Adapted from notes by ECE 5317-6351 Prof. Jeffery T ...courses.egr.uh.edu/ECE/ECE5317/Class...
Transcript of Adapted from notes by ECE 5317-6351 Prof. Jeffery T ...courses.egr.uh.edu/ECE/ECE5317/Class...
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
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 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
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)
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.
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 d d
kkkε
α δ δ′
′′= ≈ ≈
Stripline (cont.)
(TEM formula)
9
0 ck k jk ω µ ε′ ′′= − = c jσε εω
= − tan cd
c
εδε′′
=′
, ,ε µ σ
b
w
tsR
sR
sR
[ ] ( )
[ ] ( )
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
11
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
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
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.
Equivalent TEM problem:
0effrkβ ε≈
Microstrip (cont.)
14
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
⇒ =
1 1 12 2
1 12
eff r rr h
w
ε εε
+ −
= + +
Microstrip (cont.)
15
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
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.)
16
Characteristic Impedance:
Note: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:
( )
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.)
17
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
=
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 ≈
22
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
23
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.
24
TXLINEThis is a public-domain software for calculating the
properties of some common planar transmission lines.
https://www.awr.com/software/options/tx-line