Transmission lines:wave propagation

z

v+(z)

A transmission line is a physical structure with the cross-section (x-y plane) of definite shape and the developing indefinitely along the normal direction (z).At each coordinate z the voltage V(z) and the current I(z) are represented as a combination of waves propagating along the positive and negative direction of z:

i+(z)

( ) ( ) ( ), ( ) ( ) ( )v z v z v z i z i z i z+ − + −= + = +

Physical parameters of the line

Sezione ∆zL ∆z R ∆z

C ∆z G ∆z

R, L, C, G, are the physical parameters of a transmission line. They dependon the geometry of the transmission line and on the materials used for theconductors and dielectric medium .

Through the physical parameters and the Kirckoff laws the TelegraphistEquations can be derived, whose solution is the voltage (current) wavepropagating along the line:

00( ) ( )j t zv z V e eω γ+ − ⋅= ⋅

Meaning of propagation constant γ

Given γ=α+jβ, it has:

( )0( ) j t z j zv z V e e eω α β+ − ⋅ − ⋅ = ⋅ ⋅

• Attenuation constant α: it defines the rate of reduction(in space) of the wave amplitude. It is measured inNeper/m or dB/m (1Np = 8.686 dB)

• Phase constant β: it is the rate of variation of thephase of the wave along the z coordinate (for t=cost).It is related to the wavelength and to radian frequency:β=2π/λ0=ω/ν (ν represents the phase velocitydetermined by the medium). β is measured in rad/m(or °/m).

Characteristic impedance

• In general there are two waves in a transmission line, one propagating toward positive z and the other toward negative z

• The amplitude of the two waves depend on the impedance terminating the line

• There is a particular value of this impedance for which there is only the wave in positive z direction

• This impedance depends on the line and it is called Characteristic Impedance (Zc). It is a real number for sufficiently high frequencies (>10 MHz)

Secondary Parameters

• Characteristic Impedance (Zc): it is defined as the input impedanceof a line with infinite length (i.e. there is only the wave propagatingalong the positive z direction)

• Propagation constant γ=α+jβ

Formulas relating physical and secondary parameters:

1 1, ,2 2c c

c

L RZ G Z L CC Z

α β ω= = + ⋅ = ⋅ ⋅

These relations are valid for ω>> R/L, G/C; transmission lines used atmicrowave frequencies always satisfy these conditions

Voltage and current on the line

v+(z)

v-(z)

z

V(z)

I(z)

ZL

( ) ( ) ( )

( ) ( ) ( )

j z j z

j z j z

V z z z e e

I z z z e e

β β

β β

+ − − +

+ − − +

= + = +

= + = −

v v

i i

+ -0 0

+ -0 0

c c

V VV VZ Z

v+, i+: Incident waves v-, i-: Reflected waves

The sign of the reflected wave of voltage is the same of the incidentwave; the sign of reflected wave of current is instead opposite to that ofthe incident wave. In any case (incident or reflected), the ratio betweenthe wave of voltage and the wave of current is equal to Zc inmagnitude. A reflected wave is produced by a discontinuity in thetransmission line structure (as for instance the termination with a load).The reflected wave is eliminated by assigning ZL equal ZC.

Reflection Coefficient

22 2 20 0

0 00 0

( )

zj z jj z j z

j z

Reflected WavezIncident wave

V e V e e eV e V

β πβ β λ

β

− + − ++ +

+ − +

Γ = =

= = = Γ = Γ

Properties of Γ(z) with α=0:

The magnitude does not depend on z (it is constant along the line)

The phase is periodic in z (period equal to λ/2)

The magnitude is always less than 1 with a passive load (the reflected power must be less than the incident one).

|V(z)| as a function of z

z

|V| Vmax

VminVmin

V Vmax min÷ + ÷ −1 1Γ Γ,

[ ]

V(z) v (z) v (z) v (z) (z)

V z e

z

j z

= + = ⋅ +

= + ⋅ =

+ ⋅ +

+ − +

+

1

1

1 2 2

02

0 02

12

Γ

Γ

Γ Γ

( )

cos( )

V

V

0+

0+

β

β

The magnitude of V is periodic (same period of Γ) with maxima and minima given by:

The Voltage Standing Wave Ratio (VSWR) is defined as the ratio of these voltages:

max

min

VVSWRV

=

= 1 for perfect matched line (Γ=0)

= ∞ for completely mismatched line (Γ=1)

There is a biunivocal relationship between the reflection coefficient andthe impedance (normalized to Zc) seen along the line (in the loaddirection):

Inverse relation:

Impedance along the line

z

V(z)

I(z)

ZLZ(z)

( ) 1 ( ) ( ) ( ) 1 ( )( ) ( ) ( ) 1 ( )c c

Z z V z V z V z zZ Z I z V z V z z

+ −

+ −

+ + Γ= = =

− −Γ

( ) 1( )( ) 1

c

c

Z z ZzZ z Z

−Γ =

+

Impedance as a function of the load

V(z)

I(z)

ZLZc , β

L

ZVI

ZZ jZ LZ jZ Lin

in

inc

L c

c L= =

++

tan( )tan( )

ββ

Particular cases (stubs):

ZL= 0 (short circuit)

ZL= ∝ (open circuit)

0

tan( ) tan(2 )inin c c

in

V LZ jZ L jZI

β πλ

= = ⋅ =

0

cot( ) cot(2 )inin c c

in

V LZ jZ L jZI

β πλ

= = − ⋅ = −

Stub as a circuit componentAt high frequencies it is difficult to realize lumped components like inductors and capacitors. It is much easier to use stubs which realize the same reactance (susceptance) of the lumped component:

Ls 0lump sX Lω=

β0L ( )0tandist cX Z Lβ=

( )0

00tan tan

lump sc

X LZL L

v

ωωβ

= =

0lump pB Cω=

β0L ( )0tandist cB Y Lβ=

( )0

00tan tan

lump pc

B CY

L Lv

ωωβ

= =

Cp

L<λ0/4

Note: The equivalence is correct at fixed frequency. When the frequency varies the two components behavior is greatly different (lumped: monotonic, distributed: periodic)

Example

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

5

10

15

20

25

f/f0

Ohm

Xdist

Xlump

Xlump(f0)= Xdist(f0)= 10 Ω β0L=45°, Zc=10 ΩXlump(f)=10.(f/f0), Xdist(f)=10.tan(45°.f/f0)

Transmission Lines Classification

TEM Lines (Transverse Electric Magnetic)

• Coaxial

• Stripline

non-TEM Lines

• Rectangular waveguide

• Circular waveguide

• Fin-line

quasi_TEM Lines

•Microstrip

•Suspended Stripline

•Inverted Stripline

•Coplanar Lines

TEM Lines• They are constituted by two (at least) independent conductors (a

voltage can be applied among them), embedded in a homogeneousmedium

• Electric and magnetic fields of the propagating wave are orthogonaleach other and do not have components in the direction ofpropagation

• A TEM transmission line is characterized by:

– A univocal and frequency independent characteristic impedance

– A constant phase velocity lower than or equal to the light velocity c:

with εr relative dielectric constant of the medium.

vc

r

=ε

non-TEM Lines• The electromagnetic field of a generic transmission line can assume specific

configurations called modes. The modes are characterized by a minimum frequency (cutoff frequency), below of which they cannot propagate. The mode with the smallest cutoff frequency is called dominant mode and it is the most used in the practice. For line supporting TEM mode the cutoff frequency on the dominant mode (also called principal mode) is zero

• Non-TEM lines are constituted by a single hollow conductor having a section of arbitrary shape, generally called waveguide

• The modes in non-TEM lines are generally classified in:

• TE modes: the electric field has no components in the propagation direction

• TM modes: the magnetic field has no components in the propagation direction

• The characteristic impedance of non-TEM mode is not univocally defined

• Also voltage and current are not univocally defined

quasi-TEM Lines• These lines consist of two (or more) conductors, surrounded by a non

homogeneous medium

• As a consequence there is at least one component of E or H field in the direction of wave propagation. To difference of non-TEM modes, the dominant mode of quasi-TEM lines has the cutoff frequency equal to zero

• The rigorous study of this kind of lines is rather complex; has been then developed an approximate (much easier to compute) model, based on the concept of equivalent TEM line.

• The equivalent TEM model of a quasi-TEM line differs from a ideal TEM line in the fact that the characteristic impedance and the phase velocity depend on the frequency

Attenuation in transmission lines

Power along a matched line (positive z direction):

( ) 2 2

0 21 12 2

z

c c

v z vP( z ) e

Z Zα

+ +−= =

The power lost per unit length is then:

( )2

0 212 22

zdiss

c

vP e P z Pz Z

αα α+

− ∂ = − = − = − ∂

The definition of α then results:

Dissipated power p.u.l2 Carried power

dissPP

α = =

Sources of attenuation in transmission lines• Attenuation αJ due to losses in the conductors

• Attenuation αD due to losses in the medium (dielectric)

The overall attenuation α is the sum of the above contributes:

Attenuation in the conductors αJ

• It is caused by the finite conductivity of the conductor material employed. In TEMlines αJ increases with frequency as the square root of f (skin effect).

• The actual conductivity of a material depends, other that its physical properties,also on the surface roughness determined by the fabrication process adopted.Without a suitable processing of the surfaces (polishing, lapping and plating), thedegradation of conductivity due to the surface roughness may be even less 50 %of the ideal value

J Dα α α= +

Attenuation in the dielectric medium αD

• In a real dielectric medium under sinusoidal excitation, some energy must besupplied for aligning the elementary dipoles of the material along the electricfield direction. As a consequence the dielectric constant of the mediumbecome a complex number

• A parameter called tanδ is introduced for characterizing dielectric losses,which represents the ratio between the imaginary and real part of thedielectric constant. This parameter is practically independent on thefrequency (at microwave frequencies).

• Expressions of αD as function of tanδ:

0

20

2

( Lines)

( Lines)1

d

d

c

tan TEM

tan non TEMff

πα δλπ δαλ

=

= −

−

General expression of Zc and α forTEM lines

General definition of the characteristic impedance of a TEM line:2

1

.

P

tP

c w zt

cont

E dlVZ Z FI H ds

⋅∆

= = = ⋅∫

∫

Fz = Form Factor of the line (depends on the line geometry)

Attenuation of a TEM line:

2s

J Jw

R FZ

α

=

The direction n represents the direction orthogonal to the conductors surface (entering into)

377w

r

Zε

= Ω

FJ = Attenuation Factor of the line(depends on the line geometry)

1 zJ

z

FFF n

∂=

∂

Wheeler’s rule:

High order modes in TEM lines

• Also in TEM lines may exist propagating waves with components of E or H in the propagating direction (higher order modes)

• This modes are characterized by their cutoff frequency and can propagate only if the frequency is higher than this

• In practical application it is requested that only one mode can propagate (the one with the lowest cutoff frequency)

• TEM line must be dimensioned in order that the first non-TEM mode is above the maximum operating frequency

Coaxial Line

2

1

ln

2z

rr

Fπ

=

2 1

2 1 2

1

2 1

2

1

1 11 1 22

ln

1 1 ln

z z z

z z

J

F F F r rF n F r r r

rr rF

rr

ππ

∂ ∂ ∂ + = − = ∂ ∂ ∂

+=

Coaxial Line dimensioning

( ) ( )2 1 2

2 1 2 2 1 12

1

11 0 3.6 ( 76 in air)ln

Jc

F r r r Zr r r r r rr

r

∂ +∂= = ⇒ Ω

∂ ∂

2

1 0

exp 2 exp 2377

c cr

Z Zrr Z

π π ε =

Given Zc:

Zc=50 Ω for r2/r1 ≅ 2.3 in air

Single-mode propagation up to fmax:

( )2 1

max 22 1 max 2 11

r rv vf rr r f r rπ π

= ⇒ <+ +

Minimum attenuation assigned the external radius:

Minimum attenuation assigned fmax:

( ) ( ) ( )2 1 1 2 2

2 1 1 2 2 1 2 1 1

21 0 4.45 ( 97 in air)ln

Jc

F r r r r r Zr r r r r r r r r∂ + +∂

= = ⇒ Ω∂ + ∂

( ) ( )2 1

22 1

11lnJ

r rF rr r

+=

( )2 1 1 2

1 2 2 1

21lnJr r r rF

r r r r + +

= +

Other TEM lines

Parallel Plates Stripline

Slabline Bi-wires

q≅1 (a>>t)

quasi-TEM Lines

A quasi-TEM line is obtained when a inhomogeneous medium is used in a TEMline. As a consequence the electromagnetic field is no more transverse withrespect the direction of propagation.Strictly speaking it is no possible in this case to define uniquely the voltagesand currents. In the practice the quasi-TEM approximation is introduced.This consists in assuming an equivalent homogeneous medium characterizedby an effective dielectric constant εr,eff, defined as:

Cm: Capacitance p.u.l. of non-homogenous lineC0: Capacitance p.u.l. of the line with εr=1 (air

everywhere)

Note that εr,eff is in general a function of the frequency. Also Zc e vf are then functions of frequency and can be expressed as:

, 0r eff mC Cε =

, ,

377 ,c z fr eff r eff

cZ F vε ε

= =

The most important quasi-TEM line in practical applications is themicrostrip. It belongs to the category of planar structures.

hwεr

t Dielettric

Conductor

• h substrate thickness• w strip width• εr relative dielectric constant of substrate• t metallic thickness

Microstrip

Formulas for microstrip

Quasi-static (t=0):

( )

,

1

,

,

60 ln(8 0.25 ) 1

120 1.393 0.667 ln 1.444 1

1 1 300con: , cm sec2 2 1 10

r eff

c

r eff

r rr eff

eff

h w w h w h

Z

w h w h w h

h w

ε

πε

ε εε νε

−

+ ≤

=

+ + + >

+ −= + =

+

Finite metallic thickness:For taking into account the finite value of t, an effective strip width (We) is introduced:

1.25 41 ln 0.159

1.25 21 ln 0.159

e

w t w w hh h t

w h

w t h w hh h t

ππ

π

⋅ + + ≤ =

⋅ + + >

A simple model for introducing the frequency variation in εr,eff is given by:

( ) ( )( ) ( )

,, 1.33 2 3

01 0.43 0.009

r r effr eff GHz r

mm c GHz GHz

fh Z f f

ε εε ε

−= −

+ −

CAD tool for evaluating Zc

Graphic representation of Γ

Γ is a complex number in the polarplane:

•|Γ|

Φ

For the properties of Γ, assuming theline terminated with a passive load,the point in the polar plane must bewithin the circle with unit radius

Γ

1

-1

-1 1

Graphic representation of Γ

The curve in the polar planerepresenting the reflection coefficient ona lossless line terminated with a passiveload is a circle with radius equal to|Γ|. The phase changes by 2π with avariation of z of λ/2

|Γ|abcd

Characteristic points:

Matched line Γ =0 ( centre of the chart)Open circuit Γ =1 (a)Short circuit Γ =-1 (d)Maximum of voltage (b) (Γ real and positive)Minimum of voltage (c) (Γ real and negative)

Smith Chart

• On the polar plane of Γ the curves representing the locus ofpoints where the real (or imaginary) part of zn=Z/Zc is constantare defined by:

• These curves are circles, whose center and radius depend on r(or x). The Smith Chart is the graphic representation of suchcircles. It allows to solve graphically several problems related tothe transmission lines used in microwave circuits.

( ) 1Re Re ( )( ) 1

( ) 1Im Im ( )( ) 1

in

in

zZ cost rz

zZ cost xz

Γ += = Γ −

Γ += = Γ −

Angle of Γ (it is measured in degrees or in L/λ0)

Circle r = 1

Circle x = 1

Tow

ard

Sour

ce

Toward Load

Reference Axis

Admittance representation of the Smith Chart

Γ

φ

Impedance at Γ:11nz +Γ

=−Γ

−Π+φΓ’

Impedance at Γ’:

( )

( )1 11 1 1

1 1 11

j j

n jjn

e ez

e ze

π φ φ

φπ φ

− +

− +

+ Γ − Γ′+ Γ −Γ′ = = = = =′−Γ + Γ +Γ− Γ

The diametrically opposite point presents the inverse impedance (i.e. the admittance) with respect the original point. As a consequence, the Smith Chart can be used for representing either impedances or admittances

Smith ChartMoving at Γ=const

( ) ( )

( ) ( )

⋅=⋅=

⋅=⋅=⋅−∠−

⋅∠

λπΓβ

λπΓβ

ΓΓΓ

ΓΓΓdjj

Ldj

L

xjjxj

eeed

eeex

L42

40

20

0

Displacement on a lossless line:

Movement on a circle with constant radius

NOTE: Γ (z) is a periodic function of z (d) with period λ/2

ZL

LΓ0

Asse d 0x

0Γ Γ

Displacement: from source x increasing

( ) xx βΓΓ 20 +∠=∠ counterclockwise rotation from load d increasing

( ) dx L βΓΓ 2−∠=∠ Clockwise rotation

Asse xd

zL = rL + jxL

jx

zin = rL + j(xL + x)

Displacement at r costant

zL = rL + jxL

r

zin = (rL + r) + jxL

Displacement at x costant

Smith ChartDisplacement at r (or x) constant

rL

xL

ΓL

xin = xL + x

rL

xL

ΓLΓin

rin = r+rL

Γin

yL = gL + jbLjb

yin = gL + j(bL + b)

Displacement at g=const

yL = gL + jbLg

yin = (gL + g) + jbL

Displacement at b=const

Smith ChartDisplacement at g (or b) constant

Y ChartbL

ΓL

gL

gin =gL + g

Γin

bLΓL

gLΓin

xin

bB

Smith ChartExample of graphical solution

ZL

A B

jB

jX

Zcα = 0

in Zc= 50 [Ω];εr = [4];f0 = 3 [GHz];ZL= 20 + j40 [Ω];(Zn=0.4+j0.8)β=7.2 [°/mm]

d

0.62 97.1L cL

L c

Z ZZ Z

−Γ = = ∠ °

+

B= 0.05 [Ω-1]; (bn=2.5)X = -80 [Ω]; (xn=-1.6)d = 15 [mm]; (βd=108°)

[ ] °=== 0.216rad 77.3402 dd πβ

ΓL

ΓB

2βL°−∠= 6.11675.0BΓ

°∠= 4.2775.0AΓ

[ ]Ω 689611 1.3151.0 jZZ

in

incinin +≅−+

=⇒°∠=ΓΓΓ

ΓAΓin