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MICROWAVE ENGINEERINGMICROWAVE ENGINEERING

LECTURE NOTE

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PHPHẠẠM VI CM VI CỦỦA LA LĨĨNH VNH VỰỰC SIÊU C SIÊU CAO TCAO TẦẦN N

TẦN SỐ THÔNG THƯỜNG TỪ 1GHz TRỞ LÊN

BẢNG PHÂN ĐỊNH TẦN SỐ

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ĐƯĐƯỜỜNG DÂY TRUYNG DÂY TRUYỀỀN SN SÓÓNGNG. ĐIỆN ÁP VÀ DÒNG ĐIỆN PHỤ THUỘC CẢKHÔNG GIAN Ở VỊ TRÍ z VÀ THỜI GIAN TẠI

THỜI ĐIỂM t,

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Module 2:Module 2: Transmission LinesTransmission LinesTopic 1: Topic 1: TheoryTheory

OGI EE564Howard Heck

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Where Are We? Where Are We? 1. Introduction2. Transmission Line Basics

1. Transmission Line Theory2. Basic I/O Circuits3. Reflections4. Parasitic Discontinuities5. Modeling, Simulation, & Spice6. Measurement: Basic Equipment7. Measurement: Time Domain Reflectometry

3. Analysis Tools4. Metrics & Methodology5. Advanced Transmission Lines

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ContentsContentsPropagation VelocityCharacteristic ImpedanceVisualizing Transmission Line BehaviorGeneral Circuit ModelFrequency DependenceLossless Transmission LinesHomogeneous and Non-homogeneous LinesImpedance Formulae for Transmission Line StructuresSummaryReferences

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Propagation VelocityPropagation Velocity

Physical example:

Wave propagates in z direction

Circuit: L = [nH/cm]C = [pF/cm]

( )tILdzdz

zV

∂∂

∂∂

−=Total voltage change across Ldz (use ):ΔV L dI

dt= −

Total current change across Cdz (use ):dt

dVCI −=Δ( )

tVCdzdz

zI

∂∂

∂∂

−=

[2.1.1]

[2.1.2]

Simplify [2.1.1] & [2.1.2] to get the Telegraphist’s Equations [2.1.3a]

tIL

zV

∂∂

∂∂

−=

tVC

zI

∂∂

∂∂

−=

[2.1.3b]

I

V

Ldz

Cdz

dz

V+ dzdVdz

I+ dzdIdz

z

xy

V, I

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Propagation Velocity (2)Propagation Velocity (2)

Phase velocity definition: vLC

≡1

[2.1.7]

Equation in terms of current:2

2

22

2

2

2 1tI

tILC

zI

∂∂

=∂∂

=∂∂

ν[2.1.8]

Equate [2.1.4] & [2.1.5]: [2.1.6]2

2

22

2

2

2 1tV

tVLC

zV

∂∂

=∂∂

=∂∂

ν

Differentiate [2.1.3b] by z: [2.1.5]ztIL

zV

∂∂∂

−=∂∂ 2

2

2

Differentiate [2.1.3a] by t: [2.1.4]2

22

tVC

tzI

∂∂

−=∂∂

∂

Equation [2.1.6] is a form of the wave equation. The solution to[2.1.6] contains forward and backward traveling wave components, which travel with a phase velocity.

An alternate treatment of propagation velocity is contained in the appendix.

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Characteristic Impedance Characteristic Impedance (Lossless)(Lossless)

The input impedance (Z1) is the impedance of the first inductor (Ldz) in series with the parallel combination of the impedance of the capacitor (Cdz) and Z2.

Ldz

Cdx

Z1 Z2 Z3

Ldz

Cdz

Ldz

Cdz

dz dz

V1 V3V2 to ∞

a

fed

cb

dz

dz = segment length

C = capacitance per segment

L = inductance per segment

[2.1.9]( )

CdzjZCdzjZLdzjZωωω

/1/1

2

21 +

+=

( ) ( ) ( ) 0/1/1/1 2221 =−+−+ lCjZlCjZlLjlCjZZ ωωωω

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Characteristic Impedance Characteristic Impedance (Lossless)(Lossless)

Assuming a uniform line, the input impedance should be the same when looking into node pairs a-d, b-e, c-f, and so forth. So, Z2 = Z1= Z0.( ) ( ) ( ) 0/1/1/1 0000 =−+−+ CdzjZCdzjZlLdzjCdzjZZ ωωωω [2.1.10]

CdzjLdzjdzLZjZ

CdzjZ

CdzLdzdzLZj

CdzjZZ

ωωω

ωωωω

ω−−==−−−+ 0

20

00

020 0

Allow dz to become very small, causing the frequency dependent term to drop out:

0020 =−−

CLdzLZjZ ω [2.1.11]

020 =−

CLZ [2.1.12]

Solve for Z0:

CLZ =0

[2.1.13]

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Visualizing Transmission Line Visualizing Transmission Line BehaviorBehavior

Water flow– Potential = Wave

height [m]– Flow = Flow rate

[liter/sec]

I

I

V

+++++++

- - - - - - -

Transmission LinePotential = Voltage [V]Flow = Current [A] = [C/sec]

Just as the wave front of the water flows in the pipe, the voltage propagates in the transmission line. The same holds true for current.

Voltage and current propagate as waves in the transmission line.

h

f

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Visualizing Transmission Line Visualizing Transmission Line Behavior #2Behavior #2

Extending the analogy– The diameter of the pipe relates the flow rate

and height of the water. This is analogous to electrical impedance.

– Ohm’s law and the characteristic impedance define the relationship between current and potential in the transmission line.

Effects of impedance discontinuities– What happens when the water encounters a

ledge or a barrier?– What happens to the current and voltage

waves when the impedance of the transmission line changes?

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General Transmission Line General Transmission Line Model (No Coupling)Model (No Coupling)

Transmission line parameters are distributed (e.g. capacitance per unit length).A transmission line can be modeled using a network of resistances, inductances, and capacitances, where the distributed parameters are broken into small discrete elements.

R L

G C

R L

G C

R L

G C

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General Transmission Line General Transmission Line Model #2Model #2

Ω-1•cm -1

GDielectric Conductance

pF•cm-1CTotal CapacitancenH•cm-1LSelf InductanceΩ•cm-1RConductor

Resistance

UnitsSymbol

ParameterParameters

Characteristic Impedance Z R j LG j C0 =++

ωω [2.1.14]

Propagation Constant ( )( ) βαωωγ jCjGLjR +=++= [2.1.15]

α = attenuation constant = rate of exponential attenuationβ = phase constant = amount of phase shift per unit length

βων =pPhase Velocity [2.1.16]

In general, α and β are frequency dependent.

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Frequency DependenceFrequency Dependence

From [2.1.14] and [2.1.15] note that:Z0 and γ depend on the frequency content of the signal.Frequency dependence causes attenuation and edge rate degradation.

Attenuation

Edge rate degradation

Output signal from lossytransmission line

Signal at driven end oftransmission line

Output signal fromlossless transmission line

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Frequency Dependence #2Frequency Dependence #2R and G are sometimes negligible, particularly at low frequencies– Simplifies to the lossless case: no attenuation

& no dispersion In modules 2 and 3, we will concentrate on lossless transmission lines.Modules 5 and 6 will deal with lossy lines.

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Lossless Transmission LinesLossless Transmission Lines

QuasiQuasi--TEM AssumptionTEM AssumptionThe electric and magnetic fields are perpendicular to the propagation velocity in the transverse planes.

x

zy

HE

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Lossless Line ParametersLossless Line Parameters

Lossless line characteristics are frequency independent.As noted before, Z0 defines the relationship between voltage and current for the traveling waves. The units are ohms [Ω].υ defines the propagation velocity of the waves. The units are cm/ns.

S ti th ti d l

CLZ =0

vL C

=1

Characteristic ImpedanceCharacteristic Impedance

Propagation VelocityPropagation Velocity

[2.1.17]

[2.1.18]

Lossless transmission lines are characterized by the following two parameters:

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Lossless Line Equivalent CircuitLossless Line Equivalent Circuit

The transmission line equivalent circuit shown on the left is often represented by the coaxial cable symbol.

L

C

L

C

L

C

Z0, v, lengthZ0, ν, length

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Homogeneous MediaHomogeneous Media

A homogeneous dielectric medium is uniform in all directions.– All field lines are contained within the

dielectric.For a transmission line in a homogeneous medium, the propagation velocity depends only on material properties:

vLC

c cm ns

r r r

= = = =1 1 300

εμ ε μ ε/

[2.1.19]

0εεε r= Dielectric Permittivity

cmFx 14

0 10854.8 −=ε Permittivity of free space

cmHx 8

0 10257.1 −=μ Magnetic Permeability

0μμ ≅ Permeability of free space

εr is the relative permittivity or dielectric constant.

Note: only Note: only εεrris required to is required to calculate calculate νν..

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NonNon--Homogeneous MediaHomogeneous Media

A non-homogenous medium contains multiple materials with different dielectric constants.For a non-homogeneous medium, field lines cut across the boundaries between dielectric materials.In this case the propagation velocity depends on the dielectric constants and the proportions of the materials. Equation [2.1.19] does not hold:

εμ11

≠=LC

v

In practice, an effective dielectric constant, εr,eff is often used, which represents an average dielectric constant.

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Some Typical Transmission Some Typical Transmission Line StructuresLine Structures

And useful formulas for Z0

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rR

εr

TrTrởở khkháángng ccáápp đđồồngng trtrụụcc

2 3 4 5 6 7 8 9 10R/r

20

40

60

80

100

120

140

Z 0[Ω

]

εr = 1

εr= 4rε = 3.5εr = 3εr = 2.5rε = 2

Z0, v, lengthZ0, υ, length

⎟⎠⎞

⎜⎝⎛=

rRZ ln

21

0 εμ

π

[2.1.20]

⎟⎠⎞

⎜⎝⎛

=

rR

Cln

2πε

[2.1.21]

⎟⎠⎞

⎜⎝⎛=

rRL ln

2πμ

[2.1.22]

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Centered Stripline ImpedanceCentered Stripline Impedance( )⎟⎟

⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+=

wtw

hZ

r 8.067.0

4ln60 2

0πε

w

t

h1

h2

εr

Source: Motorola application note AN1051.

35.02

<− th

wValid for

25.02<h

t

0.003 0.005 0.007 0.009 0.011 0.013 0.015w [in]

10

15

20

25

30

35

40

45

50

55

60Z 0

[Ω] 0.070

0.0600.0500.0400.0300.0250.020

h2

t = 0.0007”εr = 4.0

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Dual Stripline ImpedanceDual Stripline Impedancew

t

h2

h1

εr

w

t

h1

ZYYZZ+

=2

0

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+=

wtw

hYr 8.067.0

8ln60 1

πε

( )( )⎟⎟

⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+

+=

wtw

hhZr 8.067.0

8ln60 21

πε

( ) ( )⎥⎦⎤

⎢⎣⎡

++

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++

−

=twththh

h

Zr 8.0

29.1ln4

180121

1

0 ε

.115.0 hwh ≤≤

Source: Motorola application note AN1051.

OR

0.003 0.005 0.007 0.009 0.011 0.013 0.015w [in]

10

20

30

40

50

60

70

80

90

100

110Z 0

[Ω] 0.020”

0.018”0.015”0.012”0.010”0.008”

0.005”

2h1 + h2 + 2t = 0.062”t = 0.0007”εr = 4.0

h1

[2.1.24]

[2.1.27]

[2.1.25]

[2.1.26]

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Surface Microstrip ImpedanceSurface Microstrip Impedancew

t

h

εr

ε0

[ ]Ω⎟⎠⎞

⎜⎝⎛=

dhZ

eff

4ln21

0 εμ

π

twd 67.0536.0 +=

( ) 067.0475.0 εεε += reff

[ ]Ω⎟⎠⎞

⎜⎝⎛

++=

twhZ

r 8.098.5ln

41.187

0 ε

0.003 0.005 0.007 0.009 0.011 0.013 0.015w [in]

20

40

60

80

100

120

140

160Z 0

[Ω]

0.025”0.020”0.015”0.012”0.009”0.006”0.004”

h

t = 0.0007”

εr = 4.0

[2.1.28]

[2.1.29]

[2.1.30]

[2.1.31]

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Embedded MicrostripEmbedded Microstript

h1

εr

ε0 w

h2

⎟⎠⎞

⎜⎝⎛

++=

twhKZ

r 8.098.5ln

2805.01

0 ε

6560 where ≤≤ K

⎟⎠⎞

⎜⎝⎛

++′=

twhZ

r 8.098.5ln

41.187 1

0 ε

[ ]1255.11 hhrr e−−=′ εε

67.0475.0017.1 += rετ

Or

0.003 0.005 0.007 0.009 0.011 0.013 0.015w [in]

0

20

40

60

80

100

120

140

Z 0[Ω

] 0.015”0.012”0.010”0.008”0.006”0.005”0.003”

h2 - h1 = 0.002“t= 0.0007”εr = 4.0

h1

[2.1.32]

[2.1.33]

[2.1.34]

[2.1.35]

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SummarySummary

System level interconnects can often be treated as lossless transmission lines.Transmission lines circuit elements are distributed. Voltage and current propagate as waves in transmission lines.Propagation velocity and characteristic impedance characterize the behavior of lossless transmission lines.Coaxial cables, stripline and microstrip

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ReferencesReferences

S. Hall, G. Hall, and J. McCall, High Speed Digital System Design, John Wiley & Sons, Inc. (Wiley Interscience), 2000, 1st edition.H. Johnson and M. Graham, High-Speed Signal Propagation: Advanced Black Magic, Prentice Hall, 2003, 1st edition, ISBN 0-13-084408-X.W. Dally and J. Poulton, Digital Systems Engineering, Cambridge University Press, 1998. R.E. Matick, Transmission Lines for Digital and Communication Networks, IEEE Press, 1995.R. Poon, Computer Circuits Electrical Design, Prentice Hall 1st edition 1995

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BBẢẢN CHN CHẤẤT CT CỦỦA QUA QUÁÁ TRÌNH TRÌNH TRUYTRUYỀỀN SN SÓÓNGNG

THỰC CHẤT LÀ ĐƯỜNG DÂY TRUYỀN SÓNG TRUYỀN NĂNG LƯỢNG DƯỚI DẠNG SÓNG CAO TẦNQUÁ TRÌNH TRUYỀN NÀY CÓ VẬN TỐC NHẤT ĐỊNHĐIỆN ÁP VÀ DÒNG ĐIỆN THAY ĐỔI TƯƠNG ỨNG THEO

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( )txv ,

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PHƯƠNG TRÌNH TRUYPHƯƠNG TRÌNH TRUYỀỀN SN SÓÓNG NG TRÊN ĐƯTRÊN ĐƯỜỜNG DÂYNG DÂY

HỆ PHƯƠNG TRÌNH MAXWELL

tBE∂∂

−=r

rrot

tDJH∂∂

+=r

rrrot

ρ=Dr

div

0div =Br

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MÔ HÌNH VMÔ HÌNH VẬẬT LÝT LÝ

+

-

SZ

SV LZ

x xx Δ+ l

( )txv , ( )tzxv ,Δ+

( )txi , ( )tzxi ,Δ+

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MÔ HÌNH VMÔ HÌNH VẬẬT LÝT LÝ

+

-

SZ

SV LZ

x xx Δ+ l

( )txv , ( )tzxv ,Δ+

( )txi , ( )tzxi ,Δ+

x xx Δ+

( )txv , ( )tzxv ,Δ+

( )txi ,

( )tzxi ,Δ+

( )txv , ( )tzxv ,Δ+

( )txi , ( )tzxi ,Δ+

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LZ

l

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RΔ

x xx Δ+

( )txv , ( )tzxv ,Δ+

( )txi , ( )tzxi ,Δ+

xLΔ

xGΔxCΔ

xRΔ LΔ

GΔCΔ

xΔRất nhỏ

xRR ΔΔ =xLL ΔΔ =xGG ΔΔ =xCC ΔΔ =

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x xx Δ+

( )txv , ( )tzxv ,Δ+

( )txi , ( )tzxi ,Δ+

xLΔ

xGΔ xCΔ

xRΔ

( ) ( ) ( ) ( )txxvx

txixLtxixRtxv ,,,, ΔΔΔ ++∂

∂•+•=

( ) ( ) ( ) ( )txxix

txxvxCtxxvxGtxi ,,,, ΔΔ

ΔΔΔ ++∂+∂

•++•=

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ĐĐểể ttíínhnh

ccầầnn xxéétt mmộộtt đođoạạnn nhnhỏỏ

( )txv , ( )txi ,

xΔ

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ĐiĐiỆỆNN ÁÁP VP VÀÀ DÒNG DÒNG ĐiĐiỆỆNN

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TTÍÍNH VI SAINH VI SAI

0→Δz

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MIMIỀỀN TN TẦẦN SN SỐỐ

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HHẰẰNG SNG SỐỐ SSÓÓNGNG

[ ]mradLC /2λπωβ ==

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MÔ HÌNH MMÔ HÌNH MẠẠCH ĐƯCH ĐƯỜỜNG DÂY DNG DÂY DÀÀII

+

-

SZ

SV LZ

z l

( )tzv ,

( )tzi ,

( )zV

( )zI

0

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PHƯƠNG TRÌNH TRUYPHƯƠNG TRÌNH TRUYỀỀN SN SÓÓNG NG TRÊN ĐƯTRÊN ĐƯỜỜNG DÂYNG DÂY

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TRTRỞỞ KHKHÁÁNG ĐNG ĐẶẶC TC TÍÍNH CNH CỦỦA A

ĐƯĐƯỜỜNG DÂY NG DÂY 0Z

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TTẢẢI TRÊN ĐƯI TRÊN ĐƯỜỜNG DÂY NG DÂY TRUYTRUYỀỀN SN SÓÓNGNG

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+

-

HHỆỆ SSỐỐ PHPHẢẢN XN XẠẠ

SZSV LZ

zl

+0V

−0V

d

β,0Z

SP

( ) dji eIzI β−+= 0 ( ) dj

r eIzI β−= 0

( ) dji eVzV β−+= 0

( ) djr eVzV β−= 0

−LV

+|LV

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HHỆỆ SSỐỐ PHPHẢẢN XN XẠẠ TTẠẠI TI TẢẢII

0

0

ZZZZ

VV

L

L

L

LL +

−=≡Γ +

−

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HHỆỆ SSỐỐ PHPHẢẢN XN XẠẠ TTẠẠI MI MỘỘT T ĐiĐiỂỂMM TRÊN ĐƯTRÊN ĐƯỜỜNG DÂYNG DÂY

( ) ( )( )

djL

i

r edVdVd β2−•Γ==Γ

zld −=

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+

-

HHỆỆ SSỐỐ PHPHẢẢN XN XẠẠ CÔNG SUCÔNG SUẤẤTT

SZ

SV LZ

zl

+0V

−0V

d

β,0Z

( )zPi

( )zPr

( )lPi

( )lPr

LP

SP

( )0iP

( )0rP

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HHỆỆ SSỐỐ PHPHẢẢN XN XẠẠ CÔNG SUCÔNG SUẤẤTT

( ) 22 dPP

i

r Γ==Γ

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CCÁÁC TRƯC TRƯỜỜNG HNG HỢỢP ĐP ĐẶẶC BIC BIỆỆTTTTẢẢI NGI NGẮẮN MN MẠẠCHCH

0

0

RZRZ

L

LL +

−=Γ

100

0

0 −=+−

=RR

LΓ

( ) ( )lVlV ri −=

( ) ( ) ( ) 0=+= lVlVlV ri

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+

-

TRTRỞỞ KHKHÁÁNG ĐƯNG ĐƯỜỜNG DÂYNG DÂY

SZ

SVLZ

ld( )dZ

β,0Z

INZ

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TRTRỞỞ KHKHÁÁNG TNG TẠẠI I ĐiĐiỂỂMM CCÁÁCH TCH TẢẢI I MMỘỘT KHOT KHOẢẢNG NG dd

( ) ( )( )Ω+

+=

djZZdjZZZdZ

L

L

ββ

tantan

0

00

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TRTRỞỞ KHKHÁÁNG ĐNG ĐẦẦU ĐƯU ĐƯỜỜNG DÂYNG DÂY

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STANDING WAVESTANDING WAVE

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TTỈỈ SSỐỐ SSÓÓNG ĐNG ĐỨỨNG LNG LỚỚNN

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TTỈỈ SSỐỐ SSÓÓNG ĐNG ĐỨỨNG NHNG NHỎỎ

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SSÓÓNG ĐNG ĐỨỨNG TRÊN ĐƯNG TRÊN ĐƯỜỜNG NG DÂY DÂY

TTỈỈ SSỐỐ SSÓÓNG ĐNG ĐỨỨNGNG

min

max

VVVSWR =

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CCÁÁC TRƯC TRƯỜỜNG HNG HỢỢP ĐP ĐẶẶT BIT BIỆỆTT

ĐƯỜNG DÂY NGẮN MẠCH TẢI

( ) ( )( )djZZ

djZZZdZL

Lin β

βtantan

0

00 +

+=

( ) ( )djZdZin βtan0=0=LZ

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CCÁÁC TRƯC TRƯỜỜNG HNG HỢỢP ĐP ĐẶẶT BIT BIỆỆTT

ĐƯỜNG DÂY HỞMẠCH TẢI

( ) ( )( )djZZ

djZZZdZL

Lin β

βtantan

0

00 +

+=

( ) ( )djZdZin βcotan0−=∞=LZ

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ĐƯĐƯỜỜNG TRUYNG TRUYỀỀN MN MỘỘT PHT PHẦẦN N TƯ BƯTƯ BƯỚỚC SC SÓÓNGNG

( ) ( )( )djZZ

djZZZdZL

Lin β

βtantan

0

00 +

+=

( )L

in ZZZ

204 =λ

λπβ 2

=4λ

=d

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DÂY NGẮN MẠCH:

( ) ∞=4λINZ0=LZ

DÂY HỞ MẠCH:

( ) 04 =λINZ∞=LZ

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DÂY NGẮN MẠCH:

( ) ∞=4λINZ0=LZ

DÂY HỞ MẠCH:

( ) 04 =λINZ∞=LZ

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"Threaded Neill-Concelman" connector, according to Johnson Components, it is actually a threaded BNC connector, to reduce vibration problems. Carl Concelmanwas an engineer at Amphenol.

PTFE 15 GHz TNC

Sub-miniature type A developed in the 1960s, perhaps the most widely-used microwave connector system in the universe.

PTFE 25 GHz SMA

Sub-miniature type C, a threaded subminiature connector, not widely used.

PTFE 10 GHz SMC

Micro-miniature coax connector, popular in the wire industry because its small size and cheap price.

PTFE MMCX

MCX was the original name of the Snap-On"micro-coax" connector species. Available in 50 and 75 ohms.

PTFE 6 GHz OSX, MCX, PCX

A surface mount connector PTFE 6 GHz OSMT

"Sub-miniature type B", a snap-on subminiature connector, available in 50 and 75 ohms.

PTFE 4 GHz SMB

"Bayonet type-N connector", or "Bayonet Neill-Concelman" according to Johnson Components. Developed in the early 1950s at Bell Labs. Could also stand for "baby N connector".

PTFE 4 GHz BNC

Comments and historyDielectric Frequency Limit

Connector type

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The original mass-marketed 2.92 mm connector, made by Wiltron (now Anritsu). Named the "K" connector, meaning it covers all of the K frequency bands.

Air 40 GHz K

Precision connector, developed by Mario Maury in 1974. 2.92 mm will thread to cheaper SMA and 3.5 mm connectors. Often called "2.9 mm".

Air 40 GHz 2.92 mm

Smaller than an SMA.PTFE 38 GHz SSMA

OSP stands for "Omni-Spectra subminiature push-on", a smaller version of OSP connector.

PTFE 28 GHz OSSP

A precision (expensive) connector, it mates to cheaper SMA connectors.

Air 26.5 GHz 3.5 mm

OSP stands for "Omni-Spectra push-on", a blind-mate connector with zero detent. Often used in equipment racks.

PTFE 22 GHz OSP

APC-7 stands for "Amphenol precision connector", 7mm. Developed in the swinging 60s, ironically a truly sexless connector, which provides the lowest VSWR of any connector up to 18 GHz.

PTFE 18 GHz APC-7, 7 mm

Named for Paul Neill of Bell Labs in the 1940s, available in 50 and 75 ohms. Cheap and rugged, it is still widely in use. Originally was usable up to one GHz, but over the years this species has been extended to 18 GHz, including work by Julius Botka at Hewlett Packard.

PTFE 11 GHznormal

18 GHzprecision

N

"Threaded Neill-Concelman" connector, according to Johnson Components, it is actually a threaded BNC connector, to reduce vibration problems. Carl Concelman was an engineer at Amphenol.

PTFE 15 GHz TNC

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The Rolls Royce of connectors. This connector species works up to 110 GHz. It costs a fortune! Developed at Hewlett Packard (now Agilent) by Paul Watson in 1989.

Air110 GHz1 mm

Anritsu's term for 1.85 mm connectors because they span the V frequency band.

Air60 GHzV

Mechanically compatible with 2.4 mm connectors. Air60 GHz1.85 mm

2.4 mm, and 1.85 mm will mate with each other without damage. Developed by Julius Botka and Paul Watson in 1986, along with the 1.85 mm connector.

Air 50 GHz 2.4 mm

Smaller version of OSP blind-mate connector.40 GHz OS-50P

"Gilbert push-on", "Omni-spectra microminiaturepush-on"

PTFE 40 GHz GPO, OSMP, SMP

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RETURN LOSSRETURN LOSS

dBRL Γ−= log20

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TRANSMISSION COEFICIENTTRANSMISSION COEFICIENT

Γ+=1T

00

0 21ZZ

ZZZZZT

L

L

L

L

+=

+−

+=

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INSERTION LOSSINSERTION LOSS

dBTIL log20−=

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SMITH CHARTSMITH CHART

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MMỐỐI QUAN HI QUAN HỆỆ GIGIỮỮA TRA TRỞỞKHKHÁÁNG VNG VÀÀ HHỆỆ SSỐỐ PHPHẢẢN XN XẠẠ

( ) ( )( )xxZxZ

Γ−Γ+

=11

0

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CCÁÁC GIC GIÁÁ TRTRỊỊ CHUCHUẨẨN HN HÓÓAA

( ) ( )0RxZxz =

jxrz +=

TRTRỞỞ KHKHÁÁNG CHUNG CHUẦẦN HN HÓÓAA

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0RZz L

L =

10

00 ==

RRr

( ) ( )0YxYxy =

( ) ( )xZxY 1=

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HHỆỆ SSỐỐ PHPHẢẢN XN XẠẠ

( ) ( )( ) 0

0

RxZRxZx

+−

=Γ

( ) ( )( ) 1

1

0

0

+−

=ΓRxZRxZx

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MMỐỐI QUAN HI QUAN HỆỆ GIGIỮỮA HA HỆỆ SSỐỐPHPHẢẢN XN XẠẠ VVÀÀ TRTRỞỞ KHKHÁÁNG NG

CHUCHUẨẨN HN HÓÓAA

11

+−

=Γxzxzx

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CÓ 1 GIÁ TRỊ THÌ CHỈ CÓ DUY NHẤT 1 GIÁ TRỊ

CÓ 1 GIÁ TRỊ THÌ CHỈ CÓ DUY NHẤT 1 GIÁ TRỊ

( )xΓ ( )xz

( )xz ( )xΓ

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BBẢẢN CHN CHẤẤT VT VÀÀ CCÁÁCH BICH BIỂỂU DIU DIỂỂN N HHỆỆ SSỐỐ PHPHẢẢN XN XẠẠ

060

( )xΓ

( )( )xΓRe

( )( )xΓIm

( )ΓRe

( )ΓIm

1+1−

1+

1−

0

Mặt phẳng phức

( ) 0608.0 ∠=Γ x

8.0

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HHỆỆ SSỐỐ PHPHẢẢN XN XẠẠ

( ) ( ) ( )xjxx ir Γ+Γ=Γ

ir jΓ+Γ=ΓDạng đơn giản

( )( )⎩

⎨⎧

Γ=ΓΓ=Γ

ImRe

i

r

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TRTRỞỞ KHKHÁÁNG ĐƯNG ĐƯỜỜNG DÂYNG DÂY

( ) ( ) ( )xjXxRxZ +=

jXRZ +=

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TRTRỞỞ KHKHÁÁNG CHUNG CHUẨẨN HN HÓÓAA

( ) ( ) jxxrxz +=jxrz +=

0RRr =

0RXx =

Trở kháng đườngdây chuẩn hóa

Điện trở đường dâychuẩn hóa

Điện kháng đườngdây chuẩn hóa

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ir

ir

jjjxrΓ−Γ−Γ+Γ+

=+11

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( ) 22

22

11

ir

irrΓ+Γ−Γ−Γ−

=

( ) 2212

ir

ixΓ+Γ−

Γ=

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PHƯƠNG TRÌNH ĐƯPHƯƠNG TRÌNH ĐƯỜỜNG TRÒNNG TRÒN2

22

11

1⎟⎠⎞

⎜⎝⎛+

=Γ+⎟⎠⎞

⎜⎝⎛

+−Γ

rrr

ir

⎟⎠⎞

⎜⎝⎛+

0,1 r

rtâm

bán kínhr+1

1

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( )ΓRe

( )ΓIm

1+1−

1+

1−

0

Mặt phẳng phức

0=r2.0=r

5.0=r1=r 2=r

iΓ

rΓ

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PHƯƠNG TRÌNH ĐƯPHƯƠNG TRÌNH ĐƯỜỜNG TRÒNNG TRÒN

( )22

2 111 ⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ −Γ+−Γ

xxir

⎟⎠⎞

⎜⎝⎛

x1,1tâm

bán kínhx1

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( )ΓRe

( )ΓIm

1+1−

1+

1−

0

Mặt phẳng phức

5.0=x 1=x

iΓ

rΓ

5.0−=x 1−=x

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( )ΓRe

( )ΓIm

1+1−

1+

1−

0

Mặt phẳng phức

5.0=x 1=x

iΓ

rΓ

5.0−=x

1−=x

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( )ΓRe

( )ΓIm

1+1−

1+

1−

0

Mặt phẳng phức

0=r2.0=r

5.0=r1=r 2=r

iΓ

rΓ

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( )ΓRe

( )ΓIm

1+1−

1+

1−

0

Mặt phẳng phức

5.0=x1=x

iΓ

rΓ

5.0−=x

1−=x

2.0=r5.0=r

1=r 2=r

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