Prof. David R. Jackson Dept. of ECE

Notes 11

ECE 5317-6351 Microwave Engineering

Fall 2018

Waveguiding Structures Part 6: Planar Transmission Lines

1

rε0ε

h

w

Adapted from notes by Prof. Jeffery T. Williams

2

Planar Transmission Lines

εr

Microstrip

w

h εr

Stripline

wb

w

εr

Coplanar Waveguide (CPW)

hgg

w

εr

Conductor-backed CPW

hgg

εr

Slotline

h

s

εr

Conductor-backed Slotline

h

s

3

Planar Transmission Lines (cont.)

Stripline is a planar version of coax.

Coplanar strips (CPS) is a planar version of twin lead.

εr

Coplanar Strips (CPS)

h

ww

s εr

Conductor-backed CPS

ww

hs

Stripline

Common on circuit boards

Fabricated with two circuit boards

Homogenous dielectric (perfect TEM mode)

Field structure for TEM mode:

(also TE & TM Modes)

TEM mode

4

Electric Field Magnetic Field

, , dε µ σ

wb

Analysis of stripline is not simple. TEM mode fields can be obtained from an electrostatic

analysis (e.g., conformal mapping).

Stripline (cont.)

5

A closed stripline structure is analyzed in the Pozar book by using an approximate numerical method:

, , dε µ σ

a b>>

wb

/ 2b

a

Conformal mapping solution (S. Cohn):

030 ( )

( )K kZ

K kπ

=′

sech2

tanh2

wkbwkb

π

π

= ′ =

K = complete elliptic integral of the first kind

Exact solution:

Stripline (cont.)

6

( )/2

2 20

11 sin

K k dk

π

θθ

≡−∫

Curve fitting this exact solution:

00 ln(4)4 r

e

bZw b

ηε

π

=

+

Effective width

2

0 ; 0.35

0.35 ; 0.1 0.35

e

wbw w

b b w wb b

≥= − − ≤ ≤

for

for

Stripline (cont.)

ideal 00

1 / 22 4 4 r

bb bZw w w

ηηηε

= = =

Note: The factor of 1/2 in front is from the parallel combination of two ideal PPWs.

( )ln 40.441

π=Note :

7

Fringing term

Inverting this solution to find w for given Z0:

[ ]

[ ]

0

0

0

0

; 120

0.85 0.6 ; 120

ln(4)4

r

r

r

X Zwb

X Z

XZ

ε

ε

ηπε

≤ Ω

= − − ≥ Ω

≡ −

for

for

Stripline (cont.)

8

Attenuation

Dielectric Loss:

0tan tan2 2

rd

kkkε

α δ δ′

′′= ≈ ≈

Stripline (cont.)

(TEM formula)

9

0 ck k jk ω µ ε′ ′′= − = dc j σε ε

ω= − tan c

c

εδ

ε′′

=′

, , dε µ σ

b

w

tsR

[ ]

[ ]

3 00

0

00

4(2.7 10 ) ; 120( )

0.16 ; 120

1 21 2 ln( )

1 11 0.414 ln 42 20.7

2

s rr

cs

r

R Z A Zb t

R B ZZ b

w b t b tAb t b t t

b t wBw w tt

ε εη

αε

π

ππ

− × ≤ Ω −= ≥ Ω

+ − = + + − −

= + + + + +

wher

for

for

e 2sR ωµσ

=

Stripline (cont.) Conductor Loss:

10

Note: We cannot let t → 0 when we calculate the conductor loss.

Inhomogeneous dielectric

⇒ No TEM mode

TEM mode would require kz = k in each region, but kz must be unique!

Requires advanced analysis techniques Exact fields are hybrid modes (Ez and Hz)

For h /λ0 << 1, the dominant mode is quasi-TEM.

Microstrip

11

, , dε µ σ

Note: Pozar uses (W, d)

hw

Microstrip (cont.)

12

Figure from Pozar book

Part of the field lines are in air, and part of the field lines are inside the substrate.

The flux lines get more concentrated in the substrate region as the frequency increases.

Equivalent TEM problem:

0effrkβ ε≈

Microstrip (cont.)

13

The effective permittivity gives the correct phase constant.

The effective strip width gives the correct Z0.

0 0 /air effrZ Z ε=

( )0 /Z L C=since

Actual problem

rε0ε

h

w

Equivalent TEM problem

effrεeffw

0 : 1air effrZ ε →

0Z

h

1 1 12 2

1 12

eff r rr h

w

ε εε

+ −

= + +

Microstrip (cont.)

14

1/ 0 :2

eff rrw h εε +

→ →

/ : effr rw h ε ε→ ∞ →

Effective permittivity:

Limiting cases:

(narrow strip)

(wide strip)

rε0ε

h

w

This formula ignores “dispersion”, i.e., the fact that

the effective permittivity is actually a function of

frequency.

0 0

60 8ln ; 14

; 11.393 0.667ln 1.444

effr

effr

h w ww h h

Z ww w hh h

εη

ε

+ ≤ = ≥ + + +

for

for

Microstrip (cont.)

15

Characteristic Impedance:

This formula ignores the fact that the characteristic

impedance is actually a function of frequency.

Inverting this solution to find w for a given Z0:

( )

2

8 ; 22

2 1 0.611 ln(2 1) ln 1 0.39 ; 22

A

A

r

r r

e we hw

h wB B Bh

επ ε ε

≤ −= − − − − + − + − ≥

for

for

0

0

0

1 1 0.110.3360 2 1

2

r r

r r

r

ZA

BZ

ε εε ε

η πε

+ −= + + +

=

where

Microstrip (cont.)

16

Attenuation

Dielectric loss:

( )( )

01

tan2 1

effrr r

d effr r

k εε εα δε ε

−

−

0

s sc

R RZ w h

αη

≈ ≈

“filling factor”

very crude (“parallel-plate”) approximation

Conductor loss:

Microstrip (cont.)

17

(More accurate formulas are given later.)

1: 0effr dε α→ →

0: tan2

reffr r d

k εε ε α δ→ →

0hZ

wη ≈

More accurate formulas for characteristic impedance that account for dispersion (frequency variation) and conductor thickness:

( ) ( ) ( )( )

( )( )0 0

1 00

0 1

eff effr reff effr r

fZ f Z

fε εε ε

−= −

( )( ) ( ) ( )( )

00 0

0 / 1.393 0.667ln / 1.444effr

Zw h w h

ηε

= ′ ′+ + +

( / 1)w h ≥

21 lnt hw wtπ

′ = + +

18

Microstrip (cont.)

h

tw

rε

( )2

1.5

(0)(0)

1 4

effr reff eff

r rfF

ε εε ε −

− = + +

( )( )

1 1 1 1 /02 2 4.6 /1 12 /

eff r r rr

t hw hh w

ε ε εε + − − = + − +

2

0

4 1 0.5 1 0.868ln 1rh wF

hε

λ

= − + + +

19

Microstrip (cont.) where

( / 1)w h ≥

Note: ( ) ( )

( )

0 :0

:

eff effr r

effr r

ff

ff

ε ε

ε ε

→

→

→ ∞

→

As

As

h

tw

rε

Microstrip (cont.)

20

2

0

effr k

βε

=

“A frequency-dependent solution for microstrip

transmission lines," E. J. Denlinger, IEEE Trans. Microwave Theory and

Techniques, Vol. 19, pp. 30-39, Jan. 1971.

Note: The flux lines get more

concentrated in the substrate region as the frequency increases. Quasi-TEM region

Frequency variation (dispersion)

Effe

ctiv

e di

elec

tric

cons

tant

7.0

8.0

9.0

10.0

11.0

12.0

Parameters: εr = 11.7, w/h = 0.96, h = 0.317 cm

Frequency (GHz)

rε

Note: The phase velocity is a

function of frequency, which causes pulse distortion.

p effr

cvε

=

21

Microstrip (cont.) More accurate formulas for conductor attenuation:

2

0

1 21 1 ln2 4

sc

R w h h h thZ h w w t h

απ π

′ = − + + − ′ ′

1 22

whπ

< ≤

( )2

0

/2 2ln 2 0.94 1 ln2 0.94

2

sc

w hR w w w h h h te whZ h h h w w t hh

πα π

π π

− ′ ′ ′ ′ = + + + + + − ′ ′ ′ +

2wh

≥

21 lnt hw wtπ

′ = + +

This is the number e = 2.71828 multiplying the term in parenthesis.

h

tw

rε

22

Microstrip (cont.) Note about conductor attenuation:

It is necessary to assume a nonzero conductor thickness in order to accurately calculate the conductor attenuation.

The perturbational method predicts an infinite attenuation if a zero thickness is assumed.

10 : 0szt J ss

= ∝ →as

Practical note: A standard metal thickness for PCBs is 0.7 [mils] (17.5 [µm]),

called “half-ounce copper”.

1 mil = 0.001 inch

0

(0)2l

cP

Pα =

1 2

2

0

(0)2

sl s

C C z

RP J d+ =

= ∫

szJ

h

tw

rε

s

23

TXLINE

This is a public-domain software for calculating the properties of some common planar transmission lines.

http://www.awrcorp.com/products/optional-products/tx-line-transmission-line-calculator

24

TXLINE (cont.)

TX-LINE: Transmission Line Calculator

TX-LINE* software is a FREE and easy-to-use Windows-based interactive transmission line calculator for the analysis and synthesis of transmission line structures. Register and Download Your FREE Copy of TX-LINE Software Today! TX-LINE software enables users to enter either physical or electrical characteristics for common transmission mediums: Microstrip Stripline Coplanar waveguide (WG) Grounded coplanar WG Slotline Learn more: TX-LINE Software Video Demonstration (3 minutes) *Note: TX-LINE software is embedded within NI AWR Design Environment and can be launched from the "Tools" menu.

25

Microstrip (cont.)

REFERENCES L. G. Maloratsky, Passive RF and Microwave Integrated Circuits, Elsevier, 2004. I. Bahl and P. Bhartia, Microwave Solid State Circuit Design, Wiley, 2003.

R. A. Pucel, D. J. Masse, and C. P. Hartwig, “Losses in Microstrip,” IEEE Trans. Microwave Theory and Techniques, pp. 342-350, June 1968. R. A. Pucel, D. J. Masse, and C. P. Hartwig, “Corrections to ‘Losses in Microstrip’,” IEEE Trans. Microwave Theory and Techniques, Dec. 1968, p. 1064.