Prof. David R. Jackson ECE Dept.

Spring 2016

Notes 9

ECE 6341

1

Circular Waveguide

a

z

TMz mode:

( ), ,zA zψ ρ φ=

( ) sin( )( ) cos( )

zjk zJ k

eY k

υ ρ

υ ρ

ρ υφψ

ρ υφ− =

2 2 2zk k kρ = −

2

εr

The waveguide is homogeneously filled, so we have independent TEz and TMz modes.

Circular Waveguide (cont.)

(1) φ variation [0, 2 ]φ π∈

( , 2 , ) ( , , )z zψ ρ φ π ψ ρ φ+ =

Choose

(uniqueness of solution)

cos( )nφ

nυ =

( )cos( )

( )z

n jk z

n

J kn e

Y kρ

ρ

ρψ φ

ρ− =

3

(2) The field should be finite on the z axis ( )0, , zψ φ

( )nY kρ ρ is not allowed

cos( ) ( ) zjk znn J k eρψ φ ρ −=

Circular Waveguide (cont.)

2 2 2zk k kρ = −

4

(3) B.C.’s: ( ), , 0zE a zφ =

( )

22

2

2 2

2

1

1

z

z

E kj z

k kj

kj

ρ

ψωµε

ψωµε

ψωµε

∂= + ∂

= −

=

so ( , , ) 0a zψ φ =

Circular Waveguide (cont.)

5

( ) 0nJ k aρ =Hence

( ) 0nJ k aρ =

xn1 xn2

xn3 x

Jn(x) Plot shown for n ≠ 0

npk a xρ =

Circular Waveguide (cont.)

npxk

aρ =

Note: is not included since (trivial soln.) 0 0nx = 0n npJ xaρ =

6

TMnp mode:

cos( ) 0,1,2zjk zz n npA n J x e n

aρφ − = =

1/ 222 1, 2,3,.........np

z

xk k p

a

= − =

Circular Waveguide (cont.)

7

Cutoff Frequency: TMz

npxk k

aρ= =

2 npc

xf

aπ µε =

2TM

c npr

cf xaπ ε

=

0zk =

2 2 2zk k kρ= −

8

Cutoff Frequency: TMz (cont.)

TM01, TM11, TM21, TM02, ……..

p \ n 0 1 2 3 4 5

1 2.405 3.832 5.136 6.380 7.588 8.771

2 5.520 7.016 8.417 9.761 11.065 12.339

3 8.654 10.173 11.620 13.015 14.372

4 11.792 13.324 14.796

xnp values

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TEz Modes

( ), ,zF zψ ρ φ=

cos( ) ( ) zjk znn J k eρψ φ ρ −=

2

z

kH

jρ ψ

ωµε=

( ), , 0a zψ φ ≠Note:

10

Set

( ), , 0E a zφ φ =

1Eφψ

ε ρ∂

=∂

TEz Modes (cont.)

so

( ) 0nJ k aρ′ =

Hence

11

0aρψρ =

∂=

∂

1,2,3,.....

np

np

k a xx

k pa

ρ

ρ

′=

′= =

( ) 0nJ k aρ′ =

TEz Modes (cont.)

12

x'n1 x'n2

x'n3 x

Jn' (x) Plot shown for n ≠ 1

Note: p = 0 is not included (see next slide).

1( ) ~ , 0,1,2,....2 !

nn nJ x x n

n =

Recall :

TEz Modes (cont.)

cos( ) 1,2,zjk zn npn J x e p

aρψ φ − ′= =

Note: If p = 0 0npx′ =

( )0 0n np nJ x Jaρ ′ = =

(trivial soln.) 0n ≠

0n = ( )0 0 0 1npJ x Jaρ ′ = =

zjk z jkze eψ − −= = (trivial fields)

13

0kρ =

Cutoff Frequency: TEz

npxk k

aρ

′= =

2 npc

xf

aπ µε

′=

2TE

c npr

cf xaπ ε

′=

0zk =

2 2 2zk k kρ= −

14

TE11, TE21, TE01, TE31, ……..

p \ n 0 1 2 3 4 5

1 3.832 1.841 3.054 4.201 5.317 5.416

2 7.016 5.331 6.706 8.015 9.282 10.520

3 10.173 8.536 9.969 11.346 12.682 13.987

4 13.324 11.706 13.170

x´np values

Cutoff Frequency:TEz

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TE11 Mode

TE10 mode of rectangular waveguide

TE11 mode of circular waveguide

The dominant mode of circular waveguide is the TE11 mode.

The TE11 mode can be thought of as an evolution of the TE10 mode of rectangular waveguide as the boundary changes shape.

Electric field Magnetic field

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(from Wikipedia)

Attenuation Property of TE01 Mode

2d

cf

α < >=

< >PP

212 td s

C

R H dl< > = ∫P

TEz Mode: 2

2

1 1

1

z

z z

FHj z

H k Fj

φ

ρ

ωµε ρ φ

ωµε

∂=

∂ ∂

=

Assume that Fz is order 1 as the

frequency increases.

Goal: We wish to study the high-frequency dependence of attenuation on frequency for circular waveguide modes, and show the interesting behavior of the TE01 mode (the loss decreases as frequency increases).

( )

1 12

2

sRσδ

σωµσ

ωµσ

ω

= =

=

= O

17 Recall that kρ is a constant.

Attenuation Property (cont.)

n = 0

1zH

ω =

O

(1) 00 0

nH

nφ

≠= =

O

( )( )

2

3/2

1 0d ωω

ω−

< > = +

=

O O

O

P

n ≠ 0

( ) ( )

( )2

1/2

1 1d ωω

ω

< > = +

=

P O O O

O

( ) ( )zk k ω= =O O

Note:

18

From the TEz table:

2

1(1)

1(1)

z

z

FE E

FH Hj z

φ

ρ

ε ρ

ωµε ρ

∂= =

∂

∂= =

∂ ∂

e.g.

e.g.

O

O

( )1f =P OHence

Attenuation Property (cont.)

19

If n = 0: ( )3/ 2cα ω−= O

If n ≠ 0: ( )1/ 2cα ω= O

E

Hn = 0:

Attenuation Property (cont.)

Hence

Note: The mode TE0p mode can be supported by a series of concentric rings, since there is no longitudinal

(z-directed) current (Hφ = 0).

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Usual behavior for rectangular waveguides

Decreases with frequency!

fc, TE11

f

αc

fc, TM01 fc, TE21

fc, TE01

TE01

TE21 TE11 TM11 TM01

Attenuation Property (cont.)

21

( )1/ 2cα ω= O

( )3/ 2cα ω−= O

Attenuation Property (cont.) The TE01 mode was studied extensively as a candidate for long-range communications – but was not competitive with antennas. Also, fiber-optic cables eventually became available with lower loss than the TE01 mode. It is still useful for some applications (e.g., high power).

From the beginning, the most obvious application of waveguides had been as a communications medium. It had been determined by both Schelkunoff and Mead, independently, in July 1933, that an axially symmetric electric wave (TE01) in circular waveguide would have an attenuation factor that decreased with increasing frequency [44]. This unique characteristic was believed to offer a great potential for wide-band, multichannel systems, and for many years to come the development of such a system was a major focus of work within the waveguide group at BTL. It is important to note, however, that the use of waveguide as a long transmission line never did prove to be practical, and Southworth eventually began to realize that the role of waveguide would be somewhat different than originally expected. In a memorandum dated October 23, 1939, he concluded that microwave radio with highly directive antennas was to be preferred to long transmission lines. “Thus,” he wrote, “we come to the conclusion that the hollow, cylindrical conductor is to be valued primarily as a new circuit element, but not yet as a new type of toll cable” [45]. It was as a circuit element in military radar that waveguide technology was to find its first major application and to receive an enormous stimulus to both practical and theoretical advance.

K. S. Packard, “The Origins of Waveguide: A Case of Multiple Rediscovery,” IEEE Trans. MTT, pp. 961-969, Sept. 1984.

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Attenuation Property (cont.)

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