Lesson 1

1

June 2008 © 2006 Fabian Kung Wai Lee 1

1. Advanced TransmissionLine Theory

http://pesona.mmu.edu.my/~wlkung/ADS/ads.htm

The information in this work has been obtained from sources believed to be reliable.The author does not guarantee the accuracy or completeness of any informationpresented herein, and shall not be responsible for any errors, omissions or damagesas a result of the use of this information.


Preface

• Transmission lines and waveguides are the most important elements in microwave or RF circuits and systems.

• Transmission lines and waveguides are used to connect various components together to form a complex circuit. This is similar to low frequency circuit, where we use wires or copper track to connect the various components in an electronic circuit.

• In addition, you will see later that many types of microwave components are fabricated from short sections of transmission lines or waveguides.

• For these reasons, a lot of emphasis is placed on understanding the behavior of electromagnetic fields in transmission lines and waveguides.

Transmission line

2


References

• [1] R. E. Collin, “Foundation for microwave engineering”, 2nd edition, 1992, McGraw-Hill.

• [2] D. M. Pozar, “Microwave engineering”, 2nd edition, 1998 John-Wiley & Sons. (3rd edition, 2005 is also available from John-Wiley & Sons).

• [3] S. Ramo, J.R. Whinnery, T.D. Van Duzer, “Field and waves in communication electronics” 3rd edition, 1993 John-Wiley & Sons.

• [4] C. R. Paul, “Introduction to electromagnetic compatibility”, John-Wiley & Sons, 1992.

A very advanced and in-depth book on microwave

engineering. Difficult to read but the information is

very comprehensive. A classic work. Recommended.

Easier to read and understand. Also a good book.

Recommended.

Good coverage of EM theory with emphasis on

applications.


References Cont...

• [5] F. Kung, “Modeling of high-speed printed circuit board.” Master

degree dissertation, 1997, University Malaya.

http://pesona.mmu.edu.my/~wlkung/Master/mthesis.htm

• [6] F. Kung unpublished notes and works.

3


1.0 Review of Electromagnetic (EM) Fields


Electric and Magnetic Fields (1)

• In an electronic system, such as on PCB assembly, there are electric

charges (q). To make our electronic system works, we essentially control

electric charges (the charge density and rate of flow on various point in

the circuit).

• Flow of electric charges due to potential difference (V) produces electric

current (I).

• Associated with charge is electric field (E) and with current is magnetic

field (H) *, collectively called Electromagnetic (EM) fields.

+q1

Test charge

+ q2

Test

current I2Force F

BIFrrr

×= 2

Force F

r

EqF

r

qqˆ

221

41

2

⋅=

=

πε

rr

Coulomb’s Law

r

E

*The magnetic field is B = µµµµH,

H is the magnetization.

To detect an E field we use

an electric charge. To detect

H field we use a current loop

H

I

4


Electric and Magnetic Fields (2)

+

+

+

+ +

+

++

-

--

-

-

• E fields, by convention is directed from conductor with

higher potential to conductor with less potential.

• Direction indicates force experienced by a small test charge

according to Coulomb’s Force Law.

• Density of the field lines corresponds to strength of the field.

Force on a test charge qq

• H fields, by convention is directed according

to the right-hand rule.

• Direction indicates force experienced by a small

test current according to Lorentz’s Force Law.

•Density of field lines corresponds to strength

of the field.

E fields

H fields

( )BvqF ×=

rFr

Qq

o

)

241 ⋅=

πε

+I

-I

Conductor

Conductor

Conductor

Conductor


Maxwell Equations (Linear Medium) - Time-Domain Form (1)

0=⋅∇

=⋅∇

+=×∇

−=×∇

∂∂

∂∂

B

D

DJH

BE

v

t

t

r

r

rrr

rr

ρ

Where:

E – Electric field intensityH – Auxiliary magnetic fieldD – Electric fluxB – Magnetic field intensityJ – Current densityρv – Volume charge densityεo – permittivity of free space

(≅8.85412×10-12)µo – permeability of free space

(4π×10-7)εr – relative permittivityµr – relative permeability

Gauss’s law

Modified Ampere’s law

Faraday’s law ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )tzyx

ztzyxJytzyxJxtzyxJJ

ztzyxHytzyxHxtzyxHH

ztzyxEytzyxExtzyxEE

vv

zyx

zyx

zyx

,,,

ˆ,,,ˆ,,,ˆ,,,

ˆ,,,ˆ,,,ˆ,,,

ˆ,,,ˆ,,,ˆ,,,

ρρ =

++=

++=

++=

r

r

r

x

y

z

Unit vector in x-direction

x component

No name, but can

be called Gauss’s law

for magnetic field

For linear medium

Constitutive

relations

Each parameter depends on 4 independent variables

5


Maxwell Equations (Linear Medium) -Time-Domain Form (2)

• Maxwell Equations as shown are actually a collection of 4 partial differential equations (PDE) that describe the physical relationship between electromagnetic (EM) fields, current and electric charge.

• The Del operator is a shorthand for three-dimensional (3D) differentiation:

• For instance consider the 1st and 3rd Maxwell Equations:

( )zyxzyx

ˆˆˆ∂∂

∂∂

∂∂ ++=∇

( )

ε

ρ=++=⋅∇

++−=

−+

−+

−==×∇

∂

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂∂

∂∂

∂∂

z

E

y

E

x

E

zyxt

y

E

x

E

x

E

z

E

z

E

y

E

zyx

zyx

zyx

xyzxyz

E

zByBxB

zyx

EEE

zyx

E

~

ˆˆ

ˆˆˆ

ˆˆˆ~

To truly understands this subject, and alsoRF/Microwave circuit design, one needsto have a strong grasp of Electromagnetism (EM).Read references [1], [3] or any good book on EM.

Curl

Divergence

zyxFzF

yF

xF ))

∂∂

∂∂

∂∂ ++=∇ ˆ

Gradient


Extra: Wave Function and Phasor (1)

z

z

to

to+∆t

Time t

zo

( )kfztf oo =− )( βω

( ) ( )( )ztkf

zzttf oo

∆−∆+=

∆+−∆+

βω

βω )(

zo+∆z

∆z

We see that the shape moves by within a period of ∆t, thus phase velocity vp:

( ) ( )

βω

βω

βωβω

==⇒

∆=∆⇒

∆+−∆+=−

∆∆

tz

p

oooo

v

zt

zzttzt

)( ztf βω −

A general function describing propagating

wave in +z direction

For a wave functionin –z direction:

( )ztf βω +

Direction of

travel

These 2 termsmust cancel offi.e. ∆z positive

When time increases by ∆t,

we see that we must

increase all position by ∆z to

maintain the shape. In

essence the waveform

travels in +z direction.

6



• An example: ( )zftVtzv o βπ −= 2cos),(

1 , MHz01 == β.f

z

v(z,t)

β

π

βω f

pv2==Phase Velocity:

A sinusoidal wave

βπλ 2=wavelength



• In many ways the frequency f does not carry much information. If a linear

system is excited by a sinusoidal source with frequency f, we know the

response at every point in the system will be sinusoidal with frequency f.

• It is the phase constant β which carries more information, it determines the

velocity and wavelength, of the wave.

• Thus it is more convenient if we convert the expressions for EM fields into

phasor or Time-Harmonic form, as shown below:

( )zt βω mcos zje

βm

( )zftVtzv o βπ −= 2cos),(zj

oeVzβ−=)(V

Phasor for v(z,t)

( ) zjtjeezt

βωβω mm Recos =

More compact form

Convention: small letter for time-domain

form, capital letter for phasor.

Using Euler’s formula

ααα sincos je j +=

Euler’s formula

1−=j

7



• Wave function and phasor notation is not only applicable to quantities

like voltage, current or charge. It is also applied to vector quantities like

E and H fields.

• For instance for sinusoidal E field traveling in +z direction:

• The phasor is given by:

• Finally if we substitute the phasor form for E, H, J and ρ into

time-domain Maxwell’s Equations, we would obtain the Maxwell’s

Equations in time-harmonic form.

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) tjzjozyx

zyx

eeEztzeyexe

zztyxeyztyxexztyxetzyxE

ωββω

βωβωβω

−+

+

=−++=

−+−+−=

r

r

Recosˆˆˆ

ˆcos,ˆcos,ˆcos,,,,

( ) ( ) ( ) ( ) zeyxeyeyxexeyxezyxzj

zzj

yzj

x ˆ,ˆ,ˆ,,, βββ −−−+ ++=Εr

tjzjo eeE ωβ−+r

Pattern function

(x, y dependent)

Propogating function

Eo


Maxwell Equations (Linear Medium) - Time-Harmonic Form (1)

• For sinusoidal variations with time t, we substitute the phasors for E, H, J

and ρ into Maxwell’s Equations, the result are Maxwell’s Equations in

time-harmonic form.

0B

ρD

DJH

BE

v

=⋅∇

=⋅∇

+=×∇

−=×∇

r

r

rrr

rr

ω

ω

j

j

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )zyx

zzyxyzyxxzyx

zzyxyzyxxzyx

zzyxyzyxxzyx

,,ρρ

ˆ,,Jˆ,,Jˆ,,JJ

ˆ,,Hˆ,,Hˆ,,HH

ˆ,,Eˆ,,Eˆ,,EE

vv

zyx

zyx

zyx

=

++=

++=

++=

r

r

r

Where:

For linear medium

Constitutive

relations

E – Electric field intensityH – Auxiliary magnetic fieldD – Electric fluxB – Magnetic field intensityJ – Current densityρv- Volume charge densityεo – permittivity of free space

(≅8.85412×10-12)µo – permeability of free space

(4π×10-7)εr – relative permittivityµr – relative permeability

(1.2a)

(1.2b)

(1.2c)

(1.2d)

ωjt

→∂∂

Each parameter depends on 3 independent variables

8


Electromagnetic Spectrum

ELF VLF LFMF

(MW)

HF

(SW)VHF UHF SHF EHF IR

100

Hz

10

kHz

100

kHz

1

MHz

3

MHz

30

MHz

300

MHz

1

GHz

30

GHz

300

GHzA

M

Mic

row

ave

s

FM

L S C X Ku K Ka mm

1GHz

2GHz

4GHz

8GHz

12GHz

18GHz

27GHz

40GHz

300GHz


A Little Perspective…

• Voltage/Potential

• Current

• Inductance

• Capacitance

• Resistance

• Conductance

• Kirchoff’s Voltage Law

(KVL)

• Kirchoff’s Current Law

(KCL)

0~

~

~~~

~~

=⋅∇

=⋅∇

+=×∇

−=×∇

∂∂

∂∂

B

E

EJB

HE

t

t

ε

ρ

µεµ

µ

tJ

∂∂

−=⋅∇ρ~

+

Conservation of charge

Quantum Mechanics

/PhysicsChemistry

Electronics &

Microelectronics

Information

Computer

& Telecommunication

9


2.0 Introduction –Transmission Line

Concepts


Definition of Electrical Interconnect

• Interconnect - metallic conductors that is used to transport electrical

energy from one point of a circuit to another.

• Example:

• Thus cables, wires, conductive tracks on printed circuit board

(PCB), sockets, packaging, metallic tubes etc. are all examples of

interconnect.

Conductor

InterconnectTransverse

Plane

Axial

Direction

z

y

x

1. Usually contains 2 or more

conductors, to form a closed

circuit.

2. Conductors assumed to be

perfect electric conductors (PEC)

Coaxial cable

Ribbon cable

Waveguide

Socket

PCB Traces

10


Short Interconnect – Lumped Circuit

• For short interconnect, the moment the switch is closed, a voltage will

appear across RL as current flows through it. The effect is instantaneous.

• Voltage and current are due to electric charge movement along the

interconnect.

• Associated with the electric charges are static electromagnetic (EM) field in

the space surrounding the short interconnect.

• The short interconnect system can be modeled by lumped RLC circuit.

z

y

x

EM field is static or quasi-

static.

Electric charge

Vs

+

-RL

Static EM field changes uniformly, i.e. when field at one point increases, field at the other locations also increases.

L

R

GC

Again we need to

stress that typically

the values of RLCG

are very small, at

low frequency their

effect can be simply

ignored.


Long Interconnect (1)

• If the interconnection is long, it takes some time for voltage and current

to appear on the resistor RL after the switch is closed.

• Electric charges move from Vs to the resistor RL. As the charges move,

there is an associated EM field which travels along with the charges.

• In effect there is a propagating EM field along the interconnect. The

propagating EM field is called a wave and the interconnect is guiding

the EM wave, this EM field is dynamic.

• Since any arbitrary waveform can be decomposed into its sinusoidal

components, let us consider Vs to be a sinusoidal source.

L

+

-Vs

RL

Long interconnect:

When there is an

appreciable delay

between input and

output Without the metal conductors, the EM

waves will disperse, i.e. radiated out

into space.

11



• A simple animation…

t=to

H field

E field

Positive charge

t=to+ ∆t

t=to+ 2∆t

t=to+ 3∆t

IL

VL

RL

+

-

Vs

t=to+ n∆t

Transverse

Plane

Axial

Direction

z

y

x



• The corresponding EM field generated when electric charge flows along

the interconnect is also sinusoidal with respect to time and space. The

EM fields characteristics are dictated by Maxwell’s Equations.

Electric field (E)

Magnetic field (H)

L

T

t

1/T = frequency

• Remember that current is due to the flow of free electrons.• E and H are also sinusoidal.• Behavior of E and H are dictated by Maxwell’s Equations:

12



Electric field (E)

Magnetic field (H)

L

Propagating EM fields

( ) ( ) ( )

( ) ( )( ) zjzt

zjz

zjy

zjx

ezyxeyxe

zeyxeyeyxexeyxeE

β

βββ

−

−−−+

+=

++=

ˆ,,

ˆ,ˆ,ˆ,

r

r

( ) ( ) ( )

( ) ( )( ) zjzt

zjz

zjy

zjx

ezyxhyxh

zeyxhyeyxhxeyxhH

β

βββ

−

−−−+

+=

++=

ˆ,,

ˆ,ˆ,ˆ,

r

r

By analyzing Maxwell’s

Equations or Wave

Equations (Appendix 1)

Snapshot of EM fields

at a certain instant in time

on the transverse plane

z

y

x


Voltage and Current on Interconnect (1)

• Voltage or potential difference is the energy needed to bring 1 Coulomb

of electric charge from a reference point (GND) to another (signal).

• Current is the rate of flow of electric charge across a surface.

• From Coulomb’s Law and Ampere’s Law in Electromagnetism, these two

quantities are related to E and H fields.

• We are interested in transverse voltage Vt and current It as shown.

Loop 1

( ) ∫ ⋅=

1 Loop

, ldHtzIt

rr

It(z,t)

Vt(z,t)

( ) ∫ ⋅−=b

at ldEtzVrr

,

a

b

13


Voltage and Current on Interconnect (2)

• Vt = Potential difference between two points on transverse plane and It =

Rate of flow of electric charge across a conductor surface.

• Vt and It depends on instantaneous E and H fields on the interconnect,

and correspond to how we would measure them physically with probes

• Vt and It will be unique (e.g. do not depend on measurement setup, but

only on the location) if and only if the EM field propagation mode in the

interconnect is TEM or quasi-TEM.Measuring voltage

Measuring current

Line integration

path of E field

Line integration

path of H field

See discussion in Appendix 1: Advanced Concepts – Field Theory Solutions for more information.

Transverse

plane


Demonstration - Electromagnetic Field Propagation in Interconnect (1)

• The following example simulate the behavior of EM field in a simple

interconnection system.

• The system is a 3D model of a copper trace with a plane on the bottom.

• A numerical method, known as Finite-Difference Time-Domain (FDTD)

is applied to Maxwell’s Equations, to provide the approximate value of

E and H fields at selected points on the model at every 1.0 picosecond

interval. (Search WWW or see http://pesona.mmu.edu.my/~wlkung/Phd/phdthesis.htm)

• Field values are displayed at an interval of 25.0 picoseconds.

Copper trace

GND plane

100Ω SMD resistor

z

y

x

50Ω resistivevoltage sourceconnected here

FR4 dielectric

14



Intensity

Scale

Magnitude

of Ez in

dielectric

0.5mm

PCB dielectric:FR4, εr = 4.4,σ= 5. Thickness =1.0mm.

+

-

Vs

Vo = 3.0V

tr = 50ps

tHIGH = 100ps

0.75mm

0.8mm

Show simulation using CST Microwave

Studio too

19.2mm

y

x

z

Filename: tline1_XZplane.avi



Vo = 3.0V

tr = 50ps

tHIGH = 100ps

Magnitude

of Ez in

YZ plane

Volts

Filename: tline1_YZplane.avi

y

x

z

Probe of signal generator

15


Definition of Transmission Line

• A transmission line is a long interconnect with 2 conductors – the signal

conductor and ground conductor for returning current.

• Multiconductor transmission line has more than 2 conductors, usually a

few signal conductors and one ground conductor.

• Transmission lines are a subset of a broader class of devices, known

as waveguide. Transmission line has at least 2 or more conductors,

while waveguides refer collectively to any structures that can allow EM

waves to propagate along the structure. This includes structures with

only 1 conductor or no conductor at all.

• Widely known waveguides include the rectangular and circular

waveguides for high power microwave system, and the optical fiber.

Waveguide is used for system requiring (1) high power, (2) very low

loss interconnect (3) high isolation between interconnects.

• Transmission line is more popular and is widely used in

PCB. From now on we will be concentrating on

transmission line, or Tline for short.


Typical Transmission Line Configurations

Coaxial line Microstrip line Stripline

Parallel plate line Co-planar line

Two-wire line

Conductor

Dielectric

Slot line

Shielded microstrip lineThese conductors

are physically

connected somewhere

in the circuit

16


Some Multi-conductor Transmission Line Configurations

Conductor

Dielectric


Typical Waveguide Configurations

Rectangular waveguide Circular waveguide

Optical FiberDielectric waveguide

17


Examples of Microstrip and Co-planar Lines

MicrostripCo-planar


Long or Short Interconnect? The Wavelength Rule-of-Thumb

λfvp =

Phase velocity or

propagation velocity

wavelength

frequency

f λ

f λ

Interconnect

(1.1)

We call this the 5% rule. Less conservative

estimate will use 1/10=0.10 (the 10% Rule)

• How do we determine if the interconnect is long or short, i.e. delay

between input and output is appreciable?

• Relative to wavelength for sinusoidal signals.

• Rule-of-Thumb: If L < 0.05λ, it is a short interconnect, otherwise it is

considered a long interconnect. An example at the end of this section

will illustrate this procedure clearly.L

18


Demonstration – Long Interconnect

Current

profile

+50mA

-50mA

Filename: tline1_YZplane_5_8GHz.avi

5.8GHz, 3.0V

Magnitude

of Ez in

YZ plane

19.2mm

Each frame is

displayed at

25psec interval

y

x

z

Let us assume the EM

wave travels at speed of

light, C=2.998×108, then

wavelength ≅ 52.0 mm

fC=λ

19.2 mm is greater than

5% of 52 mm


Demonstration – Short Interconnect

0.4GHz, 3.0V

Filename: tline1_YZplane_0_4GHz.avi

Magnitude

of Ez in

YZ plane

• At any instant

in time the

current profile

is almost uniform

along the axial

direction.

• Interconnect

can be considered

lumped.

Current

profile

+50mA

-50mA

19.2mmy

x

z Let us assume the EM

wave travels at speed of

light, C=2.998×108, then

wavelength ≅ 750.0 mm

19


3.0 Propagation Modes


Transverse E and H Field Patterns

x

y

z

Transverse

plane

E fields

H fields

Field patterns lie in the Transverse

Plane.

20


Non-transverse E and H Field Patterns

Field patterns does not lie in

the Transverse Plane.E fields

H fields

Field contains z-component

Field contains z-component


Propagation Modes (1)

• Assuming the transmission line is parallel to z direction. The

propagation of E and H fields along the line can be classified into 4

modes:

– TE mode - where Ez = 0.

– TM mode - where Hz = 0.

– TEM mode - where Ez and Hz are 0.

– Mix mode, any mixture of the above.

• A Tline can support a number of modes at any instance, however TE,

TM or mix mode usually occur at very high frequency.

• There is another mode, known as quasi-TEM mode, which is

supported by stripline structures with non-uniform dielectric. See

discussion in Appendix 1.

( ) ( )( ) zjzt ezyxeyxeE

βmrr ˆ,, +=±

( ) ( )( ) zjzt ezyxhyxhH βm

rr ˆ,, +=±

z

y

21


Propagation Modes (2)

H

E

Mix modes

H

E

TEM mode

H

E

TE mode

H

E

TM mode z

x

y


Examples of Field Patterns or Modes

TEM or quasi-TEM mode

E field

H field

22


Appendix 1Advanced Concepts – Field

Theory Solutions for Transmission Lines


Field Theory Solution

• The nature of E and H fields in the space between conductors can be studied by solving the Maxwell’s Equations or Wave Equations (which can be derived from Maxwell’s Equations) (See [1], [2], [3]). Assuming the condition of long interconnection, the solutions of E and H fields are propagating fields or waves.

• We assume time-harmonic EM fields with ejωt dependence and wave propagation along the positive and negative z-axis.

z

y

022 =+∇ EkE o

rr

εµω=

=+∇

o

o

k

HkH 022

rrBoundary

conditions

0=⋅∇

=⋅∇

+=×∇

−=×∇

H

E

EjJH

HjE

r

r

rrr

rr

ε

ρ

ωε

ωµ

+ +

Maxwell Equations

Wave Equations

(A.1)

For instance tangential E field

component on PEC must be zero,

continuity of E and H field components

across different dielectric material, etc.

In free space

23


Extra: Deriving the Hemholtz Wave Equations From Maxwell Equations

( ) ( ) ( )( )ε

ρωµµεω

ωµ

∇+=+∇⇒

×∇−=∇−⋅∇∇=×∇×∇

JjEE

HjEEErrr

rrrr

22

2

HjErr

ωµ−=×∇Performing curl operation on Faraday’s Law :

Note: use the well-known

vector calculus identity

( )2

2

2

2

2

22

2

zyx

AAA

∂

∂

∂

∂

∂

∂ ++=∇

∇−⋅∇∇=×∇×∇rrr

In free space there is no electric charge and current:

022 =+∇ EE

rrµεω

These are the sources for the E field

Similar procedure can be used to obtain

JHHrrr

×−∇=+∇ µεω 22

Or in free space

022 =+∇ HHrr

µεω

Note: This derivation is

valid for time-harmonic case

under linear medium only.

See more advanced text for

general wave equation. For

Example:C. A. Balanis, “Advanced engineeringElectromagnetics”, John-Wiley, 1989.


Obtaining the Expressions for E and H (1)

• Assuming an ordinary differential equation (ODE) system as shown:

• To obtain a solution to the above system (a solution means a function that when substituted into the ODE, will cause left and right hand side to be equal), many approaches can be used (for instance see E. Kreyszig, “Advance engineering mathematics”, 1998, John Wiley).

• One popular approach is the Trial-and-Error/substitution method, where we guess a functional form for y(x) as follows:

• Substituting this into the ODE:

• Since this is a 2nd order ODE, we need to introduce 2 unknown constants, A and B, and a general solution is:

( ) xexy

β= x

dx

ydx

dx

dyee

ββ ββ 22

2

, ==

( )

1 where

0

0

22

22

−=±=⇒

=+⇒

=+

jjk

k

ekx

β

β

β β

( ) jkxjkxBeAexy

−+= (1)

That the trial-and-error method

works is attributed to the

Uniqueness Theorem for

linear ODE.

( ) [ ]

( ) ( ) 21

2

and 0

,0 , , 02

2

CbyCy

bxxfyykdx

yd

==

∈==+

Boundary conditions

ODE The Domain

24



• To find A and B, we need to use the boundary conditions.

• Solving (2a) and (2b) for A and B:

• So the unique solution is:

( )

( ) 22

11

0

CBeAeCby

CBACy

jkbjkb =+⇒=

=+⇒=

−

(2a)

(2b)

( )

( )

( )kbj

CeC

kbj

CeC

jkbjkb

jkb

jkb

BCA

B

CBeeBC

sin21

sin2

21

21

21

−

−

−

−

=−=

=⇒

=+−

( ) ( ) ( )jkx

kbj

CeCjkx

kbj

CeCeexy

jkbjkb−−−

+

=

−

sin2sin22121

(q.e.d.)

(3a)

(3b)



• The same approach can be applied to Wave Equations or Maxwell Equations for Tline. Consider the Wave Equations (A.1) in time-harmonic form.

• The unknown functions are vector phasors E(x,y,z) and H(x,y,z). The differential equation for E in Cartesian coordinate is:

• This is called a Partial Differential Equation (PDE) as each Ex, Ey and Ez

depends on 3 variables, with the differentiation substituted by partial differential. There are 3 PDEs if you observed carefully. For x-component this is:

• Based on the previous ODE example, and also the fact that we expect the E field to travel along the z-axis, the following form is suggested:

( )0ˆˆˆ

0

222

22

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

=

++++

++++

+++⇒

=+∇

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ zEkyEkxEk

Ek

zozyx

yozyx

xozyx

o

( ) ( ) ( ) zjx

zjxx eyxeeyxezyxE ββ ,or ,,, −=

022

2

2

2

2

2

=

+++

∂

∂

∂

∂

∂

∂xo

zyxEk

A function of x and y The exponent e !!!

25



• Carrying on in this manner for y and z-components, we arrived at the following form for E field.

• Notice that up to now we have not solve the Wave Equations, but merely determine the functional form of its solution.

• We still need to find out what is ex(x,y), ey(x,y), ez(x,y) and β.

• Using similar approach on will yield similar expression for H field.

( ) ( ) ( )

( ) ( )( ) zjzt

zjz

zjy

zjx

ezyxeyxe

zeyxeyeyxexeyxeE

β

βββ

−

−−−+

+=

++=

ˆ,,

ˆ,ˆ,ˆ,

r

r

(A.1a)

( ) 022 =+∇ Hko

( ) ( ) ( )

( ) ( )( ) zjzt

zjz

zjy

zjx

ezyxhyxh

zeyxhyeyxhxeyxhH

β

βββ

−

−−−+

+=

++=

ˆ,,

ˆ,ˆ,ˆ,

v

v

(A.1b)


E and H fields Expressions (1)

( ) ( ) ( )

( ) ( )( ) zjzt

zjz

zjy

zjx

ezyxeyxe

zeyxeyeyxexeyxeE

β

βββ

−

−−−+

+=

++=

ˆ,,

ˆ,ˆ,ˆ,

r

r

• Thus the propagating EM fields guided by Tline can be written as:

( ) ( ) ( )

( ) ( )( ) zjzt

zjz

zjy

zjx

ezyxhyxh

zeyxhyeyxhxeyxhH

β

βββ

−

−−−+

+=

++=

ˆ,,

ˆ,ˆ,ˆ,r

r

Transverse component

Axial component

( ) ( ) ( )

( ) ( )( ) zjzt

zjz

zjy

zjx

ezyxeyxe

zeyxeyeyxexeyxeE

β

βββ

+

+++−

−=

−+=

ˆ,,

ˆ,ˆ,ˆ,

r

r

( ) ( ) ( )

( ) ( )( ) zjzt

zjz

zjy

zjx

ezyxhyxh

zeyxhyeyxhxeyxhH

β

βββ

+

+++−

+−=

+−−=

ˆ,,

ˆ,ˆ,ˆ,r

r

EM fields

Propagating

In +z direction

EM fields

Propagating

In -z direction

(A.2a)

(A.2b)

(A.3a)

(A.3b)

superscript indicates

propagation direction

26



( ) ( )

( ) ( ) ( ) ( ) ( ) ( )zztyxeyztyxexztyxe

ezyxEtzyxE

zyx

tj

ˆcos,cos,ˆcos,

,,Re,,,

βωβωβω

ω

−+−+−=

=++r

• We can convert the phasor form into time-domain form, for instance for

E field propagating in +z direction:

• Where

( ) ( ) ( ) ( )ztzyxEytzyxExtzyxEtzyxE zyx ˆ,,,ˆ,,,ˆ,,,,,, ++++ ++=r

Unit vector( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )ztyxetzyxE

ztyxetzyxE

ztyxetzyxE

zz

yy

xx

βω

βω

βω

−=

−=

−=

+

+

+

cos,,,,

cos,,,,

cos,,,,

(A.5b)

(A.4)

(A.5a)



• Usually one only solves for E field, the corresponding H field phasor

can be obtained from:

• The power carried by the EM fields is given by Poynting Theorem:

Ej

H

HjE

rr

rr

×∇=⇒

−=×∇

ωµ

ωµ

1

( ) dshesdHEP t

S

t

S z

⋅×=⋅×= ∫∫∫∫++ **

Re2

1Re

2

1 rrrrrS

( ) ( )

dshe

dshesdHEP

t

S

t

t

S

t

S

z

z

⋅×=

−⋅×−=⋅×=

∫∫

∫∫∫∫−−

*

**

Re2

1

Re2

1Re

2

1

rr

rrrrrPositive value means

that power is carried

along the propagation

direction.

(A.6)

Positive Z direction…

Negative Z direction…

27



• There are 2 reasons for choosing the sign conventions for +ve and -ve

propagating waves as in (A.2) and (A.3).

– So that for both +ve and -ve propagating E field

(consistency with Maxwell’s Equations).

– The transverse magnetic field must change sign upon reversal of

the direction of propagation to obtain a change in the direction of

energy flow.

0=⋅∇ Er

0

0

=−⋅∇⇒

=⋅∇ +

ztt eje

E

βr

r

( ) 0

0

=−+⋅∇⇒

=⋅∇ −

ztt eje

E

βr

r

For +ve direction

For -ve direction


Phase Velocity

• It is easy to show that equation (A.2a) and (A.2b) describes traveling E

field waves (also for H).

• The speed where the E and H fields travel is called the Phase Velocity,

vp.

• Phase Velocity depends on the propagation mode (to be discussed

later), the frequency and the physical properties of the interconnect.

βω=pv (A.7)

28


Wavelength

• For interconnect excited by sinusoidal source, if we freeze the time at a

certain instant, say t = to, the E and H fields profile will vary in a

sinusoidal manner along z-axis.

y

z

Ex(x,y,z,to)Wavelength λ f

vp=λ

βπ

πωβω

λ 2

2

== (A.8)


Superposition Theorem

• At any instant of time, there are E and H fields propagating in the positive and negative direction along the transmission line. The total fields are a superposition of positive and negative directed fields:

• A typical field distribution at a certain instant of time for the cross section of two interconnects (two-wire and co-axial cable) is shown below:

−++= EEE

−++= HHH

HE

Conductors

29


Field Solution (1)

• To find the value of β and the functions ex, ey, ez, hx, hy, hz, we substitute

equations (A.2a) and (A.2b) into Maxwell or Wave equations.

z

y

( ) ( ) ( )

( ) ( )( ) zjzt

zjz

zjy

zjx

ezyxeyxe

zeyxeyeyxexeyxeE

β

βββ

−

−−−+

+=

++=

ˆ,,

ˆ,ˆ,ˆ,

r

r

( ) ( ) ( )

( ) ( )( ) zjzt

zjz

zjy

zjx

ezyxhyxh

zeyxhyeyxhxeyxhH

β

βββ

−

−−−+

+=

++=

ˆ,,

ˆ,ˆ,ˆ,

r

r

022 =+∇ EkE o

rr

εµω=

=+∇

o

o

k

HkH 022

rrBoundary

conditions

0=⋅∇

=⋅∇

+=×∇

−=×∇

H

E

EjJH

HjE

r

r

rrr

rr

ε

ρ

ωε

ωµ

+ +

Maxwell Equations

Wave Equations


Field Solution (2)

HjErr

ωµ−=×∇ EjHrr

ωε=×∇

xyy

ehjejz ωµβ −=+

∂

∂

yx

ex hjej z ωµβ −=−−

∂

∂

zy

e

x

ehjxy ωµ−=−

∂

∂

∂

∂

(A.10a) xyy

hejhjz ωεβ =+

∂

∂

yx

h

x ejhj z ωεβ =−−∂

∂

zy

h

x

hejxy ωε=−

∂

∂

∂

∂

(A.10b)

(A.10c)

(A.10d)

(A.10e)

(A.10f)

• The procedure outlined here follows those from Pozar [2]. Assume the

Tline or waveguide dielectric region is source free. From Maxwell’s

Equations:

• Substituting the suggested solution for E+(x,y,z,β) of (A.2a) into (A.9),

and expanding the differential equations into x, y and z components:

(A.9)

EjJB~~~

ωµεµ +=×∇

0=Jr

30


Field Solution (3)

• From (A.10a)-(A.10f), we can express ex, ey, hx, hy in terms of ez and

hz:

−=

∂

∂

∂

∂

x

h

y

e

k

jx

zz

c

h βωε2

+=

∂

∂

∂

∂−

y

h

x

e

k

jy

zz

c

h βωε2

+=

∂

∂

∂

∂−

y

h

x

e

k

jx

zz

c

e ωµβ2

+−=

∂

∂

∂

∂

x

h

y

e

k

jy

zz

c

e ωµβ2

λπµεωβ 2222 , ==−= ooc kkk

(A.11a)

(A.11b)

(A.11c)

(A.11d)

(A.11e)

These equations

describe the x,y components

of general EM wave

propagation in a

waveguiding system.

The unknowns are

ez(x,y) and hz(x,y), called the

Potential in the literature.

See the book by Collin [1],

Chapter 3 for alternative

derivation


TE Mode Summary (1)

• For TE mode, ez= 0 (Sometimes this is called the H mode).

• We could characterize the Tline in TE mode, by EM fields:

• From wave equation for H field:

( ) zjzt ezhhH

βmrr

ˆ+±=± zjteeE

βmrr=±

εµω=

=+∇

o

o

k

HkH 022rr

( )( ) 0ˆ22

22 =+

++∇ −

∂

∂ zjzto

zt ezhhk βr

222 β−∇=∇ t

Note

Only these are needed. The other

transverse field components can

be derived from hz222

22

22

0

0

β−=

=+∇

=+∇

oc

tctt

zczt

kk

hkh

hkhrr

From (A.11e)

( ) zjzj

zee

ββ β mm 22

2−=

∂

∂Using the fact that

Transverse Laplacian

operator

31


TE Mode Summary (2)

• Setting ez = 0 in (A.11a)-(A.11d):

• These equations plus the previous wave equation for hz enable us to

find the complete field pattern for TE mode.

222

220

β−=

=+∇

oc

zczt

kk

hkh + boundary conditions

for E and H fields (A.12b)

x

h

k

jx

z

c

h∂

∂−=

2

βy

h

k

jy

z

c

h∂

∂−=

2

β

y

h

k

jx

z

c

e∂

∂−=

2

ωµx

h

k

jy

z

c

e∂

∂=

2

ωµ

From previous slide

(A.12a)


TE Mode Summary (3)

• From (A.12a) and (A.12b), we can show that:

• Therefore we cannot define a unique voltage by (but we can define a

unique current):

• Also from (A.12a) we can define a wave impedance for the TE mode.

0ˆ ≠−=×∇ zhje ztt ωµr

∫ ⋅−= 2

1

C

C tt ldeVrr

x

yoo

y

xTE

h

eZk

h

eZ

−===

β(A.13)

0=×∇ tt hr

( ) zhjzhk

zze

zzcck

j

y

h

x

h

ck

j

yxe

x

ye

tt

ˆˆ

ˆˆ

22

2

2

2

2

2

ωµωµ

ωµ

−=−=

+=

−=×∇

∂

∂

∂

∂∂

∂

∂

∂r

32


TM Mode Summary (1)

• For TM mode, hz = 0 (Sometimes this is called the E mode).

• We could characterize the Tline in TM mode, by EM fields:

• From wave equation for E field:

zjtehH

βmrr

±=± ( ) zjzt ezeeE

βmrrˆ±=±

εµω=

=+∇

o

o

k

EkE 022rr

( )( ) 0ˆ2

2

22 =+

++∇ −

∂

∂ zjzto

Zt ezeek

βr

222

22

22

0

0

β−=

=+∇

=+∇

oc

tctt

zczt

kk

eke

eke

rrOnly these are needed. The other

transverse field components

can be derived from ez


TM Mode Summary (2)

• Setting hz = 0 in (A.11a)-(A.11d):

• These equations plus the previous wave equation for ez enable us to

find the complete field pattern for TM mode.

y

e

k

jx

z

c

h∂

∂=

2

ωεx

e

k

jy

z

c

h∂

∂−=

2

ωε

x

e

k

jx

z

c

e∂

∂−=

2

βy

e

k

jy

z

c

e∂

∂−=

2

β

(A.14a)

+ boundary conditions

for E and H fields(A.14b)

222

220

β−=

=+∇

oc

zczt

kk

eke

From previous slide

33


TM Mode Summary (3)

• Similarly from (A.14a) and (A.14b), we can show that

• We cannot define a unique current by (but we can define a unique

voltage):

• Also from (A.14a) we can define a wave impedance for the TM mode.

0ˆ ≠=×∇ zejh ztt ωεr

∫ ⋅=

C

tt ldhIrr

x

y

y

x

h

e

h

eTMZ

−===

ωεβ

(A.15)

0=×∇ tt er


TEM Mode (1)

• TEM mode is particularly important, characterized by ez = hz = 0. For

+ve propagating waves:

• Setting ez= hz= 0 in (A.11a)-(A.11d), we observe that kc = 0 in order for

a non-zero solution to exist. This implies:

• Applying Hemholtz Wave Equation to E field:

zjteeE βmrr

=± zjtehH

βmrr

±=±(A.16a)

(A.16b)µεωβ == ok

( )

( ) 0

0ˆ

22

22

22

2222

=

++∇⇒

=

++∇=+∇

−−

∂

∂−

−

∂

∂−

tzj

ozj

z

zjtt

zjto

zt

zjto

eekeee

eekzeek

rr

rr

βββ

ββ

(A.17a)0 2 =∇⇒ tt er

0

34


TEM Mode (2)

• The same can be shown for ht:

• Equation (A.17a) is similar to Laplace equations in 2D. This implies

the transverse fields et is similar to the static electric fields that can

exist between conductors, so we could define a transverse scalar

potential Φ:

• Also note that:

Jhtt

rr×−∇=∇2

(A.17b)

0),(

),(

2 =Φ∇⇒

Φ−∇=

yx

yxe

t

ttr

(A.18)

Transverse potential

( ) 0=Φ∇−×∇=×∇ tttt er

(A.19)

See [1], Chapter 3

for alternative

derivation

Using an important identity in vector calculus

( ) 0=∇×∇ F Where F is arbitrary function

of position, i.e. F = F(x,y,z).


Alternative View (TEM Mode)

• Alternatively from (A.10c), and knowing that hz = 0 in TEM mode:

• From the well known Vector Calculus identity , we then

postulate the existence of a scalar function Φ(x,y) where

• From (A.10f) we can also show that (in free space):

0=−=−∂

∂

∂

∂zy

xe

x

yehjωµ

0=×∇==−∂

∂

∂

∂ttzy

xh

x

yhhejr

ωε

0

0

00 =×∇===−⇒∂∂

∂∂

∂

∂

∂

∂tt

yx

yxyxe

x

yee

ee

zyxr

( ) 0=∇×∇ F

Divergence in 2D (in XY plane)

),( yxe tt Φ−∇=r

35


TEM Mode (3)

• Normally we would find et from (A.17a), then we derive ht from et :

• An important observation is that under TEM mode the transverse field components et and ht fulfill similar equations as in electrostatic.

(A.20a)

( )yexeh

Ej

H

xyt ˆˆ

1

1

−=⇒

×∇−

=

−

εµ

ωµ

r

rr Exercise: see if you can

derive this equation

0 , 02 =×∇=∇ tt hhrr

( )tZt ezho

rr×=⇒ ˆ 1

Zo = Intrinsic impedance of free space

ZTEM = Wave impedance of TEM

mode

TEMh

e

h

eo ZZ

x

y

y

x ====−

εµ

(A.20b)

−+−=

−++−=

−∂

∂

∂

∂−

∂

∂

∂

∂

∂

∂

∂

∂−

zeyEjxEj

zyxH

zj

yxe

x

ye

xyj

yxE

x

yE

zxE

z

yE

jt

ˆˆˆ

ˆˆˆ

1

1

βωµ

ωµ

ββ

0=×∇ tt er

In free space


Voltage and Current under TEM Mode

• Due to and in the space surrounding the

conductors, we could define unique transverse voltage (Vt) and

transverse current (It) for the system following the standard definitions

for V and I. The Vt and It so defined does not depends on the shape of

the integration path.

0=×∇ tt er

0=×∇ tt hr

See the more detailed

version of this note or

see references [1] & [2].

Loop 1

( ) ∫ ⋅=

1 Loop

, ldHtzIt

rr

It(z,t)

Vt(z,t)

( ) ∫ ⋅−=b

at ldEtzVrr

,a

b

36


Extra: Independence of Vt and It from Integration Path under TEM Mode

∫ ⋅=

21 or CC

tt ldhIrr

t

L

t

L

t

L

t

L

t

L

t

L

t

L

t

S

tt

tt

Vldelde

ldelde

ldelde

ldesde

e

=⋅−=⋅−⇒

⋅−=⋅⇒

=⋅+⋅⇒

=⋅=⋅×∇⇒

=×∇

∫∫

∫∫

∫∫

∫∫∫

− 21

21

21

0

0

0

rrrr

rrrr

rrrr

rrrr

r

Loop L

Area S

sdrld

r

C1

C2

+

Since the shape of

loop L is arbitrary, as

long as it stays in the

transverse plane, paths

L1, L2 and hence the

integration path for

Vt is arbitrary.

Similar proof can be carried

out for It, using the loop as

shown and 0=×∇ tt hr

Using Stoke’sTheorem

C1

C2

Loop L


TEM Mode Summary (1)

• For TEM mode, hz= ez= 0.

• To find the EM fields for TEM mode:

– Solve with boundary conditions for the transverse

potential.

– Find E from

– Find H from

• We could characterize the Tline in TEM mode, by EM fields:

( ) 0,2 =Φ∇ yxt

( )[ ] zjt

zjtt eeeyxE ββ mm rr

=Φ∇−=±,

( ) zjt

ot eez

ZH

β±± ×=rr

ˆ1

zjt

zjt ehHeeE ββ mm

rrrr±== ±± µεωβ == ok

37


TEM Mode Summary (2)

• Or through auxiliary circuit theory quantities:

• The power carried by the EM wave along the Tline is given by

Poynting Theorem:

• Because β is always real or complex (when dielectric is lossy) for all

frequencies, the TEM mode always exist from near d.c. to extremely

high frequencies.

zjt

zjt eIIeVV ββ mm ±== ±±

( )

( ) ( )*1

21*

21

*

21*

21

ReRe

ReRe

VVZIIZP

VIdsHEP

cc

S

−==⇒

=⋅×= ∫∫rr See extra note

by F.Kung for

the proof

(A.21)

µεωβ == ok


Non-TEM Modes and Vt, It (1)

• For non-TEM modes, we cannot define both the auxiliary quantities Vt

and It uniquely using the standard definition for voltage and current

(because ).

• For instance in TE mode:

• Thus will not be unique and will depends on the line

integration path. This means if we attempt to measure the “voltage”

across the Tline using an instrument, the reading will depend on the

wires and connection of the probe!

• Furthermore for non-TEM modes:

• Thus we cannot characterize a Tline supporting non-TEM modes

using auxiliary quantities such as Vt and It.

ztt hje ωµ−=×∇r

∫ ⋅−= 2

1

C

C tt dleVr

( ) *21*

21 ReRe tt

S

IVdsHEP ≠⋅×= ∫∫rr

0or 0 ≠×∇≠×∇ tttt herr

38


Non-TEM Modes and Vt, It (2)

• As another example consider the TM mode in microstrip line:

• Using path C as shown in figure:

• In general this is true for arbitrary Tline and waveguide cross

section. If we choose integration path other than C, we still obtain

Vt= 0 due to in TM mode.

( )yxeyxk

je

k

je zyx

c

zt

c

t ,ˆˆ22

+−=∇−=

∂∂

∂∂ββr

+=⋅−= ∫∫∫ ∂

∂

∂

∂

Cy

e

Cx

e

cC

tt dydxk

jdleV zz

2

βr

( ) ( )[ ] 00,,2

02

=−=

= ∫ ∂

∂xeHxe

k

jdy

k

jV zz

c

H

y

e

c

tz ββ

0=×∇ tt er

C

x

y

Y=0

Y=H

0 because of boundary condition

Under quasi-TEM condition, whenez→ 0, kc also → 0, then Vt will bea non-zero value.


Cut-off Frequency for TE/TM Mode

• Because for TE and TM modes:

• There is a possibility that β becomes imaginary when kc > ko. When

this occur the TE or TM mode EM fields will decay exponentially from

the source. These modes are known as Evanescent and are non-

propagating.

• Thus for TE or TM mode, there is a possibility of a cut-off frequency fc ,

where for signal frequency f < fc , no propagating EM field will exist.

22co kk −=β

39


Phase Velocity for TEM, TE and TM Modes

• Phase velocity is the propagation velocity of the EM field supported by

the tline. It is given by:

• For TEM mode:

• For TE & TM mode:

• Thus we observe that TEM mode is intrinsically non-dispersive, while

TE and TM mode are dispersive.

βω=pV

µεβω 1==pV

2222

11

cco kkkpV

−−===

µεωβω

(A.22)

(A.23)


Final Note on TEM, TE and TM Propagation Modes

• Finally, note that the formulae for TEM, TE and TM modes apply to all

waveguide structures, in which transmission line is a subset.

40


Example A1 - Parallel Plate Waveguide/Tline

• The parallel plate waveguide is the simplest type of waveguide that can

support TEM, TE and TM modes. Here we assume that W >> d so that

fringing field and variation along x can be ignored.0=

∂

∂

x

x

z

y

0

d

W

Conducting plates

Propagation along z axis


Example A1 Cont...

• Derive the EM fields for TEM, TE and TM modes for parallel plate

waveguide.

• Show that TEM mode can exist for all frequencies.

• Show that TE and TM modes possess cut-off frequency fc , where for

operating frequency f less than fc , the resulting EM field cannot

propagate.

41


Example A1 – Solution for TEM Mode (1)

( )( )( ) o

t

Vdx

x

dyWxyx

=Φ

=Φ

≤≤≤≤=Φ∇

,

00,

0 , 0for 0,2

TEM mode Solution:

0 02

2

2

2

2

22 ≅⇒=+=Φ∇∂

Φ∂

∂

Φ∂

∂

Φ∂

yyxt

Boundary conditions

Solution for the transverse Laplace PDE:

( )( )

( )doV

o BVBddx

AAx

ByAyx

=⇒==Φ

=⇒==Φ

+=Φ⇒

,

0 00,

,

( ) yyxdoV

=Φ ,Thus

since ( ) ( )yyx Φ=Φ ,

General Solution

Unique solution



yyyxedoV

doV

yxtt ˆˆˆ −=

+−=Φ−∇=

∂∂

∂∂

yeE zj

doV

t ˆβ−−=

xeyezEHzj

d

Vzj

d

Vt

jt

oo ˆˆˆ11 βµεβ

ωµεµ

−−− =

−×=×∇=

Computing the E and H fields:

zjo

d zj

doV

C

tt eVydyyeldEVββ −− =⋅

−−=⋅−= ∫∫ ˆˆ

01

r

Computing the transverse voltage and current:

zj

doVW zj

doV

C

tt WexdxxeldHI βµεβ

µε −− =⋅

=⋅= ∫∫ ˆˆ

02

r

C1

C2y

x

42



Computing the power flow (power carried by the EM wave guided by the wave

-guide):

( ) ( )

( )( )

[ ] [ ]

==⋅=

=

=

⋅×−=

⋅×=

−

+−

+−

+−

∫∫

∫ ∫

∫ ∫

∫∫

d

WV

tt

zj

d

WVzj

o

zjW

d

Vzjd

d

V

W dzj

d

Vzj

d

V

W dzj

d

Vzj

d

V

S

tt

oo

oo

oo

oo

IVeeV

edxedy

dxdyee

zdxdyxeye

sdHEP

εµ

βµεβ

βµεβ

βµεβ

βµεβ

2

21*

21

21

0021

0 021

0 021

*

21

ReRe

Re

Re

ˆˆˆRe

Rer

y

x

dy

dx

ds=dxdy

S



Phase velocity vp for TEM mode:

µεµεω

ωβω 1===pv

The phase velocity is equal to speed-of-light in the dielectric.

43


Example A1 – Solution for TM Mode (1)

( ) ( )

( ) ( ) 0,0,

0 , 0for 0,

222

22

==

−=

≤≤≤≤=+∇

dxexe

kk

dyWxyxek

zz

oc

zct

β

( ) ( )yeyxe zz =,

Solution for ez:

( )( ) ( ) ( )ykBykAyx

ekek

ccz

zcy

ezct

z

cossin,e

02222

2

+=⇒

=+≅+∇∂

∂ since

Boundary conditions

General solution

( )( ) ( )

dn

c

c

cz

z

k

nndkA

dkAdx

BBAx

π

π

=⇒

==≠⇒

==

=⇒=+⋅=

L3,2,1 , and 0

0sin,e

0 000,eThus:

( ) ( )yAyxd

nnz

πsin,e =

( ) ( ) zj

dn

nz eyAyxEβπ −= sin,or

Applying boundary conditions:



( )( )

( ) ( )yye

yyAyxe

dn

dn

nAjt

dn

nyck

j

yze

ck

j

xze

ck

jt

ˆcos

ˆsinˆˆ222

ππ

β

πβββ

−=⇒

−=+=∂∂

∂

∂−

∂

∂−

r

r

or

( ) ( ) yeyEzj

dn

dn

nAjt ˆcos βπ

π

β −−=

Computing the transverse EM fields using (A.20a):

( )( ) xey

eyxH

zj

dn

dn

nAojk

zj

xze

ck

j

yze

ck

jt

ˆcos

ˆˆ22

βπ

πεµ

βωεωε

−

−∂

∂

∂

∂

=

−=

Since n is an arbitrary integer,

the TM mode is usually called

the TMn mode.

44



We can now determine β knowing kc:

( )2222

dn

con kk πµεωβ −=−=

Since the TM mode can only propagate if β is real, and the smallest value for βIs 0, then when β=0:

( )

µε

µε

π

π

πω

µεω

d

n

d

n

dn

f

f

2

22

2

0

=⇒

==⇒

=−

When n = 1, this represent the cut-off frequency for TM mode.

µεdTMcutofff

2

1_ =



For arbitrary n, phase velocity vp for TMn mode:

( )22d

npv

πµεω

ωβω

−

==

For f > fcutoff , we observe that phase velocity vp is actually greater than the

speed of light!!!

NOTE:

The EM fields can travel at speed greater than light, however we can show that

the rate of energy flow is less than the speed-of-light. This rate of energy

flow corresponds to the speed of the photons if the propagating EM wave is

treated as a cluster of photons. See the extra notes for the proof.

45


Dominant Propagation Mode

• For the various transmission line topology, there is a dominant mode.

• This dominant mode of propagation is the first mode to exist at the

lowest operating frequency. The secondary modes will come into

existent at higher frequencies.

• The propagation modes of Tline depends on the dielectric and the cross

section of the transmission line.

• For Tline that can support TEM mode, the TEM mode will be the

dominant mode as it can exist at all frequencies (there is no cut-off

frequency).


Transmission Lines Dominant Propagation Mode

• Coaxial line - TEM.

• Microstrip line - quasi-TEM.

• Stripline - TEM.

• Parallel plate line - TEM or TM (depends on homogenuity of the

dielectric).

• Co-planar line - quasi-TEM.

• Note: Generally for Tline with non-homogeneous dielectric, the Tline

cannot support TEM propagation mode.

46


Quasi TEM Mode (1)

• Luckily for planar Tline configuration whose dominant mode is not TEM,

the TM or TE dominant modes can be approximated by TEM mode at

‘low frequency’.

• For instance microstrip line does not support TEM mode. The actual

mode is TM. However at a few GHz, ez is much smaller than et and ht

that it can be ignored. We can assume the mode to be TEM without

incurring much error. Thus it is called quasi-TEM mode.

• Low frequency approximation is usually valid when wavelength >>

distance between two conductors. For typical microstrip/stripline on

PCB, this can means frequency below 20 GHz or lower.

• The Ez and Hz components approach zero at ‘low frequency’, and the

propagation mode approaches TEM, hence known as quasi-TEM.

When this happens we can again define unique voltage and current for

the system.See Collin [1], Chapter 3 for more mathematical

illustration on this.


Quasi TEM Mode (2)

H

E

Non-TEM mode

H

E

Quasi-TEM mode

0or

0

≠×∇

≠×∇

tt

tt

h

er

r

0or

0

≅×∇

≅×∇

tt

tt

h

er

r

H

E

TEM mode

0or

0

=×∇

=×∇

tt

tt

h

er

r

Vt and ItYes

Vt and ItYes

Vt and ItNo

47


Extra: Why Inhomogeneous Structures Does Not Support Pure TEM Mode (1)

• We will use Proof by Contradiction. Suppose TEM mode is supported. The propagation factor in air and dielectric would be:

• EM fields in air will travel faster than in the dielectric.

• Now consider the boundary condition at the air/dielectric interface. The E field must be continuous across the boundary from Maxwell’s equation. Examining the x component of the E field:

oair µεωβ = rodie εµεωβ =

( ) ( )die

diepvair

airp

dieair

v

βω

βω

ββ

=>=⇒

<For TEM mode

y

xDielectric

Air

( ) ( )

( )( )

( )zairdiej

zairje

zdieje

diexE

airxE

zdiejediex

zairjeairx

e

EE

βββ

β

ββ

−−−

−

−=

−

==⇒


Extra: Why Inhomogeneous Structures Does Not Support Pure TEM Mode (2)

• Since the left hand side is a constant while the right hand side is not. It

depends on distance z, the previous equation cannot be fulfilled.

• What this conclude is that our initial assumption of TEM propagation

mode in inhomogeneous structure is wrong. So pure TEM mode

cannot be supported in inhomogeneous dielectric Tline.

48


Example A2 – Minimum Frequency for Quasi-TEM Mode in Microstrip Line

• Estimate the low frequency limit for microstrip line.

H = 1.6mm

C ≈ 3.0×108

λ = C/f > 20H = 32.0mm

f < C/0.032 = 9.375GHz

Thus beyond fcritical quasi-TEM approximation cannot be applied.

The propagation mode beyond fcritical will be TM. A more conser-

vative limit would be to use 30H or 40H.

Here we replace >> sign

with the requirement that

wavelength > 20H. You can

use larger limit, as this is

basically a rule of thumb.fcritical = 9.375GHz


Summary for TEM, Quasi-TEM, TE and TM Modes

TEM:Ez = Hz = 0

Can defined uniqueVt and It.

Physical Tline can beModeled by equivalentElectrical circuit.

Phase velocity.

No cut-off frequency.

Non-dispersive

Quasi-TEM:Ez ≈0 , Hz ≈ 0

Can defined uniqueVt and It.

Physical Tline can beModeled by equivalentElectrical circuit.

Phase velocity.

No cut-off frequency.

Non-dispersive

TE:Ez = 0, Hz ≠ 0

Cannot defined unique It.

Physical Tline cannotbe modeled by equivalent electrical circuit.

Phase velocity.

Cut-off frequency.

Dispersive

TM:Ez ≠ 0, Hz = 0

Cannot defined unique Vt.

Physical Tline cannotbe modeled by equivalentelectrical circuit.

Phase velocity.

Cut-off frequency.

Dispersive

( ) zjzt ezhhH

βmrr

ˆ+±=±

zjteeE

βmrr=±

zjtehH

βmrr

±=±

( ) zjzt ezeeE

βmrrˆ±=±

zjteeE

βmrr=±

zjtehH

βmrr

±=±

zjteeE

βmrr≅±

zjtehH βmrr

±≅±

µεβω 1==pV

effpV

µεβω 1≅=

22

1

ckpv

−

==µεω

βω

22

1

ckpv

−

==µεω

βω

µεπ2

ckcf =

µεπ2

ckcf =

49


Why Vt and It is so Important ? (1)

• When we can define voltage and current along Tline or high-frequency circuit for that matter, then we can analyze the system using circuit theory instead of field theory.

• Circuit theories such as KVL, KCL, 2-port network theory are much easier to solve than Maxwell equations or wave equations.

• High-frequency circuits usually consist of components which are connected by Tlines. Thus the microwave system can be modeled by an equivalent electrical circuit when dominant mode in the system is TEM or quasi-TEM. For this reason Tline which can support TEM or quasi-TEM is very important.


Why Vt and It is so Important ? (2)

XXX.09

Amplifier

Filter

Microstrip

antenna

Antenna equivalent

circuit

Tline

A complex physical system

can be cast into equivalent

electrical circuit. Powerful

circuit simulator tools

can be used to perform

analysis on the equivalent

circuit.

50


Examples of Circuit Analysis* Based Microwave/RF CAD Software

Agilent’s Advance Design SystemApplied Wave Research’s

Microwave Office

*The software shown here also havenumerical EM solver capability, from 2D, 2.5Dto full 3D.

Ansoft’s Desinger


4.0 – Transmission Line Characteristics and

Electrical Circuit Model

51


Distributed Electrical Circuit Model for Transmission Line (1)

• Since transmission line is a long interconnect, the field and current

profile at any instant in time is not uniform along the line.

• It cannot be modeled by lumped circuit.

• However if we divide the Tline into many short segments (< 0.1λ), the

field and current profile in each segment is almost uniform.

• Each of these short segments can be modeled as RLCG network.

• This assumption is true when the EM field propagation mode is TEM

or quasi-TEM.

• From now on we will assume the Tline under discussion support

the dominant mode of TEM or quasi-TEM.

• For transmission line, these associated R, L, C and G parameters are

distributed, i.e. we use the per unit length values. The propagation of

voltage and current on the transmission line can be described in terms

of these distributed parameters.


Distributed Electrical Circuit Model for Transmission Line (2)

5.8GHz, 3.0V

Magnitude

of Ez in

YZ plane

Vt

It

y

x

z

Current profile

along conducting

trace

Within each segment

the current is more or

less constant, in and out

current is similar. Also

the EM can be considered

static.

52


Distributed Parameters (1)

• The L and C elements in the electrical circuit model for Tline is due to

magnetic flux linkage and electric field linkage between the conductors.

• See Appendix 2: Advanced Concepts – Distributed RLCG Model for

Transmission Line and Telegraphic Equations for the proofs.

∆z L ∆z

C ∆zV1 V2

I

E

H

Magnetic field

linkage

Electric

field linkage



• When the conductor has small conductive loss a series resistance R∆ zcan be added to the inductance. This loss is due to a phenomenon known as skin effect, where high frequency current converges on the surface of the conductor.

• When the dielectric has finite conductivity and polarization loss, a shunt conductance G∆ z can be added in parallel to the capacitance.

• The inclusion of R and G in the Tline distributed model is only accurate for small losses. This is true most of the time as Tline is usually made of very good conductive material and good insulator.

• The equations for finding L, C, R, G under low loss condition are given in the following slide.

L ∆z

C ∆z

R ∆z

G ∆zV1 V2

IUnder lossy condition, R, L and G

are usually function of frequency,

hence the Tline is dispersive.

Dielectric loss

Conductor loss

53



• Thus a transmission line can be considered as a cascade of many of these

equivalent circuit sections. Working with circuit theory and circuit elements

are much easier than working with E and H fields using Maxwell

equations.

• In order for this RLCG model for Tline to be valid from low to very high

frequency, each segment length must approach zero, and the number of

segment needed to accurately model the Tline becomes infinite.

• This electrical circuit model for Tline is commonly known as Distributed

RLCG Circuit Model.

Distributed

RLCG circuit

Vt

It

Vt

It

∆z→0


Finding the RLCG Parameters

∫∫∫=

V

t

t

dvHI

L2

2

rµ

∫∫∫=

V

t

t

dvEV

C2

2

' rε

dlHI

R

CC

t

tsc

∫+

=

21

2

2

1 r

δσ

∫∫∫=

V

t

t

dvEV

G2

2

" rωε

µωσδ

cs

2 depth skin ==

• See Section 3.9 of Collin [1].

• V is the volume surrounding the

conductors with length of 1 meter

along z axis.

• C1 and C2 are the paths

surrounding the surface of

conductor 1 and 2.

(4.1a)

(4.1b)

• These formulas are

derived from energy

consideration.

• Note that conductor loss

results in R, while

dielectric loss results

in G.

object metalic ofty conductivi

tan "'

=

==

c

oror

σ

δεεεεεε

Conductor 1Conductor 2

C1

C2

S

1m

This indicates the

volume enclosing

the conductors

Loss tangent of the dielectric

Skin depth

and G depend

on frequency

54


Finding RLCG Parameters From Energy Consideration

The instantaneous power absorbed by an inductor L is:

( ) ( ) ( )titvtPind =

Assuming i(t) increases from 0 at t = 0 to Io at t = to, total energy stored

by inductor is:

i(t)

v(t) L

( ) ( ) ( ) ( ) ( )

2

21

0

000

ooI

ind

ot

ddiotot

indind

LIdiLiE

diLdivdPE

=⋅=⇒

===

∫

∫∫∫ τττττττττ

This energy stored by the inductor is contained within the magnetic field

created by the current (for instance, see D.J. Griffiths, “Introductory

electrodynamics”, Prentice Hall, 1999). From EM theory the stored energy in magnetic

field is given by:

Io

H

dxdydzHE

V

H ∫∫∫=r

2

µ

Both energy are the same, hence:

dxdydzHL

dxdydzHLI

EE

VoI

V

o

Hind

2

2

2

22

21

∫∫∫

∫∫∫

=⇒

=⇒

=

r

r

µ

µ

t

i(t)

io

0 to


Multi-Conductor Transmission Line Symbol and Circuit

C12

C1G C2G

R11

L11

L22

R22

L12

Long interconnect:

l > 0.1λ

Distributed

RLCG circuit

TEM or

quasi-TEM mode

Electrical Symbol

Parameters:

Per unit length

R, L, C, G matrices.

Usually as a function

Frequency.

2212

1211

LL

LL

−

−

2212

1211

CC

CC

2221

1211

RR

RR

2221

1211

GG

GG

12111 CCC G −=

12222 CCC G −=

55


Telegraphic Equations for Vt and It

• Much like the EM field in the physical model of the Tline is governed by

Maxwell’s Equations, we can show that the instantaneous transverse

voltage Vt and current It on the distributed RLCG model are governed

by a set of partial differential equations (PDE) called the Telegraphic

Equations (See derivation in Appendix 2).

• For simplicity we will drop the subscript ‘t’ from now.

In time-harmonic form

t

VCGV

z

I

t

ILRI

z

V

∂

∂−−=

∂

∂

∂

∂−−=

∂

∂( )

( ) YVVCjGz

I

ZIILjRz

V

−=+−=∂

∂

−=+−=∂

∂

ω

ω

In time-domainFourierTransforms

Inverse FourierTransforms

Distributed

RLCG circuitV

I

(4.2b)(4.2a)


Solutions of Telegraphic Equations (1)

• The expressions for V(z) and I(z) that satisfy the time-harmonic form of

Telegraphic Equations (4.2b) are given as:

( ) ( ) ( )( )CjGLjRj ωωωβωαγ ++=+=

( ) zo

zo eVeVzV

γγ −−+ +=( ) zo

zo eIeIzI

γγ −−+ +=

Wave travelling in

+z direction

Wave travelling in

-z direction

Propagation coefficient

Attenuation factor

Phase factor

(4.3a) (4.3b)

(4.3c)

56


Solutions of Telegraphic Equations (2)

• Vo+, Vo

-, Io+, Io

- are unknown constants. When we study transmission

line circuit, we will see how Vo+, Vo

-, Io+, Io

- can be determined from the

‘boundary’ of the Tline.

• For the rest of this discussions, exact values of these constants are not

needed.

ZL

Zs

Vs

The boundary of the tline circuit

Vt

It


Signal Propagation on Transmission Line

• Considering sinusoidal sources, the expression for V(z) and I(z) can be written in time domain as:

• From the solution of the Telegraphic Equations, we can deduce a few properties of the equivalent voltage v(z,t) and current i(z,t) on a Tline structure.

– v(z,t) and i(z,t) propagate, a signal will take finite time to travel from one location to another.

– One can define an impedance, called the characteristic impedance of the line, it is the ratio of voltage wave over current wave.

– That the traveling Vt and It experience dispersion and attenuation.

– Other effects such as reflection to be discussed in later part.

( ) ( ) ( ) zo

zo eztVeztVtzv

αα φβωφβω −−−

++ ++++−= coscos,

( ) ( ) ( ) zo

zo eztIeztItzi

αα θβωθβω −−−

++ ++++−= coscos,

++ ++++ == θφ joo

joo eIIeVV

(4.4a)

(4.4b)

57


Characteristic Impedance (Zc)

• An important parameter in Tline is the ratio of voltage over current, called the Characteristic Impedance, Zc.

• Since the voltage and current are waves, this ratio can be only be computed for voltage and current traveling in similar direction.

A function of frequency

CjG

LjRZ

o

oz

o

zo

cI

V

eI

eVZ

ωω

γγ

γ

+

+

+

+

−+

−+==== (4.5)

CjG

LjR

zo

zo

ceI

eVZ

ωω

γ

γ

+

+

−

−=−=

++

−+−+

=⇒

−=−⇒

−=∂

∂

oZ

o

zo

zo

IV

eZIeV

ZIz

V

γ

γγγ

Or

From Telegraphic Equations


Propagation Velocity (vp)

• Compare the expression for v(z,t) of (4.4a) with a general expression for a traveling wave in positive and negative z direction:

• And recognizing that both v(z,t) and i(z,t) are propagating waves, the phase velocity is given by:

β

ω=pv

( ) ( ) ( ) zo

zo eztVeztVtzV

αα φβωφβω −−−

++ ++++−= coscos,

)( ztf βω − A general function describing propagating

wave in +z direction

Compare

(4.6)

58


Attenuation (αααα)

• The attenuation factor α decreases the amplitude of the voltage and current wave along the Tline.

• For +ve traveling wave:

• For -ve traveling wave:

( ) zo eztV αφβω −

++ +−cos

( ) zo eztV

αφβω −− ++cos

z

y

Z=0

z

y

Z=0


Dispersion (1)

Video

vin vout

Low dispersionTransmission Line

vinvout

• We observe that the propagation

velocity is a function of the

wave’s frequency.

• Different component of the

signal propagates at different

velocity (and also attenuate at

different rate), resulting in the

envelope of the signal being

distorted at the output.

Since ( ) ( ) ( )ωβωαωγγ j+==

( )ωβ

ω=pv

Cause of

dispersion

Components

59


Dispersion (2)

• Dispersion causes

distortion of the signal

propagating through a

transmission line.

• This is particularly evident

in a long line.

vin vout

High dispersionTransmission Line

vin vout

Video


Dispersion (3)

• At the output, the sinusoidal components overlap at the wrong ‘timing’,

causing distortion of the pulse.

vinvout

0 10 20 30 401

0.5

0

0.5

1

Vin i ∆t⋅ 1,( )

Vin i ∆t⋅ 3,( )

Vin i ∆t⋅ 5,( )

i

0 10 20 30 400.5

0

0.5

1

1.5

Vint i ∆t⋅( )

i

0 10 20 30 401

0.5

0

0.5

1

Vout i ∆t⋅ 1,( )

Vout i ∆t⋅ 3,( )

Vout i ∆t⋅ 5,( )

i

0 10 20 30 400.5

0

0.5

1

1.5

Voutt i ∆t⋅( )

i

In this example, the

higher the harmonic

frequency the larger is

the phase velocity, i.e.

higher frequency signal

takes lesser time to travel

the length of the Tline.

60


The Implications

Tline supporting TEM or quasi-TEM mode


The Lossless Transmission Line

• When the tline is lossless, R= 0 and G = 0.

• We have:

• So the lossless transmission line has no attenuation, no dispersion and the characteristic impedance is real.

• Since lossless Tline is an ideal, in practical situation we try to reduce the loss to as small as possible, by using gold-plated conductor, and using good quality dielectric (low loss tangent).

C

LZc =

LCjj ωβγ ==

µε

11==

LCvp

61


Appendix 2Advanced Concepts –

Distributed RLCG Model for Transmission Line and Telegraphic Equations


Distributed Parameters Model (1)

( ) 21

42

4321

VVdsHj

dlEdlEdsE

dlEdlEdlEdlE

dsEdlE

S

ssS

ssss

SC

−=⋅−−⇒

⋅−−⋅−=⋅×∇−⇒

⋅−⋅−⋅−⋅−=

⋅×∇−=⋅−

∫∫

∫∫∫∫

∫∫∫∫

∫∫∫

−r

rrr

rrrr

rr

µω

( )zILjVV

dsHjVV

S

∆−=−⇒

⋅−=+− ∫∫

ω

µω

12

21

r

• For TEM or quasi-TEM mode propagation along z direction:

Stoke’s

Theorem

V1 V2

L∆z

Flux linkage(Definition forInductance)

Can be representedin circuit as:

L is the inductanceper meter

s1

s2

s3

s4

Loop C

V1 V2C

Conductor

∆ z

z

y

This means therelation betweenV1 and V2 is as ifan inductor is between them

When ∆z

is small

as compared

to wavelength

( ) ( )zyxhyxhH zt ˆ,, +≅+rr

62



• C is the per unit length capacitance between the 2 conductors of the Tline.

C1

C2

I2

I1

∆ z

Volume W

Surface S

( )( ) ( ) 12

21

21

1

1

IIzCVzC

IISdE

SdJSdJSdJSdE

dWJSdE

dWSdE

dWSdE

dWdWE

tV

t

St

AASSt

WSt

Wt

St

WS

WW

−=∆=∆⇒

+−=⋅⇒

⋅−⋅−=⋅−=⋅⇒

⋅∇−=⋅⇒

=⋅⇒

=⋅⇒

=⋅∇

∂∂

∂∂

∂∂

∂∂

∂∂

∂

∂

∂∂

∫∫

∫∫∫∫∫∫∫∫

∫∫∫∫∫

∫∫∫∫∫

∫∫∫∫∫

∫∫∫∫∫∫

r

rrrrrrr

r

r

r

r

ε

ε

ε

ρε

ερ

ερ

I1 I2

C∆ztVzC

∂∂∆

V

Can be represented in circuit as

Using Divergence Theorem

See the followingslide for more proof.

A1

A2

This means therelation betweenI1 and I2 is as ifa capacitor is between them

When ∆z

is small

as compared

to wavelength



∫ ⋅−=

⋅−

⋅

∫

∫∫

BA

dlE

Q

B

A

S

dlE

dsE

rr

rε

( ) ( )

( ) ( )

⋅+⋅=

=⋅+⋅=

∫∫∫∫

∫∫∫∫

dsdsQ

dsdsE

r

rzyxsf

r

rzyxsf

r

rzyxsr

rzyxs

2Con surface2

24

,,2

1Con surface1

14

,,1

2Con surface2

24

,,2

1Con surface1

14

,,1

πεπε

πε

σ

πε

σr

Consider the ratio:

For static or quasi-static condition, the E fieldis given by:

Q is the total charge on conductors C1 or C2, σs is the surface charge density, while fsis the normalized surface charge density with respect to Q.

Hence the ratio becomes:

C does not depend onthe total charge on the conductors, we calledthis constant the ‘Capacitance’. CVdsE

S

=⋅∫∫r

ε

Potential difference between A and B

C1

C2

I2

I1

∆ zVolume W

Surface S

(x,y,z)

r2

r1

A

B

( ) ( )

( ) ( )C

dlE

dsE

BA

dldsr

rzyxsfdsr

rzyxsf

BA

dldsr

rzyxsfdsr

rzyxsfQ

Q

B

A

S

==

=⋅−

⋅

∫ ⋅

⋅+⋅−

∫ ⋅

⋅+⋅−

∫∫ ∫∫

∫∫ ∫∫∫

∫∫

1Con surface 2Con surface 2

24

,,2

1

14

,,1

1

1Con surface 2Con surface 2

24

,,2

1

14

,,1

πεπε

πεπε

ε

r

r

63



∆z

• Combining the relationship between the L, C and transverse voltages and currents, the equivalent circuit for a short section of transmission line supporting TEM or quasi-TEM propagating EM field can be represented by the equivalent circuit:

• Thus a long Tline can be considered as a cascade of many of these equivalent circuit sections. Working with circuit theory and circuit elements are much easier than working with E and H fields using Maxwell equations.

L ∆z

C ∆zV1 V2

I



• When the conductor has small conductive loss a series resistance R∆ z

can be added to the inductance.

• When the dielectric has finite conductivity, a shunt conductance G∆ z

can be added in parallel to the capacitance.

• The inclusion of constant R and G in the Tline’s distributed circuit model

is only accurate for very small losses*. This is true most of the time as

Tline is usually made of very good conductive material.

• The equations for finding L, C, R, G under low loss condition are given in

the following slide.

L ∆z

C ∆z

R ∆z

G ∆zV1 V2

I*Under lossy condition, R, L and G

are usually function of frequency,

hence the Tline is dispersive.

64


Extra: Lossy Dielectric

• Assuming the dielectric is non-magnetic, then the dielectric loss is due to leakage (non-zero conductivity) and polarization loss*.

• Polarization loss is due to the vibration of the polarized molecules in the dielectric when an a.c. electric field is imposed.

• Both mechanisms can be modeled by considering an effective conductivity σd for the dielectric at the operating frequency. This is usually valid for small electric field.

⇒+=×∇ EjJH or

rrrεωε EjEH ord

rrrεωεσ +=×∇

1 EjHorj

dor

rr

+=×∇⇒εωε

σεωε

( )

or

d

or EjjH

εωε

σδ

δεωε

=

−=×∇⇒

tan

tan1rr

This is called Loss Tangent

( )δεεεεεε

εεω

tan "'

"'

oror

EjjH

==

−=×∇rr

Note that σd is a function of frequency

Loss current density *We should also includehysterisis loss in ferromagnetic material.


Finding Distributed Parameters for Low Loss Practical Transmission Line

• When loss is present, the propagation mode will not be TEM anymore

(Can you explain why this is so?).

• However if the loss is very small, we can assume the propagating EM

field to be similar to the EM fields under lossless condition. From the E

and H fields, we could derived the RLCG parameters from equations

(4.1a) and (4.1b). Although the RLCG parameters under this condition

is only an approximation, the error is usually small.

• This approach is known as perturbation method.

65


Derivation of Telegraphic Equations (1)

Use Kirchoff’s Voltage Law (KVL):

• For Tline supporting TEM and quasi-TEM modes, the V and I on the line

is the solution of a hyperbolic partial differential equation (PDE) known

as telegraphic equations. Consider first lossless line:

I1 L∆z

C ∆zV1V2

I2I1=I

V1 V2

I2jωL∆z

LIjz

VVω−=

∆

−⇒ 12

zILjVV ∆−= ω12

LIjzV ω−=⇒

∂∂

zV

z

VVz ∂

∂∆

−→∆ =

120lim

Observing that:


Derivation of Telegraphic Equations (2)

( )

CVj

IVzCjI

z

IIω

ω

−=⇒

=∆−

∆

− 12

21

• Now considering the current on both ends, and using Kirchoff’s Current

Law (KCL):

I1 L∆z

C ∆zV1V2

I2

Using KCL here:

I1

jωL ∆z

(jωC ∆z)-1 V2=V

I2I1

Lossless

Telegraphic

Equations

CVjzI ω−=

∂∂

t

VC

z

II

∂

∂−=

∆

− 212

Again observing that:

66


Relationship Between Field Solutions and Telegraphic Equations

Interconnect

‘Long’ interconnect‘Short’ interconnect

Lumped RLCG circuit Waveguide

Transmission lineMaxwell’s Equations

Wave Equations

E & H wave TEM

TE

TM

Hybrid

Quasi-TEM

Vt, It and

Distributed

RLCG

circuit

Telegraphic

Equations

Only for striplinestructures

KVL &

KCL

Zc, vp, attenuation factor,

dispersion, power handling.


Example A3

• Find the RLCG parameters of the low loss parallel plate waveguide in

Example A1. Assuming the conductivity of the conductor is σ and the

dielectric between the plates is complex (this means the dielectric is

lossy too):

• Use the expressions for Et and Ht as derived in Example A1.

"' εεε j−=

H/m W

dL

µ= F/m

'

d

WC

ε=

/m 2

Ω=W

Rscδσ

/m C'

" 1−Ω==d

WG dσ

ε

ωε

67


5.0 - Transmission Line Synthesis On Printed Circuit

Board (PCB) And Related Structures


Stripline Technology (1)

• Stripline is a planar-type Tline that lends itself well to microwave

integrated circuit (MIC) and photolithograhpic fabrication.

• Stripline can be easily fabricated on a printed circuit board (PCB) or

semiconductor using various dielectric material such as epoxy resin,

glass fiber such as FR4, polytetrafluoroethylene (PTFE) or commonly

known as Teflon, Polyimide, aluminium oxide, titanium oxide and other

ceramic materials or processes, for instance the low-temperature co-fired

ceramic (LTCC).

• Three most common Tline configurations using stripline technology are

microstrip line, stripline and co-planar stripline.

68


Stripline Technology (2)

• A variety of substrates, Thin and Thick-Film technologies can be

employed.

• For more information on microstrip line circuit design, you can refer to

T.C. Edwards, “Foundation for microstrip circuit design”, 2nd Edition

1992, John Wiley & Sons. (3rd edition, 2000 is also available).

• For more information on stripline circuit design, you can consult H.

Howe, “Stripline circuit design”, 1974, Artech House.

• For more information on microwave materials and fabrication

techniques, you can refer to T.S. Laverghetta, “Microwave materials and

fabrication techniques”, 3rd edition 2000, Artech House.


The Substrate or Laminate for Stripline Technology

• Typical dielectric thickness are 32mils (0.80mm), 62mils (1.57mm) for

double sided board. For multi-layer board the thickness can be

customized from 2 – 62 mils, in 1 mils step.

• Copper thickness is usually expressed in terms of the mass of copper spread

over 1 square foot. Standard copper thickness are 0.5, 1.0, 1.5 and 2.0 oz/foot2.

0.5 oz/foot2 ≅ 0.7mils thick.



Dielectric thickness

Copper thickness

Dielectric

Copper (Usually gold plated

to protect against oxidation)

Standard material consistof epoxy with glassfiber reinforcement

69


Factors Affecting Choices of Substrates

• Operating frequency.

• Electrical characteristics - e.g. nominal dielectric constant, anisotropy, loss tangent, dispersion of dielectric constant.

• Copper weight (affect low frequency resistance).

• Tg, glass transition temperature.

• Cost.

• Tolerance.

• Manufacturing Technology - Thin or thick film technology.

• Thermal requirements - e.g. thermal conductivity, coefficient of thermal expansion (CTE) along x,y and z axis.

• Mechanical requirements - flatness, coefficient of thermal expansion, metal-film adhesion (peel strength), flame retardation, chemical and water resistance etc.


Comparison between Various Transmission Lines

Microstrip line Stripline Co-planar line Suffers from dispersion and non-TEM modes

Pure TEM mode Suffers from dispersion and non-TEM modes

Easy to fabricate Difficult to fabricate Fairly difficult to fabricate

High density trace Mid density trace Low density trace

Fair for coupled line structures

Good for coupled line structures

Not suitable for coupled line structures

Need through holes to connect to ground

Need through holes to connect to ground

No through hole required to connect to ground

70


Field Solution for EM Waves on Stripline Structure (1)

• In microstrip and co-planar Tline the dielectric material does not completely surround the conductor, consequently the fundamental mode of propagation is not a pure TEM mode. However at a frequency below a few GHz (<10GHz at least), the EM field propagation mode is quasi-TEM. The microstrip Tline can be characterized in terms of its approximate distributed RLCG parameters.

• For the stripline, the dominant mode is TEM hence it can be characterized by its distributed RLCG parameters to very high frequencies.

• Unfortunately there is no simple closed-form analytic expressions that for the EM fields or RLCG parameters for a planar Tline.

• A method known as ‘Conformal Mapping’ is usually used to find the approximate closed-form solution of the Laplace partial differential equation for the TEM/quasi-TEM mode fields. The expression can be very complex.

See Ramo [3], Chapter 7 for more information on ConformalMapping method. Collin [1], Chapter 3 provides mathematical derivation of the field solutions for parallel plate waveguide with inhomogeneous dielectric and the microstrip line.


Field Solution for EM Waves on Stripline Structure (2)

• Another approach is to use numerical methods to solve for the static E and H field along the cross section of the Tline. From the E and H fields, the RLCG parameters can be obtained from equations (4.1a) and (4.1b).

• There are numerous commercial and non-commercial software for performing this analysis.

• Once RLCG parameters are obtained, the propagation constant γ and characteristic impedance Zc of the Tline can be obtained. Zc can then be plotted as a function of the Tline dimensions, the dielectric constant and the operating frequency.

• Many authors have solved the static field problem for stripline structures using conformal mapping and other approaches to solve the scalar potential φ and vector potential A.

• The following slides show some useful results as obtained by researchers in the past for designing planar Tline.

• Some of the equations are obtained by curve-fitting numerically generated results.

71


Typical Iterative Flow for Transmission Line Design

Design

equations

for Tline

Yes

No

Draw Tline physical

cross section

Start

Solve for TEM mode

E and H fields at the

frequency of interest

Determine R, L, C, G

from (4.1)

Compute Zc, α and βat the frequency of

interest

Criteria met?

End


Design Equations

• By varying the physical dimensions and using the flow of the previous

slide, one can obtain a collection of results (Zc, α, β) .

• These results can be plotted as points on a graph.

• Curve-fitting techniques can then be used to derive equations that

match the results with the physical parameters of the Tline.

µ , εd

W

µo , εo

0 2 4 6 80

100

200

300263.177

15.313

Zc x 1,( )

Zc x 2,( )

Zc x 3,( )

Zc x 4,( )

Zc x 5,( )

Zc x 6,( )

80.1 xW/d

Zcεr=1

εr=2

εr=3

εr=6

72


Microstrip Line (see reference [3], Chapter 8)

1172.0

98.1377

101

11

2

11

−

+=

+

+−

+=

d

w

d

wZ

w

d

effc

reff

ε

εε

oeff

pvεµε

1=

Only valid when quasi-TEM approximation and very low loss condition applies.

Design Equations for Microstrip Line

(5.1a)

(5.1b)

(5.1c)

Effective dielectric

constant (See Appendix 3)

µ , εd

W

µo , εo

t


Stripline (see reference [3], Chapter 8):

µ , ε d

d

w

µε

1=pv

( )

1

022

2

4cosh

sin1

)(

14

2 −

=

−=

−

⋅≅

∫ d

wk

x

dxK

kK

kKZZ o

c

π

φ

φπ

Complete elliptic integral

of the 2nd kind ε

µ=oZ

Only valid when TEM, quasi-TEM approximation and very low loss condition applies.

Design Equations for Stripline

(5.2a)

(5.2b)

(5.2c)

73


Co-planar Line ([3], assume d is large compare to s):

1173.0for 1

12ln

4

173.00for 2ln

2

1

1

<<

−

+≅

<<

≅

+=

−

a

w

aw

aw

ZZ

a

w

w

aZZ

eff

oc

eff

oc

reff

ε

π

επ

εε

oeff

pvεµε

1=

a

w

d

s

Only valid when quasi-TEM approximation and very low loss condition applies.

o

ooZ

ε

µ=

Design Equations for Co-planar Line

(5.3a)

(5.3b)

(5.3c)

Effective dielectric

constant (See Appendix 3)


Dispersive Property of Microstrip and Co-planar Lines (1)

• The actual propagation mode for microstrip and co-planar lines are a combination TM and TE modes. Both modes are dispersive. The phase velocity of the EM wave is dependent on the frequency (see references [1] and [2], or discussion in Appendix 1).

• This change in phase velocity is reflected by effective dielectric constant that changes with frequency.

• At low frequency (f < fcritical), when the propagation modes for microstrip and co-planar lines approaches quasi-TEM, the phase velocity is almost constant.

• Appendix 1 shows a simple method to estimate fcritical for microstrip and co-planar lines.

• Thus fcritical is usually taken as the upper frequency limit for microstrip and co-planar lines. Typical value is 5 – 100 GHz depending on dielectric thickness.

• Stripline does not experience this effect as theoretically it can support pure TEM mode.

74


Dispersive Property of Microstrip and Co-planar Line (2)

f

1

εr

εeff

Microstrip line Stripline

f

1

εr

εeff Region where (5.1a) and (5.1b) applies.

fcritical

Limit for quasi-TEM approximation, see Example A1 and on how to estimate fcritical

Note:Beyond fcritical the concept of characteristic impedancebecomes meaningless.

For microstrip line, the EMfield is partly in the air anddielectric. Hence the effectivedielectric εeff constant is betweenthose of air and dielectric.

oeff

pvεµε

1=


Tool for Stripline Design

• You can easily refer to books, journals, magazines etc., use the

approximation equation developed by others and write your own

software tools, from a simple spreadsheet, to Visual Basic, JAVA or

C++ based programs.

• A few free software are available online, the more notable is AppCAD

by Agilent Technologies.

You can also check out

www.rfcafe.com for more

public domain tools

75


Example 5.1 - Microstrip Line Design

• (a) Design a 50ΩΩΩΩ microstrip line, given that d = 1.57 mm and

dielectric constant = 4.6 (Here it means find w).

0 1 2 3 4 520

30

40

50

60

70

80

90

100

110

120

130

140

150

160

Zo Si

Si

•Steps...

•Plot out Zc versus (w/d).

• From the curve, we see that

w/d = 1.8 for 50Ω.

• Thus w = 1.8 x 1.57 mm = 2.82 mm

50Ω

0 1 2 3 4 5 63

3.25

3.5

3.75

4

εe Si

SiW/d W/d

1.8


Microstrip Line Design Example Cont...

• (b) If the length of the Tline is 6.5 cm, find the propagation delay.

• (c) Using the l < 0.05λλλλ rule, find the frequency range where the

microstrip line can be represented by lumped RLCG circuit.

From εeff versus w/d, we see that εeff = 3.55 at w/d = 1.8. Therefore:

psv

tmsvp

delayoo

p 406065.0

10601.151.3

1 18 ≅=⇒×=⋅

= −

µε

To be represented as lumped, the wavelength λ must be > 20 x Length:

MHz 2.123

30.1

m 30.120

30.1=<⇒

>⇒

=⋅>

pv

f

pv

f

lλ

MHz 2.123=lumpedf

76



• (d) When the low loss microstrip line is considered short, derived

its equivalent LC network.

pF/m6.12410601.150

11

11

8=

××==

=×=

pc

pc

vZC

CLCC

LvZ nH/m27.3132 == CZL c

For shot interconnect we could model the Tline as:

nHzL 18.1021 =∆ nH18.10

pFzC 096.8=∆ or

nHzL 36.20=∆

pFzC 048.421 =∆pF048.4

nHL

pFC

36.20065.0

096.8065.0

=⋅

=⋅



• (e) Finally estimate fcritical , the limit where quasi-TEM

approximation begins to break down.

GHzfdpv

criticalcriticalf

pv10.5 031.020

031.0≅<⇒=>=λ

Again using the criteria that wavelength > 20d for quasi-TEM mode

to propagate:

We see that to increase fcritical, smaller thickness d

should be used.

77



µ = µo , ε = 4.6εo

µo , εo

1.57 mm

2.82 mm

65.0 mm

Zc = 50Ωvp = 1.601x108 m/s

tdelay = 406psec

Maximum usable frequency = fcritical = 5.10 GHz.

Short interconnect limit = 123.2 MHz


Example 5.2 – Estimating the Effect of Trace Width and Dielectric Thickness on ZC

• Consider the following microstrip line cross sections, assuming lossless

Tline, make a comparison of the characteristic impedance of each line.

TL1TL2 TL3

TL4

78


Example 5.3 - Stripline Design Example Using 2D EM Field Solver Program

• Here we demonstrate the use of a program called Maxwell 2D by Ansoft Inc. www.ansoft.com to design a stripline.

• The version used is called Maxwell SV Ver 9.0, a free version which can be downloaded from the company’s website.

• The software uses finite element method (FEM) to compute the two-dimensional (2D) static E and H field of an array of metallic objects.

• It is assumed that the stripline is lossless.

• Two projects are created, one is the Electrostatic problem for calculation of static electric field and distributed capacitance, the other is Magnetostatic problem, for calculation of static magnetic field and distributed inductance.

• Characteristic impedance of the stripline can then be computed from the distributed capacitance and inductance.


Example 5.3 - Screen Shot

79


Example 5.3 - Stripline Cross Section

• Draw the cross section of the model and assign material.

8.0mm

0.3mm

0.3mm

0.036mm

0.036mm

0.4mm

• Set drawing units to micron.• Set drawing size to 10000um for xand 4000um for y.

• Set grid to dU = dV = 100um.• Draw model, use direct entry modefor conducting structures like GNDplanes and signal.

• Name signal conductor “Trace1” andGND planes “GND1” and “GND2”.

• Name FR4 as “substrate”, then group both “GND1” and “GND2” as one object “GND”.

0.6mm

FR4 Substrate“substrate” Signal conductor

(PEC) “Trace1”

GND planes

(PEC)“GND2”

“GND1”


Example 5.3 - Electrostatics: Setup Boundary Conditions

• Set the boundary conditions.

• All boundary are Dirichlet type, i.e. voltages are specified.

• Let the edges of the model domain remain as Balloon Boundary.

0V 0V

0V

0V

1V

Balloon Boundaries

80


Example 5.3 - Electrostatics: Setup Executive Parameters

• Under ‘Setup Executive Parameters’ tab, select ‘Matrix…’ and proceed

to perform the capacitance matrix setup as shown.


Example 5.3 - Electrostatics: Setup Solver and Solve for Scalar Potential φφφφ

• Setup the solver and solve for the approximate potential solution. Use the ‘Suggested Values’ if you are not sure.

81


Example 5.3 - Finite Element Method (1)

• In finite-element method (FEM) an object is thought to consist of many smaller elements, usually triangle for 2D object and tetrahedron for 3D object.

• FEM is used to solve for the approximate scalar potential V (or φ) for electrostatic problem and vector potential A for magnetostatic problem at the vertex of each triangle. The partial differential equations (PDE) to solve are the Poisson’s equations.

• Potential value inside the triangle can be estimated via interpolation.• For 2D problem the PDE can be written as:

ερ

=∇ V2

JArr

µ=∇2

( ) ( )yxyxVV

yxV

tt

yxttt

, ,

2

ρρ

ερ

==

+=∇=∇∂∂

∂∂ ))

( ) ( )yxJJyxAA

JA

zzzz

zzt

, ,

2

==

=∇r

µ

x

y


Example 5.3 - Finite Element Method (2)

• 2D quasi-static E field can then be obtained by:

• Similarly magnetic flux intensity H can be obtained from:

• For more information, refer to – T. Itoh (editor), “Numerical techniques for microwave and milimeter-

wave passive structures”, John-Wiley & Sons, 1989.– P. P. Silvester, R. L. Ferrari, “Finite elements for electrical

engineers”, Cambridge University Press, 1990.– Other newer books.

( ) ( )[ ]zyxAyxH ztt)r

,, 1 ×∇−=µ

( ) ( )yxVyxE ttt ,, −∇=r

82


Example 5.3 - Electrostatics: The Triangular Mesh


Example 5.3 - Electrostatics: Plot of E field Magnitude

83


Example 5.3 - Electrostatics: Plot of Voltage Contour


Example 5.3 - Electrostatics: Capacitance

• The estimated distributed capacitance is then computed using:

• Approximation to the integration using summation is performed by the software. The result is shown below:

• C ≈ 1.9958×10-10 F/m or 199.58 pF/m.

∫∫∫=

V

t

t

dvEV

C2

2

' rε

84


Example 5.3 - Magnetostatics: Setup Boundary Conditions

Balloon Boundaries

Solid source +1A

Solid source –1A (total for 2 planes)


Example 5.3 - Magnetostatics: Setup Executive Parameters

• Under ‘Setup Executive Parameters’ tab, select ‘Matrix/Flux…’ and

proceed to perform the inductance matrix setup as shown.

85


Example 5.3 - Magnetostatics: Plot of B field Magnitude


Example 5.3 - Magnetostatics: Inductance

• The estimated distributed capacitance is then computed using:

• L ≈ 2.5093×10-7 H/m or 250.93 nH/m.

∫∫∫=

V

t

t

dvHI

L2

2

rµ

86


Example 5.3 - Derivation of Parameters for Stripline

Ω===−×

−× 819.3510

109558.1

7105093.2CL

cZ

1810427.11 −×== msLC

v p


Example 5.4 – Tline Design Using Agilent’s AppCAD V3.02

87


Example 5.4 - Derivation of LC Parameters from AppCAD V3.02 Results

( )

( )8

10998.2

466.0

0.36

×=

=

=

vacumnp

vacumnpp

c

v

vv

Z

mHCZL

mFC

c

pvcZ

/10577.2

/10988.1

72

101

−

−

×==

×==

• As we can see, both the results using EM field solver and

using closed-form solution (AppCAD) are very close.

• Bear in mind that this is only approximate solution, as

skin-effect loss and dielectric loss are ignored.


Appendix 3

88


The Origin of Effective Dielectric Constant

(εεεεeff)

• The approach can be traced to a paper by Bryant and Weiss1.

Assuming low loss and quasi-TEM mode:

LCpv 1=

CpvCL

cZ 1==

µo , εo

µo , εo

C1

With dielectric, C is the capacitanceper meter between the conductors

Without dielectric, C1 is the capacitanceper meter between the conductors

Note 1: T. G. Bryant and J. A. Weiss,”Parameters of microstrip

transmission lines and coupled pairs of microstrip lines”, IEEE

Trans. Microwave Theory Tech., vol.MTT-16, pp.1021-1027,1968.

1CC

eff =ε

eff

c

effLCpv

εε==

⋅1

1

Define

Then

1

1

1

11

Ceffcc

effC

eff

cC

eff

cc

Z

Z

⋅

⋅⋅

=⇒

==

ε

εεε

Speed of

light in vacumn

C1 is computed via numerical

methods for various W and d

µ , εd

W

µo , εo

C


• When quasi-TEM approximation is no longer valid, the preceeding

formulation is not accurate and the telegraphic equations cannot be

used.

• Also there is no longer a unique voltage and current, only E and H

fields are used. More dispersion will also be observed.

• We can resort to defining equivalent voltage and current as employed

for waveguides.

• However this is usually quite cumbersome, another option is to use full

wave analysis.

When Quasi-TEM Approximation is not Valid - Full Wave Analysis

89


Full-Wave Analysis

• Full-wave analysis is usually carried out using numerical methods such

as Method of Moments (MoM), Finite Element Methods (FEM), Finite-

Difference Time-Domain (FDTD) and Transmission Line Matrix (TLM).

• In all these methods, the Maxwell Equations are solved directly or

indirectly instead of transforming the equations into circuit theory

expressions (e.g. the telegraphic equations).


THE END

90


Example A1 – Solution for TE Mode

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) zj

dn

nz

zj

dnBj

t

zj

dnBZjk

t

eyByxH

yeyyxH

xeyyxE

dn

n

dn

noo

βπ

βπβ

βπ

π

π

−

−

−

=

=

=

cos,

ˆsin,

ˆsin,

The EM fields for TE mode are shown below:

Since n is an arbitrary integer,

the TE mode is usually called

the TEn mode.

o

o

o

o

Z

k

ε

µ

µεω

=

=

Exercise

• Suppose we have a parallel-plate transmission line with the following

parameters:

– W = 16.0 mm

– d = 1.0 mm

– εr = 2.5, µr = 1.0, dielectric breakdown at |E| = 3000 V/m.

• The length of the line is 10 meter. Find the cut-off frequency of the for

TM and TE modes.

• If TEM mode is propagating along the line, find the characteristic

impedance Zc of the line, and the maximum power that can be carried

by this line without damaging the dielectric.


91

Exercise Cont…

• Assuming this transmission line is used in the following system.

Estimate the maximum working distance R.


Transmission

Tower

Omni-directional

Antenna

Parallel-

plate Tline

Receiver

Distance = R

If the Receiver can detect the data when received poweris 2 µWatt.

Antenna gain G = 10

Lesson 1

Documents

curldivergencez y x

electric field e

fabian kung wai lee

magnetic field h

density of field lines

detecth field

xdirectionx componentno

usean electric charge