Electroacoustics and Audio Engineering: Paper ICA2016-613 · A.Y.Abdelwahed(a), A.H.Ibrahim(b),...

Buenos Aires – 5 to 9 September, 2016 Acoustics for the 21st Century…

PROCEEDINGS of the 22nd International Congress on Acoustics

Electroacoustics and Audio Engineering: Paper ICA2016-613

Effects of operation at and off-electrical resonance on the performance indices of linear alternators under

thermoacoustic-power-conversion conditions

A.Y.Abdelwahed(a), A.H.Ibrahim(b), Ehab Abdel-Rahman(c) (a)School of Sciences & Engineering, The American University in Cairo, Egypt,

[email protected] (b)School of Sciences & Engineering, The American University in Cairo. On leave from Mechanical Power

Department, Faculty of Engineering, Cairo University, Giza, Egypt, [email protected] (c)Professor of Physics, Department of Physics, The American University in Cairo, 11835 New

Cairo,Egypt, [email protected]

Abstract

Thermoacoustic power converters consist of thermoacoustic engines that convert thermal energy

into acoustic energy and linear alternators that convert the generated acoustic energy into electric

energy. The conditions required for best acoustic-to-electric power conversion include that linear

alternators operate under mechanical and electrical resonance simultaneously causing the

acoustic impedance of the linear alternator to become purely real. Electrical resonance is

achieved by balancing the linear alternator inductor’s impedance by using a power-factor-

correcting capacitor. However, the exact capacitance value depends on the mechanical stroke,

which in turn depends on the load seen by the linear alternator, including the value of the

capacitance used. Thus, if operation takes place at off-design conditions, the mechanical stroke

in operation and the capacitance used may not lead to electrical resonance. This work

experimentally investigates the linear alternator performance indices, namely the mechanical

stroke, the dynamic pressure at the face of the linear alternator’s piston, the output electric power,

the generated volt, the generated current, the acoustic-to-electric conversion efficiency, the

mechanical-motion loss, the Ohmic loss, and the fluid-seal loss when operating at electrical

resonance and when operating at different levels of off-electrical resonance for two types of loads:

a linear (resistive load) and a non-linear constant-voltage DC electronic load. Increases in the

acoustic to-electric conversion efficiencies of up to 27.8% and 54.7% can take place when

operating at electrical resonance in the linear and non-linear cases, respectively. The effects of

operation at and off-electrical resonance conditions on the harmonic generation and on the

acoustic impedance under linear and non-linear loadings are presented.

Keywords: Thermoacoustic power converter, linear alternator, electrical resonance, acoustic

impedance and power-factor correction

22nd International Congress on Acoustics, ICA 2016 Buenos Aires – 5 to 9 September, 2016

Acoustics for the 21st Century…

2

Effects of operation at and off-electrical resonance on the performance indices of linear alternators under

thermoacoustic-power-conversion conditions

1 Introduction

A thermoacoustic power converter converts thermal energy into acoustic energy. The

thermodynamic cycles employed are either Brayton cycle in standing-wave thermoacoustic

engines or Striling cycles in travelling-wave thermoacoustic engines [1-2]. The generated acoustic

energythen is converted into electrical energy in linear alternators.

Thermoacoustic engines enjoy several advantages: they mainly consist of pipes,

stack/regenerator and heat exchangers with no moving parts, because the oscillations in the

working gas replace the motion of the pistons and eliminate the need for moving seals and

lubrication typically encountered in conventional engines, thus enhancing simplicity, reliability and

durability; they can be driven by a variety of heat energy sources including fuel gas, solar energy

[3], waste heat [4]; they operate without a combustion process and without emissions; they

employ working fluids (like argon or helium and their mixtures) which have no global warming nor

ozone depletion environmental impacts; they demonstrate large conversion efficiency potential,

for example, a first-law thermal-to-acoustic conversion efficiency of 32% and a second-law

conversion efficiency of 49% were reported by [5]; they can be operated in cascade mode

[6]where the rejected heat from one engine is used to drive another engine and they can drive a

thermoacoustic refrigerator or a pulse tube refrigerator to generate cryogenic temperatures [7].

These advantages make the thermoacoustic technology a serious alternative to produce

mechanical or electrical power, cooling effects in a sustainable and environmentally-friendly way.

A recent review on thermoacoustic engines is presented in [8].

The acoustic power generated by the thermoacoustic engine, which is one form of mechanical

work, is applied to the alternator’s piston causing the piston and the permanent magnets attached

to it to oscillate, thus inducing an oscillating magnetic flux. This induces voltage in the stationary

copper coils and causes an electric current to flow into the load and electric power to be delivered

from the linear alternator to the load. These linear alternators operate with no sliding seals, no

lubrication and no wearing parts, thus they enjoy large reliability and durability. In fact, the

calculated (theoretical) mean time between failures per MIL-STD217F (military standard method)

is 129,760 hours. In practice, some units have been in continuous operation for more than eight

years without failure [9].

Consequently, integrating thermoacoustic engines with linear alternators makes a thermoacoustic

power converter with several advantages. Because of these opportunities, recent thermoacoustic

power converters have been reported in [3,10,11].

However, the integration of thermoacoustic engines and linear alternators into thermoacoustic

power converters encounters many challenges in terms of proper matching between the linear

alternator and the thermoacoustic engine on one side and between the linear alternator and the



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load on the other side. The satisfaction of all the matching conditions requires quantitative

understanding of how to control the mechanical stroke to ensure that full output power is delivered

at the full rated mechanical stroke. Delivering full output power at less than the rated mechanical

stroke leads to excessive current. On the other hand, if the full rated power is delivered at more

than the rated mechanical stroke, the thermoacoustic power converter will never be able to

achieve its rated power. Moreover, from the acoustic-to-electric conversion efficiency view-point,

operating at full rated mechanical stroke at less than the rated output power implies having the

full mechanical-motion loss while operating at only a fraction of that power, which will necessarily

lead to a reduced acoustic-to-electric conversion efficiency. Furthermore, the operating conditions

that satisfy the matching requirements must be identified without causing an over-stroke

condition, to avoid damage to the alternator pistons.

In order to fully and deeply understand the matching conditions between the supplied acoustic

power, the linear alternator and the load, the authors built a platform designed specifically to test

linear alternators under different operating conditions in a controlled environment with the

acoustic power supplied in a controlled stable form [12,13]. The setup was used to analyse the

sensitivity of the linear alternator performance indices to ± 10% variations in the operating factors

[14], to investigate the effects of mechanical stroke on the acoustic impedance of linear

alternators[15],to demonstrate the use of a passive non-linear load suitable for stable and

controllable off-grid testing and to compare operation under linear and non-linear loading [16].

In this work, the key performance indices selected to monitor the linear alternator’s performance

are the mechanical stroke, the acoustic-to-electric conversion efficiency, the output electric

power, the output volt, the output current, the resonator’s dynamic pressure, the mechanical-

motion loss, the fluid-seal loss and the Ohmic loss.

The measured signals are the mechanical stroke, the dynamic pressure at the face of the linear

alternator’s piston (hereinafter referred as the resonator’s dynamic pressure), the dynamic

pressure at the back of the linear alternator’s piston (hereinafter referred to as the enclosure’s

pressure), the generated current and the generated voltage, in addition to the dissipated voltage,

current and power in the non-linear load.

The objectives of this work are to 1-analyse the effects of operating at different levels of off-

electrical resonance conditions on the linear alternator performance indices and on the linear

alternator characteristic curves (defined as the power-stroke curves and current and voltage-

stroke curves), 2- to investigate if operation at off-electrical resonance conditions affects the

harmonic content of the main linear alternator signals (mechanical stroke, output volt and output

current) and 3- to estimate the acoustic impedance of the linear alternator at and off-electrical

resonance conditions. All the three objectives are carried-out under linear and non-linear loads.

2 Experimental Setup

The experimental setup was described in [13] and the technical specifications of the linear

alternator and the power-factor-correcting capacitors used were reported in [15]. Only the new

added features are explained in this section.



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Similar to the work presented by the authors in [13-16], the experimental conditions are selected

to match the conditions typically encountered in thermoacoustic power-conversion conditions. A

gas mixture of 60% helium and 40% argon is used because this gas mixture enjoys the least

Prandtl number (and thus the least viscous dissipation losses) amongst all argon/helium gas

mixtures [17]. The operating frequency is selected to match the natural frequency of the linear

alternator at the mean gas pressure used, while accounting for the gas spring effects in the linear

alternator enclosure to make sure that the linear alternator is operated at its mechanical

resonance frequency.

The experimental results are obtained for two different loads: a linear (resistive) load, in which the

power absorbed by the load is proportional to the square of the applied voltage and the dissipated

current and volts in the load are the same as those generated by the linear alternator. The linear

load is a 52.4-Ω resistance preceded with a 77-F power-factor-correcting capacitor, where the

capacitance used is estimated based on the results of [15]. Additionally, a non-linear load is used

which is a constant-voltage DC electronic load, set at 21 VDC and is preceded by an 85-µF power

factor correcting capacitor and a rectifier. This load is the load typically used to integrate the

outputs of several thermoacoustic engines together in a MW-scale thermoacoustic station to avoid

problems arising from different frequencies in different engines. After the outputs of all engines

are combined together in a DC form using this load, the DC outputs are inverted into AC again

for grid-connecting purposes.

The comparison between the two loads is carried-out such that both linear and non-linear loads

receive the same electric output power from the linear alternator when they are operated at

electrical resonance. This leads to operating at an input pressure ratio of 1.37% for the linear load

and at 1.47% for the non-linear load. This ratio is defined as the amplitude of the resonator’s

dynamic pressure to the mean gas pressure.

The case labelled most-inductive case refers to operation without a power-factor-correcting

capacitor and the case labelled most-capacitive case refers to operation with a 204-F capacitor.

The electrical resonance case refers to using a capacitance just enough to balance the linear

alternator’s inductance at the mechanical stroke in operation.

The different linear alternator signals are acquired simultaneously using a data acquisition card

(NI6225). The sampling parameters are selected to ensure sampling of an integer large number

of cycles with fine time and spectral resolutions without aliasing and without significant amplitude

leakage. The sampling rate is 20,400 Samples/s corresponding to sampling 400 samples/cycle

for 500 complete cycles with a total number of samples of 200,000 samples and a total sampling

time of 9.8 seconds when operating at 51 Hz, thus yielding to a quantization resolution of 0.3 mV,

a time resolution of 49 s and a spectral resolution of 0.16 Hz.

3 Results

The performance indices are estimated from the simultaneously-acquired signals, which are the

mechanical stroke, the resonator’s dynamic pressure, the enclosure’s pressure, the output volt

and the output current using the equations presented in [13, 15] and are presented in Figure 1

below. In this figure, as well as in Figure 3, the empty symbols indicate operation at the most-



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inductive case (no power-factor-correcting capacitor), while the left arrow indicates operation at

electrical resonance and the right arrow indicates operation at the most-capacitive case, where

a 204-F capacitance value is used. The results obtained using the linear load and non-linear

loads are presented in Section 3.1 and Section 3.2, respectively.

3.1 Performance of linear alternators under linear loading

The performance indices at and off-electrical resonance conditions are presented in Figure 1. In

comparison with operation at the most-inductive case, operation at electrical resonance

increases the output electric power by 40.0 %, increases the acoustic-to-electric conversion

efficiency by 27.8 %, increases the mechanical stroke by 11.6 % while increasing the

resonator’s dynamic pressure by 24 %. In comparison with operation at the most-capacitive

case, operation at electrical resonance yields increases of 15.1%, 12.9% and 12.4%, in the

same variables, respectively.

Figure 1: Linear alternator’s performance indices at and off-electrical resonance conditions. This data is for a gas mixture made of 60% helium/40% argon, a mean gas

pressure of 20 bar, an input pressure ratio to the linear alternator of 1.37%, an operating frequency of 51 Hz, and a resistive load of 52.4 Ω in series with different

power-factor-correcting capacitors, in the range of 0-204 F.

It can be seen that the introduction of a power-factor-correcting capacitor into the load seen by

the linear alternator significantly affects all of its performance indices. For example, the

mechanical stroke experiences an increase of 56.7% when operated at 10-F capacitance with

respect to operation without a power-factor-correcting capacitor. Because the capacitance value



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required to achieve electrical resonance is itself a function of the mechanical stroke [15], a trial

and error process or a simulation tool is required to identify the capacitance value that leads to

electrical resonance. This increase in the mechanical stroke causes increases in the

mechanical-motion loss, in the generated volt and in the fluid-seal loss and a decrease in the

generated current and in the Ohmic loss. The results show that the increase in the mechanical-

motion loss outweighs the decrease in the Ohmic loss causing the acoustic-to-electric

conversion efficiency to decrease. This loss in efficiency is because of operating at a large

mechanical stroke while generating low output power, which should be identified and avoided.

As the used capacitance value increases from 10 F to 204 F, the mechanical stroke

decreases at a fast rate (about 11m/F) till the point of electrical resonance, and then

decreases with a slower rate beyond this point (about 1.1 m/F). This change in rate is

observed in the rest of the performance indices as well, with the most significant change in rate

occurring in the fluid-seal loss. Spectral analysis of the acquired mechanical stroke, current and

voltage signals shows pure sinusoidal behavior with no significant harmonics, in contrast with

the non-linear loading case presented in Section 3.2.

Additionally, the presented work examines the relationships between the linear alternator’s

characteristic curves at and off-electrical resonance conditions as shown in Figure 2. The figure

presents the relationships between the output volt, output current and output power versus the

mechanical stroke. The first two curves determine the proportionality constants between the

output volt and current and the mechanical stroke, which are critical to properly control the linear

alternator to operate at its rated stroke when delivering its rated power and to allow monitoring

the output volt as an indicator of the mechanical stroke to avoid over-stroking under a given set

of conditions. The latter curve is of special interest in determining the operating point of the

system made of a linear alternator and an electric load, where this point is defined as the

intersection point between the curve of generated power versus stroke curve and the curve of

the dissipated power versus stroke [16].

Figure 2: Characteristic curves of the generated volt (RMS), current (RMS) and power versus the mechanical stroke when operating at and off-electrical resonance for three

different cases: operation at the most inductive case (empty rectangular symbols), operation at the electrical-resonance case (solid rectangular symbols) and operation at the

most capacitive case (triangular symbols).

The results show that the volt and current characteristic curves are linear in the case of linear

loading and that the proportionality constant depends on the capacitance value used. .



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3.2 Performance of linear alternators under non-linear loading

The performance indices at and off-electrical resonance conditions under the non-linear loading

case are presented in Figure 3 below.

Figure 3: Linear alternator’s performance indices at and off-electrical resonance conditions. This data is for a gas mixture made of 60% helium/40%

argon, a mean gas pressure of 20 bar, an input pressure ratio to the linear alternator of 1.47% and an operating frequency of 51 Hz. The load consists of a

85-µF power-factor-correcting capacitor followed by a rectifier and then followed by a constant-voltage DC electronic load set at 21 VDC.

It can be seen that with respect to the two most-inductive and most-capacitive cases

considered, the case of electrical resonance enjoys a lower mechanical stroke (and thus lower

mechanical-motion loss) with a larger output power. The combination of a large output power

and a low mechanical-motion loss gives rise to a large acoustic-to-electric conversion efficiency.

Similarly, in comparison with operation at the most-inductive case, operation at electrical

resonance increases the output electric power by 61.9 % while it simultaneously decreases the

mechanical stroke by 11.2% (and thus decreases the mechanical-motion loss by 21.3%). The

combined effects of generating more output power while operating at a lower mechanical stroke

give rise to a significant increase in the acoustic-to-electric conversion efficiency of 54.7%. The

resonator’s dynamic pressure is observed to increase by 44.8%.

Spectral analysis of the main linear alternator signals show that the use of non-linear loading

results in generation of harmonics in the current and voltage signals. These harmonics are



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generated only at frequencies that are odd integer multiples of the fundamental frequency. The

results show that within the range considered they do not propagate into the mechanical stroke.

The results also show that while the ratio of the first harmonic (occurring at 153 Hz) to the

fundamental current signal (occurring at 51 Hz) is 15% at electrical resonance, this ratio is 18%

at the most-inductive case and 16.2% at the most-capacitive case. Similarly, the ratio of the

second harmonics (occurring at 255 Hz) to the fundamental current signal is 5% at electrical

resonance, is 6.5% at the most-inductive case and 5.7% at the most-capacitive case. The ratios

obtained when analyzing the voltage signals reflect the same trend as in the current signal. These

observations suggest that operation off-electrical resonance does not have a significant impact

on the harmonic generation in current and voltage signals.

The characteristic curves of the linear alternator under the non-linear loading are presented in

Figure 4. It can be seen that, in contrast with the linear loading at electrical resonance, the

proportionality constant between the generated AC voltage and the mechanical stroke

increases (24.3 versus 18.2 V/mm) as well as the overall proportionality constant between the

generated AC power and the mechanical stroke in the range considered (26.9 versus 10.4

W/mm) because of the reported increase in the acoustic-to-electric conversion efficiency.

Figure 4: Characteristic volt, current and power curves versus the mechanical stroke when operating at and off-electrical resonance for three different cases: operation at the most-inductive case (empty circular symbols), operation at the electrical-resonance case (solid circular symbols) and operation at the most-capacitive case (star symbols). The top row

presents the variables measured at the output of the linear alternator (LA) and the bottom row presents the variables measured at the load.

3.2.1 Acoustic impedance at linear and non-linear load

The acoustic impedance of the linear alternator affects the acoustic coupling between the linear

alternator and the thermoacoustic engine on one side and the acoustic coupling between the

linear alternator and the electric load on another side and thus it affects the electro-acoustic

conversion efficiency and the electric power delivered to the load. Figure 5 presents

measurements of the acoustic impedances for the linear and non-linear loads at and off-electrical

resonance conditions, estimated using the equations presented in [13]. For the sake of validation,

the figure also presents a comparison between the measured and calculated acoustic impedance



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amplitudes in the linear load where the calculations are based on [15,18]. The comparison shows

reasonable agreement between measurements and calculations.

Figure 5: Linear alternator’s acoustic impedance amplitude under different tuning capacitor

values (0-204 F) for the linear load (rectangular symbols are measurements and the solid curve is calculated based on [15, 18]) and for the non-linear load (circular symbols). The

vertical arrows point to operation at electrical resonance for both loads.

4 Summary and Conclusions

The performance indicators of the linear alternator are measured at and off-electrical resonance

conditions using linear and non-linear loads. Significant increases in the output electric power and

the acoustic-to-electric conversion efficiency take place at electrical resonance in both cases. The

effects of the changes that occur in the mechanical stroke and the associated mechanical-motion

loss are empathised.

Spectral analysis of the linear alternator output current and voltage signals under non-linear

loading shows that operation at off-electrical resonance in the range considered does not have a

significant impact on the harmonic generation and that the resulting harmonics in the current and

voltage signals do not propagate into the mechanical stroke.

With respect to operation under linear loading, the use of this non-linear load at electric resonance

gives rise to a lower mechanical stroke with a larger output power resulting in an increase in the

electro-acoustic conversion efficiency.

Acknowledgments

This publication has been produced with the financial assistance of the European Union. The

contents of this document are the sole responsibility of the authors and can under no

circumstances be regarded as reflecting the position of the European Union. The authors are

thankful for Eng. Amr Sayed Taha for his help on the electric connections used in this work.

References

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[13] Ibrahim, A. H.; Abdelwahed, A. Y.; Abdou, M.; Abdel-Rahman, E. Real-time measurements of linear alternator performance indices under thermoacoustic-power-conversion conditions, Proceedings of the 23rd International Congress on Sound and Vibration, Athens, Greece, 10–14 July, 2016.

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[16] Abdelwahed, A. Y.; Ibrahim, A. H.; Abdel-Rahman, E. Performance of linear alternators using passive linear and non-linear loads under thermoacoustic-power-conversion conditions, Proceedings of the 23rd International Congress on Sound and Vibration, Athens, Greece, 10–14 July, 2016.

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