The configuration of circumferential rectangular slots on a cylindrical waveguide is shown in
The backward (B) and forward (C) scattered field amplitudes are given by [
B m n = ∫ slot E s ¯ × H ¯ t C ⋅ d s ¯ 2 ∫ S E ¯ a t × H ¯ a t ⋅ u ¯ z d s (1)
C m n = ∫ slot E s ¯ × H ¯ t B ⋅ d s ¯ 2 ∫ S E ¯ a t × H ¯ a t ⋅ u ¯ z d s (2)
where subscripts B and C represent the amplitudes of backward and forward waves, t indicates the tangential field in the cross-section, S indicates the cross- section of uniform waveguide, and “slot” shows the slot surface area.
The field components of TM01 mode in the cylindrical waveguide are:
E z = J 0 ( h ρ ) e ± j β 01 z (3)
E ρ = ± j β 01 h J ′ 0 ( h ρ ) e ± j β 01 z (4)
H φ = − j ω ϵ h J ′ 0 ( h ρ ) e ± j β 01 z (5)
E φ = H ρ = H z = 0 (6)
where β 01 is the phase constant and h = 2.405 a is cut-off number with a being
the radius of waveguide.
The tangential electric field on the n’th aperture is:
E z = V n W cos ( π ( φ − φ 0 ) α n ) { z n − W 2 < z < z n + W 2 φ 0 − α n 2 < φ < φ 0 + α n 2 W → 0 (7)
E ¯ s = E z u ¯ z , E φ = 0 (8)
where α n is the angle of n’th slot. The other field component are:
H ¯ t B = H φ u ¯ φ (9)
H ¯ t C = H φ u ¯ φ (10)
H ¯ a t = H φ u ¯ φ (11)
E ¯ a t = E ρ u ¯ ρ (12)
These field components are substituted in Equations (1) and (2) to obtain:
C 01 = B 01 = j V n h J 1 ( h a ) a β 01 π 2 α n π [ J 1 2 ( h a ) − J 0 ( h a ) J 2 ( h a ) ] (13)
Observe that the forward and backward traveling wave amplitudes are equal. Therefore the transmission line equivalent circuit consists of a parallel admittance.
The first design equation is then derived. The reflected power from the aperture is:
P r e f = 1 2 R e ∫ S ( E ¯ t × H ¯ t * ) ⋅ d s ¯ = 1 2 R e ∫ ρ = 0 a ∫ φ = 0 2π ( B 01 E 01 , t × B 01 * H 01 , t * ) ⋅ u z ¯ ρ ′ d ρ ′ d φ ′ = π β 01 ω ϵ h 2 a 2 2 [ J 1 2 ( h a ) − J 0 ( h a ) J 2 ( h a ) ] B 01 B 01 * (14)
where
E ¯ t = { B 01 E ¯ a t z < z 1 C 01 E ¯ a t z > z 2 (15)
The equations of the equivalent transmission line are:
{ V ( z ) = A e − j β z ︷ V + + B e j β z ︷ V − I ( z ) = A G 0 e − j β z ︸ I + − B G 0 e j β z ︸ I − (16)
which give the reflected power:
P r e f , T L = 1 2 V − ( z ) I − * ( z ) = − 1 2 B B * Z 0 (17)
The equality of reflected powers due to the scattered fields and the transmission line leads to the following relation:
− 1 2 B B * Z 0 = π β 01 ω ϵ 0 h 2 a 2 2 [ J 1 2 ( h a ) − J 0 ( h a ) J 2 ( h a ) ] B 01 B 01 * (18)
However, the scattered voltage amplitudes are [
B = C = − 1 2 Y n a G 0 V n (19)
The amplitudes B and B01 from Equations (19) and (13) are then substituted in (18) to obtain the first design equation:
Y n a G 0 = − j 4 ω ϵ 0 J 1 2 ( h a ) G 0 β 01 π 5 [ J 1 2 ( h a ) − J 0 ( h a ) J 2 ( h a ) ] V n s V n α n (20)
where the Bessel functions J 0 , J 1 and J 2 are calculated for ha = 2.405 for TM01 mode.
For the derivation of the second design equation, the procedure described by Elliott ( [
Y n a G 0 = η 2 2 G 0 | K | 2 α n 2 z n d , α (21)
where K = − j 4 ω ϵ 0 J 1 2 ( h a ) G 0 β π 5 [ J 1 2 ( h a ) − J 0 ( h a ) J 2 ( h a ) ] , η = 120 π is the intrinsic
impedance of medium and G 0 is characteristics admittance of cylindrical
waveguide. Equation (21) can be written as Y n a G 0 = K 1 α n 2 Z n d , a where K 1 = η 2 2 G 0 | K | 2 . Z n d , a is the active admittance of equivalent dipole that is assumed in the derivation of second design equation. we have Z n d , a = Z n n + Z n b where:
Z n n : Self impedance of circumferential slot, which is equal to K 1 α n 2 Y n s e l f / G 0
Z n b : Mutual impedance between circumferential slots on the cylindrical waveguide, which is equal to ∑ m = 1 m ≠ n N V m s V n s Z n m d .
Then
Z n d , a = Z n n + Z n b = K 1 α n 2 Y n s e l f / G 0 + ∑ m = 1 m ≠ n N V m s V n s Z n m d
Z n m d is mutual impedance between two dipole which may be obtained from the mutual admittance between two slots Y n m s by the Booker’s relation.
Z n m d = ( η 2 2 ) Y n m s (22)
The second design equation is then determined by these relations.