INSTITUTE OF PHYSICS PUBLISHING
MEASUREMENT SCIENCE AND TECHNOLOGY
doi:10.1088/0957-0233/17/7/049
Meas. Sci. Technol. 17 (2006) 2027–2034
A cylindrical capacitive sensor (CCS) for
both radial and axial motion
measurements
Hyeong-Joon Ahn
Department of Mechanical Engineering, Soongsil University, 511 Sangdo-dong,
Dongjak-Gu, Seoul 156-743, Korea
E-mail: ahj123@ssu.ac.kr
Received 3 January 2006, in final form 12 April 2006
Published 21 June 2006
Online at stacks.iop.org/MST/17/2027
Abstract
Conventional cylindrical capacitive sensors (CCS) are used only to measure
the radial error motion of rotating machinery. However, axial error motion is
also very important in rotating machines and it is very difficult to measure
the axial motion due to the limited space. This paper presents a new CCS
for both radial and axial motion measurements. The idea behind the new
sensor is that the unused axial area of the CCS is exposed to measure the
axial motion of a target. A theoretical model of the proposed CCS was
derived. Based on the derived theoretical model, compensation methods to
decouple the radial and axial motion measurements were proposed. In
addition, an error analysis of the CCS was performed and a design rule was
developed to guarantee the same accuracy in measuring both radial and axial
motion measurements. Finally, a test rig and electronics for the proposed
CCS were built and the effectiveness of the proposed CCS was verified
through experiments and simulations. The developed CCS is expected not
only to reduce the system complexity but also to afford a good balance
between the radial and axial motion measurement accuracies.
Keywords: capacitive sensor, displacement measurement, rotating machinery
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Smart sensor systems, particularly position measurement
systems, are expected to play a significant role in the
high-precision intelligent manufacturing system.
For
example, a feedback control is inevitable to ensure
sufficient repeatability for the high-resolution positioning of
semiconductor manufacturing systems [1]. In addition, usage
of active bearings like active magnetic bearings is being
increased to achieve higher precision and productivity [2].
In these systems, the position measurement systems not only
affect the system performance, but also the system size and
complexity.
Probe-type displacement sensors are highly sensitive to
the surface quality of a target and these sensors require
an additional algorithm to detect and compensate for the
unnecessary signal induced by geometric errors [3, 4]. To
0957-0233/06/072027+08$30.00
analyse an air-bearing spindle error motion, a cylindrical
capacitive sensor (CCS) was introduced by Chapman as an
alternative to the probe-type sensors [5]. It was verified that
the CCS showed much better performance in rejecting the
geometric errors of a rotor than probe-type sensors [6], and
it can minimize the effects of geometric errors by adjusting
the sensor angular size [7]. Also, an extended analysis of a
multi-segment CCS was performed [8] and a CCS was applied
to the rotor motion measurement of a small air conditioning
compressor [9].
The CCS is used only for measuring the radial error
motion of rotating machinery. However, axial error motion
is also very important in rotating machines such as hard disc
and machine tools and it is very difficult to measure the axial
motion due to the limited space. Moreover, it is very hard to
balance the accuracies of both the radial and axial error motion
measurements.
© 2006 IOP Publishing Ltd Printed in the UK
2027
H-J Ahn
New CCS
Unused axial area
Y
Radial
motion
Z
X
Axial
motion
t
δa
(a)
(a)
(b)
Figure 2. The main idea behind the novel CCS: (a) cross section of
a CCS and (b) the proposed CCS.
Figure 1. Types of CCS: (a) four-segment CCS and
(b) eight-segment CCS.
This paper presents a new CCS for both radial and axial
motion measurements. There is no hardware difference of
the new sensor from a conventional CCS, and the small axial
area of the CCS is used to measure the axial motion of a target.
A theoretical model of the new CCS was derived, which shows
that there is a significant effect of the target radial motion on the
axial measurement due to the intrinsic nonlinear nature of the
CCS. The target radial motion effect was compensated using
radial motion measurements based on the derived analytical
model. In addition, a design rule was set to guarantee the
same linearity of both radial and axial motion measurements.
Finally, a test rig and electronics for the proposed CCS were
built and the effectiveness of the proposed CCS was verified
through experiments and simulations.
2. CCS
Capacitive sensors are widely used in short-range ultraprecision and control applications because they have higher
resolutions compared with other types of sensors. The existing
four-segment CCS is designed to have the largest sensing area
for a high resolution, as shown in figure 1(a). The rotor
displacements can be approximated by equation (1) using the
capacitances of four sensing electrodes (C1 , C2 , C3 , C4 ).
XCCS4 = gain(C1 + C4 − C2 − C3 ),
YCCS4 = gain(C1 + C2 − C3 − C4 ).
(1)
Although the four-segment CCS has a high resolution,
it is sensitive to odd harmonic errors, especially the third
harmonic component in the geometric errors of a rotor [6].
To overcome this shortcoming, Jeon et al [7] proposed a new
configuration of CCS, the so-called eight-segment CCS. The
eight-segment CCS consists of four shared and four unshared
sensor segments, as shown in figure 1(b). The total angular
size 2ζ of a sensor unit is the sum of two shared and one
unshared sensor segments as follows:
2ζ = 2SS + SU .
(2)
Here, SS and SU are the angular sizes of the shared and
unshared segments, respectively.
Measured displacements of the rotor can be approximated
by equation (3). The eight-segment CCS can have an arbitrary
angular size of the sensor unit by adjusting the angular sizes
2028
(b)
of the sensor segments, and a proper angular size of the sensor
unit can minimize the effects of the geometric errors:
XCCS8 = gain(C8 + C1 + C2 − C4 − C5 − C6 ),
YCCS8 = gain(C2 + C3 + C4 − C6 − C7 − C8 ).
(3)
3. A new CCS for both radial and axial motion
measurements
3.1. Configuration
Figure 2(a) shows the cross section of a CCS. There is a small
axial area of the CCS. The main idea behind the novel CCS is
that the small axial area of the CCS is used to measure the axial
motion of the target, as shown in figure 2(b). If this concept is
feasible, a more compact design and reduction of the system
complexity can be achieved by removing the axial sensor.
In the case of the proposed CCS, the capacitance of
each electrode can be classified into the axial and radial
capacitances:
Ci = Cir + Cia .
(4)
Here, Cir is the capacitance of the radial area (radial
capacitance) and Cia is that of the axial area (axial capacitance).
Radial displacements can be approximated as equation (1)
or equation (3) because the capacitances of the axial sensing
area are removed by the differential configuration of the CCS.
If the sensor thickness is t, the sensor inner radius is b and
the axial gap is δa , the sum of the axial capacitances can be
given by
2π(b + t/2)t
Cia = ε
,
(5)
δa
i
where is the permittivity of air and the axial area is expressed
by the multiplication of the average radius of the axial area
b + t/2, the thickness t and 2π.
If the sum of the radial capacitances is constant
irrespective of the target radial motion and the offset
capacitance Coffset is properly removed, the axial motion would
be approximated by the sum of all capacitances as shown in
equation (6),
Z≈
1
1
=
.
CZ
C
−
Coffset
i
i
(6)
r
Here, Coffset =
Ci + Cstray , which means the additional
capacity due to radial and stray capacitances.
A CCS for both radial and axial motion measurements
C tr − C tr |α=0
Rotor
ζ
β
b-δ
b
α


2πεbw 
1

=
α 2 − 1 .
δ
1− δ
(10)
Hence, the additive nonlinearity can be compensated using the
radial displacement measured with the CCS as
2
2
CZ =
Ci − a XCCS
+ YCCS
− Coffset .
(11)
i
Figure 3. Measuring process of a CCS.
3.2. Total radial capacitance
Small radial capacitance of the CCS can be approximated
as [6]
εbw
θ.
(7)
C δ − α cos(θ − β)
Here, w is the sensor axial width and θ is the angle variable for
integration. Figure 3 shows the sensor radius b, the radial air
gap between the sensor and the rotor δ, the rotor eccentricity α,
the phase angle of the rotor eccentricity β and the sensor unit
angular size ζ that will be used in the appendix.
A nonlinear relationship between the rotor position and
the total radial capacitance can be calculated by integrating
equation (7) from 0 to 2π [10]. The resulting total radial
capacitance can be given by
2π
2πεbw
1
C tr =
Cir =
C =
(8)
2 .
δ
0
1 − αδ
The integration result shows nonlinear dependence of the sum
of the radial capacitances on the radial position of the rotor.
Since the total radial capacitance highly depends on the
rotor eccentricity α, it was used to measure the minimum film
thickness of a journal bearing [11]. For example, the radial
clearance of the back-up bearing is assumed as 75% of the
nominal clearance in an active magnetic bearing system, and
the total radial capacitance varies over 150%. That is, we
cannot use the total radial capacitance directly to represent the
axial motion.
3.3. Compensation method
The position dependence of the total radial capacitance can
be compensated using the measured rotor displacements. The
nonlinear radial measurement of the CCS can be expressed
by a function of the normalized rotor eccentricity, α/δ [12],
which is derived in the appendix,


1
εbw δ 
1

XCCS = 8
2 − 1 sin ζ cos β. (9)
δ α
1 − αδ
The additive nonlinearity of the total radial capacitance
can be expressed as capacitance variation due to the rotor
eccentricity as shown in equation (10).
The additive
nonlinearity of the total radial capacitances is a function of the
normalized eccentricity and it is very similar to the nonlinear
mean gain of equation (9),
Here, a is a compensation gain, and XCCS and YCCS are
measured radial displacements using the CCS.
The normalized additive compensation error (NEadd )
of equation (11) can be expressed with the normalized
eccentricity x (= α/δ) as
a
1
eq
NEadd =
−1
−1 .
(12)
√
x
1 − x2
Here, 0 x xe 1, xe is the normalized maximum
eccentricity, and aeq = 4a sin ζ /π.
The optimal gain aeq is the end point xe so that the
normalized additive compensation errors at two end points are
the same. The maximum normalized error can be calculated
by substituting the critical point at which the derivative of
equation (12) becomes zero.
The proposed compensation method of equation (11) is
too complex to be implemented with an analogue circuit.
Moreover, if the rotor is maintained to be near the centre
of the CCS, the compensation error is very small. Hence, we
can further simplify the compensation method as follows:
CZ =
Ci − am (|XCCS | + |YCCS |) − Coffset .
(13)
i
The compensation gain a is modified as am and the
compensation error obviously increases. The eccentricity
is
√ approximated as the sum of absolute displacements:
2
2
XCCS
+ YCCS
≈ (|XCCS | + |YCCS |).
The modified
compensation gain can be calculated by minimizing a modified
normalized additive compensation error such as
am
1
min
−1
−1
am
xe
1 − (xe )2
√
2am
1
(14)
− max
−1 .
−1
√
0xxe
x
1 − x2
The calculated compensation gains and relative
compensation errors are shown in figure 4. Two compensation
gains are same when the normalized eccentricity is less
than 0.5 and the optimal modified compensation gain has
discontinuity as the normalized eccentricity is larger than 0.9.
In addition, although the normalized compensation error using
the modified compensation method of equation (13) becomes
large as the rotor eccentricity increases, the compensation error
is not too severe where the normalized rotor eccentricity is less
than 0.8.
3.4. Finite element analysis
The proposed CCS is modelled under the assumption that
there is no coupling between radial and axial capacitances.
However, this assumption is not valid at the edge of the CCS.
Therefore, an electrostatic analysis using the finite element
2029
H-J Ahn
0.8
0.8
a =xe
Modi. gain a
Normalized additive error
Optimal compensation gain
1
m
0.6
0.4
0.2
0.7
0.6
a =x
e
Modi. gain am
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
Normalized eccentricity
0.8
1
0
0
0.2
0.4
0.6
Normalized eccentricity
(a)
0.8
1
(b)
Figure 4. Compensation method: (a) compensation gain and (b) normalized errors.
Cr
Sensor
Sensor
Car
Ca
Target
-50
(a)
(b)
Figure 5. FEM analysis: (a) electric field and (b) capacitance variation.
method (FEM) is performed to investigate the radial and axial
capacitance coupling. Figure 5(a) shows the results of the
FEM analysis in a case where the radial gap is 0.5 mm and the
axial gap is 0.9 mm. The axial electric field is bent at the edge
of the CCS.
To investigate the edge effect of the CCS, capacitance
variations from the reference axial gap (0.1 mm) to
various gaps (up to 0.9 mm) are calculated and shown in
figure 5(b). The ideal case of neglecting axial and radial
capacitance coupling the capacitance variation is calculated
using equation (5), while the real case considering axial and
radial capacitance coupling is calculated using the FEM. The
capacitance between the axial area and the target radial area,
Car in figure 5(b), contributes to the increase of the axial
measurement sensitivity. The edge effect is good for the axial
motion measurement and it is not significant, so we neglect
the effect in this paper.
shown in equation (15) since the linearized gain at the centre
α
linear
= 4 εbw
sin ζ cos β,
is XCCS
δ δ
1
2
− 1 − 1.
(15)
RDr = 2 √
x
1 − x2
4. Error analysis and design rule
4.2. Error analysis in the axial measurement
4.1. Error analysis in the radial measurement
There are harmonic errors of the radial displacements
measured with the proposed CCS due to its intrinsic
nonlinearity, as shown in equation A.5 [12]. The harmonic
errors of the measured radial displacements are transmitted
into the axial measurement during the compensations of the
total radial capacitance. However, the nonlinear compensation
The nonlinear characteristic of the radial motion measurement
is expressed as equation (9) and its linearized normalized
gain near the CCS centre is 1. The relative deviation of the
nonlinear gain from the linearized gain at the centre (RDr ) is
2030
Here, x is α/δ.
The CCS gain has severe nonlinearity as the normalized
eccentricity increases. Thus, the CCS is generally calibrated
using a regression within a given operating range. The relative
deviation and regressed relative error are shown in figure 6(a),
and their ratio is shown in figure 6(b). The ratio is nearly
constant (≈2.5) if the normalized rotor eccentricity varies
from 0.1 to 0.8. Therefore, the relative error of the radial
measurement (REr ) can be approximated as
1
1
2
REr (x) −
1
−
1
.
(16)
√
2.5 x 2
1 − x2
A CCS for both radial and axial motion measurements
10
3.5
2
0.6
10
10
Width ratio (t/w)
10
1
Ratio of two errors
Normalized additive error (%)
3
0
−1
2.5
2
1.5
10
10
−2
0.5
Normalized eccentricity
0.3
0.2
0
0
0.5
−3
0
0.4
0.1
1
Deviation
Regres. error
0.5
1
0.2
0.4
0.6
0
0
1
0.5
Figure 7. Desired width ratios (t/w).
1
Normalized eccentricity
(a)
width ratio t/w. That is, the width ratio should satisfy
NEadd |max (x) δδa max + REr (x) δδa max − δδa |min
t
δ
=
.
− δ
w
REr (x) δ (b)
Figure 6. Relative error of radial measurement: (a) relative
deviation and error and (b) their ratio.
δa min δa max
error is much bigger than the harmonic errors and the nonlinear
harmonic error is ignored in this analysis.
If NEadd |max is the maximum normalized additive
compensation error, the axial sensor output can be simplified
by
2πεt (b + t/2) 2πεbw
NEadd |max
−
δa
δ
t
2πεbw t δ
− NEadd |max .
1+
=
δ
w δa
2b
CZ =
ZCCS = 1/CZ − 1/CZ0 ,
δa min
(20)
The desired width ratio t/w of equation (20) is calculated
with increasing radial measurement range in the case when the
axial measurement range is given by 0.5 δ/δa 2 and the
result is shown in figure 7.
5. Simulation and experiment
(17)
The sum of the total capacitance should be inverted in
order to obtain the axial displacement from the CCS output,
(18)
and the relative error of the axial motion measurement, REax ,
can be represented by the ratio of the measurement error
to the ideal sensor output. The axial measurement error of
the developed CCS depends on its geometry, as shown in
equation (17): radial thickness, axial width and axial and
radial air gaps. Since t/2b is very small (1), the linearity
error depends on the ratio of the radial thickness to the axial
width (width ratio), t/w, and the axial and radial gaps ratio,
δ/δa . As a result, the relative measured error of the developed
CCS is calculated by
REax =
0.8
Normalized eccentricity (α/δ)
ideal
ZCCS − ZCCS
ideal
ZCCS
NEadd |max · δδa max
t δ . (19)
= δ
− δ
− NEadd |max
δa max
δa min w δa min
4.3. Design rule
Obviously, the relative errors of both radial and axial
measurements should be equalized in a given measurement
range in order to balance the accuracies of both radial and
axial measurements. The relative errors of both radial and
axial measurements (equations (16) and (19)) are equalized.
Then, we can determine an important sensor geometry or the
5.1. Experimental set-up
To verify the performance of the proposed CCS, an
experimental set-up was built as shown in figure 8(a). The
experimental set-up consists of the proposed CCS, a motorized
XY stage (Newport M-VP-25XA), a manual Z stage (Sigma
Koki TSD-603), a target and a sensor supporter. Figures 8(b)
and (c) show the proposed CCS and the target assembly that
consists of the motorized XY stage, the manual Z stage and the
target. The CCS specifications are as follows: the diameter is
54 mm, the radial air gap is 0.5 mm, the sensor axial length
is 6 mm and the sensor thickness is 2.5 mm. The proposed
CCS uses the same circuit as an existing CCS for the radial
measurement. An axial measurement circuit is composed of a
sum of all capacitances and an offset adjustment.
The schematic diagram of the experimental set-up is
shown in figure 9. First, offsets of the proposed CCS are set
electrically through setting the X and Y sensor outputs to zero
without the target and the Z offset is adjusted through setting
the Z sensor output zero only with the radial target. Then,
X, Y motions are actuated using a GPIB communication with
PC and Z motion is adjusted manually (±0.3 mm by 0.05 mm
steps in the X and Y directions and from 0.3 to 0.95 mm by
0.05 mm steps in the Z direction), and the X, Y and Z sensor
outputs were measured synchronously with the stage motion.
The maximum normalized eccentricity is 0.6 and the gap ratio
(δ/δa ) varies from 0.5 to 1.67.
5.2. Results
Experimental and simulation results are shown in
figure 10. Simulations were performed under the assumption
2031
10
0
5
5
−1
−5
−10
0.2
−4
0.2
Y
Target
)
Y
−0.2
m
)
m
0
−0.2
di
−0.2
0.2
0
Y
di
sp
− 0.2
.(m
0
sp
.(m
m
)
0
)
mm
p.(
dis
− 0.2
X
− 0.2
0.2
)
mm
p.(
dis
)
mm
p.(
dis
0
0.2
X
X
0.2
0
(a) Experimental setup
−3
.(m
−10
−2
sp
−5
0
di
0
Z sensor output (V)
10
Y sensor output (V)
X sensor output (V)
H-J Ahn
(a)
(b) Sensor
(c) Target assembly
Figure 8. Experimental set-up.
10
− 0.8
Z sensor output (V)
Motorized
XY stage
Complex sensor output (V)
Manual
Z stage
5
0
−5
−1
− 1.2
− 1.4
− 1.6
− 1.8
− 10
−2
0.2
0
− 0.2
− 0.2
)
)
m
m
Y
.(
p
dis
0.2
0
.(m
.(m
− 0.2
)
mm
sp
sp
0
di
di
0.2
0
− 0.2
X
X
0.2
Y
m
.(
isp
m
)
d
(b)
Figure 10. Sensor outputs at Z = 0.65 mm: (a) experiment and (b)
simulation.
Figure 9. Schematic of experiment.
2032
5
Y sensor output (V)
5
X sensor output (V)
that radial and axial capacitances are independent of each other.
Experiments were performed in Cartesian coordinates while
the simulations were performed in cylindrical coordinates.
The Z sensor output before the compensation is highly
dependent on the rotor position.
At the centre of the CCS, the X, Y sensor outputs due to
the axial motions are shown in figure 11. The X, Y sensor
outputs did not change and the gain near the centre decreases
about 2%, which is caused by the coupling effect of the axial
and radial capacitances.
Compensations are performed using both equations (11)
and (13) and the results are shown in figure 12. The radial
position dependence of the axial measurement is reduced
significantly (about five times), compared with figure 10.
The normalized additive errors due to the radial motion are
shown in figure 13(a). Axial measurement errors due to radial
motions are normalized using the theoretical sensor gain as
in equation (12), and the compensations 1 and 2 in figure 12
denote methods of equations (11) and (13), respectively. In
addition, the normalized additive errors from experiments,
simulations and theory are compared in table 1. The results
0
−5
0
−5
0.3
−0.2
0.5
Xd
isp. 0 0.2
(mm
)
0.7
0.9
is
Zd
p.(
)
mm
0.3
−0.2
0.5
Yd
isp. 0 0.2
(mm
)
m
p.(m
0.7
0.9
)
is
Zd
Figure 11. Radial sensor variation due to axial motion.
of experiment, simulation and theory agree well. After the
conditioning of equation (18), the measured axial sensor
outputs and their variations are shown in figure 13. The width
ratio t/w of the designed CCS is 0.417. The relative error in
the radial measurement is about 15% and the measured error
of the axial measurement is about 25%, as expected. The
A CCS for both radial and axial motion measurements
− 0.6
− 0.8
−1
− 1.2
− 0.4
− 0.6
− 0.8
−1
− 1.2
− 0.3
Compensated Z output (V)
− 0.4
− 0.3
Compensated Z output (V)
− 0.2
Compensated Z output (V)
Compensated Z output (V)
− 0.2
− 0.4
− 0.5
− 0.6
− 0.7
− 0.8
− 0.9
X 0.2
dis
0
p.( − 0.2
mm
)
− 0.2
0
0.2
0.2 X d
isp 0− 0.2
)
− 0.2
.(m
m)
m
isp.(m
Yd
0.2
0
X
(mm)
p.
Y dis
− 0.4
− 0.5
− 0.6
− 0.7
− 0.8
− 0.9
dis 0.2 0
p.(
mm − 0.2
)
− 0.2
0.2
0
X
(mm)
p.
Y dis
(a)
0.2
dis
p.(
0
− 0.2
mm
− 0.2
0.2
0
p.(mm
Y dis
)
)
(b)
Figure 12. Compensation results: (a) experiment: equations (11) and (13), (b) simulation: equations (11) and (13).
Normalized sensor output (1/Z)
Normalized additive error
0
0.25
w/o compen. (Exp.)
w/o compen. (Theory)
Compen. 1 (Exp.)
Compen. 1 (Theory)
Compen. 2 (Exp.)
Compen. 2 (Theory)
0.2
0.15
0.1
0.05
0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Z displacement (mm)
− 0.2
− 0.4
− 0.6
− 0.8
−1
Ideal
Compen. 1
Compen. 2
w/o compen.
− 1.2
− 1.4
− 1.6
0.3
0.4
0.5
0.6
0.7
0.8
Z displacement (mm)
(a)
0.9
(b)
Figure 13. Axial motion measurement: (a) normalized additive error due to the radial motion and (b) conditioned axial sensor output.
Table 1. Comparison of the normalized addition errors.
w/o
Compensation 1: Compensation 2:
compensation equation (11)
equation (13)
Theory
0.250
Simulation 0.2395
Experiment 0.2438
0.0486
0.050
0.0513
0.1075
0.1033
0.0946
measurements. Finally, a test rig for the proposed CCS was
built, and the performances of the proposed CCS were verified
with simulations and experiments. The developed CCS is
expected not only to reduce the system complexity but also
to afford a good balance between the radial and axial motion
measurement accuracies.
Acknowledgments
sensor gain increases about 10% due to the radial and axial
capacitance coupling.
6. Conclusion
This paper proposed a cylindrical capacitive sensor (CCS)
for both radial and axial motion measurements. Although
the developed CCS has the same geometric configuration
as a conventional CCS, an unused axial area of the CCS
is utilized to measure the axial motion of a rotor. First, a
theoretical model of the proposed CCS was derived. Based
on the derived theoretical model, compensation methods to
decouple the radial and axial motion measurements were
proposed. In addition, an error analysis of the CCS was
performed and a design rule was developed to guarantee the
same accuracy in measuring both radial and axial motion
This work was supported by the Brain Korea 21 Project in
2005 and by the Soongsil University Research Fund.
Appendix. Nonlinear analysis of the radial motion
measurement [12]
The measured X displacement using a CCS of angular size 2ζ
can be expressed as follows:
ζ
π +ζ
XCCS =
C −
C.
(A.1)
−ζ
π −ζ
The small capacitance, equation (7), can be expressed using a
power series as follows:
∞
n
εbw α
cos(θ − β) θ.
(A.2)
C δ n=0 δ
2033
H-J Ahn
If equation (A.2) is substituted into equation (A.1), the
measured X displacement can be calculated as follows:
∞
2εbw ζ α 2l+1
cos2l+1 (θ − β)θ. (A.3)
XCCS =
δ
δ
−ζ l=0
Using the trigonometric functional relation of equation (A.4),
the measured displacement of the rotor can be simplified as
equation (A.5):
2l+1
cos
XCCS
l 1 2l + 1
ϑ= l
cos (2l + 1 − 2k)ϑ
k
4 k=0
(A.4)
l ∞
εbw α 2l+1 2l + 1
=8
δ l=0 2δ
k
k=0
×
sin (2l + 1 − 2k)ζ
cos (2l + 1 − 2k)β.
2l + 1 − 2k
(A.5)
There appear harmonic errors in the radial motion
measurement of the CCS and the biggest nonlinear harmonic
error is the third harmonic one that can be removed if ζ is
60◦ . In addition, the nonlinear mean gain of the CCS can be
calculated through summing the first harmonic terms:
∞
εbw α 2l+1 2l + 1
1
XCCS
=8
sin ζ cos β. (A.6)
l
δ l=0 2δ
Equation (A.6) can be simplified with the power series of
equation (A.7),
1·3 2 1·3·5 3
1
1
x +
x + ···
=1+ x+
√
2
2·4
2·4·6
1−x
∞
∞
2k x k
2k + 1
1 k+1
=
=1+
x .
2k+1
k
k
4
2
k=0
k=0
(A.7)
That is, the nonlinear mean gain can be expressed by
1
XCCS
∞
εbw δ α 2l+2 1 2l + 1
=8
sin ζ cos β
δ α l=0 δ
22l+1
l
(A.8)
2034
and the simplified nonlinear mean gain is given by (A.9), which
is a function of the normalized rotor eccentricity, α/δ.
∞
εbw δ α 2l+2 1 2l + 1
1
=8
sin ζ cos β
XCCS
l
δ α l=0 δ
22l+1


1
εbw δ 

(A.9)
=8
2 − 1 sin ζ cos β.
δ α
1 − αδ
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