Electric Sail Thrust Model from a Geometrical Perspective

Full text

Journal of Guidance, Control, and Dynamics. doi: https://doi.org/10.2514/1.G003169
Electric Sail Thrust Model from a Geometrical Perspective
Mingying Huo,∗Giovanni Mengali†and Alessandro A. Quarta‡
(∗)Department of Aerospace Engineering, Harbin Institute of Technology, Harbin 150006, China
(†,‡)Department of Civil and Industrial Engineering, University of Pisa, I-56122 Pisa, Italy
Nomenclature
A, B, C = auxiliary variables, see Eq. (12)
a= propulsive acceleration vector (with a=kak), [ mm/s2]
ac= characteristic acceleration, [ mm/s2]
bi= cone angle curve-fit coefficient
ci= dimensionless acceleration curve-fit coefficient
ds= tether unit length, [ km]
F= thrust vector, [ N]
ˆ
iB= unit vector of axis xB
ˆ
jB= unit vector of axis yB
ˆ
kB= unit vector of axis zB
J= functional, [ mm/s2]
L= tether length, [ km]
m= spacecraft total mass, [ kg]
mp= solar wind proton mass, [ kg]
N= number of tethers
n= local solar wind number density, [ cm−3]
ˆn= unit vector normal to the sail nominal plane
O= spacecraft center-of-mass, origin of TB
r= Sun-spacecraft vector (with r=krk), [ au]
s= tether root-tip vector, [ km]
TB= body reference frame
ˆ
t= transverse unit vector
u= solar wind velocity, [ km/s]
Vk=k-th tether voltage, [ kV]
Vw= solar wind electric potential, [ kV]
xB, yB, zB= axes of TB
α= cone angle, [ deg]
αn= pitch angle, [ deg]
αp= angle between ˆpand ˆr, [ deg]
αr= polar angle of ˆr, [ deg ]
δr= azimuthal angle of ˆr, [ deg]
0= vacuum permittivity, [ F/m]
τ= switching parameter
γ= dimensionless propulsive acceleration
σ⊕= tether maximum thrust magnitude per unit length, [ N/m]
ζ= tether angular position, [ deg]
Subscripts
0 = initial, reference value
k=k-th tether
⊥k= perpendicular to the k-th tether
r= radial
t= transverse
max = maximum
⊕= Earth
Superscripts
?= optimal
∧= unit vector
∼= dimensionless
∗Research assistant, huomingying123@gmail.com.
†Professor, g.mengali@ing.unipi.it. Senior Member AIAA.
‡Associate Professor, a.quarta@ing.unipi.it. Associate Fellow AIAA (corresponding author).
1

2ELECTRIC SAIL THRUST MODEL FROM A GEOMETRICAL PERSPECTIVE
Introduction
The Electric Solar Wind Sail (E-sail) is a propellantless propulsion system concept that consists of thin centrifugally
stretched tethers, charged by an onboard electron gun. The interaction of the artificial electric field generated by the tethers
with the solar wind deflects the proton flow and generates a propulsive thrust. In the last years, much efforts have been
dedicated to estimate the E-sail propulsive acceleration for mission analysis purposes, as is thoroughly discussed in Ref. [1].
In all of the available mathematical models [2–5] the E-sail thrust vector is written as a function of the Sun-spacecraft
distance and the sail attitude, the latter being the angle between the direction of the local solar wind and the normal to the
E-sail nominal plane. The accuracy and complexity of the available mathematical models vary depending on several factors,
including the different plasmadynamic simulations and/or the inclusion of the curvature effect of the generic loaded tether.
The aim of this Note is to propose a compact vectorial description of the E-sail propulsive acceleration that takes into
account the thrust contribution generated by any single tether. The obtained results may be translated into an elegant
geometrical interpretation, which is particularly effective for optimal control law design.
Thrust Vector Analytical Model
Consider an E-sail that consists of N≥2 tethers, each one of length L, which are centrifugally stretched by the spacecraft
spin and radially displaced from the vehicle main body, see Fig. 1(a). All tethers are assumed to belong to the same plane,
referred to as sail nominal plane, orthogonal to the spacecraft spin axis.
B
x
B
y
B
z
ˆ
B
i
ˆ
B
j
ˆ
B
k
sail nominal plane
spin axis
main body
a) E-sail sketch.
B
x
O
main body
B
y
1 tether
st
N-th tether
2/Np
k-th tether
k
z
ˆ
k
s
L
2nd tether
b) Tethers position in TB(O;xB, yB, zB).
Figure 1 E-sail conceptual scheme and body-axis reference frame.
To identify the position of the generic k-th tether (with k= 1,2,...,N), introduce a body-axis reference frame TB(O;xB, yB, zB),
with origin Oat the spacecraft center-of-mass and unit vectors
ˆ
iB,
ˆ
jB, and
ˆ
kB, see Fig. 1(b). The plane (xB, yB) coincides
with the sail nominal plane, xBis aligned with the first tether (corresponding to k= 1), and
ˆ
kBcoincides with the spacecraft
spin velocity unit vector. Also, let ˆskbe the unit vector aligned with the k-th tether and directed from its root to the tip.
Assuming all tethers to be the same angle apart from each other (equal to 2 π/N rad), the components of ˆskin the body-axis
reference frame TBare
[ˆsk]TB= [cos ζksin ζk0]Twith ζk,2π(k−1)
N(1)

ELECTRIC SAIL THRUST MODEL FROM A GEOMETRICAL PERSPECTIVE 3
where ζkis the angle between the direction of
ˆ
iBand that of ˆsk, measured counterclockwise from the xBaxis, see Fig. 1(b).
According to Refs. [3, 5, 6], when the Sun-spacecraft distance is on the order of 1 au, the force dFper unit length ds
generated by the k-th tether can be written as
dF
ds= 0.18 max (0, Vk−Vw)√0mpnu⊥k(2)
where Vkis the k-th tether voltage (on the order of 20−40 kV ), Vwis the electric potential corresponding to the kinetic energy
of the solar wind ions (with a typical value of about 1 kV), 0is the vacuum permittivity, mpis the solar wind ion (proton)
mass, nis the local solar wind number density, and u⊥kis the component of the solar wind velocity uperpendicular to the
k-tether direction. Assuming the solar wind to propagate radially from the Sun, the vectors uand u⊥kcan be written as
u=uˆr(3)
u⊥k=u(ˆsk׈r)׈sk≡u[ˆr−(ˆsk·ˆr)ˆsk] (4)
where uis the solar wind flow speed (about 400 km/s), and ˆris the Sun-spacecraft unit vector, see Fig 2.
O
to Sun
k-th tether
ˆ
r
ˆ
k
s
ˆˆ
(, )plane
k
rs
u
k
^
u
to Sun
Figure 2 Components of solar wind velocity u.
Recall that the local solar wind number density nis proportional to the inverse square distance from the Sun, i.e. n=
n⊕(r⊕/r)2where n⊕is the solar wind number density at r=r⊕,1au. Therefore, substituting Eq. (4) into Eq. (2), the
force per unit length becomes dF
ds=σ⊕kr⊕
r[ˆr−(ˆsk·ˆr)ˆsk] (5)
where
σ⊕k
,0.18 max (0, Vk−Vw)u√0mpn⊕(6)
represents the maximum modulus of the specific force kdF/dskgiven by the k-th tether at a Sun-spacecraft distance r=r⊕.
Besides Eq. (2), other analytical models of the thrust force have been proposed for positively biased tethers, see e.g. Ref. [7].
The differences between the available models are however confined to the expression of σ⊕kand, therefore, the structure of
Eq. (5) remains unchanged.
Since dF/dsis nearly constant along the tether, the total force Fkprovided by the k-th tether of length Lis
Fk=L σ⊕kr⊕
r[ˆr−(ˆsk·ˆr)ˆsk] (7)
Note that σ⊕kis a function of the tether voltage Vk, which may be slightly changed in any tether using a suitable potential
control [3,5]. As a result, σ⊕kmay not be the same for all tethers, see Eq. (6), and the general expression for the E-sail total
thrust Fis
F=τ
N
X
k=1
Fk≡τ L r⊕
rˆr
N
X
k=1
σ⊕k−τ L r⊕
rN
X
k=1
σ⊕k(ˆsk·ˆr)ˆsk(8)
where Fkis given by Eq. (7). The dimensionless parameter τ∈ {0,1}is introduced in Eq. (8) to model the fact that the
E-sail thrust can be turned off (τ= 0) at any time by simply switching off the onboard electron gun.
An interesting form of the E-sail propulsive acceleration vector a=F/m, where mis the total spacecraft mass, is obtained
when all of the tethers have the same voltage Vk=V0and, therefore, the same maximum modulus of specific force σ⊕k. In
that case, substituting σ⊕k=σ⊕into Eq. (8), the propulsive acceleration vector becomes
a=τ N L
mσ⊕r⊕
rˆr−τ L
mσ⊕r⊕
rN
X
k=1
(ˆsk·ˆr)ˆsk(9)
Note that both Eqs. (8) and (9) neglect the sheath interference between tethers, which depends on several parameters, such
as the voltage and the number of tethers.
The components of ˆrare now written in the body-axis reference frame TBas a function of angles αr∈[0, π] rad and
δr∈[0,2π] rad, see Fig. 3, as
[ˆr]TB= [sin αrcos δrsin αrsin δrcos αr]T(10)

4ELECTRIC SAIL THRUST MODEL FROM A GEOMETRICAL PERSPECTIVE
to Sun
O
B
x
B
y
B
z
ˆ
r
sail nominal plane
main body
r
a
r
d
Figure 3 Components of the Sun-spacecraft position unit vector ˆr.
Accordingly, the summation on the right-hand side of Eq. (9) becomes
N
X
k=1
(ˆsk·ˆr)ˆsk= sin αrh(Acos δr+Bsin δr)
ˆ
iB+ (Bcos δr+Csin δr)
ˆ
jBi(11)
where
A,
N
X
k=1
cos2ζk, B ,
N
X
k=1
cos ζksin ζk, C ,
N
X
k=1
sin2ζk(12)
with ζkgiven by Eq. (1). It can be shown, see Appendix, that
A≡C=N/2, B = 0 (13)
Recalling Eq. (10), the summation in Eq. (11) reduces to
N
X
k=1
(ˆsk·ˆr)ˆsk=N
2sin αrcos δr
ˆ
iB+ sin δr
ˆ
jB≡N
2hˆr−ˆr·
ˆ
kBˆ
kBi(14)
Substituting this last relation into Eq. (9), the spacecraft propulsive acceleration vector becomes
a=τ N L σ⊕
2mr⊕
rhˆr+ˆr·
ˆ
kBˆ
kBi(15)
Note that ˆr·
ˆ
kBˆ
kB≡(ˆr·ˆn)ˆnwith ˆr·ˆn≥0, where ˆnis the unit vector normal to the sail nominal plane in the
direction opposite to the Sun. A more compact expression of amay be obtained using the concept of spacecraft characteristic
acceleration ac, that is, the maximum modulus of the propulsive acceleration at a distance r=r⊕from the Sun. Since kakis
maximum for a purely radial propulsive acceleration (that is, when ˆr=
ˆ
kB), from Eq. (15) the characteristic acceleration can
be written as a function of σ⊕as
ac=N L σ⊕
m(16)
Substituting Eq. (16) into Eq. (15), the final expression of the E-sail propulsive acceleration abecomes
a=τac
2r⊕
r[ˆr+ (ˆr·ˆn)ˆn] (17)
In the general case when ˆn6=ˆrand τ= 1, abelongs to the plane spanned by (ˆr,ˆn) and its direction is between ˆn
and ˆr, see Fig. 4. Furthermore, the propulsive acceleration modulus can be written as a function of the sail pitch angle
αn,arccos(ˆr·ˆn)∈[0, π/2] rad, that is, the angle between ˆnand ˆr, as
a,kak=τac
2r⊕
rp1 + 3 cos2αn(18)
The pitch angle αnis related to the more common sail cone angle α∈[0, π/2] rad, defined as the angle between ˆrand
ˆa, see Fig. 4. In fact, taking the scalar product of both sides of Eq. (17) by ˆr, and substituting Eq. (18) into the resulting
relation, it is found that
α= arccos a·ˆr
a= arccos 1 + cos2αn
√1 + 3 cos2αn(19)

ELECTRIC SAIL THRUST MODEL FROM A GEOMETRICAL PERSPECTIVE 5
Sun
ˆ
n
ˆ
r
ˆ
a
a
n
a
Figure 4 Pitch (αn) and cone (α) angle.
Equation (19) states that α≤αn/2, that is, the direction of the propulsive acceleration is closer to ˆrthan to ˆn. In particular,
when the sail nominal plane is orthogonal to the radial direction (αn=α= 0), Eq. (17) reduces to
a=τ acr⊕
rˆr(20)
which is the classical expression used to study the performance of a Sun-facing E-sail in an interplanetary mission scenario [8,9].
For small values of the pitch angle, that is, when αn≤20deg, a good approximation of the cone angle is α'αn/2, which is
in accordance with the simplified model used to characterize the thrust vector [2]. The latter point is clearly highlighted in
Fig. 5, where the dimensionless propulsive acceleration modulus a/(acr⊕/r) and the sail cone angle αare drawn as a function
of the sail pitch angle αn, see Eqs. (18)-(19). Note that acr⊕/r is the local maximum value of aobtained with αn= 0 and
τ= 1.
The maximum modulus of the cone angle αmax and the corresponding value of the sail pitch angle αn(αmax) are obtained
by enforcing the necessary condition ∂α/∂αn= 0 in Eq. (19). The result is
αmax = arcsin (1/3) rad '19.5 deg , αn(αmax) = arccos (1/√3) rad '35.3 deg (21)
Geometrical Interpretation
In the general case when ˆr6=ˆnand τ= 1, there exists an interesting graphical representation of the E-sail propulsive
acceleration vector. To that end, assuming αn6= 0, introduce the transverse unit vector ˆ
t,[(ˆr׈n)׈r]/sin αn. From the
vector triple product rule
ˆ
t=
ˆn
sin αn−
ˆr
tan αn
(22)
Substituting Eq. (22) into Eq. (17), the propulsive acceleration vector can be rewritten in terms of radial (ar) and transverse
(at) components as
a=ar+at(23)
where
ar,τac
4r⊕
r[3 + cos (2 αn)] ˆr(24)
at,τac
4r⊕
rsin (2 αn)ˆ
t(25)
In particular, from Eq. (25), the maximum modulus of the transverse propulsive acceleration is reached at a pitch angle
αn=π/4 rad, that is, when the cone angle is α= arccos 3/√10rad '18.5 deg, see Eq. (19). The importance of this result
is that the transverse component of the propulsive acceleration changes the specific orbital angular momentum vector hand,
therefore, a sail pitch angle of αn=π/4 rad maximizes the (local) variation of khk. The latter result is consistent with the
analysis of Toivanen and Janhunen [5], who modelled the actual tether curvature using a simplified approach.
Notably, assuming τ= 1, the E-sail propulsive acceleration vector may be described in graphical form using the polar plot
of Fig. 6, in which
ear,kark
acr⊕/r =3 + cos (2 αn)
4(26)
eat,katk
acr⊕/r =sin (2 αn)
4(27)
According to Fig. 6, the function ear=ear(eat) describes a half-circle in the plane (eat,ear), with radius R= 1/4 and center
C= (0,3/4). Therefore, a vector from the axes’ origin oto a generic point Pon the half-circle represents (when τ= 1) the
dimensionless propulsive acceleration vector e
a,a/(acr⊕/r) whose cone angle αis equal to the angle between the ordinate
axis and the oP segment, while the pitch angle αncoincides with one-half the angle between the ordinate axis and the CP
segment, see Fig. 6 and Eqs. (26)-(27).

6ELECTRIC SAIL THRUST MODEL FROM A GEOMETRICAL PERSPECTIVE
0 15 30 45 60 75 90
0.5
0.6
0.7
0.8
0.9
1
a/(a
c
r
)
/r)
0 15 30 45 60 75 90
0
5
10
15
20
,n[deg]
,[deg]
2
n
a
a=
Figure 5 Dimensionless propulsive acceleration modulus and cone angle as a function of pitch angle (with τ= 1).
Optimal Steering Law
The graphical representation of the propulsive acceleration vector of Fig. 6 is now used to find the value of the switching
parameter τand the optimal direction of ˆnthat maximize the projection of aalong a given unit vector ˆp. This amounts to
maximizing the scalar functional
J,a·ˆp(28)
The solution to this problem is important in the context of optimal heliocentric transfers, in particular when an indirect
approach (based on the calculus of variations) is used to evaluate the minimum flight time to reach a prescribed target
orbit. The steering law that maximizes Jalso provides the optimal thrust vector in a locally-optimal transfer, i.e. when the
performance index to be maximized is a function of the (local) time variation of the spacecraft osculating orbit parameters.
In the special case when ˆp≡ˆr, the solution is simply a Sun-facing E-sail (i.e., ˆn=ˆr≡ˆp) with τ= 1. In that case, the
propulsive acceleration modulus takes its maximum value, see Eq. (20), with τ= 1. In the general case when ˆp6=ˆr, the unit
vector ˆnmust belong to the plane spanned by ˆrand ˆp, since a/τ depends on the sail attitude through the pitch angle only,
see Eq. (18). Let αp,arccos ( ˆp·ˆr)∈[0, π] rad be the angle between ˆpand ˆr. Recalling Eq. (17), the functional Jbecomes
J=τac
2r⊕
r[cos αp+ cos αncos (αp−αn)] (29)
Enforcing the necessary condition ∂(J/τ )/∂αn= 0, the last relation provides a compact expression for the optimal pitch angle
α?
n, that is
α?
n=αp/2 (30)
whereas Eq. (29) gives the maximum value of the ratio J/τ as a function of the angle αp, viz.
max (J/τ ) = ac
4r⊕
r(1 + 3 cos αp) (31)
In other terms, according to Eq. (30), the optimal direction of ˆncoincides with the bisector of the angle between ˆrand ˆp.

ELECTRIC SAIL THRUST MODEL FROM A GEOMETRICAL PERSPECTIVE 7
0.25 0.5
0.25
0.5
0.75
1
eat
ea
r
C
o
P
%
a
a
2n
a
R
Figure 6 Polar plot of the dimensionless propulsive acceleration components (with τ= 1).
The optimal value τ?can be found by observing that Jis a linear function of τ, see Eq. (29). Therefore, the resulting
bang-bang optimal control law is τ?= 1 when max (J/τ)≥0, and τ?= 0 when max (J/τ )<0. In other terms, from Eq. (31)
τ?=sign (1 + 3 cos αp) + 1
2≡sign (1 + 3 ˆp·ˆr) + 1
2(32)
where sign (·) is the signum function.
The results of Eqs. (30) and (32) may also be recovered with a geometric approach, using the polar plot shown in Fig. 7.
In fact, the maximum of the projection of a/τ along ˆpis found by looking for the P H segment that is both tangent to the
half-circle (which describes the locus of points where the tip of vector e
alies) and orthogonal to ˆp. Since C P and oH are
parallel segments, it follows that αp= 2 α?
n. Note that, taking into account the expression of αmax given by Eq. (21), the
condition τ?= 0 described by Eq. (32) is obtained when the direction of ˆplies in the shaded area of Fig. 7.
From Eq. (30), the optimal sail orientation defined by the unit vector ˆn?,ˆn(α?
n) is
ˆn?=cos (αp/2)
1 + cos αp
(ˆr+ˆp) (33)
When Eqs. (32) and (33) are substituted into Eq. (17), the optimal propulsive acceleration a?=a(α?
n, τ ?) may be written as
a function of ˆp,rand acas
a?=ac
8r⊕
r(3 ˆr+ˆp) [sign (1 + 3 ˆp·ˆr) + 1] (34)
In the special case when αp= 0, that is ˆp=ˆr, Eq. (33) simply states that ˆn?=ˆr, while Eq. (34) reduces to Eq. (20) with
τ= 1.
Comparison with the Literature Results
The previous thrust vector model provides an elegant analytical proof of the numerical model proposed by Yamaguchi and
Yamakawa [4, 10] in which the cone angle αand the propulsive acceleration modulus aare both functions of the sail pitch
angle αn.
In particular, according to Refs. [4,10], when the thruster is on (τ= 1), the sail cone angle depends on the sail pitch angle
through a sixth-order polynomial equation that fits the numerical results, viz.
α=b6α6
n+b5α5
n+b4α4
n+b3α3
n+b2α2
n+b1αn+b0(35)
where the coefficients bi, with i= 1,2,...,6 are summarized in Tab. 1. The model of Yamaguchi and Yamakawa [4, 10] also
provides the following expression for the propulsive acceleration modulus
a=acτr⊕
rγ(36)
where γis a sort of dimensionless propulsive acceleration. The value of γdepends, again, on the sail pitch angle αnthrough
a best-fit sixth-order polynomial equation in the form
γ=c6α6
n+c5α5
n+c4α4
n+c3α3
n+c2α2
n+c1αn+c0(37)

8ELECTRIC SAIL THRUST MODEL FROM A GEOMETRICAL PERSPECTIVE
0.25 0.5
0.25
0.5
0.75
1
eat
ea
r
C
o
P
a
%å
a
å
2n
a
å
ˆ
p
H
p
a
max
a
0t=
å
max
2
p
a-
Figure 7 Graphical approach to find the optimal steering law.
i0 1 2 3 4 5 6
bi0 4.853 ×10−13.652 ×10−3−2.661 ×10−46.322 ×10−6−8.295 ×10−83.681 ×10−10
ci1 6.904 ×10−5−1.271 ×10−47.027 ×10−7−1.261 ×10−81.943 ×10−10 −5.896 ×10−13
Table 1 Best-fit interpolation coefficients of thrust model (angles in degrees) by Yamaguchi and Yamakawa [4, 10].
Table adapted from Ref. [1].
whose coefficients ciare collected in Tab. 1.
The results of Fig. 5 are in perfect agreement with the plot of the function α=α(αn) in Eq. (35) with coefficients bitaken
from Tab. 1. Also, upon comparing Eq. (36) and Eq. (18), the dimensionless propulsive acceleration γof Refs. [4, 10] can be
written as a function of the sail pitch angle as
γ=√1 + 3 cos2αn
2(38)
The correctness of the last relation is confirmed by a plot of the polynomial function γ=γ(αn) given by Eq. (37), which
overlaps to the upper curve drawn in Fig. 5.
Starting from the model by Yamaguchi and Yamakawa [4, 10], Ref. [1] discusses the optimal control law using both an
analytical and a graphical approach. In the latter case, the curve ear=ear(eat) calculated with Eqs. (35) and (37) is approximated
with a circle of radius ρ'0.2523 and center C= (0, d) with d'0.7477. The resulting expressions for γand αare [1]
γ=pd2+ρ2+ 2 ρ d cos(2 αn) (39)
α= arctan ρsin(2 αn)
d+ρcos(2 αn)(40)
Notably, these last two relations coincide with Eqs. (19) and (38) when ρ=R= 1/4 and d= 3/4, see Fig. 6. Moreover, the
results obtained in Ref. [1] show that the optimal pitch angle is about one half of αp, as is stated by Eq. (30), and the optimal
switching parameter is zero when αpis greater than a critical value of about 110 deg. In fact, according to Fig. 7 and Eq. (32),
the critical value of αpis arccos (−1/3) ≡(αmax +π/2) '110 deg.
Finally, recalling that ˆr·
ˆ
kBˆ
kB≡(ˆr·ˆn)ˆnwith ˆr·ˆn≥0, the vectorial model of Eq. (17) gives the same result of Ref. [5]
when the tethers are assumed to be straight. However, it is worth noting that the analysis of Toivanen and Janhunen [5] is
based on the assumption of a continuous angular distribution of the tethers and, as such, on a sufficiently high number Nof
available tethers. The model of this Note, instead, can be applied to an arbitrary number of tethers with N≥2 and provides
a vectorial relation that is very useful for a preliminary analysis of an orbital transfer.

ELECTRIC SAIL THRUST MODEL FROM A GEOMETRICAL PERSPECTIVE 9
Conclusions
The propulsive acceleration generated by an electric solar wind sail may be described in vectorial form as a function of the
angle between the propagation direction of the solar wind and the spacecraft spin axis. The acceleration vector has a simple
and effective geometrical interpretation that illustrates the relation existing between the sail pitch angle and the cone angle.
Likewise, the optimal sail attitude for minimum time transfer trajectories can be easily visualized in graphical form.
The new results, which are very useful for preliminary mission analyses, have been compared to existing models obtained
using numerical simulations. The limits of the new model are confined to the assumptions adopted, in particular to the fact
that the tethers must be straight and all belonging to the same plane. A more refined analysis should account for the curvature
of each tether due to the solar wind load and the tension due to the centrifugal force. The latter point requires a dynamical
analysis of the tethers and a substantial complication of the mathematical model.
Appendix
Using standard trigonometric identities, the coefficients A,Band Cof Eqs. (12) may be rewritten as
A=
N−1
X
`=0
1 + cos (2 ζ`)
2, B =
N−1
X
`=0
1
2sin (2 ζ`), C =
N−1
X
`=0
1−cos (2 ζ`)
2(41)
where, from Eq. (1), ζ`= 2 π `/N. The evaluation of the three coefficients reduces to calculating the two summations in
cos (2 ζ`) and sin (2 ζ`). To that end, note that
N−1
X
`=0
cos (2 ζ`) =
N−1
X
`=0
ej2ζ`+ e−j2ζ`
2=1
2
N−1
X
`=0 he(j4π/N)i`+1
2
N−1
X
`=0 he(−j4π/N)i`(42)
The last two summations in Eq. (42) are in the form of a geometric series. Recalling that
N−1
X
`=0
x`=1−xN
1−x(43)
where xis the common ratio of the series, Eq. (42) becomes
N−1
X
`=0
cos (2 ζ`) = 1
2
1−ej4π
1−ej4π/N +1
2
1−e−j4π
1−e−j4π/N = 0 (44)
Likewise, using a similar approach, it can be verified that
N−1
X
`=0
sin (2 ζ`) = 0 (45)
The final result A=C=N/2 and B= 0, see Eq. (13), is immediately obtained by substituting Eqs. (44) and (45) into
Eq. (41).
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