Adapted from notes by ECE 5317-6351 Prof. Jeffery T. Williams Microwave Engineering

Fall 2018 Prof. David R. Jackson Dept. of ECE

Notes 11 Waveguiding Structures Part 6: Planar Transmission Lines

w ε0

ε r h

1 Planar Transmission Lines

w w ε h b r εr

Microstrip Stripline

w w

εr g g h εr g g h

Coplanar Waveguide (CPW) Conductor-backed CPW

s s

εr h εr h

Slotline Conductor-backed Slotline 2 Planar Transmission Lines (cont.)

w w w w

ε h εr s h r s

Coplanar Strips (CPS) Conductor-backed CPS

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

3 Stripline

. Common on circuit boards ε,, µσd b . Fabricated with two circuit boards w . Homogenous dielectric (perfect TEM mode) TEM mode (also TE & TM Modes)

Field structure for TEM mode:

Electric Field Magnetic Field 4 Stripline (cont.)

. Analysis of stripline is not simple. . TEM mode fields can be obtained from an electrostatic analysis (e.g., conformal mapping).

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

b w b /2 ε,, µσd

a ab>> 5 Stripline (cont.)

Conformal mapping solution (S. Cohn):

Exact solution:

30π Kk ( ) Z0 = K = complete elliptic Kk()′ integral of the first kind

π /2 1 Kk( ) ≡ dθ ∫ 22 0 1− k sin θ π w k = sech 2b π w k′ = tanh  2b

6 Stripline (cont.)

Curve fitting this exact solution:

η b Z = 0 ln( 4) 0 ε ln(4) Note : = 0.441 4 r wb+ π e π

Effective width Fringing term

 w  0 ;for ≥ 0.35 wwe  b = −  2 bbww 0.35− ;for 0.1≤≤ 0.35 bb

1bb /2 η η b ideal 0 The factor of 1/2 in front is from the parallel Note: Z0 =η  = = 24ww4w ε r combination of two ideal PPWs. 7 Stripline (cont.)

Inverting this solution to find w for given Z0:

 XZ; for ε r 0 ≤Ω120 [ ] w  =  b  0.85−− 0.6XZ ; for ε r 0 ≥Ω120 [ ]

η ln(4) X ≡−0 4 ε r Z0 π

8 Stripline (cont.)

Attenuation

Dielectric Loss:

k′ k ε α=≈≈k′′ tan δδ0 r tan (TEM formula) d 22

σ ε′′ ′ ′′ d c k=−= k jk ω µε0 c εεc = − j tanδ = ω ε c′

Rs t b w ε,, µσd

9 Stripline (cont.)

Conductor Loss:

 − 4RZε ×3 sr 0 ε ≤Ω (2.7 10 )AZ ; for r 0 120 [ ]  η0 ()bt− α =  c   Rs 0.16BZ ; for ε ≥Ω120 [ ]   r 0  Zb0 ωµ Rs = where 2σ w12 bt+−  bt  A =++1 2  ln  ()bt−−π  bt  t  b11 tw B =+1 ++0.414 + ln 4π w 22wtπ  + 0.7t 2

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

10 Microstrip

w ε,, µσd h . Inhomogeneous dielectric

⇒ No TEM mode Note: Pozar uses (W, d)

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.

11 Microstrip (cont.)

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

Figure from Pozar book

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

12 Microstrip (cont.)

Equivalent TEM problem:

eff βε≈ k0 r Actual problem w ε0

ε r h

air eff Z0 :1ε r → . The effective permittivity gives the correct phase constant. eff eff Z0 w ε r . The effective strip width gives the correct Z0. h air eff ZZ00= / ε r Equivalent TEM problem (since Z0 = LC/ )

13 Microstrip (cont.)

Effective permittivity:

This formula ignores “dispersion”, i.e., the fact that eff εεrr+−1 11 ε r = + the effective permittivity is 22h actually a function of 1+ 12 w frequency.

Limiting cases:

ε +1 wh/→→ 0: ε eff r (narrow strip) r 2

eff wh/:→∞εεrr → (wide strip) w ε0

ε r h

14 Microstrip (cont.)

Characteristic Impedance:

 60 8hw w  ln+≤ ;for 1 ε eff wh4 h  r Z0 =  η w  0 ;1for ≥  eff ww h ε r ++1.393 0.667ln + 1.444  hh

This formula ignores the fact that the characteristic impedance is actually a function of frequency.

15 Microstrip (cont.)

Inverting this solution to find w for a given Z0:

 8ewA ≤  2 A ;2for w  eh− 2 =  ε −  hw 2 r 1 0.61 BB−−1 ln(2 − 1) + ln( B − 1) + 0.39 − ;for ≥ 2  π εε  2 rrh

where

Z εε+−1 1 0.11 A =0 rr ++0.33 60 2εεrr+ 1 ηπ B = 0 2Z0 ε r

16 Microstrip (cont.) Attenuation

Dielectric loss: “filling factor”

eff k ε ε (ε r −1) αδ 0 r tan r d εεeff − 21rr( )

eff εαrd→→1: 0 Conductor loss: k ε εεαeff →→: 0 r tan δ rrd 2 RRss αc ≈≈ Zw0 η h very crude (“parallel-plate”) approximation ηh Z0 ≈ (More accurate formulas are given later.) w

17 Microstrip (cont.)

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

εεeff ( f ) −10eff ( ) Zf= Z0 rr 00( ) ( )eff eff εεrr(01) − ( f )

η = 0 Z0 (0) ≥ ε eff ′′++ + (wh / 1) r (0) (wh /) 1.393 0.667ln((wh /) 1.444)

th2 ww′ =++1 ln  π t w t

ε r h

18 Microstrip (cont.) where

2 εε− eff (0) εεeff ( f ) =eff (0) + rr (wh /≥ 1) rr+ −1.5 14F

εε+−1 1 1ε− 1/th eff rrr ε r (0) =+−  2 2 4.6 1+ 12(hw / ) wh/

2 hw As f → 0: F =4ε r − 1 0.5 ++ 1 0.868ln 1 + λ h effNote: eff 0 εεrr( f ) → (0)

→∞ w t As f : eff εεrr( f ) → ε r h

19 Microstrip (cont.) 2 β ε eff = 12.0 r  k0 11.0 ε r c 10.0 vp = eff ε r 9.0 Frequency variation 8.0 (dispersion) “A frequency-dependent solution for microstrip transmission lines," E. J. 7.0

Effectivedielectric constant Denlinger, IEEE Trans. Microwave Theory and Frequency (GHz) Techniques, Vol. 19, pp. 30-39, Jan. 1971. Parameters: εr = 11.7, w/h = 0.96, h = 0.317 cm

Note: The flux lines get more concentrated in the Note: substrate region as the The phase velocity is a frequency increases. function of frequency, which Quasi-TEM region causes pulse distortion. 20 Microstrip (cont.)

More accurate formulas for conductor attenuation:

2 1 w Rs 12  w′  hh ht <≤2 αc =1 −  1 ++ln  − ππ′′ 2π h hZ0 24 h w w t h

−2  ′′′ ′ π  w Rs w22 w   wwh/ ( ) hh ht απc = +ln 2e + 0.94    +1 ++ln  −  ≥ 2 hZ h ππ2h  h w′ w′′ w t h h 0   + 0.94   2h

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

th2 ww′ =++1 ln  π t w t 

ε r h

21 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. Pl (0) αc = 1 2P0 tJ=∝→0: sz as s0 R 2 s = s Pls(0) ∫ Jd 2 + CC12 z=0

J sz Practical note: s A standard metal thickness for PCBs is 0.7 [mils] (17.5 [µm]), w t called “half-ounce copper”.

1 mil = 0.001 inch ε r h

22 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

23 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.

24 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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