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Original Article
Clearance seal compressors with linear
motor drives. Part 2: Experimental
evaluation of an oil-free compressor
Kun Liang, Mike Dadd and Paul Bailey
Abstract
Oil-free compressors are required for various applications, including cryocoolers and refrigeration systems that use
compact evaporators. A novel low-cost moving magnet oil-free linear compressor is described here, and follows a
companion paper on the linear motor system. The experimental apparatus to evaluate the thermodynamic performance
of the linear compressor is comparatively straightforward, but the data analysis is involved, since key measurements have
to be corrected for the phase response of the transducers. Harmonic fitting, using fast Fourier transform with zero-
padding and minimisation algorithms, have been used for the voltage, current and displacement signals. This enables
corrections to be made for the phase lag in the voltage, current and displacement measurements. The experiments using
nitrogen, with a constant pressure ratio, show that the linear motor system under resonance achieves a high overall
efficiency. For the design point, the motor efficiency is 74%, while for a pressure ratio of 3.0, the average overall adiabatic
efficiency is 54% and the isothermal efficiency is 46%, all of which are within a reasonable range for a small compressor.
Keywords
Oil-free compressor, linear motor, experimental evaluation, harmonic fitting, fast Fourier transform, phase lag,
resonance, overall efficiency
Date received: 19 July 2012; accepted: 13 December 2012
Introduction
The companion paper has provided a literature review
and a description of the moving magnet compressor
design, as well as an analysis of the component per-
formance. The compressor has been designed for a
refrigeration system and the test rig was established
for future tests with refrigerant. To provide baseline
experimental data tests were carried out using
nitrogen.
Experimental apparatus
Figure 1 presents the linear compressor test system
that is also suitable for use with refrigerants. The
hot and pressured gas from compression in the cylin-
der is discharged into a heat exchanger (condenser).
A needle valve controls the pressure ratio, and this is
followed by another heat exchanger with a heater that
would act as an evaporator. This system comprises
the main flow loop. Meanwhile, the bleed control
valve controls pressure in the compressor body so as
to maintain the correct mean piston position.
Pressure transducers are used to measure the com-
pressor discharge pressure, evaporator inlet pressure,
compressor suction pressure and compressor body
pressure, while thermocouples are employed to meas-
ure temperatures including the compressor discharge,
heat exchanger outlet, evaporator inlet, compressor
suction, evaporator outlet, motor coil and compressor
body. Two mass flow meters are used to measure the
main flow and bleed flow.
The control system and associated instrumentation
for the linear compressor experiments is shown in
Figure 2, which shows the power flow in bold lines,
together with the instruments to control and measure
the compressor performance.
The drive frequency is adjusted by the signal gen-
erator while the amplitude control is used to set the
piston stroke. A power amplifier finally amplifies the
analogue signal in order to drive the linear compres-
sor. A capacitance box is employed to permit adjust-
ment of the capacitance in response to different
resonant frequencies – this power factor correction
reduces the voltage requirement for the amplifier.
Department of Engineering Science, University of Oxford, Oxford, UK
Corresponding author:
Kun Liang, Department of Engineering Science, Parks Road, University
of Oxford, Oxford OX1 3PJ, UK.
Email: kun.liang@eng.ox.ac.uk
Proc IMechE Part A:
J Power and Energy
227(3) 252–260
!IMechE 2013
Reprints and permissions:
sagepub.co.uk/journalsPermissions.nav
DOI: 10.1177/0957650913475620
pia.sagepub.com
The power input, overall root mean square (RMS)
current and voltage can be read from the power
analyser.
For the measurements, two NI 6251 data acquisi-
tion (DAQ) cards are used for both the high-speed
DAQ (HDAQ) and the low-speed DAQ (LDAQ) sys-
tems. The HDAQ instruments include four pressure
transducers, two displacement transducers (linear
variable differential transformers (LVDTs)), a voltage
sensor and two channel current transducers.
Signal
Generator Amplitude
Control
Power
Amplifier
Panic Box
Voltage
Sensor
Power
Analyser
Current
Transducer
Capacitance
Box
Compressor 1 Compressor 2
Channel 1 Channel 2
In In
Out Out
Top Coils
Bottom Coils
Top Coils
Bottom Coils
LVDT 1 LVDT 2
0.1
W
0.1
W
0.1
W
0.1
W
Figure 2. Experimental apparatus (power supply and instruments) for the linear compressor, including the displacement transducers
(LVDT).
LVDT: linear variable differential transformer.
PT
TC
PT TC
PT
PT
TC
TC
TC
TC
Compressor 1
Discharge
Compressor 2
Discharge
SuctionSuction
BodyBody
Bleed Flow
Main Flow
Suction
Suction Suction
Discharge Discharge
Heat
Exchanger
(Condenser)
Heat
Exchanger
(Evaporator)
Main Mass
Flow Meter
Bleed Mass
Flow Meter
Bleed
Valve
Main Flow Valve
Figure 1. Linear compressor test loop.
PT: pressure transducer; TC: thermocouple.
Liang et al. 253
The LDAQ instruments are four pressure transducers,
eight thermocouples and two mass flow meters.
A LabVIEW program was written as an interface to
collect, display and save the data from both the
HDAQ and LDAQ. Some measurements (such as
from the power meter) had to be recorded manually.
Data analysis
A MATLAB code was written specifically to analyse
the data from both HDAQ and LDAQ. Figure 3 illus-
trates the process of data analysis. One of the chal-
lenges with the current, voltage and displacement
signals is the phase lag introduced by these instru-
ments. In order to calculate the shaft power and
power input accurately, the phase lag characteristics
need to be quantified so that these signals can be
reconstructed. Based on fast Fourier transform
(FFT) with zero-padding and minimisation, harmonic
analysis was applied to these current, voltage and dis-
placement signals. Shaft power can then be derived
from the calibration map of the motor force and the
piston velocity (derived from the displacement). The
motor efficiency, volumetric and overall efficiency are
then calculated to evaluate the performance of the
linear compressor system.
Phase lag characteristics of the LVDT
The measurements of the motor position are used to
calculate the volume in the compressor and hence the
PV loop, and for the calculation of motor force and
power. It is therefore important that they are accurate
in terms of stroke and phase. The LVDT phase dis-
tortion was measured by applying a sine wave with a
DC bias to the inputs (the demodulator essentially
rectifies the input according to the modulating signal
and therefore purely positive signals pass through
unaffected). The overall setup for LVDT phase lag
test is shown in Figure 4. The input connection
(with cable) of the LVDT was connected from Pin 1
to the function generator and Pin 2 was connected to
ground. Therefore, the primary coil in the LVDT will
induce the voltage difference between two secondaries
which was recorded by the HDAQ. A phase meter
was used to measure the phase shift of the signal
(Signal A) from the LVDT compared to the original
signal (Signal B) from the function generator.
The phase lag against frequency within a range of
30–350 Hz is given in Figure 5. The two methods
(measurement and MATLAB calculation using the
HDAQ data) give almost the same results, indicating
an apparent phase lag of 4.8for a frequency of
50 Hz. The phase lag tests were also carried out for
the current transducer and voltage sensor.
Harmonic fitting
A crucial part of the data analysis is the harmonic
fitting which is used to reconstruct the voltage, current
and displacement signals. Figure 6 is a flowchart for
the signal reconstruction in MATLAB.
The FFT with zero-padding gave the fundamental
frequency, amplitude and phase of each harmonic
term in the signal. The fundamental frequency from
the calculation (as expected) was exactly the same as
the drive frequency. However, the other parameters
(amplitude and phase) of each harmonic term from
the FFT calculation were not accurate enough due to
spectral leakage and discretisation effects, but they
can be used as initial values for minimisation in the
harmonic fitting procedure.
1
The next step in the analysis is to minimise the
RMS error between the original signal and its recon-
struction using a pre-determined number of harmonic
terms with uncertain values of amplitude and phase.
The accuracy of the minimisation algorithm depends
on the number of harmonic terms extracted from
the signal.
2
Figure 3. Overview of the experimental data analysis
strategy for the motor/compressor system.
HDAQ: high-speed data acquisition; LDAQ: low-speed data
acquisition; FFT: fast Fourier transform.
Phase Meter
HDAQ
LVDT Connector
LVDT
B
Channel 4
Channel 8
A
1
23456 7 89
Function
Generator Signal A
Signal B
Figure 4. Phase lag test setup for the LVDT.
LVDT: linear variable differential transformer; HDAQ:
high-speed data acquisition.
254 Proc IMechE Part A: J Power and Energy 227(3)
For instance, take the measured displacement
signal (D
m
). The harmonic form of the signal with
six terms is given as
Dm¼a0þX
6
n¼1
ðancosðn!0tþ1nÞÞ ð1Þ
where a0is the DC offset, nis the order of the har-
monic term, anis the amplitude of term n,!0is the
fundamental frequency and 1nis the optimised har-
monically determined phase of term n.
By adding the phase lag characteristic (vs. fre-
quency) into each harmonic term of the signal, the
true signal for displacement (D
t
) can be written as
Dt¼a0þX
6
n¼1
ðancosðn!0tþ1n’fÞÞ ð2Þ
where ’fis the phase lag in respect of frequency fof
each harmonic.
The same process for signal reconstruction was
also applied to the voltage and current signals.
Take one set of test data as an example (with a
stroke of 13 mm and pressure ratio of 3.36). Using a
window function (Hanning) and FFT with zero-
padding, the spectral distribution is as given in
Figure 7. Several frequency components existed in dis-
placement signal. The DC offset value is the FFT
amplitude when the frequency is zero.
A comparison of the FFT results (with zero-pad-
ding) and the harmonic fitting results for the displace-
ment signal of Figure 7 (at 36 Hz) is shown in Table 1.
The fundamental frequency calculated is 35.996 Hz.
The FFT with zero-padding produces almost the
same amplitude for each harmonic term as the har-
monic fitting. However, the phase values are not
accurate, and this can be seen in Figure 8. With six
harmonic terms, the RMS error via harmonic fitting is
0.0065 while seven terms makes it 0.0058. The differ-
ence is not significant, but seven terms takes twice the
computation time.
Comparison between the two reconstructed dis-
placement signals shows that the harmonic signal is
very well fitted to the original data as the RMS error
is over 30 times lower with the harmonic fitting than
with the FFT. Figure 8 gives the error of the two
reconstructed current signals from the original. This
verifies that the harmonic fitting method produces
high accuracy reconstructed signals.
A high accuracy is important since the motor force
depends on the displacement and current, and the
power is calculated by integrating the instantaneous
product of the force and the velocity (which is
obtained by differentiation of the displacement).
This is discussed further in the section of results.
Figure 9 shows the improvement of accuracy in the
shaft power calculation using reconstructed displace-
ment and current signals (that have been corrected for
the phase response of the transducers) compared to
the original signals for a pressure ratio of 3.0. The
average error for the shaft power calculation using
the original signals compared with reconstructed sig-
nals is approximately 4.0%.
It is noted that the effect of phase errors on the
calculated shaft power is dependent on the phase
angle ’between the velocity vand motor force F.
For waveforms dominated by the first harmonic
_
Wshaft cosð’Þ:maxFmax =2ð3Þ
Hence the error is given by
dð_
WshaftÞ
d’sinð’Þ:max :Fmax=2ð4Þ
If the compressor is run at resonance then the
phase angle between velocity and motor force is 0,
and sin(’)is 0. The error of 4% is relatively low
because the compressor was run fairly close
Figure 5. Phase lag characteristic of the LVDT.
LVDT: linear variable differential transformer.
Figure 6. Overview of the signal reconstruction.
FFT: fast Fourier transform.
Liang et al. 255
to resonance. If it was being run some way from res-
onance the error would be expected to be significantly
higher.
Compressor evaluation using nitrogen
Experimental method
To investigate the performance of the linear compres-
sor, it was operated under resonance with different
strokes at various constant pressure ratios using gas.
The bleed valve and main valve were adjusted to keep
the pressure ratio constant (observed from the
LabVIEW interface) and the mean position of the
piston (observed from the oscilloscope) constant.
The resonant frequency is measured by changing the
drive frequency of the function generator and chan-
ging the amplifier gain to keep the stroke constant –
resonance was assumed to be the frequency at which
the power is a minimum for the specified stroke.
Operation of the compressor under off-resonance
was also carried out for comparison.
The specific test conditions using nitrogen are
shown in Table 2. Table 3 gives the design specifica-
tions of the compressor.
3
Prior to operation, the com-
pressor is filled with gas to a ‘fill pressure’, and this
defines the nominal mean pressure in the operating
system. The two compressor halves are nominally
identical, but due to the varying DC offset of the
piston and for reasons of safety (not hitting the cylin-
der head), the maximum stroke used was limited to
13 mm. The design point for the compressor is 200 W
of power input at 50 Hz (pressure ratio of 3 and fill
pressure of 7 bar, absolute).
Results
The resonant frequency of the moving assembly is
determined by the moving mass, and the combined
stiffness of the suspension (flexure) spring and the
gas spring.
4
Figure 10 shows the measured resonant
frequency changing with stroke and pressure ratio.
It can be seen that a higher pressure ratio produces
a higher resonant frequency due to larger induced gas
spring stiffness. It is also noticeable that for a set pres-
sure ratio the resonant frequency decreases as the
stroke increases. For a variation in stroke from 8
to 13 mm, the resonant frequency varies by 10%.
For a fixed pressure ratio, increasing the stroke means
that the valves will be open for a greater fraction of
the cycle, so that the ‘effective stiffness’ associated
with compressing the gas is reduced.
The relation between pressure ratio and mass
flow (per cycle) for five different strokes is shown
Figure 7. Spectrum of the displacement signal from a fast Fourier transform with zero-padding.
Table 1. Comparison of the FFT results (with zero-padding)
and the harmonic fitting of some typical displacement data with
a drive frequency of 36 Hz.
Harmonic
term
FFT with zero-padding Harmonic fitting
Amplitude
(mm)
Phase
(rad)
Amplitude
(mm)
Phase
(rad)
1 6.3856 1.693 6.3886 1.6478
2 0.4307 1.6054 0.4311 1.5139
3 0.0312 0.5541 0.0313 0.4142
4 0.021 2.3366 0.021 2.1574
5 0.0129 1.9025 0.0129 1.6758
6 0.0093 1.1367 0.0093 0.8663
RMS error 0.2062 0.0065
7 0.0042 1.1367 0.0041 0.2245
RMS error 0.2042 0.0058
DC offset 0.331 0.3318
FFT: fast Fourier transform; RMS: root mean square.
256 Proc IMechE Part A: J Power and Energy 227(3)
in Figure 11. For a pressure ratio of 3.0, the mass flow
is 0.80 g/s with a 13 mm stroke. An interesting aspect
of the results shown in Figure 11 is that the general
form of the curves is very consistent suggesting that
the results could be expressed by a single curve, the
variation with stroke being accounted for by a single
mass delivered term Figure 12 shows how the mass
flow rate increases with stroke for different pressure
ratios and confirms that the mass flow is proportional
to stroke, as one would expect.
Power input (Pinput) is calculated using the current
and voltage signals after their reconstruction through
harmonic fitting and the phase lag correction.
Figure 13 shows the power supplied to the compressor
when operated at four different pressure ratios. It can
be seen that the power varies fairly linearly with
Figure 8. Comparison between the errors of the two reconstructed signals.
FFT: fast Fourier transform.
Figure 10. Resonant frequency variation with stroke at
different pressure ratios.
Figure 9. Comparison of the shaft power calculation using
the original and the reconstructed signals.
Table 3. Design specifications for each compressor half.
Total mass of magnet/piston (kg) 0.66
Piston diameter (mm) 18.99
Total series
a
resistance of coils ()14
Peak shaft force (N) 84
Peak current at peak force (A) 1.29
Flexure stiffness (total) (kN/m) 17
Maximum stroke (mm) 14
Note:
a
The four motor coils were connected with two coils in series
then each pair in parallel to give a resistance of 3.5 for each
compressor half, as seen in Figure 2.
Table 2. Experimental conditions.
Gas Nitrogen
Pressure ratio 1.5, 2.0, 2.5, 3.0, 3.5
Stroke (mm) 6, 8, 10, 11, 12, 13
Fill pressure, absolute (bar) 7, 8
HDAQ sampling rate (Hz) 5000
HDAQ sampling number 5000
LDAQ sampling rate (Hz) 1000
HDAQ: high-speed data acquisition; LDAQ: low-speed data acquisition.
Liang et al. 257
Figure 11. Pressure ratio against mass flow per cycle of nitrogen for different strokes with a fill pressure of 7bar, absolute.
Figure 13. Power input against stroke with four different
pressure ratios.
Figure 12. Mass flow of nitrogen against stroke for four dif-
ferent pressure ratios with a fill pressure of 7 bar, absolute.
Figure 14. Electrical, isentropic and adiabatic efficiencies
plotted against stroke for different pressure ratios
(2.0 <PR <3.5).
Figure 15. Theoretical volumetric efficiency of the linear
compressor (PR ¼3.0) with different values of the polytropic
index.
CL: clearance.
258 Proc IMechE Part A: J Power and Energy 227(3)
stroke and the gradient increases noticeably with an
increasing pressure ratio.
For an adiabatic process with an ideal gas, the the-
oretical power can be rewritten as a function of the
suction temperature T1(typically 22C in operation),
mass flow rate _
mand pressure ratio
5
_
Wadiabatic ¼
1_
mRT1
P2
P1
1
1
"#ð5Þ
In contrast, for an isothermal process, the theoret-
ical power is
_
Wisothermal ¼_
mRT1ln P2
P1
ð6Þ
Given that the compressor operates at a relatively
high speed, and that the cylinder is not attached to a
heat sink, the compressor tends to operate closer to
the adiabatic than to the isothermal case.
Shaft power is the experimentally determined
power required to run a compressor and is calculated
from the measured current and displacement, ignor-
ing all the frictional losses of the compressor.
Let the shaft force Fbe a function of current Iand
armature position x
F¼BðxÞIð7Þ
which was calibrated and given in Part I of this article
where BðxÞis the magnetic density when moving
assembly is at the position of x.
Therefore, the shaft power can be written as
_
Wshaft ¼1
TZ
T
0
F_
xdtð8Þ
where _
xis the velocity of shaft which is the derivative
of the armature position xmeasured by the LVDT,
and Tis the period of one oscillation.
In order to characterise the volumetric, energetic
and thermal efficiencies of the linear compressor,
the following efficiency definitions are used.
Overall efficiencies are defined as
adiabatic ¼
_
Wadiabatic
Pinput
ð9Þ
isothermal ¼
_
Wisothermal
Pinput
ð10Þ
Electrical efficiency (or motor efficiency) is given as
electrical ¼
_
Wshaft
Pinput
ð11Þ
The efficiencies for the compressor operated at a fill
pressure of 7 bar are shown in Figure 14 along with
the power input (Figure 13). The motor efficiency
trend shows that the motor efficiency is reduced as
the stroke increases. This is due to the conductor
losses (Isquared Rlosses) increasing more rapidly
than the shaft power as the drive current increases.
For the same stroke the higher pressure ratios give
higher motor efficiencies primarily because the oper-
ating frequency is higher (as determined by reson-
ance). For the design point (stroke of 13 mm and
pressure ratio of 3.0), the electrical efficiency is
about 74% which agrees closely with the prediction
of motor performance in Part I of this article.
However, since the additional eddy current losses
can be reduced, the motor efficiency with a revised
design will be higher (approximately 86%).
Overall, the adiabatic and isothermal efficiencies
decrease with increasing pressure ratio, and show
the same trend against stroke (with a smaller value
for the isothermal efficiency than the adiabatic effi-
ciency). For the pressure ratio of 3, the average adia-
batic efficiency is about 54% while the isothermal
efficiency is about 46%.
Considering the losses associated with seal leakage,
DC offset and heat transfer, these two efficiencies are
plausible values of compressor performance. However,
in practical use, the compressor should be operated at
its best efficiency with sufficient shaft power output.
In contrast to operation at resonance, a series of
experiments with the compressor running off-reso-
nance were carried out. Consider a 12 mm stroke
and pressure ratio of 2.0 for instance, when operated
at resonance (33.5 Hz), the motor efficiency, overall
adiabatic and isothermal efficiency are 72.8%,
56.5% and 51.1%, respectively, while if operated
off-resonance (34.5 Hz), they are 70.1%, 53.2% and
48.2%, respectively.
To determine how the theoretical mass flow rate _
m
depends on compressor geometry and its operation,
the theoretical volumetric efficiency is given by
V¼1CL P2
P1
1
n
1
"# ð12Þ
Figure 16. Comparison between theoretical and
experimental volumetric efficiency plotted against stroke
for a PR of 3.0 and a polytropic index of 1.3.
Liang et al. 259
where CL is the clearance (the ratio of clearance
volume to swept volume) and nis the polytropic
index. The ratio of clearance volume to swept
volume is calculated from the piston displacement
measurements.
The actual mass flow is further limited by the
imperfect nature of the valves, as well as wall friction,
seal leakage and other frictional losses. The experi-
mental volumetric efficiency, V, e, can be defined as
V,e¼
_
mRT1
P1
SAfð13Þ
where Sis the stroke of piston, Ais the area of piston
and fis the operation frequency.
Figure 15 shows the theoretical volumetric effi-
ciency against polytropic index (n) for a stroke of
13 mm and pressure ratio of 3.0. It rises when pressure
ratio increases. The clearances for comparison are 5%
and 10%, respectively. Figure 16 compared experi-
mental volumetric efficiency to theoretical values for
the polytropic index (n) of 1.3.
Additional experiments were carried out to inves-
tigate the compressor performance with a fill pressure
of 8 bar compared to 7 bar. For pressure ratio of 2.0
and 3.0, the fill pressure of 8 bar results in a lower
motor efficiency than 7 bar (the designed operation
condition) as the current in each coil is higher which
reaches saturation point of the magnet.
Conclusions
A test rig has been built to evaluate the performance
of an oil-free clearance-seal moving magnet linear
compressor. The key step in the data analysis is the
reconstruction of the current, voltage and displace-
ment signals which all have phase lag errors. The
phase lag response for the range of operating frequen-
cies was investigated separately for three instruments.
The measured value from a phase meter agreed well
with the MATLAB calculations. Harmonic fitting has
been employed to reconstruct the signals using a mini-
misation algorithm, with the initial estimates based on
the results of an FFT with zero-padding. Unlike the
FFT results, there is negligible difference between
the original signal and its reconstruction when using
the harmonic fitting method.
Using nitrogen, a series of compressor experiments
has been carried out at constant pressure ratios under
resonance. The experimental results show the following.
1. Oil-free linear compressors are capable of a good
performance, and there is scope for significant
improvement over the performance reported here
by improvements to the motor design.
2. The variation of efficiency with stroke and pres-
sure ratio shows that the efficiency is maintained
over a fairly wide range of operating parameters.
This suggests that systems using this type of com-
pressor could have good part load efficiency.
3. For the design point of the compressor, the elec-
trical efficiency is about 74% and this agrees well
with the prediction of motor performance in Part
I. Additional tests have verified that operation
under resonance produces a higher motor
efficiency.
4. When the pressure ratio is 3.0, the average overall
adiabatic and isothermal efficiencies are about
54% and 46%, respectively, which is reasonable
and within a plausible range for a small compres-
sor (less than 70%).
5. A small difference in clearance volume between
the two compressor halves arises from a different
DC offset during operation; this results in differ-
ences in the theoretical volumetric efficiency.
6. Higher pressure ratios cause a higher resonant fre-
quency due to the larger induced gas spring stiff-
ness. The resonant frequency decreases with
stroke.
7. A fill pressure of 8 bar results in a lower motor
efficiency than at 7 bar, as higher current reaches
the saturation point of the magnet (leading to
more copper losses).
Funding
Construction of the linear compressor was funded by The
Engineering and Physical Sciences Research Council
(EPSRC).
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