lesson 1
TRANSCRIPT
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.
June 2008 © 2006 Fabian Kung Wai Lee 2
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
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June 2008 © 2006 Fabian Kung Wai Lee 3
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.
June 2008 © 2006 Fabian Kung Wai Lee 4
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• [6] F. Kung unpublished notes and works.
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June 2008 © 2006 Fabian Kung Wai Lee 5
1.0 Review of Electromagnetic (EM) Fields
June 2008 © 2006 Fabian Kung Wai Lee 6
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
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June 2008 © 2006 Fabian Kung Wai Lee 7
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
o
)
241 ⋅=
πε
+I
-I
Conductor
Conductor
Conductor
Conductor
June 2008 © 2006 Fabian Kung Wai Lee 8
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
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June 2008 © 2006 Fabian Kung Wai Lee 9
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
June 2008 © 2006 Fabian Kung Wai Lee 10
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.
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June 2008 © 2006 Fabian Kung Wai Lee 11
Extra: Wave Function and Phasor (2)
• 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
June 2008 © 2006 Fabian Kung Wai Lee 12
Extra: Wave Function and Phasor (3)
• 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
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June 2008 © 2006 Fabian Kung Wai Lee 13
Extra: Wave Function and Phasor (4)
• 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
June 2008 © 2006 Fabian Kung Wai Lee 14
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
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June 2008 © 2006 Fabian Kung Wai Lee 15
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
June 2008 © 2006 Fabian Kung Wai Lee 16
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
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June 2008 © 2006 Fabian Kung Wai Lee 17
2.0 Introduction –Transmission Line
Concepts
June 2008 © 2006 Fabian Kung Wai Lee 18
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
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June 2008 © 2006 Fabian Kung Wai Lee 19
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.
June 2008 © 2006 Fabian Kung Wai Lee 20
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.
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June 2008 © 2006 Fabian Kung Wai Lee 21
Long Interconnect (2)
• 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
June 2008 © 2006 Fabian Kung Wai Lee 22
Long Interconnect (3)
• 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:
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June 2008 © 2006 Fabian Kung Wai Lee 23
Long Interconnect (4)
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
June 2008 © 2006 Fabian Kung Wai Lee 24
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
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June 2008 © 2006 Fabian Kung Wai Lee 25
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
June 2008 © 2006 Fabian Kung Wai Lee 26
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
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June 2008 © 2006 Fabian Kung Wai Lee 27
Demonstration - Electromagnetic Field Propagation in Interconnect (2)
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
June 2008 © 2006 Fabian Kung Wai Lee 28
Demonstration - Electromagnetic Field Propagation in Interconnect (3)
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
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June 2008 © 2006 Fabian Kung Wai Lee 29
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.
June 2008 © 2006 Fabian Kung Wai Lee 30
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
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June 2008 © 2006 Fabian Kung Wai Lee 31
Some Multi-conductor Transmission Line Configurations
Conductor
Dielectric
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Typical Waveguide Configurations
Rectangular waveguide Circular waveguide
Optical FiberDielectric waveguide
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Examples of Microstrip and Co-planar Lines
MicrostripCo-planar
June 2008 © 2006 Fabian Kung Wai Lee 34
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
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June 2008 © 2006 Fabian Kung Wai Lee 35
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
June 2008 © 2006 Fabian Kung Wai Lee 36
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
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June 2008 © 2006 Fabian Kung Wai Lee 37
3.0 Propagation Modes
June 2008 © 2006 Fabian Kung Wai Lee 38
Transverse E and H Field Patterns
x
y
z
Transverse
plane
E fields
H fields
Field patterns lie in the Transverse
Plane.
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June 2008 © 2006 Fabian Kung Wai Lee 39
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
June 2008 © 2006 Fabian Kung Wai Lee 40
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
June 2008 © 2006 Fabian Kung Wai Lee 41
Propagation Modes (2)
H
E
Mix modes
H
E
TEM mode
H
E
TE mode
H
E
TM mode z
x
y
June 2008 © 2006 Fabian Kung Wai Lee 42
Examples of Field Patterns or Modes
TEM or quasi-TEM mode
E field
H field
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June 2008 © 2006 Fabian Kung Wai Lee 43
Appendix 1Advanced Concepts – Field
Theory Solutions for Transmission Lines
June 2008 © 2006 Fabian Kung Wai Lee 44
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
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June 2008 © 2006 Fabian Kung Wai Lee 45
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.
June 2008 © 2006 Fabian Kung Wai Lee 46
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
June 2008 © 2006 Fabian Kung Wai Lee 47
Obtaining the Expressions for E and H (2)
• 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)
June 2008 © 2006 Fabian Kung Wai Lee 48
Obtaining the Expressions for E and H (3)
• 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
June 2008 © 2006 Fabian Kung Wai Lee 49
Obtaining the Expressions for E and H (4)
• 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)
June 2008 © 2006 Fabian Kung Wai Lee 50
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
June 2008 © 2006 Fabian Kung Wai Lee 51
E and H fields Expressions (2)
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )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)
June 2008 © 2006 Fabian Kung Wai Lee 52
E and H fields Expressions (3)
• 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
June 2008 © 2006 Fabian Kung Wai Lee 53
E and H fields Expressions (4)
• 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
June 2008 © 2006 Fabian Kung Wai Lee 54
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
June 2008 © 2006 Fabian Kung Wai Lee 55
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)
June 2008 © 2006 Fabian Kung Wai Lee 56
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
June 2008 © 2006 Fabian Kung Wai Lee 57
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
June 2008 © 2006 Fabian Kung Wai Lee 58
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
June 2008 © 2006 Fabian Kung Wai Lee 59
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
June 2008 © 2006 Fabian Kung Wai Lee 60
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
June 2008 © 2006 Fabian Kung Wai Lee 61
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)
June 2008 © 2006 Fabian Kung Wai Lee 62
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
June 2008 © 2006 Fabian Kung Wai Lee 63
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
June 2008 © 2006 Fabian Kung Wai Lee 64
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
June 2008 © 2006 Fabian Kung Wai Lee 65
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
June 2008 © 2006 Fabian Kung Wai Lee 66
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
June 2008 © 2006 Fabian Kung Wai Lee 67
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).
June 2008 © 2006 Fabian Kung Wai Lee 68
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
June 2008 © 2006 Fabian Kung Wai Lee 69
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
June 2008 © 2006 Fabian Kung Wai Lee 70
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
June 2008 © 2006 Fabian Kung Wai Lee 71
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
June 2008 © 2006 Fabian Kung Wai Lee 72
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
June 2008 © 2006 Fabian Kung Wai Lee 73
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
June 2008 © 2006 Fabian Kung Wai Lee 74
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
June 2008 © 2006 Fabian Kung Wai Lee 75
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.
June 2008 © 2006 Fabian Kung Wai Lee 76
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
June 2008 © 2006 Fabian Kung Wai Lee 77
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)
June 2008 © 2006 Fabian Kung Wai Lee 78
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
June 2008 © 2006 Fabian Kung Wai Lee 79
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
June 2008 © 2006 Fabian Kung Wai Lee 80
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
June 2008 © 2006 Fabian Kung Wai Lee 81
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
June 2008 © 2006 Fabian Kung Wai Lee 82
Example A1 – Solution for TEM Mode (2)
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
June 2008 © 2006 Fabian Kung Wai Lee 83
Example A1 – Solution for TEM Mode (3)
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
June 2008 © 2006 Fabian Kung Wai Lee 84
Example A1 – Solution for TEM Mode (4)
Phase velocity vp for TEM mode:
µεµεω
ωβω 1===pv
The phase velocity is equal to speed-of-light in the dielectric.
43
June 2008 © 2006 Fabian Kung Wai Lee 85
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:
June 2008 © 2006 Fabian Kung Wai Lee 86
Example A1 – Solution for TM Mode (2)
( )( )
( ) ( )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
June 2008 © 2006 Fabian Kung Wai Lee 87
Example A1 – Solution for TM Mode (3)
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_ =
June 2008 © 2006 Fabian Kung Wai Lee 88
Example A1 – Solution for TM Mode (4)
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
June 2008 © 2006 Fabian Kung Wai Lee 89
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).
June 2008 © 2006 Fabian Kung Wai Lee 90
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
June 2008 © 2006 Fabian Kung Wai Lee 91
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.
June 2008 © 2006 Fabian Kung Wai Lee 92
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
June 2008 © 2006 Fabian Kung Wai Lee 93
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
−−−
−
−=
−
==⇒
June 2008 © 2006 Fabian Kung Wai Lee 94
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
June 2008 © 2006 Fabian Kung Wai Lee 95
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
June 2008 © 2006 Fabian Kung Wai Lee 96
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
June 2008 © 2006 Fabian Kung Wai Lee 97
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.
June 2008 © 2006 Fabian Kung Wai Lee 98
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
June 2008 © 2006 Fabian Kung Wai Lee 99
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
June 2008 © 2006 Fabian Kung Wai Lee 100
4.0 – Transmission Line Characteristics and
Electrical Circuit Model
51
June 2008 © 2006 Fabian Kung Wai Lee 101
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.
June 2008 © 2006 Fabian Kung Wai Lee 102
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
June 2008 © 2006 Fabian Kung Wai Lee 103
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
June 2008 © 2006 Fabian Kung Wai Lee 104
Distributed Parameters (2)
• 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
June 2008 © 2006 Fabian Kung Wai Lee 105
Distributed Parameters (3)
• 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
June 2008 © 2006 Fabian Kung Wai Lee 106
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
June 2008 © 2006 Fabian Kung Wai Lee 107
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
June 2008 © 2006 Fabian Kung Wai Lee 108
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
June 2008 © 2006 Fabian Kung Wai Lee 109
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)
June 2008 © 2006 Fabian Kung Wai Lee 110
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)
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June 2008 © 2006 Fabian Kung Wai Lee 111
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
June 2008 © 2006 Fabian Kung Wai Lee 112
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
June 2008 © 2006 Fabian Kung Wai Lee 113
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
June 2008 © 2006 Fabian Kung Wai Lee 114
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)
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June 2008 © 2006 Fabian Kung Wai Lee 115
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
June 2008 © 2006 Fabian Kung Wai Lee 116
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
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June 2008 © 2006 Fabian Kung Wai Lee 117
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
June 2008 © 2006 Fabian Kung Wai Lee 118
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.
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June 2008 © 2006 Fabian Kung Wai Lee 119
The Implications
Tline supporting TEM or quasi-TEM mode
June 2008 © 2006 Fabian Kung Wai Lee 120
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
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June 2008 © 2006 Fabian Kung Wai Lee 121
Appendix 2Advanced Concepts –
Distributed RLCG Model for Transmission Line and Telegraphic Equations
June 2008 © 2006 Fabian Kung Wai Lee 122
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
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June 2008 © 2006 Fabian Kung Wai Lee 123
Distributed Parameters Model (2)
• 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
June 2008 © 2006 Fabian Kung Wai Lee 124
Distributed Parameters Model (3)
∫ ⋅−=
⋅−
⋅
∫
∫∫
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
June 2008 © 2006 Fabian Kung Wai Lee 125
Distributed Parameters (4)
∆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
June 2008 © 2006 Fabian Kung Wai Lee 126
Distributed Parameters Model (5)
• 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
June 2008 © 2006 Fabian Kung Wai Lee 127
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.
June 2008 © 2006 Fabian Kung Wai Lee 128
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.
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June 2008 © 2006 Fabian Kung Wai Lee 129
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:
June 2008 © 2006 Fabian Kung Wai Lee 130
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
June 2008 © 2006 Fabian Kung Wai Lee 131
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.
June 2008 © 2006 Fabian Kung Wai Lee 132
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
June 2008 © 2006 Fabian Kung Wai Lee 133
5.0 - Transmission Line Synthesis On Printed Circuit
Board (PCB) And Related Structures
June 2008 © 2006 Fabian Kung Wai Lee 134
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
June 2008 © 2006 Fabian Kung Wai Lee 135
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.
June 2008 © 2006 Fabian Kung Wai Lee 136
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.
1.0 oz/foot2 ≅ 1.4mils thick.
2.0 oz/foot2 ≅ 2.8mils thick.
Dielectric thickness
Copper thickness
Dielectric
Copper (Usually gold plated
to protect against oxidation)
Standard material consistof epoxy with glassfiber reinforcement
69
June 2008 © 2006 Fabian Kung Wai Lee 137
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.
June 2008 © 2006 Fabian Kung Wai Lee 138
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
June 2008 © 2006 Fabian Kung Wai Lee 139
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.
June 2008 © 2006 Fabian Kung Wai Lee 140
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
June 2008 © 2006 Fabian Kung Wai Lee 141
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
June 2008 © 2006 Fabian Kung Wai Lee 142
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
June 2008 © 2006 Fabian Kung Wai Lee 143
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
June 2008 © 2006 Fabian Kung Wai Lee 144
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
June 2008 © 2006 Fabian Kung Wai Lee 145
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)
June 2008 © 2006 Fabian Kung Wai Lee 146
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
June 2008 © 2006 Fabian Kung Wai Lee 147
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=
June 2008 © 2006 Fabian Kung Wai Lee 148
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
June 2008 © 2006 Fabian Kung Wai Lee 149
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
June 2008 © 2006 Fabian Kung Wai Lee 150
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
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June 2008 © 2006 Fabian Kung Wai Lee 151
Microstrip Line Design Example Cont...
• (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
=⋅
=⋅
June 2008 © 2006 Fabian Kung Wai Lee 152
Microstrip Line Design Example Cont...
• (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.
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Microstrip Line Design Example Cont...
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
June 2008 © 2006 Fabian Kung Wai Lee 154
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
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June 2008 © 2006 Fabian Kung Wai Lee 155
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.
June 2008 © 2006 Fabian Kung Wai Lee 156
Example 5.3 - Screen Shot
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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”
June 2008 © 2006 Fabian Kung Wai Lee 158
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
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Example 5.3 - Electrostatics: Setup Executive Parameters
• Under ‘Setup Executive Parameters’ tab, select ‘Matrix…’ and proceed
to perform the capacitance matrix setup as shown.
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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.
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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
June 2008 © 2006 Fabian Kung Wai Lee 162
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
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Example 5.3 - Electrostatics: The Triangular Mesh
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Example 5.3 - Electrostatics: Plot of E field Magnitude
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Example 5.3 - Electrostatics: Plot of Voltage Contour
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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
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June 2008 © 2006 Fabian Kung Wai Lee 167
Example 5.3 - Magnetostatics: Setup Boundary Conditions
Balloon Boundaries
Solid source +1A
Solid source –1A (total for 2 planes)
June 2008 © 2006 Fabian Kung Wai Lee 168
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.
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Example 5.3 - Magnetostatics: Plot of B field Magnitude
June 2008 © 2006 Fabian Kung Wai Lee 170
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
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June 2008 © 2006 Fabian Kung Wai Lee 171
Example 5.3 - Derivation of Parameters for Stripline
Ω===−×
−× 819.3510
109558.1
7105093.2CL
cZ
1810427.11 −×== msLC
v p
June 2008 © 2006 Fabian Kung Wai Lee 172
Example 5.4 – Tline Design Using Agilent’s AppCAD V3.02
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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.
June 2008 © 2006 Fabian Kung Wai Lee 174
Appendix 3
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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
June 2008 © 2006 Fabian Kung Wai Lee 176
• 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
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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).
June 2008 © 2006 Fabian Kung Wai Lee 178
THE END
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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.
June 2008 © 2006 Fabian Kung Wai Lee 180
91
Exercise Cont…
• Assuming this transmission line is used in the following system.
Estimate the maximum working distance R.
June 2008 © 2006 Fabian Kung Wai Lee 181
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