lecture 6 transmission lines – part i: tem and quasi-tem tls and

Lecture 6

Transmission Lines – Part I:TEM and Quasi-TEM TLs

and PUL ParametersOptional Reading: Steer – Section 1.7

Transmission Lines You Know: Coaxial Line

2a2b

EH

The instantaneous field vectors in the cross-section of a coaxial line at a certain moment in time are given in the plot. What is the direction of the instantaneous Poynting vector at this time, in or out of the plot?

ElecEng4FJ4 Nikolova 2LECTURE 06: TRANSMISSION LINES - PART I

Static Field Distribution in the Coaxial Line Cross-section

1( ) ˆln( / )

Vb a

E ρ

1ˆ( )2I

H


PUL Parameters of the Coaxial Line • obtained through 2D static analysis

2 (F/m)ln( / )

Cb a

2 (S/m)ln( / )

dGb a

ln (H/m)2

bLa

1 1 , /m2 2

1c

Ra b

a

bI cI

metalIdielI

CG

LR

tand d

ElecEng4FJ4 Nikolova 4

R l L l

G l C l

l

Nv

z

1Nv

Ni 1Ni N N

sR

see Tutorial #2, Problem #6, p.15

Resistance PUL : DC vs. AC

• reminder: DC static analysis gives

2 21 1 1 1c c

RAa b


,w ,w

,sh ,sh

22

eff eff

eff eff

A a wA b w

1 1 1 , /m1c c

LRRL L A A

• high-frequency Rꞌ is higher since current is confined to skin layer

• effective cross-sectional areas through which current flows

inner wire

L

2,w ,weff effA w A a

,w 2effw a2aI

s s

,w ,sheff eff

R RRw w

effective width of wireeffective width of shield

sR

Resistance PUL at High Frequencies: Effective Cross-section

• current flows in a very thin (skin) layer δ of the conductors of the TL, with an effective area Aeff much smaller than the physical cross-section A

L

2eff effA w A a

2effw a2aI

strip line

the inner conductor of a coax or one of the leads of a twin-lead line

eff effA w A tw effw w

L

th


Resistance PUL – General Formula


• start from loss power per unit area (see L05, sl. 7 and L03, sl. 32)2 2

s0.5 | | , W/m where , A/mˆs s sp R J J J z

• find loss power per unit length by integrating over the contour C of the TL cross-section (assume Rs the same over entire contour)

m

2s0.5 | | , W/m

C sCR J dP p l ld

portion of C that is metal

• introduce current flowing through Cm

m, AsC

I J dl

m

2 2s00.5 | . | || 5 sC

RP R dI J l • express loss power PUL through the resistance PUL

Resistance PUL – The Case of Uniform Current Density


• express resistance PUL to obtain the general formula

m m

m

2 2

s s 22

| | | |

| |

s sC C

sC

J dl J dlR R R

I J dl

• if surface current density is the same over Cm

m m

mm

2 2

s s2 22

s| | | |

, /m| |

effw

C

eff

sC

s C

s

sC

J dl JR R R

J

dl R

l dld J w

Coaxial Line: Applications


• the coaxial line is the preferred interconnect in measurement instrumentation up to about 110 GHz

• the higher the frequency the smaller the cross-section to avoid non-TEM modes

• great variety of coaxial connectors and adaptors exists; they ensure reflection-free interconnections between cables and devices (precision adaptors typically have SWR ≤ 1.15)

• microwave connectors/adaptors industry is more than a 2-billion-dollar-per-year industry; most of these are coaxial types

Some Common Coaxial Connectors


Connector Freq. Range CommentsBNC ≤ 4 GHz Bayonet type-N connector, Bayonet Neill-Concelman

connector, aka baby N connector; PTFE

TNC ≤ 15 GHz aka Threaded Neill-Concelman connector; this is a threaded BNC connector; PTFE

N-type ≤ 18 GHz Named for Paul Neill of Bell Labs; PTFE; cheap and rugged; very common

SMA ≤ 25 GHz Sub-miniature type A; very common; PTFE; cheap; mates with 3.5 mm and 2.92 mm

3.5 mm ≤ 26 GHz Precision connector; mates with SMA and 2.92 mm; air

2.92 mm ≤ 40 GHz aka 2.9 mm; precision connector; mates with SMA and 3.5 mm; air

2.4 mm ≤ 50 GHz mates with 1.85 mm; air

1.85 mm ≤ 60 GHz mates with 2.4 mm; air

1 mm ≤ 110 GHz air; very expensive

Notes: 1) The name of a connector (e.g., 1.85 mm) is determined by the inner diameter of the shield (outer diameter of air insulator). 2) PTFE stands for Polytetrafluoroethylene (Teflon)

Some Common Coaxial Connectors – 2

11

• connectors can be ruggedized or not; female (F, aka jack or socket) or male (M, aka plug)

• sexless connectors, e.g., APC-7 (expensive, very low SWR)

type N connectors

http://en.wikipedia.org/wiki/N_connector

75 Ω

50 Ω M

MF

F

BNC connectors

F F

M Mhttp://www.citruscables.com/Products/AntennaandRFCableAssemblies/RFConnectorIdentificationChart.aspx

ElecEng4FJ4 Nikolova LECTURE 06: TRANSMISSION LINES - PART I

Some Common Coaxial Connectors – 3


F

3.5 mm connectors

M

SMA connectors

F F

M Mhttp://www.citruscables.com/Products/AntennaandRFCableAssemblies/RFConnectorIdentificationChart.aspx

http://www.microwaves101.com/encyclopedia/connectorsprecision.cfm#35mm

ruggedized

Mpasternack.com

F

Notes on Ruggedized Coaxial Connectors


F

M

• ruggedized connectors are also referred to as NMD connectors

• NMD stands for Network Measurements Division, an outdated term for the original Hewlett-Packard division that produced the first vector network analyzers

• used to connect to microwave-instrument ports (e.g., vector network analyzers)

• 3.5mm and 2.4mm male connectors need to be ruggedized if frequent connection/disconnection of devices is expected

• ruggedized connectors include a large threaded body which stabilizes the test port cable when attached to the front of the instrument

Coaxial Adaptors: Some Jargon

ElecEng4FJ4 Nikolova 14LECTURE 06: TRANSMISSION LINES - PART I F

• male-to-male adaptor; aka barrel

• female-to-female adaptor; aka bullet

• there are also elbow adapters (M/M, F/F, F/M)

SMA

• adaptors are also used to interconnect different types of connectors, e.g., N/F-to-BNC/M, N/M-to-SMA/M, etc.

SMA


Coaxial Cables and Connectors: Why 50 Ω or 75 Ω?

coaxial cables and connectors are designed to have characteristic impedance of 50 Ω or 75 Ω (TV cables)• air-filled coax has minimum attenuation when Z0 ≈ 77 Ω• air-filled coax has maximum power capacity when Z0 ≈ 30 Ω• the 50-Ω line seems to be a compromise between the two values

s 1 12 ln( / )c

Rb a a b

the attenuation of the air-filled coaxial line entirely depends on its conductors

αc is minimized for conductor radii such that

ln 1 , where /x x x x b a 03.5911 ( 1) 77 rx Z

you will derive this formula yourselves later in an assignment

0 ln 2

L bZC a

attenuation constant


Coaxial Cables and Connectors: Why 50 Ω or 75 Ω?

the maximum power capacity of a coaxial line depends on its breakdown voltage

2 2

max lnda E bPa

dielectric strength

the value of x = b/a which maximizes Pmax is ≈ 1.65

0 ( 1) 30 rZ

(a) Derive the expression for the maximum allowed power Pmax in a coaxial cable given above (see *). Hint: See Ex. 2 in Tutorial 2.

(b) Show that, for a fixed b, indeed the ratio b/a ≈ 1.65 maximizes Pmax.

( )

Transmission Lines You Know: Twin-lead Line

braided antenna cable Z0 ≈ 300 Ω

PUL parameters (2D statics)

(F/m)F

C

(S/m)d

FG

12 ( /m)2eff

cA

Rr

LI

IA

A2h

r

2

ln 1 arccoshh h hFr r r

(H/m)FL

( ) or( tan )

d

d d


1s

cR

Transmission Lines You Know: Twin-lead Line (2)Which of the plots represents the static E field and which one represents the static H field in the cross-section of a twin-lead line?


AnsoftMaxwell SV

Transmission Lines You Know: Parallel-plate Line

PUL parameters (2D statics)

(F/m)wCh

(S/m)dwGh

(H/m)hLw

12 ( /m)

effAc

Rw

l

wh

t

( ) or( tan )

d

d d


s1c

R

static H-field

Can you draw the E-field lines of a parallel-plate line on top of the H-field plot above? Assume the upper plate is at a higher potential than the lower plate.

Transmission Lines You Know: Parallel-plate Line


AnsoftMaxwell SV

the resistances PUL in the two leads of a TL add up

• coaxial line

• twin-lead line

• strip line

s 1 1 , /m2RR

a b

s s2 , /m2R RR

a a

s2 , /mRRw


Resistance PUL at High Frequencies – Summary

(negligible proximity effect)

a

bc,w

,sh

22

eff

eff

w aw b

2effw a

effw wl

wh

t


Other Transmission Lines (TEM and quasi-TEM)• in hybrid and monolithic microwave ICs (MICs), a variety of planar

TLs are used – easy to fabricate, not so easy to model analytically

• to avoid higher-order modes, cross-section must be small

microstrip line

cross-section


Microstrip Line: Quasi-static E-field Distribution

AnsoftMaxwell SV


Microstrip Line: Quasi-static H-field Distribution

AnsoftMaxwell SV


Microstrip Line: Applications

• used in both hybrid and monolithic MICs

• upper frequency limit depends on how small a cross-section the fabrication process can handle (tolerances)

• typical frequency range in HMICs: about 1 GHz to 15 GHz

• convenient for series surface mount of lumped devices

• inconvenient for shunt mount (requires via-holes)

1 1 21 2

(b)(a) (c)[Steer, Microwave and RF Design]


(1) analyze the cross-section of the line (with the substrate, if any) using 2D electrostatic solver and obtain charge PUL ρl

(2) calculate capacitance PUL (most simulators will do this for you)

(3) repeat the above 2 steps for the same line but with “air” substrate, i.e., set εr of the substrate equal to 1, and get

(4) calculate the TEM line parameters as follows

0

lCV

0air

1 1impedance Zc C C

airC

airphase velocity pCv cC

2

eair

effective dielectric constant rp

c Cv C

air

e

guide wavelength gr

Arbitrary TL Cross-section: 2D Static Analysis

27

Example: 2D Static Analysis of Microstrip Line• 2D electrostatic simulation of a microstrip line on alumina substrate

6 mm6 mm

hw

101.6964 10 F/mC Ansoft

Maxwell SV

• 2D electrostatic simulation of the same microstrip line on air substrate

11air 2.7210 10 F/mC

• calculations

0air

1 49.06 Zc C C

e

air6.2347r

CC

8

e1.2015 10 m/sp

r

cv

air(3GHz)(3GHz)

e

10 4 cm6.2347

gr

ElecEng4FJ4 Nikolova LECTURE 06: TRANSMISSION LINES - PART I


Explanation of Parameter Formulas from 2D Static Analysis

• the procedure outlined above applies to the analysis of any TEM line, not only a microstrip line

• for a TEM field wave,

• for a voltage/current wave

• the TEM field and the voltage/current wave have the same propagation constant

k

L C

k L C • apply the above formula to the case when the line cross-section

contains only air as dielectric

air air 0 0 air2 2air

1 1 L C L Lc c C


2air

0air

11 1ZC C cC CL

Cc

calculating the characteristic impedance

calculating the effective dielectric constant via

0 e0

Sr

dC

V

E s

• assuming homogeneous dielectric

• if the dielectric is air

r

0

h

wt

(a)

We

h

t

(b)

air 00

S

dC

V

E s

e air/r C C

0 er r

[Steer, Microwave and RF Design]

Explanation of Parameter Formulas from 2D Static Analysis – 2

divide


calculating the phase velocity – TEM waves slow down in dielectrics

air

e

1p

r

C cv cCL C L C

calculating the guided wavelength – λ shortens in dielectrics

air

air air airair

e

pg

r

v c C CC C

2D static analysis of TEM TLs is only approximate – errors grow as frequency increases!

Explanation of Parameter Formulas from 2D Static Analysis – 3

2air

1Lc C


Coplanar Waveguide

cross-section

may or may not have ground plane at bottom

signal

ground


Coplanar Waveguide: Quasi-static E-field Distribution

AnsoftMaxwell SV


Coplanar Waveguide: Quasi-static H-field Distribution

AnsoftMaxwell SV


Coplanar Waveguide: Applications• preferred above 10 GHz as it has reduced loss and dispersion

compared to microstrip• takes up more area than microstrip which is less of an issue at

high frequencies where the lines are small anyway• usually implemented as finite-ground CPW (FG-CPW)• ground planes on both sides need

to be bridged with wire bonds every tenth of λ or so

• if a bottom ground plane is used, the top-layer ground strips can be bridged to the bottom ground through via-holes

• convenient for both shunt and series surface mount of devices

https://awrcorp.com/download/faq/english/docs/Elements/cpwabrgx.htm

air bridge


Menagerie of Printed Quasi-TEM TLs

stripline (embedded) differential lineHOMOGENEOUS

INHOMOGENEOUSdifferential line

slot line

finline

1

2 21

21 1

36

Menagerie of Quasi-TEM TLs – 2

inverted microstrip line

inverted trapped microstrip line


Substrates for Printed TLs

[Steer, Microwave and RF Design]


Summary

• the PUL inductance, capacitance and conductance of TEM TLs are the same as those obtained using static analyses

• the PUL resistance at HF is not the same as that at DC because the effective cross-section through which current flows is much smaller than the actual conductor cross-section:

• most of the losses in TEM TLs are caused by the conducting leads

• hybrid and monolithic RF and microwave ICs use a variety of quasi-TEM TLs such as microstrip line, coplanar waveguide, etc.

• the PUL parameters of such lines are easily determined by 2 electrostatic analyses producing Cꞌ and Cꞌair

actualeff effA w A

lecture 6 transmission lines – part i: tem and quasi-tem tls and

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