudspeakers for the linear alternator applications in low-cost thermoacoustic electricity generators. Secondly, a commercially available loudspeaker was selected (mainly from the point of view of high Bl, low electrical resistance and low mechanical resistance) to serve as a case study to demonstrate the measurement of the acoustic-to-electrical efficiencies. The experimental results show that the selected candidate loudspeaker (model B&C 3PS38) can reach acoustic–electric transduction efficiency of around 60%, when an optimum value of the load resistance is chosen. Even when the load resistance approaches to the coil resistance, the measured transduction efficiency can reach around 40%. The experimental results also show that the displacement amplitude does not affect the acoustic– electric transduction efficiency significantly. This is important for applying audio loudspeakers to develop low-cost thermoacoustic electricity generators, where the high stroke and low pressure drop across the diaphragm are preferred. Therefore, from the practical performance point of view, it is feasible and possible to utilise the commercially available loudspeakers as alternators for such applications. Furthermore, this work also shows that the measurement methodology developed is useful and reliable for characterising acoustic–electric alternator efficiencies, and that the linear model is still quite useful to describe the behaviour of such alternators. However, to fully understand the experimental results shown in this paper, a nonlinear model should be developed to account for frequency and amplitude dependent parameters. Acknowledgements The authors wish to thank the Engineering and Physical Sciences Research Council UK for supporting this work under grants GR/T04502/01, GR/T04519/01 and EP/E044379/1 (SCORE Project). Xiaoan Mao is acknowledged for his help in measuring Thiele/ Small parameters of the loudspeaker used in this study. References [1] Swift GW. Analysis and performance of a large thermoacoustic engine. J Acoust Soc Am 1992;92:1551–63. [2] Backhaus S, Swift GW. A thermoacoustic-stirling heat engine: detailed study. J Acoust Soc Am 2000;107:3148–66.

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