Transmission Lines Fundamentals

VII

1

Transmission Lines

(a) Parallel-plate transmission line

(b) Two-wire transmission line

Metal strip Metal strip Dielectric subtrate Grounded conducting plane

Grounded conducting plane

Dielectric subtrate Grounded conducting plane

Two types of microstrip lines

(c) Coaxial transmission line

VII

2

TEM-Waves along a Parallel-Plate Transmission Line

Lossless case:

y

d

x

z

w

( term e jωt always omitted) r r r E = Ey ⋅ ey = E0 ⋅ e− γz ⋅ ey r r r E H = Hx ⋅ ex = − 0 ⋅ e− γz ⋅ ex Γ µ γ = jω µε Γ = ε

in order to find the charge density and the current density we use: r r Dn2 − Dn1 = σ → D ⋅ ey = σ → σ = ε ⋅ Ey = ε E0 ⋅ e− γz σ: free surface charge r r r r r r E Ht 2 − Ht1 = Js → ey × H = Js → Js = − ez ⋅ Hx = ez ⋅ 0 e− γ z Γ Js: free surface current d Ι ds

VII

3

Fields, Charge and Current Distribution along a Coaxial Transmission Line B

+ +++ + x x x xx x

-

E

E -

---

+++ x

+

+++

x x x xx x

x xx

-

+ +++ + x x x xx x

+

-

---

+

-

---

-

---

+

B -

-

---

x xx

-

+ +++ + x x x xx x

x xx

-

E

+ +++ +

-

---

x xx

-

λ

B

Current

Displacement Current

VII

4

Parallel-Plate Transmission Line in Terms of L and C Lossless case

(term e

jωt

)

always omitted

r r ∇ × E = − jωµH

+

dEy = jωµHx dz

r r ∇ × H = jωε E dHx = jωε Ey dz

d d d Eydy = jωµ ∫ Hxdy ∫ dz 0 0

w d w Hxdx = jωε ∫ Eydx dz ∫0 0

( ) = jωµ J (z) ⋅ d sz dz

dV z

 d = jω  µ  Jsz z ⋅ w  w

( () )

() [H m]

= jω L ⋅ Ι z L = µ⋅

d w

−

d Ι ( z) = − jωε Ey ( z) ⋅ w dz w = jω  ε  −Ey ( z) ⋅ d  d = jω CV( z)

(

C=ε

w d

[F m]

)

VII

5

( ) = −ω 2LCΙ(z)

d2V( z) 2 2 = − ω LCV ( z) dz V( z) = V0 ⋅ e− jω

LC z

d2Ι z d z2

= V0 ⋅ e− jω

µε z

Ι( z) = Ι 0 ⋅ e− jω

Phase velocity:

up =

ω 1 1 = = ω µε µε LC

Characteristic impedance:

Z0 =

V( z) L = Ι( z) C

LC z

= Ι 0 ⋅ e− jω

µε z

VII

6

Lossy Parallel-Plate Transmission Line

Conductance between the two conductors: Compare with the analogy of resistance and capacitance ε c case a

= R⋅κ case b

1 κ κ w w = C = ⋅ε = κ ⋅ R ε ε d d w G= κ⋅ [S m] d ⇒G=

VII

7

Ohmic power dissipated in the plates r r r SLoss = ezEz × ex ⋅ Hx * Def.

Surface impedance Zs =

r Power flux density flowing into the plates (ey ) Et Js

Js = free surface current

dΙ z dx

Zs = Rs + j ⋅ Xs Rs =

=

1 length 1 l ⋅ = κ c cross sec tion κ w ⋅ d

1 l κ c µ cω l µ cω ⋅ ⋅ = κc w 2 w 2κ c

effective series resistance per unit length 2 µ cω 2 µ c f ⋅ π R= = [Ω / m] κc w 2κ c w

d = penetration depth =

2 κ c µ cω

VII

Equivalent Circuit of a Differential Length ∆ z of a Two-Conductor Transmission Line

R ∆z •

L ∆z •

G ∆z •

C ∆z •

8

VII

9

Distributed Parameters of Transmission Lines Parameter

Parallel Plate 2 πfµ c w κc

Two-Wire Line Rs πa

Coaxial Line R s  1 1 + 2π  a b 

R

Ω/m

L

d µ w

µ D cosh −1   2a π

µ b ln 2π a

H/m

G

w κ d

πκ cosh−1(D / 2a)

2πκ ln(b / a)

S/m

w d

πε cosh−1(D / 2a)

2πε ln(b / a)

F/m

ε C

w=width d=separation

Rs =

πfµ c κc

a=radius D=distance cosh −1(D / 2a) ≈ ln (D / a) if (D / 2a) >> 1 2

Rs =

πfµ c κc

a=radius center cond. b=radius outer cond.

Unit

VII

10

Wave Equation for Lossy Transmission Lines

dV( z) = (R + jωL) Ι ( z) dz d Ι ( z) − = (G + jωC) V ( z) dz d2V( z) = γ 2V( z) 2 dz d2Ι ( z) 2 2 = γ Ι ( z) dz −

γ = α + j β = (R + jωL)(G + jωC)

VII

11

Waveguides y x

z

A uniform waveguide with an arbitrary cross section Time-harmonic waves in lossless media: r r ∆E + ω 2µ ε E = 0 r r0 j ωt −k z ⋅z ) E ( x, y, z, t ) = E ( x, y) ⋅ e (

(∇ ∇ xy

2

∇ xy

2

2 xy

+ ∇z

2

)

r r r 2 2 E = ∇ xy E − k z E

r r 2 2 E + ω µε − k z E = 0

(

)

r r 2 2 H + ω µε − k z H = 0

(

)

VII

r r From ∇ x E = − jωµH we get: ∂E0z + jk zEy0 = − jωµH0x ∂y − jk zE0x

∂E0z = − jωµH0y − ∂x

∂E0y ∂E0x − = − jωµH0z ∂x ∂y

H0x

1 ∂H0z ∂E0z  = − 2  jk z − jωε  ∂x ∂y  h 

H0y

∂H0z ∂E0z  1 = − 2  jk z + jωε  ∂y ∂x  h 

E0x

∂E0z ∂H0z  1 = − 2  jk z + jωµ  ∂x ∂y  h 

E0y

∂E0z ∂H0z  1 = − 2  jk z − jωµ  ∂y ∂x  h 

r r From ∇ xH = jωεE we get: ∂Hz0 + jk zHy0 = jωεE0x ∂y − jk zH0x

∂Hz0 = jωεEy0 − ∂x

∂Hy0 ∂H0x − = jωεE0z ∂x ∂y

h2 = ω 2µε − k z2

12

VII

13

Three Types of Propagating Waves

Transverse electromagnetic waves TEM

:

EZ = 0 & HZ = 0

Transverse magnetic waves TM

:

EZ - 0 & HZ = 0

Transverse electric waves TE

:

EZ = 0 & HZ - 0

VII

14

TEM - Waves Hz = 0 & Ez = 0 → − k z2TEM + ω 2µε = 0 → k z TEM = ω µε 1 ω = µε kz

Phase velocity

upTEM =

Wave impedance

ωµ µ E0x Z TEM = 0 = = ε Hy k z TEM

for hollow single-conductor

waveguides:

Hz = 0 → there is only Hx and Hy r div H = 0 → H − fields must form closed loops ∂Dz Ez = 0 → =0 ∂t r r rot H = J → TEM waves cannot exist in sin gle − conductor hollow waveguides

VII

15

TM-Waves

Ex =

− jk z ∂Ez ω 2µε − k z2 ∂x

Ey =

− jk z ∂Ez ω 2µε − k z2 ∂y

Hx =

jωε ∂Ez ω 2µε − k z2 ∂y

Hy =

− jωε ∂Ez ω 2µε − k z2 ∂x

Wave equation ∂2Ez ∂2Ez 2 2 2 + 2 + ω µε − k z Ez = 0 ∂x ∂y

(

)

VII

16

TM-Modes in Rectangular Waveguides

boundary conditions

x

Ez (0, y) = 0

and

Ez (a, y) = 0

in the x direction

Ez ( x, 0) = 0

and

Ez ( x, b) = 0

in the y direction

separation of variables

a z

( )

Ez ( x, y) = E0 sin(k x x) sin k y y y b

kx =

mπ a

and

ky =

nπ b

( m, n are int egers)

VII

17

Solution

Ex ( x, y) =

− jk z mπ  mπ  x sin nπ  y E cos 0  a   b a ω 2µε − k z2

TM13 mode means m=1, n=3 (if m=0 or n=0 then E=H=0)

Ey ( x, y) =

Hx ( x, y) =

Hz ( x, y) =

nπ  mπ  − jk z  nπ  y E x sin cos 0  a   b b ω 2µε − k z2

mπ  nπ = ω µε −  −   a   b 2

kz

2

2

2

jωε nπ  mπ   nπ  y E x sin cos 0  a   b b ω 2µε − k z2

 mπ  2  nπ  2  ω c µε −  + =0     a b  

mπ − jωε  mπ  x sin nπ  y E cos 0  a   b a ω 2µε − k z2

 mπ  +  nπ  fc =  b 2π µε  a 

2

1

if f < fc then jkz is real

2

2

cut off frequency

no wave propagation

VII

18

Field Lines for TM11 Mode in Rectangular Waveguide

y/b

y/b 1,0

x x

0,5 x

x/a

O Electric field lines

0 0

x

x x x x x π/2

x

x

x x x

x

π

Magnetic field lines

3π/2

2π βz

VII

19

TE-Waves

Ex =

− jωµ ∂Hz ω 2µε − k z2 ∂y

Ey =

jωµ ∂Hz ω 2µε − k z2 ∂x

− jk z ∂Hz Hx = 2 ω µε − k z2 ∂x Hy =

− jk z ∂Hz ω 2µε − k z2 ∂y

wave equation ∂2Hz ∂2Hz 2 2 2 + 2 + ω µε − k z Hz = 0 ∂x ∂y

(

)

VII

20

TE-Modes in Rectangular Waveguides

boundary condition ∂Hz 0, y = 0 ∂x ∂Hz x,0 = 0 ∂y

( )

and

( )

and

∂Hz a, y = 0 ∂x ∂Hz x,b = 0 ∂y

( )

in the x direction

(Ey ) = 0

( )

in the y direction

(Ex ) = 0

separation of variables mπ  nπ Hz ( x, y) = H0 cos x cos  y  a   b

VII

Solution: Ex ( x, y) =

21

jωµ nπ  mπ  x sin nπ  y H cos 0  a   b b ω 2µε − k z2

− jωµ mπ  mπ   nπ  y H x sin cos 0  a   b a ω 2µε − k z2 jk mπ  mπ  nπ Hx ( x, y) = 2 z 2 H0 sin x cos  y  a   b a ω µε − k z Ey ( x, y) =

Hy ( x, y) =

TE01 mode means m = 0, n = 1 mπ  nπ = ω µε −  −   a   b 2

kz

2

2

2

nπ jk z  mπ  x sin nπ  y H cos 0  a   b b ω 2µε − k z2

 mπ  +  nπ  cut off frequency fc =  b 2π µε  a  no wave propagation if f < fc then jkz is real 1

2

2

VII

22

Field Lines for TE10 Mode in Rectangular Waveguide y/b

y/b 1,0

x x x x x

0,5

O

x

x x x x x x

x

x/a

x

x x

0 0

x x

π/2

π

3π/2

2π βz

π

3π/2

2π βz

x/a 1,0 x x

Electric field lines x

Magnetic field lines

x x

x

x x

x x

x

x

x

x x

x 0 0

π/2