transmission lines in mmic technology - unirc.it · 4 propagation in transmission lines definition...

1

Transmission lines in MMIC technology

Evangéline BENEVENT

Università Mediterranea di Reggio Calabria

DIMET

2


Propagation in transmission lines

Topologies of transmission lines

Planar transmission lines

Microstrip line

Coplanar waveguide

Coaxial cable

Waveguides

3





Microstrip line

Coplanar waveguide

Coaxial cable

Waveguides

4


Definition of propagation modes:

TE modes (Transverse Electric) have no electric field in the direction of

propagation.

TM modes (Transverse Magnetic) have no magnetic field in the direction of

propagation.

TEM modes (Transverse ElectroMagnetic) have no electric nor magnetic field

in the direction of propagation.

Hybrid modes have both electric and magnetic field components in the

direction of propagation.


Generally, transmission lines are used respecting certain conditions in order to only have TEM modes of propagation.

5


At low frequency:

At one time “t”, the points of the circuit linked by a conductor are at the same

potential:


∼∼∼∼EG

ZS

ZL

A

B

C

D

v (t) v (t)

))(())(( tVVtVV DCBA −=−

6


At high frequency:

When the frequency “f” of the signal increases enough, the differences of

potential between two points linked by a conductor have not equal values any

more:


))(())(( tVVtVV DCBA −≠−

∼∼∼∼EG

ZS

ZL

A

B

C

D

v (t) v‘ (t)

ℓ

7



At high frequency:

We can define the wavelength of the

signal which propagates at the

frequency “f” by the following

expression:

where c is the speed of the light in

vacuum:

)(mf

c=λ

)/(10.3 8 smc = 6 mm50 GHz

6 cm5 GHz

600 m500 kHz

60 km5 kHz

6000 km50 Hz

Wavelength λλλλFrequency f

Examples of wavelengths

at different frequencies:

8

Propagation in transmission lines:

Each time the wavelength is in the same order of the physical dimensions of

the conductor linking to points, it is necessary to take into account the

propagation effects:

If ℓ ≈ λ ⇒ propagation effects !

⇒ Theory of the transmission lines


Electrical length ≠ Physical length !

9


Equivalent circuit of a transmission line with a TEM (or quasi-TEM) propagation:

A transmission line is a structure with distributed elements (due to propagation effects),

The elements of the equivalent circuit are given in units of length:

Resistance: R (Ω/m),

Inductance: L (H/m),

Capacitance: C (F/m),

Conductance: G (S/m = Ω-1/m).


Associated to the conductors

Associated to the substrate

R ⇒ conductor losses ! G ⇒ substrate losses !

10


Two equivalent circuits for transmission lines:

Line without losses: Line with losses:

Z = jLωωωω

Y = jCωωωω Y = G + jCωωωω

Z = R + jLωωωω


Ldx

Cdx

x x + dx

Rdx Ldx

CdxGdx

x x + dx

11



Coupled differential equations ⇒ Equation of telegraphers:

By combining these two equations, we can obtain two independent equations:

Where γ is the propagation constant:

With α is the attenuation constant (Nepers/m) and β is the phase constant(rad/m).


ijLRx

v)( ω+−=

∂

∂vjCG

x

i)( ω+−=

∂

∂

vx

v²

²

²γ=

∂

∂i

x

i²

²

²γ=

∂

∂

( )( )ωωγ jCGjLR ++= βαγ j+= ZY=γ

12



The previous equations in voltage and current have as solutions a

combinations of waves propagating on the transmission line in opposite

directions:

Where Vi and Vr, and Ii and Ir are linked to the incident and reflected waves,

respectively.

ZC is the characteristic impedance:


)()()( xr

xi eVeVxV γγ += −

Y

Z

jCG

jLR

I

V

I

VZ

r

r

i

iC =

+

+=−==

ω

ω

( ))()()()( 1)( x

rx

i

C

xr

xi eVeV

ZeIeIxI γγγγ −=−= −−

13


Incident and reflected waves:

From the previous expressions of voltage and current through the transmission

lines:

We can define the expressions in voltage and current of the incident and

reflected waves:


)()()( xr

xi eVeVxV γγ += −

( ))()()()( 1)( x

rx

i

C

xr

xi eVeV

ZeIeIxI γγγγ −=−= −−

)()( xii eVxV γ−=

)()( x

C

ii e

Z

VxI γ−=

)()( xrr eVxV γ=

)()( x

C

rr e

Z

VxI γ−=

Incident wave Reflected wave

14


Reflection coefficient and impedance at each point of the transmission line:

Two systems of axis are considered:

Towards the load,

Towards the generator.


ZL

0

0

Lx

sL

Towards the load

Towards the generator

V

I

15



Reflection coefficient at each point of the transmission line:

Impedance at each point of the transmission line:

Reduced impedance:


)2(

)(

)()( x

i

r

i

r eV

V

xV

xVx γ==Γ

( ) )(1

)(1

1

1

1)(

)()(

)2(

)2(

)()(

)()(

x

xZ

eV

V

eV

V

Z

eVeVZ

eveV

xI

xVxZ C

x

i

r

x

i

r

Cx

rx

i

C

xr

xi

Γ−

Γ+=

−

+

=

−

+==

−

−

γ

γ

γγ

γγ

)(1

)(1)()(

x

x

Z

xZxz

C Γ−

Γ+==

16



The reflection coefficient can be expressed from the impedance:

At the abscissa x = L, the impedance is equal to the load impedance ZL:

Then:

If ZL = ZC then the reflection coefficient is null (ΓΓΓΓ = 0) and the transmission line is matched.


LZxZ =)(

C

C

ZxZ

ZxZ

xz

xzx

+

−=

+

−=Γ

)(

)(

1)(

1)()(

L

CL

CL

ZZ

ZZL Γ=

+

−=Γ )(

17



In the plane of the load (s=0), the reflection coefficient is:

We have seen that at each point of the transmission line:

So:

From these expressions, one can obtain the expression of the reduced impedance brought back at the entry:


i

r

i

rL

V

Ve

V

Vs ===Γ=Γ )0()0(

)2(

)(

)()( s

i

r

i

r eV

V

sV

sVs γ−==Γ

)2()( sLes γ−Γ=Γ

)(1

)()(

sthz

sthzsz

L

L

γ

γ

+

+=

18



In the case of a transmission line without losses, the propagation constant is

the following:

From the trigonometric properties, and replacing the propagation constant in the

expression of the impedance, one can obtained, for the reduced impedance brought back at the entry:


βγ j=

)(1

)()(

stgjz

sjtgzsz

L

L

β

β

+

+=

19


Stationary waves ratio:

Generally, the reflection coefficient is not null, they exist stationary waves along

the transmission lines, and we can define the stationary waves ratio:


L

L

V

VSWR

Γ−

Γ+==

1

1

min

max

20


Transmission lines terminated by particular load:

Transmission lines terminated by a matched load: ZL = ZC

ΓL = 0 ⇒ no reflection,

SWR = 1 ⇒ progressive wave,

z (s) = 1 or Z (s) = ZC ⇒ the impedance at each point is equal to the characteristic impedance of the transmission line,

The wave propagates along the transmission line at the phase speed:


)/( smvβ

ωϕ =

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Transmission lines terminated by a short-circuit: ZL = 0

ΓL = -1 ⇒ total reflection,

SWR = ∞,

The impedance at each point of the transmission line is equal to:


)()( sjtgsz β=

22



Transmission lines terminated by a open-circuit: ZL = ∞∞∞∞

ΓL = 1 ⇒ total reflection,

SWR = ∞,

The impedance at each point of the transmission line is equal to:


)(

1)(

sjtgsz

β=

23





Microstrip line

Coplanar waveguide

Coaxial cable

Waveguides

24

Topologies of transmission lines:

Choice of the topology of a transmission line:

Frequency of the signal,

Technology employed,

Power to transmit,

…Application…

Lines with several conductors:

Twisted pair line,

Coaxial cable,

Planar lines.

Transmission with only one conductor:

Waveguides.


25

Topologies of transmission lines:


Twisted pair line

Coaxial cable

Waveguides

26





Microstrip line

Coplanar waveguide

Coaxial cable

Waveguides

27

Planar transmission lines:

Advantages of planar transmission lines:

technology of integrated circuit,

planar conductors,

little dimensions,

propagation with a TEM or quasi-TEM mode.

Structure of planar transmission lines:

Generally, one conductor supports the signal,

Another is the ground plane,

A (dielectric) substrate is used as mechanical support.


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Planar transmission lines:

Different topologies of planar transmission lines:


Microstrip line Stripline Coplanar line Slotline

Coupled microstrip lines Coplanar stripsCoupled striplines

Etc…

The microstrip line and the coplanar waveguide are the most used transmission lines.

29





Microstrip line

Coplanar waveguide

Coaxial cable

Waveguides

30


Microstrip line:

Structure:

W: width of the conductor strip,

t: thickness of the conductor strip,

h: thickness of the substrate.

Characteristics of materials:

σ: conductivity of the conductor,

ε: permittivity of the substrate,

tgδ: dielectric loss tangent of the substrate.

EM field:

Quasi-TEM propagation.

W

h

t

εεεε, tgδδδδ

σσσσ

E

H

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Microstrip line:

Characteristic impedance:

Effective permittivity:

W

h

t


σσσσ

++=

22

1ln2 W

h

W

hC

ZZ

eff

vaccumc

επ

ab

rreff

W

h−

+

−+

+= 101

2

1

2

1 εεε

−−+=

7528.0

666.30exp)62(6W

hC π

[1] E. Hammerstad, O. Jensen, « Accurate models for microstrip computer-aided design », 1980MTT-S International Microwave Symposium Digest, Vol. 80, N°1, May 1980, pp. 407-409.

++

+

+

+=

3

4

2

4

1.181ln

7.18

1

432.0

52ln

49

11

u

u

uu

a

053.0

3

9.0564.0

+

−=

r

rbε

ε

h

Wu =

πε

1200

0 ≈=µ

Zvacuum)/(10.4 7

0 mHµ −= π

)/(10.842.8 120 mF−=ε

32

Microstrip line:

Example: εr = 10 (alumina), W = 5 µm to 1.54 mm.


0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016

5.6

5.8

6.0

6.2

6.4

6.6

6.8

7.0

7.2

7.4

Effective permittivity of a microstrip line vs. W

W (m)

Eff

ective

pe

rmittivity

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016

20

40

60

80

100

120

140

160

180

200

Characteristic impedance of a microstrip l ine vs. W

W (m)

Zc (

Oh

ms)

For Zc ≈ 50 Ω ⇒ W ≈ h

W = h = 635 µm

or

W = h = 1.54 mm

h = 1.54 mm

h = 635 µm

h = 1.54 mm

h = 635 µm

33

Microstrip line:

Total attenuation constant: αt

Attenuation due to the dielectric losses of the substrate: αd

Attenuation due to the losses in the conductors: αc

Rs is the sheet resistance. δ is the skin depth. σ is the conductivity of the strip.

Ki is the current distribution factor.

Kr is a correction term due surface roughness. ∆ is the effective (rms) surface roughness of the substrate.


cdt ααα +=

e

r

reff

reff

rd δ

λ

π

ε

ε

ε

εα tan

1

1

0−

−=

ir

c

sc KK

WZ

R..=α ( ) 1−

= δσsR

−=

7.0

2.1expvacuum

ci

Z

ZK

∆+=

2

4.1arctan2

1δπ

rKσω

δcµµ0

2=

34

Microstrip line:

Attenuation constant:

Example:

εr = 10

tanδe = 0.001

σ = 45 MS/m

W = 50 µm

h = 635 µm


0 2 4 6 8 10 12 14 16 18 20

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

Attenuation constant of a microstrip line vs. frequency

f (GHz)

Att

en

ua

tio

n c

on

sta

nt

(dB

/mm

)

ααααt

ααααd

ααααc

⇒ The conductor losses are the preponderant factor of losses for microstrip lines.

35

Microstrip line:

Operating frequency:

In order to prevent higher-order transmission modes, the thickness of the

microstrip substrate should be limited to 10 % of the wavelength.


36





Microstrip line

Coplanar waveguide

Coaxial cable

Waveguides

37


Coplanar waveguide:

Structure:

W: width of the conductor strip,

S: width of slots between central conductor and ground planes,

Wg: width of the ground planes,

t: thickness of the conductor strip,

h: thickness of the substrate.

Characteristics of materials:

σ: conductivity of the conductor,

ε: permittivity of the substrate,

tgδ: dielectric loss tangent of the substrate.

EM field:

Quasi-TEM propagation.

W

h

t


σσσσ

Wg S

H EE

38

)(

)('

4 1

1

kK

kKZZ

eff

vacuumc

ε=


Coplanar waveguide:


Effective permittivity:

W

h

t


σσσσ

Wg S

[2] C.P. Wen, “Coplanar waveguide: a surface strip transmission line suitable for non-reciprocal gyromagnetic device applications”, IEEE Trans. MTT, Vol. 17, p. 1087, 1969.

[3] K.C. Gupta, R. Garg, I.J. Bahl, “Microstrip lines and slotlines”, Artech House, Dedham, 1979.

)()('

)(')(

2

11

12

12

kKkK

kKkKreff

−+=

εε

d

Wk =1

K: elliptic integral

=

h

d

d

W

k

4sinh

4sinh

2π

π

SWd 2+=

−

+=

'1

'12ln

)('

)(

k

kkK

kK πFor 0 ≤ k ≤ 0.707

²1' kk −=

π

−

+

=k

k

kK

kK 1

12ln

)('

)( For 0.707 ≤ k ≤ 1

39

0 20 40 60 80 100 120 140 160 180 200

5.42

5.43

5.44

5.45

5.46

5.47

5.48

5.49

5.50

Effective permittivity of a colanar waveguide vs. W

W (µm)

Eff

ective

pe

rmittivity

0 20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

140

Characteristic impedance of a coplanar waveguide vs. W

W (µm)

Zc (

Oh

ms)

Coplanar waveguide:

Example: h = 635 µm, εr = 10 (alumina), S = 5 to 100 µm, W = 5 µm to 200 µm.


50

εeff ≈ (εr + 1)/2 = 5.5Several values of (W,S) for Zc = 50 Ω

S = 5 µm

S = 10 µm

S = 20 µm

S = 50 µm

S = 100 µm

40

Coplanar waveguide:

Propagation constant:

Different origins of losses:

Losses due to the dielectric substrate: αd

Losses due to the conductors: αc

Two additional phenomena at high frequencies:

Dispersion: εeff = f (f)

Losses by radiation: αr


⇒ Total attenuation αt = αd + αc + αr

41

Coplanar waveguide:

Phase constant:


)/(22

0

mradeff

g

ελ

π

λ

πβ ==

)(10.31 8

00

0 mfµff

c===

ελ

0 2 4 6 8 10 12 14 16 18 20

0

100

200

300

400

500

600

700

800

900

1000

Phase constant of a coplanar waveguide vs. frequency

f (GHz)

Ph

ase

co

nsta

nt

(ra

d/m

)

Example:

W = 20 µm

S = 10 µm

h = 635 µm

εr = 10

εeff = 5.5

Zc = 51 Ω

42

Coplanar waveguide:

Dispersion of the effective permittivity:


)/()(22

)(0

mradff eff

g

ελ

π

λ

πβ ==

[4] G. Hasnain, A. Dienes, J.R.Whinnery, “Dispersion of picosecond pulses in coplanar transmission lines”, IEEE MTT, Vol. MTT-34, N°6, June 1986, pp. 738-741.

2

1

)(

+

−+=

−b

TE

effr

effeff

f

fa

fεε

εε

fTE is the cutoff frequency of the lowest TE mode.

εeff is the initial effective permittivity calculated from the previous expressions.

εr is the relative permittivity of the substrate.

h is the height of the substrate.

²015.064.054.0 xxu +−=

²540.086.043.0 xxv +−=

8.1=b14

0

−=

r

TEh

cf

ε

vS

Wua +

= ln.)ln(

=

h

Wx ln

43

Coplanar waveguide:

Dispersion of the effective permittivity:


0 100 200 300 400 500 600 700 800 900 1000

5.4

5.6

5.8

6.0

6.2

6.4

6.6

6.8

7.0

7.2

7.4

Effective permittivity of a coplanar waveguide vs. frequency

f (GHz)

Eff

ective

pe

rmittivity

0 100 200 300 400 500 600 700 800 900 1000

0

10000

20000

30000

40000

50000

60000

Phase constant of a coplanar waveguide vs. frequency

f (GHz)

Ph

ase

co

nsta

nt

(ra

d/m

)

Example:

W = 70 µmS = 40 µmh = 635 µmεεεεr = 10

ββββ (f)

ββββ

εεεεeff (f)

εεεεeff

⇒ Very high frequency phenomena: f > 200 GHz

⇒ Not so high frequency phenomena if W ≈ h

44

Coplanar waveguide:


Losses due to the dielectric substrate: αd

εr is the relative permittivity of the dielectric substrate.

tanδe is the dielectric loss tangent of the dielectric substrate.

εeff is the effective permittivity of the transmission line calculated from the previous expressions.

The attenuation constant is expressed in Nepers/m. To be expressed in

dB, one can multiply the attenuation constant by 8.686 (1 Np = 8.686 dB).


)/(tan686.8)/(tan00

mdBqmNepersq e

eff

re

eff

rd δ

ε

ε

λ

πδ

ε

ε

λ

πα ==

[5] Rainee N. Simons“Coplanar waveguide circuits, components, and systems”, 2001 John Wiley and Sons, Inc.

)(

)('

)('

)(

2

1

1

1

2

2

kK

kK

kK

kKq =

45

Coplanar waveguide:


Losses due to the conductors: αC

R linked to the central conductor:

R linked to the ground planes:

Sheet resistance: and skin effect:


−

+−

+

−=

1

11

11 1

14ln

)²(²)1(4 k

kk

t

W

kKkW

RR s

c

ππ

−

+−

++

−=

1

1

111 1

1ln

1)2(4ln

)²(²)1(4 k

k

kt

SW

kKkW

RR s

g

ππ

σωδ

cµµ0

2=

)/(2

686.8)/(2

mdBZ

RRmNepers

Z

RR

C

gc

C

gc

c

+=

+=α

( ) 1−= δσsR

46

Coplanar waveguide:


Losses due to the radiation: αr

c0 = 3.108 m/s (light velocity)

K is an elliptic function.

εeff (f) can be evaluated from the previous expressions of the dispersion.

εr is the relative permittivity of the substrate.

W and S are the width of the central strip and of the slot of the CPW,

respectively.


[6] M.Y. Frankel, S. Gupta, J.A. Valdmanis, G.A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanartransmission lines”, IEEE MTT, Vol. 39, N°6, June 1991, pp. 910-916

SW

Wk

20

+=

²1' 00 kk −=

( ) 3

00

3

0

2/32

2

5

)()'(

2

)(

)(1

.2.2

fkKkKc

SW

f

f

r

r

eff

r

eff

r

ε

ε

ε

ε

ε

πα

+

−

=

47

Coplanar waveguide:

Total attenuation constant: αt


0 10 20 30 40 50 60 70 80 90 100

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Attenuation constant of a coplanar waveguide vs. frequency

f (GHz)

Att

en

ua

tio

n c

on

sta

nt

(dB

/mm

)

Example:

W = 40 µmS = 20 µmh = 635 µmεεεεr = 10tanδδδδe = 0.0001t = 4 µmσσσσ = 45 MS/m

ααααt

ααααc

ααααd

ααααr

-60 -40 -20 0 20 40 60 80 100 120 140

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

Attenuation constant of a coplanar waveguide vs. frequency

f (GHz)

Att

en

ua

tio

n c

on

sta

nt

(dB

/mm

)⇒ The losses due to the conductors are the main

factor of losses in planar transmission lines !

48





Microstrip line

Coplanar waveguide

Coaxial cable

Waveguides

49

Coaxial cable:

Features:

Widely used for high frequency measurements.

Ultra-large bandwidth.

The manufacturer guarantees that the coaxial cable correctly operates up to

a certain frequency.

Extremely low attenuation, depending on the dielectric material.

Various topologies:

choice depending on the frequency,

connections,

rigid or flexible cable…


50

Coaxial cable:

Structure:


TEM propagation:

[7] P-F. Combes, R. Crampagne, “Circuits passifs hyperfréquences, guides d’ondes métalliques”, Techniques de l’Ingénieur.

51

Coaxial cable:

a is the inner radius and b the outer radius.


From the transmission line theory, one can obtain the total attenuation:

Attenuation due to the dielectric material:

Attenuation due to the conductors:


=

a

bZ

r

c ln60

ε

dcC

C

t

GZ

Z

Rααα +=+=

22

fab

ab

bmdB

r

c)/ln(

)/(110.03.3)/(

9+

=− ε

α

fmdB erd .tan10.94.90)/( 9 δεα −=

52

Coaxial cable:

RLCG elements:

δ is the skin depth:


)/ln(2

)/( 0 abµ

mHLπ

=

)/ln(

2)/( 0

abmFC rεπε

=

δσπab

bamR

2)/(

+=Ω

eCmSG δω tan)/( =

σπδ

0

1

fµ=

53

Coaxial cable:

Cut-off frequency:

One can use a coaxial cable until the apparition of the first higher order

mode, the TE01 propagation mode.

The TE01 cut-off frequency is:


επ µba

fc

+=

2

1

54





Microstrip line

Coplanar waveguide

Coaxial cable

Waveguides

55

Waveguides:

Features:

Propagation into hollow (empty) waveguide.

Rectangular or circular section.

TEmn or TMmn mode of propagation:


Rectangularwaveguide

Circularwaveguide

⇒ Theory of transmission lines !

56

Waveguides:

Features:

Extremely high frequency.

High power applications.

Advantage of waveguides: not disturbed by the EM environment.

Even if a waveguide operates as a high-pass filter, the manufacturer guarantees that the waveguide correctly operates for a certain bandwidth because of disturbing higher frequency modes.

The bandwidth is directly linked to the size of the waveguide.


57


Waveguides:

Conventional rectangular waveguides:

1.27002.540075.0 – 110.0WR10

1.54943.098860.0 – 90.0WR12

1.8803.75950.0 – 75.0WR14

2.3884.77540.0 – 60.0WR18

…………

43.1886.362.20 – 3.30WR340

54.61109.221.70 – 2.60WR430

64.77129.541.45 – 2.20WR510

82.55165.101.12 – 1.70WR650

Height b (mm)Width a (mm)Frequency range (GHz)Waveguide type

58


Waveguides:

Propagation modes: TEmn or TMmn

m is the number of ½ wavelength variations of field in the “a” direction.

n is the number of ½ wavelength variations of field in the “b” direction.

Rectangular waveguide: Circular waveguide:

TE10 mode TE11 mode

a

b

59


Waveguides:

Rectangular waveguide: TE10 mode

The field expressions are normalized respect to the maximal value of Ey which is obtained at the center of the waveguide at x = a/2:

The fields of the fundamental mode are:

λ

λcEjE −=0µ

EHε

00 =

−=

g

y

zj

a

xEE

λ

ππ 2expsin0

−−=

gg

x

zj

a

xHH

λ

ππ

λ

λ 2expsin0

−=

gc

z

zj

a

xHjH

λ

ππ

λ

λ 2expcos0

a

ccf

c

c2

00 ==λ

ac 2=λ

2/1

²4

²1

−

−=

ag

λλλ

60


Waveguides:

Rectangular waveguide:

Attenuation in rectangular waveguide:

Pw is the power lost in the walls of the waveguide.

Pd is the power transmitted by the dielectric material.

ε is the permittivity of the material filling the waveguide.

c is the speed in the material filling the waveguide.

fc is the cutoff frequency of the waveguide.

2

1

22

1

2

1)/(

2/3

2/3

−

+

−==

f

f

f

f

b

a

f

f

caP

PmNepers

c

c

c

d

wt ρπεα

µc

ε

1=

61


Waveguides:

Circular waveguide: TE modes (fundamental = TE11 mode)

The fields of the TE modes are:

u

uJn

µHE nc )(

λ

λ

ερ −=

)(' uJµ

HjE nc

λ

λ

εθ =

)(' uJHjH n

g

c

λ

λρ −=

u

uJnHH n

g

c )(

λ

λθ −=

ρcku =

The value of H is fixed by measuring the

power transmitted by the waveguide.

a is the radius of the circular waveguide.

Jn is the Bessel’s function of first kind

and at the order n.

c

ckλ

π2=

mn

cu

a

'

2πλ =

U’0,1 = 3.831U2,1 = 5.136

U’2,1 = 3.054U1,1 = 3.832

U’1,1 = 1.841U0,1 = 2.405

u’mnumn

Roots and their derivatives of the Bessel’s functions

62


Waveguides:

Circular waveguide:

Attenuation in a circular waveguide for the TE11 mode:

(fundamental mode)

fc is the cutoff frequency of the circular waveguide.

a is the radius of the circular waveguide.

1

38.2

1

10.5.5)/(

2

2/32/1

2/3

5

−

+

=

−

−

c

cc

t

f

f

f

f

f

f

amNepersα

63


Evangéline BENEVENT

Università Mediterranea di Reggio Calabria

DIMET

Thank you for your attention !

transmission lines in mmic technology - unirc.it · 4 propagation in transmission lines definition...

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