Adapted from notes by ECE 5317-6351 Prof. Jeffery T ...courses.egr.uh.edu/ECE/ECE5317/Class...

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Prof. David R. Jackson Dept. of ECE Notes 11 ECE 5317 - 6351 Microwave Engineering Fall 2019 Waveguiding Structures Part 6: Planar Transmission Lines 1 Adapted from notes by Prof. Jeffery T. Williams , , εµσ h w t

Transcript of Adapted from notes by ECE 5317-6351 Prof. Jeffery T ...courses.egr.uh.edu/ECE/ECE5317/Class...

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Prof. David R. JacksonDept. of ECE

Notes 11

ECE 5317-6351 Microwave Engineering

Fall 2019

Waveguiding Structures Part 6:Planar Transmission Lines

1

Adapted from notes by Prof. Jeffery T. Williams

, ,ε µ σ hw

t

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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

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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

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Stripline

Common on circuit boards

Fabricated with two circuit boards

Homogenous dielectric(perfect TEM mode*)

Field structure for TEM mode:

(also TE & TM Modes at high frequency)

TEM mode

4

Electric FieldMagnetic Field

* The mode is a perfect TEM mode if there is no conductor loss.

, ,ε µ σw b

t

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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:

, ,ε µ σ

a b>>

wb

/ 2b

a

(to simulate stripline)

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Conformal Mapping Solution (R. H. T. Bates)

030 ( )

( )K kZ

K kπ

=′

sech2

tanh2

wkbwkb

π

π

= ′ =

K = complete elliptic integral of the first kind

Exact solution (for t = 0):

Stripline (cont.)

6

( )/2

2 20

11 sin

K k dk

π

θθ

≡−∫

R. H. T. Bates, “The characteristic impedance of the shielded slab line,” IEEE Trans. Microwave Theory and Techniques, vol. 4, pp. 28-33, Jan. 1956.

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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

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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

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Attenuation

Dielectric Loss:

0tan tan2 2

rd d d

kkkε

α δ δ′

′′= ≈ ≈

Stripline (cont.)

(TEM formula)

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0 ck k jk ω µ ε′ ′′= − = c jσε εω

= − tan cd

c

εδε′′

=′

, ,ε µ σ

b

w

tsR

sR

sR

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[ ] ( )

[ ] ( )

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

+ − = + + − − = + + + + +

r

wher

fo

for

e

wider strips

narrower strips

2sR ωµσ

=

Stripline (cont.)Conductor Loss:

10

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

σ = conductivity of metal

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Stripline (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 on strip

bt

wrε

s

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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

12

Note: Pozar uses (W, d)

, ,ε µ σ hw

t

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Microstrip (cont.)

13

Figure from Pozar book

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

Note:The flux lines get more concentrated in the substrate

region as the frequency increases.

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Equivalent TEM problem:

0effrkβ ε≈

Microstrip (cont.)

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The effective permittivity gives the correct phase constant.

The effective strip width gives the correct Z0.

Equivalent TEM problem

0 0 /air effrZ Z ε=

( )0 /Z L C=since

effrεeffw0Z

h

Actual problem

rε0ε

h

w2

0

effr k

βε

⇒ =

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1 1 12 2

1 12

eff r rr h

w

ε εε

+ −

= + +

Microstrip (cont.)

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1/ 0 :2

eff rrw h εε +

→ →

/ : effr rw h ε ε→∞ →

Effective permittivity:

Limiting cases:

(narrow strip)

(wide strip)

Note:This formula ignores

“dispersion”, i.e., the fact that the effective permittivity is

actually a function of frequency.

rε0ε

h

w

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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.)

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Characteristic Impedance:

Note:This formula ignores the fact

that the characteristic impedance is actually a function of frequency.

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Inverting this solution to find w for a given Z0:

( )

28 ; 2

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

2

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.)

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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

′ = + +

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Microstrip (cont.)

h

tw

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( )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

λ

= − + + +

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Microstrip (cont.)where

( / 1)w h ≥

Note: ( ) ( )

( )

0 :0

:

eff effr r

effr r

ff

ff

ε ε

ε ε

→∞

As

As

h

tw

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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)

Note:The phase velocity is a

function of frequency, which causes pulse distortion.

p effr

cvε

=

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Attenuation

Dielectric loss:

( )( )

01

tan2 1

effrr r

d 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.)

21

(More accurate formulas are given on next slide.)

1: 0effr dε α→ →

0: tan2

eff rr r d d

k εε ε α δ→ →

0hZ

wη ≈

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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

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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.

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TXLINEThis is a public-domain software for calculating the

properties of some common planar transmission lines.

https://www.awr.com/software/options/tx-line