1

Review of Transmission lines & Guiding structures

1) Theory of Transmission Lines

Telegraph equationsTransmission line parametersSmith chart

2) Guiding Structures

Coaxial cableWaveguideMicrostrip

Theory of transmission lines (eg coaxial cable)

I(z)

V(z)

Z’ d z I(z + d z)

V(z + d z)Y’ d z

dz z

a

bln

'2''C

0

V

Q

C '= Parallel capacity per unit of length (F / m)

L = Series inductance per unit length (H / m)

G '= Parallel conductance per unit of length (S / m)

R '= Series resistance per unit of length (Ω / m)

Z '= R' + jL ‘ Impedance per unit of length (Ω / m)

Y '= G ’+ j C' Admittance per unit length (S / m)

a

bln

2

''L

0

I

ψ

' '

' '

2

Theory of transmission lines(coaxial cable)

=====

tan'C

a

bln

''2

a

bln

2

V

'I'G D

0

c

CS

2R

=

tan = ''/'

D = dielectric Conductivity

C = metal conductivity

1

2𝑅′𝐼0

2 =1

2𝑅𝑒

𝑙1+𝑙2𝐸𝑡 ×𝐻𝑡

∗ ∙ 𝑛0 𝑑𝑙 = 1

2𝑅𝑆

𝑙1+𝑙2𝐻𝑡

2 𝑑𝑙

𝑅′ =𝑅𝑆 𝑙1+𝑙2

𝐻𝑡2 𝑑𝑙

𝑙1𝐻𝑡 𝑑𝑙

2= 𝑅𝑆 𝑏+𝑎

2𝜋 𝑎𝑏

Telegraph equations

z'Z

dz

zdI

V

z'Y

dz

zdV

I

zz eez VVV zz

C

eeZ

z

VVI

1

'Cj'G'Lj'R'Y'Zj

''

''

'

'

CjG

LjR

Y

ZjZZZ CjCrC

Direct and reflected voltage wave Direct and reflected current wave

Propagation constant

(attenuation and phase)

Characteristic impedance

Secondary parameters of the line

3

Coaxial cable

low losses ''''

2

1CLjZG

Z

RC

C

n

fcv

smc

v

cCL

rrr

rr

rr

000

8

00

/103

''≈

(phase speed)

'

2

1

4

12

2

1'

2

1≈

CCC

S

C Zab

ab

Z

ab

abR

Z

R(m-1)

(m-1)

n = index of refraction

Coaxial cable

a

b

a

b

C

LZ

r

C ln60

ln2

1

'

'

Typical value for cables and microstrips ZC = Z0= 50

]dB[l686.8eP

Plog10

P

Plog10A

l2IN

IN

OUT

INdB

=== Attenuation

]m/dB[686.8A m/dB =for l = 1m

4

zz

zz

ee

eeZ

z

zzZ

VV

VV

I

V0

z2

z

z

ee

ez

V

V

V

V

0

0

ZzZ

ZzZz

z

zZzZ

1

10

Impedance along the line

reflection coefficient along the line

Transmission lines terminated on loads

Z0

z = - l z = 0 z

ZL

j

L

LL e

ZZ

ZZ

0

0

LLL

L0L jXR

-1

1ZZ +=+

=

ljZlZ

ljZlZZlZ

L

L

sincos

sincos-

0

00

lZlZ

lZlZZlZ

L

L

sinhcosh

sinhcosh-

0

00

( ) lj2-l2-jl2-L eel- +==

For a "long" line with losses it results ( ) 0l- =

with losses without losses

5

Line terminated on a matched load

z = 0 z

ZL=Z0 Z0

z = - l

ZL = Z00

ZZ

ZZ

0L

0LL

Impedance, voltage and current

V(z)

I(z)

z z=0

z

R(z)

z=0

Z0

6

Line terminated on a resistive load

z = 0 z

ZL = R Z0

z = - l

ZL = R0L

0LL

ZR

ZR

Impedance, voltage and current

7

Line terminated on a short

ItanjZIZ 0

z = 0 z

ZL = 0 Z0

z = - l

ZL = 01

Z0

Z0

0

0L

Impedance, voltage and current

Series resonance

Parallel resonance

8

Resonant circuits

Short circuited Line

For l < lambda / 4

Leq = X(-l) / = (Z0 / ) tan l

For l < lambda / 12

Leq = Z0l / = Z0l / c = L'l

9

Line terminated on an open circuit

ljYlY tan0

z = 0 z

Z0

z = - l

ZL =

ZL = 1Z

Z

0

0L

ljZlZ cot0

Impedance, voltage and current

Series Resonance

Parallel Resonance

10

Line terminated on an open circuit

For l < lambda / 4

Ceq = B(-l)/ = (Y0 / ) tan l

For l < lambda / 12

Ceq = Y0 l / c = C'l

SWR

-1

1

V

VSWR

MIN

MAX +==

Standing Wave Ratio

(SWR) 1

1-WR

SWR

S

Matched load ZL = Z0 = 0 SWR = 1

Short Circuit ZL = 0 = 1 SWR =

Open Circuit ZL = = 1 SWR =

with no

losses

11

Impedance Smith Chart

CC 0

0.5

1.0

2.0

-0.5

-1.0

-2.0

0.5 1.0 2.0

P

CM CA

ZL = 100 + j50

Inductive Region

Capacitive Region

Load as a function of the frequency

Ideal Inductance Ideal Capacity

12

Coaxial Cable

Commercial productsflexible coaxial cables

13

Commercial productssemi-rigid coaxial cables

Commercial productsspline coaxial cables

14

EM field in the cable

00

0tt

00

tt

20

212

r

1

a

bln

V1ze

1h

rr

1

a

bln

Ve

C)rln(

a

bln

VC)rln(

a

bln

),r(

(r0,0,z0)

15

EM field in the cable

Unimodal frequency region

ba

vfc

)11(

The first higher order mode is the TE11

For this mode it results:

16

Cable RG 58/U

Unimodal band

a = 0.44 mm b = 1.46 mm (2a=0.035 inches, 2b=0.116 inches)

ft(TEM) = 0.0 GHz

ft(11) = v/((a+b)) = (3108/2.1)/(3.14 1.9 10-3) = 34.7 GHz

Characteristic impedance

506.49

44.0

46.1ln

21.2

120

a

bln

2Z0

Cable RG 58/U

Wire attenuation (copper)

3

7

99

10016.0107.5368

101.2102

g8

f =1 GHz

040.0

44.0

46.1ln

46.1

10

44.0

10

10016.0

a

bln

b

1

a

1

g8

33

3c

]dB[6,105,30686,8040.0 ft100c AdB

1 foot = 30,5 cm

100 ft = 30,5 m

17

rectangularwaveguide

18

Geometry of the Rectangular Waveguide

Commercial Products

19

Study of the EM Field in the Waveguide

Purely transverse electric field (TE): EZ = 0

Purely transverse magnetic field (TM): HZ = 0

TE Mode Transverse Solution (Rectangular Waveguide)

2

22

2

222t

b

n

a

mk

yb

ncosx

a

mcosA)y,x(hz

m = 0, 1, .. n = 0, 1, ..

the case m = n = 0 null field is excluded

eigenfunction

eigenvalue

20

EM field Longitudinal Solution

z

fz

1zzz

1tjzzjk

1e

dt

dzv0dzdt

ztsenePeePIm)t,z(Z

Progressive wave

Regressive wave

z

fz

1zzz

2tjzzjk

2e

dt

dzv0dzdt

ztsenePeePIm)t,z(Z

constant phasephase

velocity

phase

velocity

Phase Constant

2c

2tk

20

2

2

2c

z 111

22z

2t

2 kkk

2

222 1

c

czzz jk

(lossless waveguide)

For KZ = 0 we set: = C for which

For > C

Longitudinal transversal

f

fc

Condition of separability

21

Dispersion Curves

k

Unimodal Region

2

22

2

22

cb

n

a

m

2

cf

a

cff

a2

cff )0,2(c2c)0,1(c1c

2

22

2

222c

2t

b

n

a

mk

m = 0, 1, .. n = 0, 1, ..

with a > 2b

Unimodal region 1.25 fc1 < f < 0.95 fc2

1c

22

Example WR-90 waveuide

a = 22.86 mm b = 10.16 mm

(a=0.9 inches, b=0.4 inches)

ft(10) = c/2a = 3108/(2 22.86 10-3) = 6.56 GHz

ft(20) = c/a = 3108/(22.86 10-3) = 13.12 GHz

Banda unimodale

6.56 1.25 = 8.2 GHz 13.12 0.95 = 12.4 GHz

RF/microwave SystemsDenominations Frequency Interval GHz (109 Hz)

HF 0.003 - 0.030

VHF 0.030 - 0.300

P (Previous) UHF 0.300 - 1.000

L (Long) Band 1.0 - 2.0

S (Short) Band 2.0 - 4.0

C (Compromise) Band 4.0 - 8.0

X (Cross*) Band 8.0 - 12.0

Ku ( Under K) Band 12.0 - 18.0

K Band 18.0 - 26.5

Ka (Above K) Band Ka 26.5 - 40.0

Q Band 40.0 - 50.0

V Band 50.0 - 75.00

millimeter 40.0 - 300.0

Terahertz > 300.0

*Used in WW II for fire control, X for cross (as in crosshair).

23

Field in Rectangular Waveguide TE10 Mode

xa

cosA)x(hz

xa

jBsen)x(hx

x

ajCsen)x(ey

Transverse Electric Field

24

Longitudinal Electric Field

circularwaveguide

25

Geometry of the Circular Waveguide

TE Modes

ncosra

CJ,rh

nsinr

ra

J

nk

kC,rh

ncosra

Jak

kC,rh

0,re

ncosra

Jak

jC,re

nsinr

ra

J

nk

jC,re

]m,n[nz

]m,n[n

2]m,n[t

]n,m[z

]m,n[n

]m,n[

2]m,n[t

]n,m[zr

z

]m,n[n

]m,n[

2]n,m[t

]m,n[n

2]n,m[t

r

m-th zeros of the

n-th Bessel function derivative

m,n′

a

vf

mn

nmc

,

,,2

Cutoff frequency

Bessel function of the first kind

26

0,rh

ncosra

Jak

jC,rh

nsinr

ra

J

nk

jC,rh

ncosra

CJ,re

nsinr

ra

J

nk

kC,re

ncosra

Jak

kC,re

)m,n(n

]m,n[

2)m,n(t

c

)m,n(n

2)m,n(t

cr

)m,n(nz

)m,n(n

2)m,n(t

)n,m(z

)m,n(n

)m,n(

2)m,n(t

)n,m(zr

TM Modes

m,n

a

vf

mn

nmc

,

,,2

m-th zeros of the

n-th Bessel function

Cutoff frequency

Mode Spectrum

27

Fundamental TE11 Mode

Electric field lines

Magnetic field lines

Electrical Circular Modes TE0m

TE01 mode TE02 mode

Electric field lines

28

microstrip

Geometry of the Microstrip

metallic strip

dielectric substrate

ground plane

29

Quasi TEM Mode

Substrates

material

surface

finish (m)

104.tan

(10 GHz) r

Thermal

conductivity

(W/cm2/°C)

Alumina 99 % 0.25 1 - 2 10 0.37

Alumina 96 % 20 6 9 0.28

Alumina 85 % 50 15 8 0.20

Sapphire 0.025 0.7 9.4 0.4

Glass 0.025 20 5 0.01

Polyolefin 1 1 2.3 0.001

Duroid (Roger) 0.75-8.75 5-60 2-10 0.0026

Quartz 0.025 1 3.8 0.01

Beryllium 1.25 1 6.6 2.5

GaAs (high-res) 0.025 6 13 0.3

Silicio (high-res) 0.025 10-100 12 0.9

Air (dry) - 0 1 0.00024

30

Analysis Equations

'0

'

effC

Cε

effeff0c

effeff

CC

Cc

ZZ

'

0

0 1

Characteristic Impedance (t = 0)

0

50

100

150

200

250

300

0.1 1 10 w/h

r

1

2 3 4 6

16

10

Z0 [

31

Phase Constant

f

propagazione in aria

Modo dominante

modo quasi-TEM

propagazione nel dielettrico

modi di ordine

superiore

/c0

0

r

c

dielectric

propagationdominant

mode

quasi-TEM

mode

air

propagation

higher

order

modes

Higher Order Modes

x

y

x

y weff

a) b)

weff

eff

eff10c

w2

/cTEf

eff

eff20c

w

/cTEf

32

APPENDIX

wave impedance

good conductors

APPENDIX