cable&waveguide.pdf

7/29/2019 cable&waveguide.pdf

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2006 by Fabian Kung Wai Lee

Chapter 5: CoaxialComponents and

Rectangular WaveguideComponents

The information in this work has been obtained from sources believed to be reliable.The author does not guarantee the accuracy or completeness of any informationpresented herein, and shall not be responsible for any errors, omissions or damagesas a result of the use of this information.

November 2008 1

November 2008 2006 by Fabian Kung Wai Lee 2

References

[1] D.M. Pozar, Microwave engineering, 2nd edition, 1998 John-Wiley& Sons. (3rd edition, 2005 is also available from John-Wiley & Sons).

[2] R.E. Collin, Foundation for microwave engineering, 2nd edition,1992, McGraw-Hill.

[3] C.A. Balanis, Advanced engineering electromagnetics, 1989, John-

Wiley & Sons.


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5.1 Coaxial Components

Introduction

The microstrip and stripline structures are important for guidingelectromagnetic waves on printed circuit board (PCB).

For system-to-system or board-to-board, a cable is used for guidingelectromagnetic waves.

The most common cable type for this purpose is the coaxial cable, which

consists of two circular conductors, one is hollow and the other isusually solid, sharing a similar center axis (hence the name coaxial).

Although generally used for transporting high-frequency electrical signal,the coaxial cable can also be used for low-frequency signal by virtue of itbeing a two-conductor interconnection. One conductor would serve asthe signal and the other for the return current.

The coaxial cable can support TEM, TE and TM modes of propagatingelectromagnetic waves.



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Pictures of Coaxial Cables

Rigid coax cable

Semi-rigid coax cable

Flexible coax cable

Flexible Coax Cable Semi-rigid Coax Cable

Inner conductor

Outer conductor

Dielectric(usually

uniform) fillingthe gapbetweenconductors


Flexible Coaxial Cables

Velocity of Propagation:

Solid Dielectric = 66.5 - 69% of the speed of light in vacuum

Foamed Dielectric = 72 - 85% of the speed of light

centerconductordielectricinnershieldinnerjacketoutershieldouterjacket

centerconductor

dielectricbraid outerconductor

outerjacket

Single braid

Double braid


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Semi-Rigid Coaxial Cablesouter

conductordielectric center

conductor

outerconductor

dielectric centerconductor

SolidDielectric

AirArticulated

Typical dielectric typesare PTFE (Polytetraflouro-ethylene), foam and air.

Coaxial Cable Parameters (1)

The dominant propagation mode for electromagnetic waves in coaxialcable is TEM mode.

In this mode the RLCG parameters (under low-loss condition) are givenas follows:


)/ln(

'2

dDC

=

L D d=

2ln( / )

d

D

HE

''' j=

( )dDsR11

1

+=

)/ln(

''2

dDG

=

(1.1)

(1.2a)

(1.2b)

(1.2c)

(1.2d)

November 2008 8


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Coaxial Cable Parameters (2) From these basic RLCG parameters under TEM mode, other parameters

of interest such as characteristic impedance, attenuation factor, velocity ofpropagation, maximum power handling, cut-off frequency (when non-TEMmodes start to propagate) can be derived.

Other parameters which are influenced by the mechanical aspects of thecoaxial cable are flexibility of the cable, operating temperature range,connector type, cable diameter, cable noise or shielding effectiveness etc.

Of these, the most importance is the characteristic impedance Zc. Underlossless approximation, with R = G = 0, the characteristic impedance isgiven by:

Typical Zc values are 50, 75 and 93. Of these Zc = 50 is the mostcommon.


)/ln('2

1 dDCLZc

== (1.3)

Why Zc = 50 Ohms ? (1)

Most coaxial cables have Zc = 50 under low loss condition, with the75 being used in television systems. The original motivation behindthese choice is that an air-filled coaxial cable has minimum attenuationfor Zc = 75, while maximum power handling occurs for a cable with Zc= 30.

A cable with Zc = 50 thus represents a compromise between minimumattenuation and maximum power capacity.

Bear in mind this is only true for air-filled coaxial cable, but the traditionprevails for coaxial cable with other type of dielectric.


In the old days coaxial cable with Zc = 93 is also manufactured, theseare mainly used for sending digital signal, between computers. The capacitanceper unit length C is minimized for this impedance value.


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Why Zc = 50 Ohms ? (2)


1.0

0.9

0.8

0.7

0.6

0.5

1.1

1.2

1.3

1.4

1.5

20 30 40 50 60 70 80 90

NormalizedValue

Characteristic Impedance ()

50 standard

Attenuation islowest at 77

Power handling

capability is

highest at 30

Cable Specifications (1)

A series of standard types of coaxial cable were specified for military usesin the United States, in the form "RG-#" or "RG-#/U". These are datedback to World War II and were listed in MIL-HDBK-216 (1962).

These designations are now obsolete. The current US military standard isMilitary Specifications MIL-C-17. MIL-C-17 numbers, such as "M17/75-

RG214," are given for military cables and manufacturer's catalog numbersfor civilian applications.

However, the RG-series designations were so common for generationsthat they are still used today although the handbook is withdrawn.



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Cable Specifications (2)


Cable type Zo () Dielectric OverallDiameter

(inch)

Attenuation

(dB/100ft@

3GHz)

Maximum

power

(W@3GHz)

Capacitance

(pF/ft)

RG-8A 52 Polyethylene 0.405 16 115 29.5

RG-58C 50 Polyethylene 0.195 54 25 30.0

RG-174A 50 Polyethylene 0.100 64 15 30.0

RG-196A 50 Teflon 0.080 85 40 29.4

RG-179B 75 Teflon 0.100 44 100 19.5

RG-401 50 Teflon 0.250 (S) 14 750 28.5

RG-402 50 Teflon 0.141 (S) 21.5 250 28.5RG-405 50 Teflon 0.086 (S) 34 90 28.5

Upper Usable Frequency

Since a coaxial cable supports TEM, TE and TM electromagnetic wavepropagation modes, the TE and TM modes will come into existent forsufficiently high operating frequency.

The Upper Usable Frequency (UUF) for coaxial cable refers to thefrequency where the first non-TEM mode comes into existent.

For coaxial structure, the non-TEM mode with the lowest cut-offfrequency (fc) is the TE11 mode.

The UUF can be estimated by [1]:


dDdkc + 1

4

UUFf cCk

c==

'2

(1.4a)

(1.4b)d

D


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Connectors and Adapters (1)

The ends of co-axial cable are connected to connectors.

Typical RF/microwave connectors are:

BNC (Baby N connector), for d.c. to 300MHz.

N connector, for d.c. to 6GHz.

3.5mm or SMA connectors, for d.c. to 20GHz.

2.5mm connectors, for d.c. to 40GHz.

BNC connectors N connectors

3.5mm/SMA connectors


Connectors & Adapters (2)

BNC male

BNC female

TNC male

TNC femaleAPC-7

SMA male

SMA female

APC-3.5 male SMC male Type-N male

Type-N female

2 GHz

50 or 75 ohm

Threaded BNC

12 GHz

sexless, 18 GHz

low , repeatable

24 GHz 34 GHz

low VSWR

7 GHz

small size

12 - 18 GHz

50 or 75 ohm

Examples of RF/microwave connectors and adapters from variouscompanies


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Attenuators (1)

T-section

-section


Example of programmable PIN Diode attenuator

Attenuators (2)


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Terminations

20-dB attenuator

Short-circuit plate

Short-circuit plate

Tapered load

A range of terminations


5.2 - Rectangular Waveguide


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Ex

Ey

Ez

~ ~$E E E at z z= +

Rectangular Waveguide (1)

Examples ofrectangular waveguides

November 2008 21


EM Waves Propagating Modes inWaveguide

Ey

x

Ex

y

Ey

x

y

Ex

Ey

x

TE1,0 TE1,1 TE2,1

y

Ex

( ) ( )fc

m a n bc = +2

2 2/ /

( ) ( )c

m a n b=

+

2

2 2/ /

Cutoff frequency for TEmnor TMmn mode

The corresponding cutoff wavelengthfor TEmn or TMmn mode

Thus we see that for m=1,n=0, the TE10 has the lowestcut-off frequency, and is thedominant mode.

November 2008 22


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Dominant Mode for Rectangular

Waveguide

TE10

mode

a

b

E

( ) ( )c

m a n ba=

+=

22

2 2/ /

vg =

c=

Recommended operatingfrequency range forrectangular waveguide

For TE10 mode, m=1, n=0,thus the cut-off wavelengthis:

November 2008 23


Waveguide Wall Currents

g/2g/2

The current on the wall of the rectangular waveguide at a certain instance intime for TE10 mode.

We can create a slotparallel to the centeraxis of the waveguide

without disturbing thecurrent flow, hence theinternal EM fields.

November 2008 24


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Slotted-Line Probe

A

Variable Short-

circuit tuner

Slotted

waveguide

SWR meter

A probe can be inserted into the slot with minimal disturbance to the EM fieldsof the TE10 mode. This probe can be used to measure the relative strength of theelectric field in the waveguide cavity. Usually the microwave EM fields will bemodulated by a low-frequency signal, and the diode/capacitor pair acts as envelopedetector to demodulate this low-frequency signal.

November 2008 25


a

b

E

E

a

b

H H

Bends and Twists

E-bend H-bend

Twist

November 2008 26


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Matched Terminations

short circuit

l 1

l2

l2

l1

Lossy material

The gradual transition from the rectangular waveguide to the lossy materialensures minimal reflection of the electromagnetic wave.

Usually heatsink will beattached to the exterior for

heat dissipation

WaveguideHeatsink

November 2008 27


Attenuators

An example of attenuator, here the lossy material is shaped so as to providegradual change in the waveguide internal geometry, resulting in small reflectionof incident wave.

Tapered lossy material

November 2008 28


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1

2

3

1

2

3

1

23

1

23

EH

E-plane Tee H-plane Tee

=

+

+

+

3

2

1

3

2

1

011

101

110

2

1

V

V

V

V

V

V

=

+

+

+

3

2

1

3

2

1

011

101

110

2

1

V

V

V

V

V

V

Often used aspower dividers

Waveguide Tees

November 2008 29


Bdc

ferrite

post

Circulator

Example of a rectangular waveguide circulator (see Chapter 4):

November 2008 30


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Waveguide to Co-axial Adapter

Coaxial cable

Waveguide

Shortcircuit

November 2008 31


Appendix 1.0 Solution ofElectromagnetic Fields for

Rectangular Waveguide


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Introduction Rectangular waveguides are one of the earliest waveguide structures

used to transport microwave energies.

Because of the lack of a center conductor, the electromagnetic fieldsupported by a waveguide can only be TM or TE modes.

For rectangular waveguide, the dominant mode is TE, which has thelowest cut-off frequency.

Why No TEM Mode in Waveguide?

From Maxwells Equations, the magnetic flux lines always close uponthemselves. Thus if a TEM wave were to exist in a waveguide, the fieldlines of B and H would form closed loop in the transverse plane.

However from the modified Amperes law:

The line integral of the magnetic field around any closed loop in atransverse plane must equal the sum of the longtitudinal conduction anddisplacement currents through the loop.

Without an inner conductor and with TEM mode there is no longtitudinalconduction current and displacement current inside the waveguide.Consequently this leads to the conclusion that there can be no closedloops of magnetic field lines in the transverse plane.


+=+=

S

t

C

tsdDIldHDJHrrrrrrrr


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Rectangular Waveguide The perspective view of a rectangular waveguide is referred below.

The following slides shall illustrate the standard procedures of obtainingthe electromagnetic (EM) fields guided by this structure.

z

y

x0 a

b

Assume to be very

good conductor (PEC)and very thin


TE Mode Solution (1)

To obtain the TE mode electromagnetic (EM) field pattern, we use thesystematic procedure developed in Chapter 1 Advanced TransmissionLine Theory.

We start by solving the pattern function for z-component of the magneticfield:

Problem (1.1) is called Boundary Value Problem (BVP) in mathematics.

Once we know the function of hz(x,y), the EM fields are given by:

22222 ,0 ==+ oczczt kkhkh

zehyexeH zjzzj

yzh

ck

jzjxzh

ck

j

22

+

+

=

r

yexeE zjxzh

ck

jzjyzh

ck

j

22

+

=

r

and boundary conditions (1.1)

(1.2b)

(1.2a)

Note: Here we only considerpropagation in positive

direction, treatment for

negative propagationis similar.


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TE Mode Solution (2) Expanding the partial differential equation (PDE) of (1.1) in cartesian

coordinates:

Using the Separation of Variables Method, we can decompose hz(x,y)into the product of 2 functions and kc

2 to be the sum of 2 constants:

Putting these into (1.2), and after some manipulation we obtain 2

ordinary differential equations (ODEs):

( ) ( ) ( )yYxXyxhz =,

( ) 0,22

2

2

2=

++

yxhk zcyx

002

2

21

2

212

2

2

2

2=++=++

c

x

YYx

XXcx

Y

x

X kXYkXY

222yxc kkk +=

22

21

xx

XX

k=

22

21

yx

YY

k=

(1.2)

(1.4a) (1.4b)

222

21

2

21

yxx

YYx

XX

kk =+

(1.3a) (1.3b)



From elementary calculus, we know that the general solution for (1.4a)and (1.4b) are:

Thus hz(x,y) is given by:

( ) ( ) ( )

( ) ( ) ( )ykDykCyYxkBxkAxX

yy

xx

sincos

sincos

+=

+= (1.5a)

(1.5b)

( ) ( ) ( )[ ] ykDykCxkBxkAyxh yyxxz sincossincos, ++= (1.6)


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TE Mode Solution (4) A, B, C and D in (1.6) are unknown constants, to be determined by

applying the boundary conditions that the tangential electric field mustvanish on the conductive walls of the waveguide. From (1.2b):

yexeyExEEzj

xzh

ck

jzj

yzh

ck

jyx 22

++

+

=+=

r

z

y

x0 a

b

(1.7a)

( ) ( )

( ) ( )0

0,0,

,0,==

==

y

bxzh

y

xzh

xx bxExE

( ) ( )

( ) ( )0

0,,0

,,0==

==

x

yazh

x

yzh

yy yaEyE

(1.7b)



Using (1.6) and applying the boundary condition (1.7a):

Using (1.6) and applying the boundary condition (1.7b):

In the above equations, we can combine the product of AC, lets call itR. Common sense tells us that R would be different for each pair ofinteger (m,n), thus we should denote R by: Rmn

( ) ( )[ ] ( ) ( )[ ]ykDykCxkBxkAk yyxxyyzh cossinsincos ++=

( )00

0,==

D

y

xzh ( )L3,2,1,0,0

,===

nk

bn

yy

bxzh

( ) ( )[ ] ( ) ( )[ ]ykDykCxkBxkAk yyxxxxzh sincoscossin ++=

( )00

,0==

B

x

yzh ( )L3,2,1,0,0

,===

mk

am

xx

yazh


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TE Mode Solution (6) From (1.3b), kc and the propagation constant are given by:

Since kc and also depends on the integer pairs (m,n), it is moreappropriate to write these as:

( ) ( )22b

na

mmnck

+=

( ) ( )2222a

mb

nyxc kkk

+=+=

( ) ( )222

22

bn

am

o

co

k

kk

=

=

( ) ( ) == obn

am

omn kk ,222

(1.7a)

(1.7b)



With these information, and using (1.2a) and (1.2b), we can write out thecomplete mathematical expressions for the EM fields under TEpropagation mode for a rectangular waveguide:

( ) ( ) ( ) zmnjb

na

mmnb

n

mnck

jx eyxRE

+

= sincos2

( ) ( ) ( ) zmnjb

na

mmna

m

mnck

jy eyxRE

+

= cossin

2

( ) ( ) ( ) zmnjb

na

mmna

m

mnck

mnjx eyxRH

+

= cossin2

( ) ( ) ( ) zmnjb

na

mmnb

n

mnck

mnjy eyxRH

+

= sincos2

zmnj

bn

am

mnz eyxRH + = sincos

(1.8a)

(1.8b)

(1.8c)

(1.8d)

(1.8e)


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Cut-Off Frequency for TE Mode (1) Notice from (1.7b) that the propagation constant mn is real when:

When mn is imaginary the EM fields cannot propagate.

Since =2f, we can define a limit for the frequency f as follows:

The lower limit of this frequency is called the Cut-off Frequency fc.

( ) ( )222b

na

mok

+>

( ) ( )2222b

na

mok

+>=

( ) ( )222

1b

na

mf

+>

( ) ( )222

1b

na

m

mnTEcf

+= (1.9)


Cut-Off Frequency for TE Mode (2)

The TE mode electromagnetic field is usually labeled as TEmn since themathematical function of the field components depend on the integerpair (m,n).

The pair (m,n) cannot be both zeros, otherwise from (1.8a) to (1.8e), Ex+,

Ey+, Hx

+, Hy+, Hz

+ are all zero, no fields at all! This is a trivial solution,

although a valid one. The smallest combination of (m,n) are (m,n) = (1,0) or (0,1).

Since a > b (the lateral dimensions of the rectangle), we see that (m,n) =(1,0) produces a smaller fc, thus lower cutoff frequency. Therefore theTE propagation mode TE10 is the dominant mode for TE waves. Itscorresponding cut-off frequency is given by:

Only excitation frequency greater than will cause EMwaves to propagate within the rectangular waveguide.

( )

aaTEcf

2

12

2

1

10==

aTEcf

2

1

10=


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Example Consider a rectangular waveguide, a = 45.0mm, b = 35.0mm, filled with

air ( = o, = o).

7

12

104

10854.8

=

=

o

o

9

045.02

1

1010331.3 ==

ooTEc

f


Phase Velocity

Since phase velocity vp depends on propagation constant mn, it toodepends on the integer pair (m,n) hence the property of the TE modefields.

We thus observe that the phase velocity of TE mode has the peculiarproperty of traveling faster than the speed of light!

( ) ( ) ok

bn

am

okmn

pv

>==

222

(1.10)

Speed of light in dielectric of (,)


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TM Mode Solution (1) The procedure for obtaining the EM field solution for TM propagation is

similar to the TE procedure.

We start by solving the pattern function for the z-component of theelectric field:

As in solving TE mode problem, the Separation of Variables Method isused in solving (1.11), and integer pair (m,n) needs to be introduced inthe TM mode solution.

The mathematical expressions for the EM field components thusdepends on the integer pair (m,n), and is denoted by TMmn field.

The derivation details will be omitted here due to space constraint. Youcan refer to reference [1] for the procedure.

22222 ,0 ==+ oczczt kkeke and boundary conditions (1.11)


TM Mode Solution (2)

The complete expressions for the TMmn field components are shownbelow:

( ) ( ) ( ) zmnjb

na

mmna

m

mnck

mnjx eyxRE

+

= sincos2

( ) ( ) ( )zmnj

b

n

a

m

mnb

n

mnck

mnj

y eyxRE

+

= cossin2

( ) ( ) ( ) zmnjb

na

mmnb

n

mnck

jx eyxRH

+

= cossin2

( ) ( ) ( ) zmnjb

na

mmnb

n

mnck

mnjy eyxRH

+

= sincos2

( ) ( ) zmnjb

na

mmnz eyxRE

+ = sinsin

(1.12a)

(1.12b)

(1.12c)

(1.12d)

(1.12e)


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TM Mode Solution (3) Where

( ) ( )22b

na

mmnck

+=

( ) ( ) == obn

am

omn kk ,222

(1.13a)

(1.13b)


Cut-Off Frequency for TM Mode

Since the propagation constant mn is similar for both TEmn and TMmnmode, a cut-off frequency also exists for TMmn :

Observe that from (1.12a) to (1.12e) the EM field components become 0if either m or n is 0. Thus TM00 , TM10 and TM01 do not exist. Thelowest order mode is TM11.

It is for this reason that we consider TE10 to be the dominant mode ofrectangular waveguide.

( ) ( )222

1b

na

m

mnTMcf

+= (1.14)

( ) ( ) ( )10

2

2

122

2

1

11 TEcabaTMc

ff =>+=


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Appendix 2.0 Plots ofElectromagnetic Fields for

Rectangular Waveguide


TE10 Mode

E

H


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TE01

E

H


TE11

E

H


28/28


TM11

E

H

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