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*

--------------------------------------------------------------------------------------- *Originally published in Radiotekhnika i elektronika, No. 4, 1994, pp. 552-559. --------------------------------------------------------------------------------------------------------------------

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, ,mr r m r r e ϕϕ= = (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

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= :

( ) ( ) ( ) ( )( )

2 222,

n s

r rr rd nT

ϕ π

ϕ

ϕ ϕϕ π ϕϕ

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

( ) ( ) ( ) ( )( )

( ) ( )( )

2 2 2 22 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

,nn s

r rrT

ϕ ϕϕϕ

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..

( )21

2 n nT d Tϕ π

ϕ

ϕ ϕπ

+

=∫ % % (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 22 2 2

21 1 1 0nn

n s n s

Tr r r Tυ 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 21 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

( )20 1M r rʹ′− = ± , (7)

having an exponential solution of the form (1) for ( )201/ 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)

and sign sign / nRM n υ= is the sign of the product RM . Then the equation of synchronous spirals becomes

( )2 2 21 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 22

1 1 11

M MM

ρ ρʹ′ = − ± + −−

(10)

2. Equations and Forms of Synchronous Spirals

We will consider different cases. Slowed spatial harmonics, 1M > . By introducing the parameter

21/ 1m M= − (11) we reduce the differential Eq. (10) to

( )2 2 21m m mρ ρʹ′ = − + ± + (12)

and we integrate

2 2 2

1 .1d

m m mρ

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 mm

ϕ ρ ρ⎡= − + + ±⎢⎣m

0

2 2 22

2 2 2

1 11ln

1 1

m mmm m

ρ

ρ

ρ ρ

ρ ρ

⎤− + + + −⎥± +⎥− + + + + ⎦

mm

(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

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

21/ 1m M= − (16) we obtain the differential equation

( )2 2 21m m mρ ρʹ′ = − − ± − (17)

the integral of which

2 2 2

11d

m m mρ

ϕρ

=− − ± −∫ (18)

is also calculated by means of the Euler formulation. As a result, we have

0

22

2

1 12arctg 1ln1

m mmm m m

ρ

ρ

τϕ 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 = .

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.

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 = .

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 2m ρ τ ρ+ = + (A. l)

Converting to the new variable τ , we obtain

( )

2 21 mm Q

τϕ

τ τ+

= ∫ (A. 2)

where

( ) 2 2, , 2 1, 1Q a b c a m b m cτ τ τ= + + = = + =m (A.3)

and 24 4 0ac bΔ = − = − < . We calculate the integrals [3] entering into Eq. (A. 2) in the following manner:

( ) ( )1 ln ,2 2

d b dQQ c c Qτ τ ττ τ

= −∫ ∫

( ) ( ) ( )

21 ln ,2 2

d b dQ a Q a Qτ τ τ ττ τ τ

= −∫ ∫

( )1 2ln ,

2d b cQ b cτ τ ττ τ

+ − −Δ=

−Δ + + −Δ∫

as a result, we obtain 2

2

2

1 1 1ln 1ln ,1 1

mmm m

τϕ τ

τ

⎛ ⎞+ −⎜ ⎟= ± +⎜ ⎟+ +⎝ ⎠

mmm

(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

( )( )2

2 2

121

dτϕ τ

τ α βτ

−= −

+ +∫ (A. 6)

where 2 21 , 1 1/ .m m m mα β α= − ± = − =m

By representing the integrand in the form of the sum

( )( ) ( )( ) ( )( )2 2 2 2

1 11 1

β

τ α βτ α β τ α β α βτ= −

+ + − + − +

we arrive at the standard integrals:

21 2arctg 1lnmm

α τϕ 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 1ln 1ln1

mm

τ β τϕ

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.