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Some Remarks on Electrostatic Influence Machines
Some Remarks on Electrostatic Machines
Injuence
hJ>D.SCHIEBER
Department ofElectrical Haifa 32000, Israel
ABSTRACT:
A simpl$ed
Engineering,
model
Technion-Israel
qf an influence
analysed by means of a corzformul truruformalion
generutor
Institute
is put
of’ Technology,
f&ward.
The model is
and the output current is determined.
high harmonic content is traced to the inherent geometry
Its
of the device.
I. Introduction Influence machines are quite well known, although their application is at present rather limited due to their inherently low energy densities--a feature closely associated with the hazard of dielectric breakdown. Accordingly, avoidance of this hazard is the primary consideration in the design and construction of such machines, while achievement of a so-called “clean” sinusoidal time-dependence of voltage or current takes secondary place. Although crude electrostatic generators made their appearance in the 17th century, a comprehensive theory of such devices has not been put forward to date ; one of the first steps in this direction was taken in 1923 by Ollendorff (l), and followed up by the same author’s analysis of the Wimshurst machine (2).t Recent research on liquid insulating materials with high breakdown gradient (e.g. 3) opened new possibilities for electrostatic machines. In particular, with the advent of outer space propulsion with the attendant conditions of extremely high vacuum, vacuum-insulated generators hold promise as power sources ; furthermore, atomic and X-ray research, generation of high-energy electron beams used in cancer treatment etc., have led-two decades ago (e.g. 4, 5) and again more recently (6)-to renewed interest in electrostatic machines. The present paper proposes a very simple model for analysis of some forms of electrostatic influence machines. This model is based on the schematic configuration shown in Fig. 1 : the rotor R, externally charged, moves inside the circumference of the machine casing (mean radius r,,) and influences the stator S, which is directly connected to a load resistor (not shown).
tThis machine, however, never gained widespread USC: its electrodes, having thin metal edges, are incapable of achieving very high voltages.
D. Schieher
FIG. 1. Schematic representation
of influence machine: electrode S represents the stator; electrode R the rotor.
ZZ.Machine Model In order to calculate the primary, exciting the circular periodicity of the real machine is the horizontal axis of a rectangular (Argand) 2. The basic “wavelength” of one member denoted by
electric field, we proceed as follows : replaced by a linear periodicity along system of coordinates X, y ; see Fig. of the entire “machine” manifold is
p = 2nro. Each machine
is therefore
restricted
(1)
(e.g. 2, 7) to the region
*POI
FIG. 2. Basic mathematical model of influence machine.
172
Remarks
on Electrostatic
Influence Machines
-++modp) comprising residing at
a moving
(exciting)
(2)
cylindrical
electrode
x=x,(modp); as well as a static (working)
cylindrical x=x,
(radius
p,,) instantaneously
y=y, electrode
(3)
(radius p ,) located at
=O(modp),y=y,
(4)
and grounded through a load resistance R,. For further simplicity, the machine is assumed to extend “infinitely” along an axis perpendicular to the plane of the drawing, but only a finite stretch h of this “infinite” extent is considered. We now supplement the geometrical model with kinematic considerations: denoting the instantaneous velocity of the working electrode by U, we assume for each instant of time t n() = z?t.
(9
Hence, X0 -=P
Introducing the mechanical circular machine, we approximate v by fir,, i.e. x0
-_= P
vt
(6)
27rr0 frequency
Q of the original,
rotating
!Sr,t _ 2rcr”
(7)
so that, obviously, x(J 1 _ = _~ nt. 271 P
We conclude inequalities
these geometric
and kinematic
PO~YO;
considerations
PI CYI
by assuming
the
(9)
as well as PO << lY,,-“VII;
PI << IJ’o-Y,l.
(10)
III. Field Analysis It is now assumed that the moving electrode carries an externally imposed electric charge density A0 per unit length. Introducing the primary (superscript p) complex electric potential xc”), we readily obtain [e.g. 8, 91 at each point Vol. 328, No. 1, pp. 171-178. Printed in Great Britain
1991
173
D. Schieber z=x+iy
(11)
that
The real primary
potential 4 (/I) = Re X’“’
reduces to zero along the J’ = 0 plane ; it exhibits, the exciting electrode z = (xu+iyo) (vO is the angle expression
of azimuth
We now turn our attention
across
however, a finite value 4::
across
+p,e”‘O
electrode),
towards
(13)
(14)
which
the stationary
we approximate
electrode
by the
of contour
I’ = (O+iy,)+p,e”lI (y , being the counterpart [see Eq. (12)] :
of qo). The potential
(16)
4xJ arising along this “stator”
reads
(17)
With the formerly that
mentioned
approximations
and kinematic
constraints,
we find
However, as the stator is connected to the ground through a “working resistance” R,, it acquires a finite charge. Denoting the axial, linear density of the latter by A,, we approximate-as per Eq. (15)-the so-called “secondary” (superscript s) potential due to this charge across the working electrode, i.e. we take (19) The total potential
rise V of the “stator”
is therefore
given by the expression
Remarks
on Electrostatic
Influence Machines
(20) f-0 This potential rise, in turn, forces a current I to the ground through the resistance R,. Determination of this current is the object of the following section. IV. Output Cuwent The negative time rate of the electrical charge q, residing on the stator electrode determines the load current I; on the other hand, this current is readily obtained once the load resistance R, is known. Thus for a “machine” of length 11we have on the one hand
while on the other
I=;./
(22)
Hence cash ‘A?! rn
- cos Clt
coshl)O--!?
-cosQt
+
*A&!! 0
PI
r.
Defining
now the time constant
I.
(23)
(24)
we are able to rewrite Eq. (23) in the more amenable
form
(25) r.
PI Finally,
differentiating
Eq. (25) we obtain
the equation
sin SZt dl R&h 1 I+zdt=p~2Yl 2 cosr”+~! _cosfJt In ~~~ rl) PI Fourier
expansion
for the current sin fit (26)
cosy”_-~~ rO
-cos&
.
yields quite readily the result
Vol. 328, No. I, pp. 171~178. 1991 Printcd in Great Bnlam
175
D. Schieber
.3 .l
-. 1
-. 3
-. 5
c 0
I‘ 8
6
4
6
FIG. 3. Time-variation of dimensionless current y Jr0 = 0.40 ; R = 15 s- ’ ; z = 0.02 s ; the wave-form
10
for the comprises
i-it
specific case y,/u,, = 0.80; the first ten harmonics [Eq.
(2811. 1
sin fit
2 i
Yo+Yl cos ~ r.
sin Rt
- cos nt
cos ELI!? r.
_ l.os Qt i (27)
so that I is obtained
in the form
of an infinite
series,
i.e.
where a, = arctan
The dimensionless
(n&).
(29)
i(t)
(30)
current I 2CUoh -
is reproduced, for a specific case, in Fig. 3 ; the relatively high percentage of harmonics, see Table I, is evident. While not always detrimental, such an abundance of harmonics may prove undesirable in certain applications, and appropriate measures of filtering must be undertaken where needed. 176
Journalof the
Frankhn Pergamon
lnstaute Press plc
Remarks TABLE
on Electrostatic
Influence
Machines
I
Relative current amplitude, p%, as dependent on term order, n n 1 2 3 4 5 6 7 8 9 10
p% 100 86.97 57.58 35.07 20.84 12.36 7.37 4.43 2.68 1.64
IV. Conclusion A simple model of an influence machine was analysed by means of conformal transformations. The output current for a resistive load was found, and shown to comprise a relatively high percentage of harmonics ; these are obviously due to the geometry of the device. In modern electrical engineering, electrostatic machines will find an ever-widening field of application; research on such machines should be undertaken along two main avenues, namely : (a) development of suitable insulating material, and (b) design of configurations conductive to low percentages of current or voltage harmonics.
References (1) F. Ollendorff, “ iiber Kapazitltsmaschinen”,
Archiv ,fir Elektrotechnik, Vol. 12, pp. 2977319, 1923. (2) F. Ollendorff, “Field theory of self-excited influence machines”, in “Topics in Applied
(3)
(4) (5) (6)
Mechanics” (Edited by D. Abir, F. Ollendorff and M. Reiner), Elsevier, Amsterdam, 1965. C. M. Cooke, “New insulating materials and their use to achieve high operating stresses in electrostatic machines”, Nucl. Instrum. Methods Phys. Res., Sect. A, Vol. 244, pp. 6472, 1986. A. W. Bright and B. Makin, “Modern electrostatic generators”, Confemp. Phys., Vol. 10. pp. 331-353, 1969. M. W. Layland, “Generalized electrostatic-machine theory”, Proc. ZEE, Vol. 116, pp. 403405, 1969. P. T. Krein and J. M. Crowley, “Harmonic effects in electrostatic induction motors”, Electric Mach. Power Syst., Vol. 10, pp. 4799497, 1985.
Vol. 328, No. I, pp. 171-178, 1991 Printed m Great Britain
177
D. Schieber (7) F. Ollendorff, 94, 1959. (8) J. C. Maxwell, York, 1954 (9) L. V. Bewley, New York.
178
“ijber
unipolare
Induktion”,
Archia,fiir Elektrotechnik,
Vol. 44,‘~~. 8-’
“A Treatise on Electricity and Magnetism”, Vol. I, p. 31 1, Dover, New (reproduction of 1891 edition). “Two-dimensional fields in electrical engineering”, pp. 52-54, Dover, 1963.
Injuence
hJ>D.SCHIEBER
Department ofElectrical Haifa 32000, Israel
ABSTRACT:
A simpl$ed
Engineering,
model
Technion-Israel
qf an influence
analysed by means of a corzformul truruformalion
generutor
Institute
is put
of’ Technology,
f&ward.
The model is
and the output current is determined.
high harmonic content is traced to the inherent geometry
Its
of the device.
I. Introduction Influence machines are quite well known, although their application is at present rather limited due to their inherently low energy densities--a feature closely associated with the hazard of dielectric breakdown. Accordingly, avoidance of this hazard is the primary consideration in the design and construction of such machines, while achievement of a so-called “clean” sinusoidal time-dependence of voltage or current takes secondary place. Although crude electrostatic generators made their appearance in the 17th century, a comprehensive theory of such devices has not been put forward to date ; one of the first steps in this direction was taken in 1923 by Ollendorff (l), and followed up by the same author’s analysis of the Wimshurst machine (2).t Recent research on liquid insulating materials with high breakdown gradient (e.g. 3) opened new possibilities for electrostatic machines. In particular, with the advent of outer space propulsion with the attendant conditions of extremely high vacuum, vacuum-insulated generators hold promise as power sources ; furthermore, atomic and X-ray research, generation of high-energy electron beams used in cancer treatment etc., have led-two decades ago (e.g. 4, 5) and again more recently (6)-to renewed interest in electrostatic machines. The present paper proposes a very simple model for analysis of some forms of electrostatic influence machines. This model is based on the schematic configuration shown in Fig. 1 : the rotor R, externally charged, moves inside the circumference of the machine casing (mean radius r,,) and influences the stator S, which is directly connected to a load resistor (not shown).
tThis machine, however, never gained widespread USC: its electrodes, having thin metal edges, are incapable of achieving very high voltages.
D. Schieher
FIG. 1. Schematic representation
of influence machine: electrode S represents the stator; electrode R the rotor.
ZZ.Machine Model In order to calculate the primary, exciting the circular periodicity of the real machine is the horizontal axis of a rectangular (Argand) 2. The basic “wavelength” of one member denoted by
electric field, we proceed as follows : replaced by a linear periodicity along system of coordinates X, y ; see Fig. of the entire “machine” manifold is
p = 2nro. Each machine
is therefore
restricted
(1)
(e.g. 2, 7) to the region
*POI
FIG. 2. Basic mathematical model of influence machine.
172
Remarks
on Electrostatic
Influence Machines
-++modp) comprising residing at
a moving
(exciting)
(2)
cylindrical
electrode
x=x,(modp); as well as a static (working)
cylindrical x=x,
(radius
p,,) instantaneously
y=y, electrode
(3)
(radius p ,) located at
=O(modp),y=y,
(4)
and grounded through a load resistance R,. For further simplicity, the machine is assumed to extend “infinitely” along an axis perpendicular to the plane of the drawing, but only a finite stretch h of this “infinite” extent is considered. We now supplement the geometrical model with kinematic considerations: denoting the instantaneous velocity of the working electrode by U, we assume for each instant of time t n() = z?t.
(9
Hence, X0 -=P
Introducing the mechanical circular machine, we approximate v by fir,, i.e. x0
-_= P
vt
(6)
27rr0 frequency
Q of the original,
rotating
!Sr,t _ 2rcr”
(7)
so that, obviously, x(J 1 _ = _~ nt. 271 P
We conclude inequalities
these geometric
and kinematic
PO~YO;
considerations
PI CYI
by assuming
the
(9)
as well as PO << lY,,-“VII;
PI << IJ’o-Y,l.
(10)
III. Field Analysis It is now assumed that the moving electrode carries an externally imposed electric charge density A0 per unit length. Introducing the primary (superscript p) complex electric potential xc”), we readily obtain [e.g. 8, 91 at each point Vol. 328, No. 1, pp. 171-178. Printed in Great Britain
1991
173
D. Schieber z=x+iy
(11)
that
The real primary
potential 4 (/I) = Re X’“’
reduces to zero along the J’ = 0 plane ; it exhibits, the exciting electrode z = (xu+iyo) (vO is the angle expression
of azimuth
We now turn our attention
across
however, a finite value 4::
across
+p,e”‘O
electrode),
towards
(13)
(14)
which
the stationary
we approximate
electrode
by the
of contour
I’ = (O+iy,)+p,e”lI (y , being the counterpart [see Eq. (12)] :
of qo). The potential
(16)
4xJ arising along this “stator”
reads
(17)
With the formerly that
mentioned
approximations
and kinematic
constraints,
we find
However, as the stator is connected to the ground through a “working resistance” R,, it acquires a finite charge. Denoting the axial, linear density of the latter by A,, we approximate-as per Eq. (15)-the so-called “secondary” (superscript s) potential due to this charge across the working electrode, i.e. we take (19) The total potential
rise V of the “stator”
is therefore
given by the expression
Remarks
on Electrostatic
Influence Machines
(20) f-0 This potential rise, in turn, forces a current I to the ground through the resistance R,. Determination of this current is the object of the following section. IV. Output Cuwent The negative time rate of the electrical charge q, residing on the stator electrode determines the load current I; on the other hand, this current is readily obtained once the load resistance R, is known. Thus for a “machine” of length 11we have on the one hand
while on the other
I=;./
(22)
Hence cash ‘A?! rn
- cos Clt
coshl)O--!?
-cosQt
+
*A&!! 0
PI
r.
Defining
now the time constant
I.
(23)
(24)
we are able to rewrite Eq. (23) in the more amenable
form
(25) r.
PI Finally,
differentiating
Eq. (25) we obtain
the equation
sin SZt dl R&h 1 I+zdt=p~2Yl 2 cosr”+~! _cosfJt In ~~~ rl) PI Fourier
expansion
for the current sin fit (26)
cosy”_-~~ rO
-cos&
.
yields quite readily the result
Vol. 328, No. I, pp. 171~178. 1991 Printcd in Great Bnlam
175
D. Schieber
.3 .l
-. 1
-. 3
-. 5
c 0
I‘ 8
6
4
6
FIG. 3. Time-variation of dimensionless current y Jr0 = 0.40 ; R = 15 s- ’ ; z = 0.02 s ; the wave-form
10
for the comprises
i-it
specific case y,/u,, = 0.80; the first ten harmonics [Eq.
(2811. 1
sin fit
2 i
Yo+Yl cos ~ r.
sin Rt
- cos nt
cos ELI!? r.
_ l.os Qt i (27)
so that I is obtained
in the form
of an infinite
series,
i.e.
where a, = arctan
The dimensionless
(n&).
(29)
i(t)
(30)
current I 2CUoh -
is reproduced, for a specific case, in Fig. 3 ; the relatively high percentage of harmonics, see Table I, is evident. While not always detrimental, such an abundance of harmonics may prove undesirable in certain applications, and appropriate measures of filtering must be undertaken where needed. 176
Journalof the
Frankhn Pergamon
lnstaute Press plc
Remarks TABLE
on Electrostatic
Influence
Machines
I
Relative current amplitude, p%, as dependent on term order, n n 1 2 3 4 5 6 7 8 9 10
p% 100 86.97 57.58 35.07 20.84 12.36 7.37 4.43 2.68 1.64
IV. Conclusion A simple model of an influence machine was analysed by means of conformal transformations. The output current for a resistive load was found, and shown to comprise a relatively high percentage of harmonics ; these are obviously due to the geometry of the device. In modern electrical engineering, electrostatic machines will find an ever-widening field of application; research on such machines should be undertaken along two main avenues, namely : (a) development of suitable insulating material, and (b) design of configurations conductive to low percentages of current or voltage harmonics.
References (1) F. Ollendorff, “ iiber Kapazitltsmaschinen”,
Archiv ,fir Elektrotechnik, Vol. 12, pp. 2977319, 1923. (2) F. Ollendorff, “Field theory of self-excited influence machines”, in “Topics in Applied
(3)
(4) (5) (6)
Mechanics” (Edited by D. Abir, F. Ollendorff and M. Reiner), Elsevier, Amsterdam, 1965. C. M. Cooke, “New insulating materials and their use to achieve high operating stresses in electrostatic machines”, Nucl. Instrum. Methods Phys. Res., Sect. A, Vol. 244, pp. 6472, 1986. A. W. Bright and B. Makin, “Modern electrostatic generators”, Confemp. Phys., Vol. 10. pp. 331-353, 1969. M. W. Layland, “Generalized electrostatic-machine theory”, Proc. ZEE, Vol. 116, pp. 403405, 1969. P. T. Krein and J. M. Crowley, “Harmonic effects in electrostatic induction motors”, Electric Mach. Power Syst., Vol. 10, pp. 4799497, 1985.
Vol. 328, No. I, pp. 171-178, 1991 Printed m Great Britain
177
D. Schieber (7) F. Ollendorff, 94, 1959. (8) J. C. Maxwell, York, 1954 (9) L. V. Bewley, New York.
178
“ijber
unipolare
Induktion”,
Archia,fiir Elektrotechnik,
Vol. 44,‘~~. 8-’
“A Treatise on Electricity and Magnetism”, Vol. I, p. 31 1, Dover, New (reproduction of 1891 edition). “Two-dimensional fields in electrical engineering”, pp. 52-54, Dover, 1963.