This document summarizes a research article about planar spiral systems that allow for cylindrical waves to have a constant radial phase velocity. It begins by introducing planar spiral systems used in antennas and slow-wave structures, which are typically logarithmic or Archimedean spirals. The document then derives the differential equation that describes a new class of "synchronous spirals" that maintain a constant initial radial phase velocity regardless of radial or azimuthal location. It provides the solutions to this equation for both slowed and accelerated spatial harmonics of the radial wave. These solutions describe six types of synchronous spirals that vary in their limiting radii and direction of winding. The document concludes by discussing potential applications of synchronous spirals in slow-wave structures and antennas.

1. UDC 621.385.6.32
Journal of Communications Technology and Electronics 39(8), 1994
ISSN 1064-2269/94/0008-0042 ® 1994 Scripta Technica, Inc
Planar Spiral Systems with Waves of Constant Radial Phase
Velocity*
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*Originally published in Radiotekhnika i elektronika, No. 4, 1994, pp. 552-559.
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V. A. SOLNTSEV
Planar spiral systems are shown to exist, in which a cylindrical wave has a
constant radial phase velocity. The equation of such spirals is derived and its
solution is obtained in elementary functions. It is established that these spirals
include the logarithmic spirals as a special case; in the general case, however,
they have a large variety of forms and in particular they have a limiting inside or
outside radius. This makes it possible to use such spirals for building new slow-
wave structures and antennas.
Key words: Planar spiral systems with constant wave radial phase velocity;
solution of equation of spiral system; varieties of spirals; slow-wave structures
and antennas based on spirals.
INTRODUCTION
Planar spiral systems are used as ultrawide-band antennas [1] and they can be
employed as slow-wave structures for traveling-wave tubes [2]. Two types of
planar spirals are considered for these purposes: Archemedian and logarithmic
spirals, defined in a polar coordinate system by
0 0, ,m
r r m r r e ϕ
ϕ= = (1)
respectively, where 0r is the initial radius for 0ϕ = , m is a constant
characterizing the properties of the spiral. The angle ψ between the tangent to the
spiral and the normal to the radius vector determines the radial phase velocity 0υ
of a cylindrical wave.
For the Archimedean spiral 0ψ → as the radius r increases. For the
logarithmic spiral cot 1/m constψ = = , which results in the phase velocity of the
radial wave being independent of r and ϕ and depending only slightly on the
frequency. For example, for a calculation in the approximation of a plane that is

2. anisotropically conducting along the turns of the spiral 0 sincυ ψ= , where c is
the velocity of light. The use of logarithmic spirals as ultrawide-band antennas and
slow-wave systems is based on these properties.
A new class of planar spiral systems is found in this paper that have an initial
phase velocity nυ of the spatial harmonics of a cylindrical wave that is
independent of r and ϕ . If these spirals are used as the slow-wave systems of
TWT with a radial electron motion, then synchronism of the electrons with the
wave is possible regardless of their radial and azimuthal location. Therefore, this
class of planar curves can be called "synchronous spirals" SS.
Synchronous spirals include the logarithmic spiral as a general case; however,
in the general case they have a large variety of forms making it possible to carry
out a more effective optimization of the characteristics of slow-wave systems and
antennas based on them.
1. Differential Equation of Synchronous Spiral
Let us assume a wave with phase velocity sυ is moving along the turns of a
spiral formed by a single conductor, by a twisted, two-conductor line, by a twisted
waveguide or in some other fashion (Fig. 1). The phase velocity of a radial wave
and its spatial harmonics nυ ( 0, 1, 2,...n = ± ± ) is determined by the condition
that the travel time
Fig. 1. Derivation of equation of synchronous spiral.
of the radial wave between adjacent turns of the spiral and the travel time of the
wave along the turns of the spiral are equal or differ by an integer number of
oscillation periods 2 /T π ω= :
( ) ( ) ( ) ( )
( )
2 22
2
,
n s
r rr r
d nT
ϕ π
ϕ
ϕ ϕϕ π ϕ
ϕ
υ υ ϕ
+
ʹ′++ −
= +∫
% %
%
%
(2)
where the phase velocity along the turns of the spiral can be assumed to depend on
ϕ in the general case. By differentiating both sides of Eq. (2) with respect to ϕ ,
we arrive at

3. ( ) ( ) ( ) ( )
( )
( ) ( )
( )
2 2 2 2
2 22
,
2n s s
r r r rr r ϕ π ϕ π ϕ ϕϕ π ϕ
υ υ ϕ π υ ϕ
ʹ′ ʹ′+ + + +ʹ′ ʹ′+ −
= −
+
which is satisfied if
( ) ( ) ( )
( )
( )
2 2
,n
n s
r rr
T
ϕ ϕϕ
ϕ
υ υ ϕ
ʹ′+ʹ′
= + (3)
where ( )nT ϕ is some periodic function, ( ) ( )2n nT Tϕ π ϕ+ = . By integrating the
resulting Eq. (3) over a period, we see that the original Eq. (2) is satisfied if the
average value of ( )nT ϕ is a multiple of the oscillation period T , i.e..
( )
2
1
2
n nT d T
ϕ π
ϕ
ϕ ϕ
π
+
=∫ % % (4)
By transferring ( )nT ϕ to the left side of Eq. (3) and squaring both sides, we obtain
the following differential equation of the synchronous spiral:
2 2 2
2 2 2
21 1 1
0n
n
n s n s
T
r r r T
υ υ υ υ
⎛ ⎞
ʹ′ ʹ′− + − + =⎜ ⎟
⎝ ⎠
. (5)
In the general case for variable coefficients of ( )sυ ϕ and ( )nT ϕ this equation
can only be solved numerically. Let us consider the case of constant coefficients
when Eq. (5) can be integrated. We write this equation as
( )2 2 2 2
1 2 0,n n n nM r R M r r Rʹ′ ʹ′− + − + = (6)
where /n s nM υ υ= is the slowing (for 1nM > ) or acceleration (for 1nM < ) of
the n-th spatial harmonic of the radial wave, 2 n n s sR T nTπ υ υ= = is a
constant, which for given oscillation period T and wave velocity sυ along - spiral
can be assumed to be an infinite series of positive and negative discrete values
corresponding to different spatial harmonics.
For the fundamental spatial harmonic 00, 0n R= = and we obtain from Eq.
(6) the equation of a normal logarithmic spiral
( )2
0 1M r rʹ′− = ± , (7)
having an exponential solution of the form (1) for ( )2
01/ 1m M= ± − .
For the other spatial harmonics the constant 0nR ≠ and it can be eliminated
from Eq. (6). We will omit the index n for simplicity and introduce the
dimensionless radius
/r Rρ = (8)

4. and sign sign / nRM n υ= is the sign of the product RM . Then the equation of
synchronous spirals becomes
( )2 2 2
1 2 sign 1 0M M RMρ ρ ρʹ′ ʹ′− + − + = (9)
The sign of the product RM entering into Eq. (9) has no effect on the form of the
solution. A change in this sign only leads to a change in the sign of the derivative
ρʹ′, i.e., to the substitution ϕ ϕ→ − and to a reversal of the direction of the spiral
coil. Therefore, we will assume for the sake of being specific that 0RM > .
Solving in this case Eq. (9), which is quadratic in terms of ρʹ′, we obtain a
differential equation in explicit form:
( )( )2 2
2
1
1 1
1
M M
M
ρ ρʹ′ = − ± + −
−
(10)
2. Equations and Forms of Synchronous Spirals
We will consider different cases.
Slowed spatial harmonics, 1M > . By introducing the parameter
2
1/ 1m M= − (11)
we reduce the differential Eq. (10) to
( )2 2 2
1m m mρ ρʹ′ = − + ± + (12)
and we integrate
2 2 2
1
.
1
d
m m m
ρ
ρ− + ± +
∫ (13)
By means of one of the Euler formulations [3] we reduce this integral to tabulated
integrals (see Appendix) and we obtain the equation of spirals in logarithms:
2 21
ln m
m
ϕ ρ ρ⎡= − + + ±
⎢⎣
m
0
2 2 2
2
2 2 2
1 1
1ln
1 1
m m
m
m m
ρ
ρ
ρ ρ
ρ ρ
⎤− + + + −
⎥± +
⎥− + + + + ⎦
m
m
(14)
The differential Eq. (12) and its solution (14) lead to different forms of
synchronous spirals depending on the choice of signs and values of the radius ρ .
For the upper sign the steady-state radius
1,стац стацr Rρ = = (15)
exists, corresponding to 0ρʹ′ = . If 1 ρ< < ∞, then the spiral approaches this
radius from the outside; we denote this type of spiral by the symbol SS1. If

5. 0 1ρ< < , then the spiral approaches this radius from the inside and we denote it
by SS2.
For the lower sign there is no steady-state radius, 0 ρ< < ∞, and we denote such
a spiral by SS3.
As pointed out above, each of these types of spirals can have two coil directions
depending on the sign of RM . The SS1, SS2, SS3 spirals, unwinding
counterclockwise in the direction of increasing ϕ , which corresponds to
sign 1, 1, 1RM = + − − in Eq. (9), are plotted in Figs. 2,a-2,c. As 0R → , the
dimensioned steady-state radius 0str → , i.e., the existence region of SS2
vanishes, and SS1 and SS3 are converted into the normal logarithmic spiral.
Accelerated spatial harmonics, 1M < . By assuming
2
1/ 1m M= − (16)
we obtain the differential equation
( )2 2 2
1m m mρ ρʹ′ = − − ± − (17)
the integral of which
2 2 2
1
1
d
m m m
ρ
ϕ
ρ
=
− − ± −
∫ (18)
is also calculated by means of the Euler formulation. As a result, we have
0
2
2
2
1 1
2arctg 1ln
1
m m
m
m m m
ρ
ρ
τ
ϕ τ
τ
⎡ ⎤− ± +
⎢ ⎥= + −
⎢ ⎥− ± −⎣ ⎦
m (19)
where
( )
2
/ / 1m mτ ρ ρ= + − (20)
It is obvious that the dimensionless radius ρ can vary here only within the limits
0 mρ< < . Just as in the case of slowed spatial harmonics, the steady-state radius
(15) exists for the upper sign. If 1 mρ< < , then the spiral curls from the outside
to this radius and we denote it by SS4. We denote the spiral lying within the limits
0 1ρ< < by SS5. For the lower sign there is no steady-state radius and the spiral,
denoted by SS6, lies within the limits 0 mρ< < . These types of synchronous
spirals, in which an acceleration of the spatial harmonics occurs, are shown in Figs.
2,d – d,f, respectively. Let us point out that the dimensioned radius r can vary
only within the limits 0 r m R< < therefore, the spirals SS4, SS5, SS6 do not
exist for 0R = .

6. 3. Synchronous Spiral as a "Pseudoperiodic" System with Spatial
Harmonic Selection
An important property of synchronous spirals is the following: each spatial
harmonic corresponds to its own curl density and shape of the spiral, defined by
the limiting radius nR and the slowing nM , or in a given spiral there is only one
spatial harmonic (or wave) with a phase velocity that is constant with radius. This
property provides for efficient wave selection even with an appreciable increase in
the size of the spiral, exceeding the wavelength. This property of the spiral is a
result of the fact that the period of the system is not constant with radius. The
distance between adjacent turns can be considered as a variable "period"
( ) ( ) ( )2L r rϕ ϕ π ϕ= + − , which varies along the radius in accordance with the
arc length along the spiral, providing for a constant phase velocity of the wave of
one spatial harmonic and destroying all other spatial harmonics.
This same principle can be taken as the basis of wave selection in other periodic
slow-wave systems also: by varying the period of the system along it and at the
same time one or several other parameters of the system, which determine the
phase change of the wave over a period, one can provide for a constant phase
velocity of one of the spiral harmonics along the system and can destroy the other
spatial harmonics. In such pseudoperiodic systems one can increase their
transverse dimensions without increasing the number of waves. This principle can
be used, for example, for systems in the form of a chain of coupled resonators or
spiral waveguides wound in accordance with the volume spiral law.
The winding for the synchronous spiral becomes more and more dense near the
limiting radius and this can be utilize to improve the properties of corresponding
slow-wave systems and antennas. Thus, one can use a waveguide wound in
accordance with the law of a planar synchronous spiral and pierced radially by
electron streams that are synchronous with the corresponding spatial harmonic of
the radial wave as the slow-wave structure in a radial TWT. In this case the
interaction efficiency of the electrons with the field is increased compared with the
case when a conventional logarithmic spiral is used because of the higher density
of turns.

7. Fig. 2. Spiral with slowed (a-c) and accelerated (d-f) spatial harmonics of radial wave for SSI (a),
SS2 (b), SS3 (c) for 0,1m = ; SS4 (d), SS5 (e) for 10m = ; SS6 (f) for 1,001m = .

8. CONCLUSION
Various synchronous spirals, corresponding to constant values of sυ and nT ,
have been considered in this paper. A further broadening of the class of
synchronous spirals is possible through the use of different functions ( )sυ ϕ and
( )nT ϕ , entering into the basic Eq. (5), which makes it possible to optimize the
properties of spirals when applied to specific systems.
The author is grateful to V. V. Stepanchuk for performing the calculations.
APPENDIX
CALCULATION OF INTEGRALS
To calculate the integral (13) we use the Euler formation:
2 2
m ρ τ ρ+ = + (A. l)
Converting to the new variable τ , we obtain
( )
2 2
1 m
m Q
τ
ϕ
τ τ
+
= ∫ (A. 2)
where
( ) 2 2
, , 2 1, 1Q a b c a m b m cτ τ τ= + + = = + =m (A.3)
and 2
4 4 0ac bΔ = − = − < .
We calculate the integrals [3] entering into Eq. (A. 2) in the following manner:
( ) ( )
1
ln ,
2 2
d b d
Q
Q c c Q
τ τ τ
τ τ
= −∫ ∫
( ) ( ) ( )
2
1
ln ,
2 2
d b d
Q a Q a Q
τ τ τ τ
τ τ τ
= −∫ ∫
( )
1 2
ln ,
2
d b c
Q b c
τ τ τ
τ τ
+ − −Δ
=
−Δ + + −Δ
∫
as a result, we obtain
2
2
2
1 1 1
ln 1ln ,
1 1
m
m
m m
τ
ϕ τ
τ
⎛ ⎞+ −
⎜ ⎟= ± +
⎜ ⎟+ +⎝ ⎠
m
m
m
(A. 4)
which leads to Eq. (14) after the formulation (A. l).
We calculate the integral (18) by means of another Euler formulation:
2 2
,m mρ ρτ− = − (A.5)
in this case

9. ( )( )
2
2 2
1
2
1
d
τ
ϕ τ
τ α βτ
−
= −
+ +∫ (A. 6)
where
2 2
1 , 1 1/ .m m m mα β α= − ± = − =m
By representing the integrand in the form of the sum
( )( ) ( )( ) ( )( )2 2 2 2
1 1
1 1
β
τ α βτ α β τ α β α βτ
= −
+ + − + − +
we arrive at the standard integrals:
21
2arctg 1lnm
m
α τ
ϕ τ
α τ
⎛ + ⎞
= + −⎜ ⎟−⎝ ⎠
m (A.7)
By means of the formulation (A. 5) we can also calculate the integral (13), in this
case we obtain
21 1
ln 1ln
1
m
m
τ β τ
ϕ
τ β τ
⎛ ⎞+ +
= − + +⎜ ⎟
− −⎝ ⎠
m (A. 8)
If one converts from the variable 2τ τ= , determined by the formulation (A.5),
to the variable 1,τ τ= determined by the formulation (A. 1), then we again obtain
Eq. (A.4).
REFERENCES
1. Ramsey, W. [name not verified]. Frequency-Independent Antennas [Russian
translation, title not verified]. Mir, Moscow, 1968.
2. Silin, R. A. and V. P. Sazonov. Zamedlyayushchiye sistemy (Slow-Wave
Structures). Sov. Radio, Moscow, 1966.
3. Gradshteyn, I. S. and I. M. Ryzhik. Tablitsy integralov, summ, ryadov i
proizvedeniy (Tables of Integrals, Sums, Series and Products). Nauka, Moscow,
1971.