1. Advanced Transmission
Line 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 information
presented herein, and shall not be responsible for any errors, omissions or damages
as a result of the use of this information.
June 2008
© 2006 Fabian Kung Wai Lee
1
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
June 2008
© 2006 Fabian Kung Wai Lee
2
1
References
•
[1] R. E. Collin, “Foundation for microwave engineering”, 2nd edition,
1992, McGraw-Hill.
A very advanced and in-depth book on microwave
engineering. Difficult to read but the information is
very comprehensive. A classic work. Recommended.
•
[2] D. M. Pozar, “Microwave engineering”, 2nd edition, 1998 John-Wiley
& Sons. (3rd edition, 2005 is also available from John-Wiley & Sons).
Easier to read and understand. Also a good book.
Recommended.
•
[3] S. Ramo, J.R. Whinnery, T.D. Van Duzer, “Field and waves in
communication electronics” 3rd edition, 1993 John-Wiley & Sons.
Good coverage of EM theory with emphasis on
applications.
•
[4] C. R. Paul, “Introduction to electromagnetic compatibility”, JohnWiley & Sons, 1992.
June 2008
© 2006 Fabian Kung Wai Lee
3
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[6] F. Kung unpublished notes and works.
June 2008
© 2006 Fabian Kung Wai Lee
4
2
1.0 Review of
Electromagnetic (EM) Fields
June 2008
© 2006 Fabian Kung Wai Lee
5
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.
Force F
Test charge
r
+ q2
r
F = q2 E
E
+q1
r
qq
= 1 ⋅ 1 2 rˆ
4πε r 2
r r
rH
F = I2 × B
Test
current I2
Force F
Coulomb’s Law
To detect an E field we use
an electric charge. To detect
H field we use a current loop
June 2008
I
© 2006 Fabian Kung Wai Lee
*The magnetic field is B = µH,
H is the magnetization.
6
3
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.
E fields
Conductor
+
+
+ +
++
++
q
Force on a test charge q
Conductor
-
-
-
F=
1 ⋅ Qq r)
4πε o r 2
Conductor
H fields
+I
• 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.
Conductor
-I
( )
F = q v× B
•Density of field lines corresponds to strength
of the field.
June 2008
© 2006 Fabian Kung Wai Lee
7
Maxwell Equations (Linear Medium) - TimeDomain Form (1)
Each parameter depends on 4 independent variables
Faraday’s law
r
Where: E = E x (x, y , z, t )xˆ + E y (x, y, z , t ) yˆ + E z (x, y , z , t )zˆ
r
H = H x ( x, y , z , t )xˆ + H y (x, y, z , t ) yˆ + H z ( x, y, z , t )zˆ
r
J = J x ( x, y , z , t )xˆ + J y (x, y, z, t ) yˆ + J z (x, y, z, t )zˆ
r
r
∇×E = − ∂ B
∂t
r r ∂ r
∇×H = J + D
ρ v = ρ v ( x, y , z , t )
∂t
Unit vector in x-direction
r
Modified Ampere’s law
x
component
∇ ⋅ D = ρv
Gauss’s law
r
E – Electric field intensity
No name, but can
∇⋅B = 0
H – Auxiliary magnetic field
be called Gauss’s law
for magnetic field
z
y
x
Constitutive
relations
For linear medium
June 2008
© 2006 Fabian Kung Wai Lee
D – Electric flux
B – Magnetic field intensity
J – Current density
ρv – Volume charge density
εo – permittivity of free space
(≅8.85412×10-12)
µo – permeability of free space
(4π×10-7)
εr – relative permittivity
µr – relative permeability
8
4
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:
∇ = ∂ xˆ + ∂ yˆ + ∂ zˆ
•
(
∂x
∂y
∂z
)
For instance consider the 1st and 3rd Maxwell Equations:
•
Curl
xˆ
~
∂
∇ × E = ∂x
yˆ
zˆ
∂
∂y
∂
∂z
Ex
Ey
Ez
= − ∂∂t
∂E
∂E
∂E
∂E
∂E
∂E
=  ∂yz − ∂zy  xˆ +  ∂zx − ∂xz  yˆ +  ∂xy − ∂yx  zˆ


 


Gradient
(Bx x + B y yˆ + Bz zˆ )
)
)
∇F = ∂F x + ∂F yˆ + ∂F z
∂x
∂E
ρ
~ ∂E
∂E
∇ ⋅ E = ∂xx + ∂yy + ∂zz =
ε
Divergence
June 2008
∂y
∂z
To truly understands this subject, and also
RF/Microwave circuit design, one needs
to have a strong grasp of Electromagnetism (EM).
Read references [1], [3] or any good book on EM.
© 2006 Fabian Kung Wai Lee
9
Extra: Wave Function and Phasor (1)
f (ω t o − β z o ) = f (k )
A general function describing propagating
wave in +z direction
When time increases by ∆t,
f (ωt − β z )
to
we see that we must
increase all position by ∆z to
maintain the shape. In
essence the waveform
travels in +z direction.
z
zo
f (ω (t o + ∆ t ) − β ( z o + ∆ z ))
= f (k + ω ∆ t − β ∆ z )
∆z
to+∆t
For a wave function
in –z direction:
f (ωt + βz )
June 2008
z
zo+∆z
Direction of
travel
Time t
These 2 terms
must cancel off
i.e. ∆z positive
We see that the shape moves by within
a period of ∆t, thus phase velocity vp:
ωto − β zo = ω (to + ∆t ) − β ( z o + ∆z )
⇒ ω∆t = β ∆z
⇒ vp =
© 2006 Fabian Kung Wai Lee
∆z
∆t
=
ω
β
10
5
Extra: Wave Function and Phasor (2)
•
An example:
v( z , t ) = Vo cos(2πft − β z )
f = 1.0MHz , β = 1
A sinusoidal wave
v(z,t)
z
Phase Velocity:
June 2008
v p = ωβ =
2πf
wavelength
β
λ = 2π
β
© 2006 Fabian Kung Wai Lee
11
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:
Euler’s formula
e j α = cos α + j sin α
j=
{
cos(ωt m βz ) = Re e jωt e m jβz
}
Using Euler’s formula
−1
cos(ωt m β z )
e m jβ z
v ( z , t ) = Vo cos(2πft − β z )
Convention: small letter for time-domain
form, capital letter for phasor.
June 2008
© 2006 Fabian Kung Wai Lee
More compact form
V ( z ) = Vo e − jβz
Phasor for v(z,t)
12
6
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:
•
Propogating function
r+
E (x, y , z , t ) = e x ( x, y )cos(ωt − β z )xˆ + e y ( x, y )cos(ωt − β z ) yˆ + e z ( x, y )cos(ωt − βz )zˆ
r
= e x xˆ + e y yˆ + e z zˆ cos(ωt − β z ) = Re Eo+ e − jβ z e jωt
Pattern function
(
{
)
}
(x, y dependent)
Eo
•
•
The phasor is given by:
r+
Ε (x, y, z ) = e x (x, y )e − jβ z xˆ + e y (x, y )e − jβz yˆ + e z (x, y )e − jβz zˆ
r
Finally if we substitute the phasor form Eo+ e − jβz e jωt for E, H, J and ρ into
time-domain Maxwell’s Equations, we would obtain the Maxwell’s
Equations in time-harmonic form.
June 2008
© 2006 Fabian Kung Wai Lee
13
Maxwell Equations (Linear Medium) - TimeHarmonic 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
Each parameter depends on 3 independent variables
time-harmonic form.
r
∂ → jω
Where: E = E x (x, y , z )xˆ + E y (x, y , z ) yˆ + E z (x, y , z )zˆ
∂t
r
r
(1.2a)
∇ × E = − jωB
r r
r
∇ × H = J + jωD (1.2b)
r
∇ ⋅ D = ρv
(1.2c)
r
(1.2d)
∇⋅B = 0
r
H = H x (x, y, z )xˆ + H y (x, y, z ) yˆ + H z (x, y , z )zˆ
r
J = J x (x, y , z )xˆ + J y (x, y , z )yˆ + J z (x, y, z )zˆ
ρ v = ρ v (x, y , z ) E – Electric field intensity
Constitutive
relations
For linear medium
June 2008
© 2006 Fabian Kung Wai Lee
H – Auxiliary magnetic field
D – Electric flux
B – Magnetic field intensity
J – Current density
ρv- Volume charge density
εo – permittivity of free space
(≅8.85412×10-12)
µo – permeability of free space
(4π×10-7)
εr – relative permittivity
µr – relative permeability
14
7
ELF
100
Hz
VLF
10
kHz
LF
100
kHz
MF
(MW)
1
MHz
L
1
GHz
HF
(SW)
3
MHz
S
2
GHz
June 2008
Microwaves
FM
AM
Electromagnetic Spectrum
VHF
30
MHz
C
4
GHz
UHF SHF
300
MHz
X
8
GHz
1
GHz
30
GHz
K
Ku
12
GHz
EHF
18
GHz
300
GHz
Ka
27
GHz
IR
mm
40
GHz
300
GHz
© 2006 Fabian Kung Wai Lee
15
A Little Perspective…
~
~
∇ × E = −µ ∂∂t H
~ ~
~
∇ × B = µJ + µε ∂∂t E
~ ρ
∇⋅ E =
ε
~
∇⋅ B = 0
+
~
∂ρ
∇ ⋅ J = − ∂t
• Voltage/Potential
• Current
• Inductance
• Capacitance
• Resistance
• Conductance
• Kirchoff’s Voltage Law
(KVL)
• Kirchoff’s Current Law
(KCL)
Electronics &
Microelectronics
Information
Computer
& Telecommunication
Conservation of charge
Quantum Mechanics
/Physics
June 2008
© 2006 Fabian Kung Wai Lee
Chemistry
16
8
2.0 Introduction –
Transmission Line
Concepts
June 2008
© 2006 Fabian Kung Wai Lee
17
Definition of Electrical Interconnect
•
•
Interconnect - metallic conductors that is used to transport electrical
energy from one point of a circuit to another.
y
Example:
Conductor
x
Axial
Direction
z
Transverse
Plane
Interconnect
•
Thus cables, wires, conductive tracks on printed circuit board
(PCB), sockets, packaging, metallic tubes etc. are all examples of
interconnect.
Socket
Ribbon cable
1. Usually contains 2 or more
conductors, to form a closed
circuit.
2. Conductors assumed to be
perfect electric conductors (PEC)
June 2008
PCB Traces
Waveguide
Coaxial cable
© 2006 Fabian Kung Wai Lee
18
9
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.
•
•
•
Electric charge
y
Vs
x
z
+
RL
-
EM field is static or quasistatic.
June 2008
Static EM field changes
uniformly, i.e. when field at one
point increases, field at the
other locations also increases.
Again we need to
stress that typically
the values of RLCG
are very small, at
low frequency their
effect can be simply
ignored.
R
L
C
G
© 2006 Fabian Kung Wai Lee
19
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.
Long interconnect:
When there is an
appreciable delay
between input and
output
June 2008
L
Vs
+
RL
Without the metal conductors, the EM
waves will disperse, i.e. radiated out
into space.
© 2006 Fabian Kung Wai Lee
20
10
Long Interconnect (2)
•
y
A simple animation…
Positive charge
H field
E field
x
Axial
Direction
z
Transverse
Plane
IL
VL
+
Vs
RL
-
t=to t=to
+ ∆t
June 2008
t=to t=to
+ 2∆t + 3∆t
© 2006 Fabian Kung Wai Lee
t=to
+ n∆t
21
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.
L
• 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:
Electric field (E)
t
T
Magnetic field (H)
1/T = frequency
June 2008
© 2006 Fabian Kung Wai Lee
22
11
Long Interconnect (4)
y
L
x
z
By analyzing Maxwell’s
Equations or Wave
Equations (Appendix 1)
r
E + = ex ( x, y )e − jβz xˆ + e y ( x, y )e − jβz yˆ + ez ( x, y )e − jβz zˆ
r
= (et ( x, y ) + e z ( x, y )zˆ ) e − jβz
Electric field (E)
Propagating EM fields
Magnetic field (H)
r+
− jβz ˆ
− jβz ˆ
− jβz ˆ
H = hx ( x, y )e
x + h y ( x, y )e
y + h z ( x, y )e
z
r
= ht ( x, y ) + hz ( x, y )zˆ e − jβz
Snapshot of EM fields
at a certain instant in time
on the transverse plane 23
June 2008
© 2006 Fabian Kung Wai Lee
(
)
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.
It(z,t)
Loop 1
b
Vt(z,t)
r
I t (z , t ) =
r
∫ H ⋅ dl
a
Loop 1
br r
Vt ( z , t ) = − ∫ E ⋅ dl
a
June 2008
© 2006 Fabian Kung Wai Lee
24
12
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
Line integration
path of H field
See discussion in Appendix 1:
Advanced Concepts – Field Theory
Solutions for more information.
June 2008
Line integration
path of E field
Transverse
plane
Measuring current
© 2006 Fabian Kung Wai Lee
25
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.
FR4 dielectric
100Ω SMD resistor
y
x
Copper trace
50Ω resistive
voltage source
connected here
June 2008
z
GND plane
© 2006 Fabian Kung Wai Lee
26
13
Demonstration - Electromagnetic Field
Propagation in Interconnect (2)
z
x
PCB dielectric:
FR4, εr = 4.4,
σ= 5.
Thickness
=1.0mm.
y
Magnitude
of Ez in
dielectric
0.75mm
Intensity
Scale
19.2mm
0.5mm
0.8mm
+
Filename: tline1_XZplane.avi
June 2008
Vo = 3.0V
tr = 50ps
tHIGH = 100ps
Vs
-
Show simulation using CST Microwave
Studio too
© 2006 Fabian Kung Wai Lee
27
Demonstration - Electromagnetic Field
Propagation in Interconnect (3)
z
Probe of signal generator
x
y
Vo = 3.0V
tr = 50ps
tHIGH = 100ps
Magnitude
of Ez in
YZ plane
Volts
Filename: tline1_YZplane.avi
June 2008
© 2006 Fabian Kung Wai Lee
28
14
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
29
Typical Transmission Line
Configurations
Coaxial line
Microstrip line
Stripline
Dielectric
Conductor
Shielded microstrip line
Two-wire line
These conductors
are physically
connected somewhere
in the circuit
Parallel plate line
June 2008
Co-planar line
© 2006 Fabian Kung Wai Lee
Slot line
30
15
Some Multi-conductor Transmission
Line Configurations
Dielectric
Conductor
June 2008
© 2006 Fabian Kung Wai Lee
31
Typical Waveguide Configurations
Rectangular waveguide
Circular waveguide
Optical Fiber
Dielectric waveguide
June 2008
© 2006 Fabian Kung Wai Lee
32
16
Examples of Microstrip and Co-planar
Lines
Microstrip
June 2008
Co-planar
© 2006 Fabian Kung Wai Lee
33
Long or Short Interconnect? The
Wavelength Rule-of-Thumb
•
•
•
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
We call this the 5% rule. Less conservative
estimate will use 1/10=0.10 (the 10% Rule)
Interconnect
v p = fλ
wavelength
λ
f
Phase velocity or
propagation velocity
June 2008
frequency
© 2006 Fabian Kung Wai Lee
(1.1)
f
λ
34
17
Demonstration – Long Interconnect
z
x
5.8GHz, 3.0V
y
19.2mm
Magnitude
of Ez in
YZ plane
Each frame is
displayed at
25psec interval
Let us assume the EM
wave travels at speed of
light, C=2.998×108, then
wavelength ≅ 52.0 mm
λ=C
f
19.2 mm is greater than
5% of 52 mm
+50mA
Current
profile
-50mA
Filename: tline1_YZplane_5_8GHz.avi
June 2008
© 2006 Fabian Kung Wai Lee
35
Demonstration – Short Interconnect
z
x
0.4GHz, 3.0V
y
19.2mm
Let us assume the EM
wave travels at speed of
light, C=2.998×108, then
wavelength ≅ 750.0 mm
Magnitude
of Ez in
YZ plane
• At any instant
in time the
current profile
is almost uniform
along the axial
+50mA
direction.
• Interconnect Current
can be considered
profile
lumped.
-50mA
Filename: tline1_YZplane_0_4GHz.avi
June 2008
© 2006 Fabian Kung Wai Lee
36
18
3.0 Propagation Modes
June 2008
© 2006 Fabian Kung Wai Lee
37
Transverse E and H Field Patterns
Field patterns lie in the Transverse
Plane.
E fields
H fields
Transverse
plane
y
x
z
June 2008
© 2006 Fabian Kung Wai Lee
38
19
Non-transverse E and H Field Patterns
Field patterns does not lie in
the Transverse Plane.
E fields
Field contains
z-component
H fields
Field contains
z-component
June 2008
© 2006 Fabian Kung Wai Lee
39
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.
y
– TEM mode - where Ez and Hz are 0.
– Mix mode, any mixture of the above.
z
r
r
H ± = ht ( x, y ) + hz ( x, y )zˆ e m jβz
(
)
r
r
E ± = (et ( x, y ) + e z ( x, y )zˆ ) e m jβz
•
•
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.
June 2008
© 2006 Fabian Kung Wai Lee
40
20
Propagation Modes (2)
E
E
H
H
y
x
TM mode
z
TEM mode
E
E
H
H
Mix modes
TE mode
June 2008
© 2006 Fabian Kung Wai Lee
41
Examples of Field Patterns or Modes
TEM or quasi-TEM mode
E field
H field
June 2008
© 2006 Fabian Kung Wai Lee
42
21
Appendix 1
Advanced Concepts – Field
Theory Solutions for
Transmission Lines
June 2008
© 2006 Fabian Kung Wai Lee
43
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.
y
Maxwell Equations
r
r
∇ × E = − j ωµ H
r
r
r
∇ × H = J + j ωε E
Boundary
r ρ
conditions
∇⋅E =
ε
r
For instance tangential E field
∇⋅H = 0
+
z
Wave Equations
+
component on PEC must be zero,
continuity of E and H field components
across different dielectric material, etc.
June 2008
© 2006 Fabian Kung Wai Lee
r
r
∇ 2 E + ko2 E = 0
r
r
∇ 2H + ko2H = 0
ko = ω
εµ
(A.1)
In free space
44
22
Extra: Deriving the Hemholtz Wave
Equations From Maxwell Equations
r
r
Performing curl operation on Faraday’s Law ∇ × E = − jωµH :
r
r
r
r
∇ × ∇ × E = ∇ ∇ ⋅ E − ∇ 2 E = − jωµ ∇ × H
r
r
r
⇒ ∇ 2 E + ω 2 µεE = jωµJ + ∇ ερ
(
) (
)
(
)
()
These are the sources for the E field
Note: use the well-known
vector calculus
identity
r
r
r
(
)
∇ × ∇ × A = ∇ ∇ ⋅ A − ∇2 A
2
∇ =
∂2
∂x 2
+
∂2
∂y 2
+
∂2
∂z 2
In free space there is no electric charge and current:
r
r
∇ 2 E + ω 2 µεE = 0
Note: This derivation is
valid for time-harmonic case
under linear medium only.
See more advanced text for
general wave equation. For
Example:
Similar procedure can be used to obtain
r
r
r
∇ 2 H + ω 2 µεH = −∇ × J
Or in free space
C. A. Balanis, “Advanced engineering
Electromagnetics”, John-Wiley, 1989.
r
r
∇ H + ω 2 µεH = 0
2
June 2008
© 2006 Fabian Kung Wai Lee
45
Obtaining the Expressions for E and H
(1)
•
Assuming an ordinary differential equation (ODE) system as shown:
ODE
d 2y
dx 2
+ k 2 y = 0 , y = f (x ) , x ∈ [0, b]
y (0) = C1 and y (b ) = C2
•
•
•
The Domain
Boundary conditions
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
dy
d 2y
guess a functional form for y(x) as follows:
= β e βx , 2 = β 2 e βx
y(x ) = e βx
dx
dx
Substituting this into the ODE:
2
2 βx
(β
)
+k e
=0
⇒ β 2 + k2 = 0
⇒ β = ± jk
•
where j = − 1
That the trial-and-error method
works is attributed to the
Uniqueness Theorem for
linear ODE.
Since this is a 2nd order ODE, we need to introduce 2 unknown constants, A and
B, and a general solution is:
jkx
− jkx
y (x ) = Ae
June 2008
+ Be
© 2006 Fabian Kung Wai Lee
(1)
46
23
Obtaining the Expressions for E and H
(2)
•
To find A and B, we need to use the boundary conditions.
y (0 ) = C1 ⇒ A + B = C1
(2a)
y (b ) = C2 ⇒ Ae jkb + Be − jkb = C2 (2b)
•
Solving (2a) and (2b) for A and B:
(C1 − B )e jkb + Be− jkb = C2
C e jkb −C
(3a)
⇒ B = 21j sin (kb2)
− jkb
Ce
−C
A = C1 − B = 21 j sin (kb )2 (3b)
•
So the unique solution is:
 C e − jkb −C 
 C e jkb −C 
y (x ) =  21 j sin (kb )2 e jkx +  21j sin (kb2) e − jkx




June 2008
(q.e.d.)
© 2006 Fabian Kung Wai Lee
47
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:
(∇ 2 + ko2 )E = 0
2
2
2
2
2
2
 2

 2

 2

⇒  ∂ 2 + ∂ 2 + ∂ 2 + k o2  E x xˆ +  ∂ 2 + ∂ 2 + ∂ 2 + ko2  E y yˆ +  ∂ 2 + ∂ 2 + ∂ 2 + k o2  E z zˆ = 0
∂y
∂z
∂y
∂z
∂y
∂z
 ∂x

 ∂x

 ∂x

•
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:
 ∂2
2
∂2
∂2
 2 + 2 + 2 + ko  E x = 0
∂y
∂z
 ∂x

•
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:
E x ( x, y, z ) = e x ( x, y )e − jβz or e x ( x, y )e jβz
June 2008
A function of x©and
y
2006 Fabian Kung Wai Lee
The exponent e !!!
48
24
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.
r
E + = ex ( x, y )e − jβz xˆ + e y ( x, y )e − jβz yˆ + ez ( x, y )e − jβz zˆ
(A.1a)
r
= (et ( x, y ) + ez (x, y )zˆ ) e − jβz
•
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 ∇ 2 + ko2 H = 0 will yield similar expression for
H field.
•
•
(
)
v
H + = hx (x, y )e − jβz xˆ + h y (x, y )e − jβz yˆ + hz (x, y )e − jβz zˆ
v
= ht (x, y ) + hz ( x, y )zˆ e − jβz
(
)
June 2008
(A.1b)
© 2006 Fabian Kung Wai Lee
49
E and H fields Expressions (1)
•
Thus the propagating EM fields guided by Tline can be written as:
Transverse component
r
E + = ex ( x, y )e − jβz xˆ + e y ( x, y )e − jβz yˆ + ez ( x, y )e − jβz zˆ
r
= (et ( x, y ) + ez (x, y )zˆ ) e − jβz
Axial component
r
H + = hx ( x, y )e − jβz xˆ + h y ( x, y )e − jβz yˆ + h z ( x, y )e − jβz zˆ
r
superscript indicates
propagation direction
= ht ( x, y ) + hz (x, y )zˆ e − jβz
(
)
r
E − = e x (x, y )e + jβz xˆ + e y ( x, y )e + jβz yˆ − e z (x, y )e + jβz zˆ
r
= (et (x, y ) − e z (x, y )zˆ ) e + jβ z
r
H − = −hx ( x, y )e + jβz xˆ − h y ( x, y )e + jβz yˆ + hz ( x, y )e + jβz zˆ
r
= − ht ( x, y ) + hz ( x, y )zˆ e + jβz
(
June 2008
)
© 2006 Fabian Kung Wai Lee
(A.2a)
EM fields
Propagating
In +z direction
(A.2b)
(A.3a)
EM fields
Propagating
In -z direction
(A.3b)
50
25
E and H fields Expressions (2)
•
We can convert the phasor form into time-domain form, for instance for
E field propagating in +z direction:
r
 +

E + (x, y, z , t ) = Re  E (x, y, z )e jωt 
(A.4)


ˆ
= e x (x, y )cos(ωt − β z )x + e y ( x, y )cos(ωt − β z ) y + ez ( x, y )cos(ωt − β z )zˆ
•
Where
r
E + (x, y , z , t ) = E x + (x, y, z , t )xˆ + E y + (x, y, z , t ) yˆ + E z + (x, y, z , t )zˆ
E x + ( x, y, z , t ) = e x (x, y )cos(ωt − βz )
(A.5a)
Unit vector
E y + (x, y, z , t ) = e y ( x, y )cos(ωt − βz )
(A.5b)
E z + (x, y , z , t ) = e z (x, y )cos(ωt − β z )
June 2008
© 2006 Fabian Kung Wai Lee
51
E and H fields Expressions (3)
•
Usually one only solves for E field, the corresponding H field phasor
r
r
can be obtained from:
∇ × E = − jωµH
r
r
1
(A.6)
⇒H =
∇× E
jωµ
•
The power carried by the EM fields is given by Poynting Theorem:
S
Positive Z direction…
P=
r
r * r 1
r r
1
Re ∫∫ E + × H + ⋅ ds = Re ∫∫ et × ht * ⋅ ds
2
2
( )
S
Sz
Negative Z direction…
Positive value means
that power is carried
along the propagation
direction.
June 2008
P=
r
r * r 1
r r
1
Re ∫∫ E − × H − ⋅ ds = Re ∫∫ − et × ht * ⋅ − ds
2
2
( )
S
=
( )
Sz
r r
1
Re ∫∫ et × ht * ⋅ ds
2
S
© 2006 Fabian
Kung Wai Lee
z
52
26
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).
r
– So that ∇ ⋅ E = 0 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.
r
∇ ⋅ E+ = 0
r
⇒ ∇ t ⋅ et − jβe z = 0
r
∇ ⋅ E− = 0
r
⇒ ∇ t ⋅ et + jβ (− ez ) = 0
June 2008
For +ve direction
For -ve direction
© 2006 Fabian Kung Wai Lee
53
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.
vp = ω
β
June 2008
© 2006 Fabian Kung Wai Lee
(A.7)
54
27
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.
Ex(x,y,z,to)
λ=
Wavelength λ
λ=
vp
f
ω
β
ω
2π
= 2βπ
(A.8)
y
z
June 2008
© 2006 Fabian Kung Wai Lee
55
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:
+
E =E +E
•
−
+
H =H +H
−
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:
Conductors
H
E
June 2008
© 2006 Fabian Kung Wai Lee
56
28
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.
•
r
E + = ex ( x, y )e − jβz xˆ + e y ( x, y )e − jβz yˆ + ez ( x, y )e − jβz zˆ
r
= (et ( x, y ) + ez ( x, y )zˆ ) e − jβz
r+
H = hx ( x, y )e − jβz xˆ + h y ( x, y )e − jβz yˆ + h z ( x, y )e − jβz zˆ
r
= ht ( x, y ) + hz ( x, y )zˆ e − jβz
(
y
)
z
Maxwell Equations
r
r
∇ × E = − j ωµ H
r
r
r
∇ × H = J + j ωε E
r ρ
∇⋅E =
ε
r
∇⋅H = 0
June 2008
Wave Equations
+
Boundary
conditions
+
r
r
∇ 2 E + ko2E = 0
r
r
∇ 2 H + ko2 H = 0
ko = ω
εµ
© 2006 Fabian Kung Wai Lee
57
Field Solution (2)
•
The procedure outlined here follows those from Pozar [2]. Assume the
Tline or waveguide dielectric region is source free. From Maxwell’s
Equations:
r
r
r
r
∇ × E = − jωµH
∇ × H = jωεE
(A.9)
r
J =0
~
~
~
∇ × B = µJ + jωµεE
•
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:
∂ez
∂y
+ jβe y = − jωµ hx (A.10a)
∂e
− jβe x − ∂xz = − jωµh y
∂e y
∂x
∂e
− ∂yx = − jωµ hz
June 2008
∂hz
∂y
+ jβh y = jωεe x
(A.10d)
(A.10b)
− jβ hx − ∂∂hxz = jωεe y (A.10e)
(A.10c)
∂h y
∂x
∂h
− ∂yx = jωεe z
© 2006 Fabian Kung Wai Lee
(A.10f)
58
29
Field Solution (3)
•
From (A.10a)-(A.10f), we can express ex, ey, hx, hy in terms of ez and
hz:
These equations
∂e
∂h
j
hx = 2  ωε ∂yz − β ∂xz 
(A.11a)
describe the x,y components

kc 
of general EM wave
∂e
∂h
−j
propagation in a
h y = 2  ωε ∂xz + β ∂yz 
(A.11b)


k
waveguiding system.
c
The unknowns are
∂e
∂h
−j
e x = 2  β ∂xz + ωµ ∂yz 
ez(x,y) and hz(x,y), called the
(A.11c)

kc 
Potential in the literature.
 − β ∂ez + ωµ ∂hz 
∂y
∂x 

(A.11d)
k c2 = ko 2 − β 2 , ko = ω µε = 2λπ
(A.11e)
ey =
See the book by Collin [1],
Chapter 3 for alternative
derivation
June 2008
j
kc2
© 2006 Fabian Kung Wai Lee
59
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:
r
r
r
r
E ± = et e m jβz
H ± = ± ht + hz zˆ e m jβz
(
•
)
From wave equation for H field: Transverse Laplacian
r
r
∇ 2 H + ko 2 H = 0
k o = ω εµ
From (A.11e)
∇ t 2 hz + kc 2 hz = 0
r
r
∇ t 2 ht + k c 2 h t = 0
k c 2 = k o2 − β 2
June 2008
Note
operator
2

 ∇ t 2 + ∂ + k o2
∂z 2

∇ 2 = ∇ t2 − β 2
((
)
 r
 ht + h z zˆ e − j β z = 0

Using the fact that
)
(
)
∂ 2 e m jβz = − β 2 e m jβ z
∂z 2
Only these are needed. The other
transverse field components can
be derived from hz
© 2006 Fabian Kung Wai Lee
60
30
TE Mode Summary (2)
•
Setting ez = 0 in (A.11a)-(A.11d):
hx =
− jβ ∂hz
k 2 ∂x
hy =
− jωµ ∂hz
k 2 ∂y
ey =
c
ex =
c
(A.12a)
jωµ ∂hz
k 2 ∂x
c
c
•
− jβ ∂hz
k 2 ∂y
These equations plus the previous wave equation for hz enable us to
find the complete field pattern for TE mode.
∇ t 2 hz + k c2 hz = 0
k c 2 = ko 2 − β 2
+ boundary conditions
for E and H fields
(A.12b)
From previous slide
June 2008
© 2006 Fabian Kung Wai Lee
61
TE Mode Summary (3)
•
From (A.12a) and (A.12b), we can show that:
r
∇t × et = − jωµ hz zˆ ≠ 0
r  ∂e y ∂e x 
jωµ  ∂ 2 h ∂ 2 h 
∇ t × et = 
−
zˆ =
+

 zˆ
∂y 
kc2  ∂x 2 ∂y 2 
 ∂x
jωµ
=
− kc2 hz zˆ = − jωµ hz zˆ
k c2
•
(
r
∇ t × ht = 0
)
Therefore we cannot define a unique voltage by (but we can define a
unique current):
r
C r
Vt = − ∫
2
e
C1 t
•
⋅ dl
Also from (A.12a) we can define a wave impedance for the TE mode.
− ey
e
k Z
Z TE = x = o o =
hy
β
hx
June 2008
© 2006 Fabian Kung Wai Lee
(A.13)
62
31
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:
r
r
r
r
E ± = (et ± ez zˆ )e m jβz
H ± = ± ht e m jβz
•
From wave equation for E field:
r
r
∇ 2 E + ko2 E = 0
ko = ω
εµ
2

 ∇ t 2 + ∂
+ k o2
2
∂Z

∇ t 2ez + kc 2ez = 0
r
r
∇ t 2 et + k c 2 et = 0
)
Only these are needed. The other
transverse field components
can be derived from ez
k c 2 = k o2 − β 2
June 2008
(
 r
 (e t + e z zˆ )e − j β z = 0

© 2006 Fabian Kung Wai Lee
63
TM Mode Summary (2)
•
Setting hz = 0 in (A.11a)-(A.11d):
hx =
ex =
jωε ∂ez
kc2 ∂y
hy =
c
− jβ ∂ez
k 2 ∂x
ey =
(A.14a)
− jβ ∂ez
k 2 ∂y
c
c
•
− jωε ∂ez
k 2 ∂x
These equations plus the previous wave equation for ez enable us to
find the complete field pattern for TM mode.
∇ t 2e z + kc2e z = 0
2
2
k c = ko − β
2
+ boundary conditions
for E and H fields
(A.14b)
From previous slide
June 2008
© 2006 Fabian Kung Wai Lee
64
32
TM Mode Summary (3)
•
Similarly from (A.14a) and (A.14b), we can show that
r
∇ t × ht = jωε ez zˆ ≠ 0
r
∇ t × et = 0
•
We cannot define a unique current by (but we can define a unique
voltage):
r r
I t = ∫ ht ⋅ dl
C
•
Also from (A.14a) we can define a wave impedance for the TM mode.
β
e
−e
ZTM = hx = ωε = h y
y
x
June 2008
© 2006 Fabian Kung Wai Lee
(A.15)
65
TEM Mode (1)
•
TEM mode is particularly important, characterized by ez = hz = 0. For
+ve propagating waves:
r
r
E ± = et e m jβz
•
r
r
H ± = ± ht e m jβz
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:
β = ko = ω µε
•
(A.16a)
(A.16b)
Applying Hemholtz Wave Equation to E field:
(∇ 2 + ko2 )ert e− jβz =  ∇t2 + ∂∂z22 zˆ + ko2 ert e− jβz = 0
0
 2
r
r
⇒ ∇ t2 (et e − jβ z )+  ∂ e − jβ z + ko2e − jβz et = 0
 ∂z 2

r
⇒ ∇ t2et = 0
June 2008
(A.17a)
© 2006 Fabian Kung Wai Lee
66
33
TEM Mode (2)
•
The same can be shown for ht:
r
r
∇ h = −∇ × J
2
t t
(A.17b)
•
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
r
potential Φ:
et = −∇ t Φ ( x, y ) Transverse potential
(A.18)
⇒ ∇ t 2 Φ ( x, y ) = 0
•
Also note that:
r
∇ t × et = ∇ t × (− ∇ t Φ ) = 0
See [1], Chapter 3
for alternative
derivation
(A.19)
Using an important identity in vector calculus
Where F is arbitrary function
∇ × (∇F ) = 0
of position, i.e. F = F(x,y,z).
June 2008
© 2006 Fabian Kung Wai Lee
67
Alternative View (TEM Mode)
•
Alternatively from (A.10c), and knowing that hz = 0 in TEM mode:
∂e y ∂e x
−
= − jωµhz = 0
∂x
∂y
Divergence in 2D (in XY plane)
⇒
•
∂e y
∂x
x
−
y
∂e x
=0= ∂
∂y
∂x
∂
∂y
ex
ey
z
r
0 = ∇ t × et = 0
0
From the well known Vector Calculus identity ∇ × (∇F ) = 0 , we then
postulate the existence of a scalar function Φ(x,y) where
r
et = −∇t Φ ( x, y )
•
From (A.10f) we can also show that (in free space):
∂h y
∂x
June 2008
−
r
∂hx
= jωεe z = ∇ t × ht = 0
∂y
© 2006 Fabian Kung Wai Lee
68
34
TEM Mode (3)
•
Normally we would find et from (A.17a), then we derive ht from et :
r
r
∇ 2 ht = 0 , ∇ × ht = 0
In free space
r
r
−1
H=
∇× E
jωµ
r
⇒ ht =
µ
ε
−1
(e y xˆ − ex yˆ )
r
r
⇒ ht = Z1 ( zˆ × et )
o
•
 ∂E y
 ∂E y ∂E x  
∂E
 zˆ
H t = −1 −
xˆ + x yˆ + 
−
jωµ
∂z
∂z
∂y  
 ∂x


∂
e



y ∂e x − jβ z 
e
= −1  jβE y xˆ − jβE x yˆ + 
−
zˆ 
jωµ
∂y 
 ∂x


r
∇ t × et = 0
Exercise: see if you can
derive this equation
Zo = Intrinsic impedance of free space
ZTEM = Wave impedance of TEM
mode
(A.20a)
−e
e
µ
= hx = h y = ZTEM (A.20b)
ε
y
x
Zo =
An important observation is that under TEM mode the transverse field
components et and ht fulfill similar equations as in electrostatic.
June 2008
© 2006 Fabian Kung Wai Lee
69
Voltage and Current under TEM Mode
•
r
r
Due to ∇ t × et = 0 and ∇ t × ht = 0 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.
It(z,t)
Vt(z,t)
Loop 1
r r
∫ H ⋅ dl
I t (z , t ) =
See the more detailed
version of this note or
see references [1] & [2].
June 2008
b
Loop 1
br r
Vt ( z , t ) = − ∫ E ⋅ dl
a
a
© 2006 Fabian Kung Wai Lee
70
35
Extra: Independence of Vt and It from
Integration Path under TEM Mode
r
∇ t × et = 0
C2
Using Stoke’s
Theorem
S
L
r r
r r
⇒ ∫ et ⋅ dl + ∫ et ⋅ dl = 0
L1
L2
r r
r r
⇒ ∫ et ⋅ dl = − ∫ et ⋅ dl
Area S
L1
r
dl
r
ds
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.
r r
r r
⇒ ∫∫ ∇ t × et ⋅ ds = ∫ et ⋅ dl = 0
C1
L2
r r
r r
⇒ − ∫ et ⋅ dl = − ∫ et ⋅ dl = Vt
− L1
L2
Loop L
Similar proof can be carried
out for It, using the
r loop as
shown and ∇ t × ht = 0
Loop L
r
It =
+
r
∫ ht ⋅ dl
C1 or C 2
C2
C1
June 2008
© 2006 Fabian Kung Wai Lee
71
TEM Mode Summary (1)
•
•
For TEM mode, hz= ez= 0.
To find the EM fields for TEM mode:
2
– Solve ∇ t Φ( x, y ) = 0 with boundary conditions for the transverse
potential.
r
r
– Find E from
Et ± = [− ∇t Φ ( x, y )]e m jβz = et e m jβz
– Find H from
r
1
Ht± =
( zˆ × ert )e ± jβz
Zo
•
We could characterize the Tline in TEM mode, by EM fields:
r
r
E ± = et e m jβz
June 2008
r
r
H ± = ± ht e m jβz
© 2006 Fabian Kung Wai Lee
β = ko = ω µε
72
36
TEM Mode Summary (2)
•
Or through auxiliary circuit theory quantities:
V ± = Vt e m jβz
•
I ± = ± I t e m jβz
The power carried by the EM wave along the Tline is given by
Poynting Theorem:
( ) 2
(A.21)
2
S
⇒ P = 1 Re(Z c II * ) = 1 Re(Z c −1VV * )
2
2
r r
P = 1 Re ∫∫ E × H * ⋅ ds = 1 Re VI *
•
See extra note
by F.Kung for
the proof
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.
β = ko = ω µε
June 2008
© 2006 Fabian Kung Wai Lee
73
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
r
r
(because ∇ t × et ≠ 0 or ∇ t × ht ≠ 0 ).
r
For instance in TE mode: ∇t × et = − jωµhz
C2 r
Thus Vt = − ∫C1 et ⋅ dl 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!
r r
Furthermore for non-TEM modes: P = 1 Re ∫∫ E × H * ⋅ ds ≠ 1 Re Vt I t *
2
(
S
)
2
Thus we cannot characterize a Tline supporting non-TEM modes
using auxiliary quantities such as Vt and It.
June 2008
© 2006 Fabian Kung Wai Lee
74
37
Non-TEM Modes and Vt, It (2)
•
As another example consider the TM mode in microstrip line:
r
jβ
jβ
et = −
∇ e =−
2 t z
kc
kc2
•
 ∂ xˆ + ∂ yˆ e ( x, y )
∂y  z
 ∂x
y

r
jβ  ∂e z
∂e
Vt = − ∫ et ⋅ dl =
dx + ∫ z dy 
∫
∂y

kc2  C ∂x
C
C

Using path C as shown in figure:
Under quasi-TEM condition, when
ez→ 0, kc also → 0, then Vt will be
a non-zero value.
Vt =
Y=H
C
Y=0
x
H
jβ  ∂e z  jβ
∫ dy = [ez (x, H ) − ez (x,0)] = 0
kc2  0 ∂y  kc2
0 because of boundary condition
•
In general this is true for arbitrary Tline and waveguide cross
section. If we choose
r integration path other than C, we still obtain
Vt= 0 due to ∇ t × et = 0 in TM mode.
June 2008
© 2006 Fabian Kung Wai Lee
75
Cut-off Frequency for TE/TM Mode
β = ko 2 − kc 2
•
Because
•
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 nonpropagating.
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.
•
June 2008
for TE and TM modes:
© 2006 Fabian Kung Wai Lee
76
38
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: V p = ω
β
V p = ωβ = 1
µε
•
For TEM mode:
•
For TE & TM mode:
•
Thus we observe that TEM mode is intrinsically non-dispersive, while
TE and TM mode are dispersive.
June 2008
(A.22)
1
1
V p = ωβ =
=
ω 2 µε − kc2
ko 2 − kc 2
(A.23)
© 2006 Fabian Kung Wai Lee
77
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.
June 2008
© 2006 Fabian Kung Wai Lee
78
39
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
y
Conducting plates
d
0
W
x
Propagation along z axis
z
June 2008
© 2006 Fabian Kung Wai Lee
79
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.
June 2008
© 2006 Fabian Kung Wai Lee
80
40
Example A1 – Solution for TEM Mode
(1)
TEM mode Solution:
∇ t 2Φ (x, y ) = 0 for 0 ≤ x ≤ W , 0 ≤ y ≤ d
Φ ( x, 0 ) = 0
Boundary conditions
Φ ( x, d ) = Vo
Solution for the transverse Laplace PDE:
2
2
∇ t2 Φ = ∂ Φ + ∂ Φ = 0
∂x 2
∂y 2
2
⇒ ∂ Φ ≅0
⇒ Φ (x, y ) = A + By
∂y 2
since
Φ( x, y ) = Φ( y )
General Solution
Φ ( x ,0 ) = A = 0 ⇒ A = 0
V
Φ( x, d ) = Bd = Vo ⇒ B = o
d
Thus
V
Φ ( x, y ) = o y
d
June 2008
Unique solution
© 2006 Fabian Kung Wai Lee
81
Example A1 – Solution for TEM Mode
(2)
Computing the E and H fields:
V
V
et = −∇t Φ = − ∂ xˆ + ∂ yˆ  o y  = − o yˆ
∂y  d 
d
 ∂x
V
E t = − o e − jβ z yˆ
d
1 ∇ × E = 1  zˆ ×  − Vo e − jβ z yˆ   = ε Vo e − jβz xˆ
H t = j−ωµ

t
  d
µ d
µ 


ε
y
C2
Computing the transverse voltage and current:
r
V
d
Vt = − ∫ E t ⋅ dl = − ∫  − o e − jβz yˆ  ⋅ dyyˆ = Vo e − jβz
0 d

C
C1
x
1
r
It =
W  ε Vo − jβz 
xˆ  ⋅ dxxˆ =
 µ d e


∫ H t ⋅ dl = ∫0
C2
June 2008
© 2006 Fabian Kung Wai Lee
ε Vo We − jβz
µ d
82
41
Example A1 – Solution for TEM Mode
(3)
Computing the power flow (power carried by the EM wave guided by the wave
-guide):
y

r
*
P = 12 Re  ∫∫ E t × H t ⋅ ds 
S

∫ (−
Vo
d
= 12 Re  ∫
 0
∫(
e − jβ z
d
0
W
d V
o
0 d
= 12 Re  ∫
 0
d V
o
d
[
e − jβz yˆ ×
)(
ε Vo
µ d
ε Vo
µ d
W
ε VoW
µ d
)
e + jβz xˆ ⋅ dxdyzˆ 

)
e + jβz dxdy 

dy e − jβz  ∫

 0
= 12 Re Vo e − jβz ⋅
June 2008
x
)(
= 12 Re  ∫
 0
W
S
ε Vo
µ d
]
dx e + jβz 


dy
ds=dxdy
2
e jβz = 12 Re Vt I t* = 12  VoµW 
 εd 
[ ]
© 2006 Fabian Kung Wai Lee
dx
83
Example A1 – Solution for TEM Mode
(4)
Phase velocity vp for TEM mode:
vp = ω = ω = 1
β ω µε
µε
The phase velocity is equal to speed-of-light in the dielectric.
June 2008
© 2006 Fabian Kung Wai Lee
84
42
Example A1 – Solution for TM Mode (1)
(∇t 2 + kc2 )ez (x, y) = 0 for 0 ≤ x ≤ W , 0 ≤ y ≤ d
kc2 = ko2 − β 2
e z ( x,0) = e z ( x, d ) = 0
Boundary conditions
Solution for ez:
(∇t 2 + kc2 )ez ≅ ∂∂ye
2
z
2
since e z (x, y ) = e z ( y )
+ kc2e z = 0
⇒ e z (x, y ) = A sin (kc y ) + B cos(kc y )
General solution
Applying boundary conditions:
Thus:
e z (x,0 ) = A ⋅ 0 + B = 0 ⇒ B = 0
(d )
e z ( x, y ) = An sin nπ y
e z (x, d ) = A sin (kc d ) = 0
⇒ A ≠ 0 and kc d = nπ , n = 1,2,3L
⇒ k = nπ
c
(d )
E z ( x, y ) = An sin nπ y e − jβz
or
d
June 2008
© 2006 Fabian Kung Wai Lee
85
Example A1 – Solution for TM Mode (2)
Computing the transverse EM fields using (A.20a):
(
( d ))
r − jβ ∂e z
− jβ ∂e z
jβ ∂
et =
xˆ +
yˆ = −
An sin nπ y yˆ
k c2 ∂x
k c2 ∂y
k c2 ∂y
(d ) ( )
r
jβAn
⇒ et = −
cos nπ y yˆ
nπ
d
( ) ( )
jβ An
or E t = −
cos nπ y e − jβz yˆ
nπ
d
d
 jωε ∂e

z xˆ − jωε ∂e z yˆ e − jβz
Ht =
 k 2 ∂y
kc2 ∂x 
 c
jk A
= o n cos nπ y e − jβ z xˆ
d
µ nπ
ε d
( )
June 2008
Since n is an arbitrary integer,
the TM mode is usually called
the TMn mode.
( )
© 2006 Fabian Kung Wai Lee
86
43
Example A1 – Solution for TM Mode (3)
We can now determine β knowing kc:
β n = k o2 − kc2 = ω 2 µε − ( ndπ )
2
Since the TM mode can only propagate if β is real, and the smallest value for β
Is 0, then when β=0:
2
ω 2 µε − nπ = 0
d
⇒ ω = nπ = 2πf
d µε
( )
⇒ f =
n
2d µε
When n = 1, this represent the cut-off frequency for TM mode.
f cutoff _ TM =
June 2008
1
2d µε
© 2006 Fabian Kung Wai Lee
87
Example A1 – Solution for TM Mode (4)
For arbitrary n, phase velocity vp for TMn mode:
vp = ω =
β
ω
( )
2
ω 2 µε − nπ
d
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.
June 2008
© 2006 Fabian Kung Wai Lee
88
44
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
89
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.
June 2008
© 2006 Fabian Kung Wai Lee
90
45
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
91
Quasi TEM Mode (2)
E
H
r
∇ t × et ≅ 0
Vt and It
r
or ∇ t × ht ≅ 0 Yes
r
∇ t × et ≠ 0
r
or ∇ t × ht ≠ 0
Vt and It
No
E
H
Non-TEM mode
E
r
∇ t × et = 0
r
or ∇ t × ht = 0
Vt and It
Yes
June 2008
Quasi-TEM mode
H
TEM mode
© 2006 Fabian Kung Wai Lee
92
46
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:
β die = ω µε oε r
β air = ω µε o
•
EM fields in air will travel faster than in the dielectric.
β air < β die
⇒v
•
For TEM mode
p (air )= ω > v p (die ) = ω
β air
β die
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:
Air
E
y
Dielectric
June 2008
x
E
x(air )e − jβ air z = x (die )e − jβ die z
⇒
E x (air )
E x (die )
− jβ die z
=e
= e − j (β die − β air )z
−
j
z
β
air
e
© 2006 Fabian Kung Wai Lee
93
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.
June 2008
© 2006 Fabian Kung Wai Lee
94
47
Example A2 – Minimum Frequency for
Quasi-TEM Mode in Microstrip Line
•
Estimate the low frequency limit for microstrip line.
H = 1.6mm
Here we replace >> sign
with the requirement that
wavelength > 20H. You can
use larger limit, as this is
basically a rule of thumb.
C ≈ 3.0×108
λ = C/f > 20H = 32.0mm
f < C/0.032 = 9.375GHz
fcritical = 9.375GHz
Thus beyond fcritical quasi-TEM approximation cannot be applied.
The propagation mode beyond fcritical will be TM. A more conservative limit would be to use 30H or 40H.
June 2008
© 2006 Fabian Kung Wai Lee
95
Summary for TEM, Quasi-TEM, TE and
TM Modes
TEM:
Ez = Hz = 0
r
r
E ± = et e m jβ z
r
r
±
m jβ z
H = ± ht e
Can defined unique
Vt and It.
Quasi-TEM:
Ez ≈0 , Hz ≈ 0
TE:
Ez = 0, Hz ≠ 0
r
r
H ± = ± ht + hz zˆ e m jβz
r
r
E ± = et e m jβz
Cannot defined
unique It.
H = ± ht e
r
r
E ± = (et ± ez zˆ )e m jβz
Physical Tline cannot
be modeled by
equivalent electrical
circuit.
Physical Tline cannot
be modeled by
equivalent
electrical circuit.
Phase velocity.
1
Vp = ω ≅
β
µε eff
Phase velocity.
1
vp = ω =
Phase velocity.
1
vp = ω =
No cut-off frequency.
Cut-off frequency.
Cut-off frequency.
r
r
E ± ≅ et e m jβ z
r
r
H ± ≅ ± ht e m jβ z
Can defined unique
Vt and It.
Physical Tline can be Physical Tline can be
Modeled by equivalent Modeled by equivalent
Electrical circuit.
Electrical circuit.
Phase velocity.
V p = ωβ = 1
µε
No cut-off frequency.
(
)
β
ω 2 µε − k c2
fc =
Non-dispersive
June 2008
Non-dispersive
Dispersive
© 2006 Fabian Kung Wai Lee
kc
2π µε
TM:
Ez ≠ 0, Hz = 0
r m jβz
r±
Cannot defined
unique Vt.
β
ω 2 µε − k c2
fc =
kc
2π µε
Dispersive
96
48
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
97
Why Vt and It is so Important ? (2)
Tline
XXX.09
Filter
Amplifier
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.
Antenna equivalent
circuit
Microstrip
antenna
June 2008
© 2006 Fabian Kung Wai Lee
98
49
Examples of Circuit Analysis* Based
Microwave/RF CAD Software
Agilent’s Advance Design System
Applied Wave Research’s
Microwave Office
Ansoft’s Desinger
*The software shown here also have
numerical EM solver capability, from 2D, 2.5D
to full 3D.
June 2008
© 2006 Fabian Kung Wai Lee
99
4.0 – Transmission Line
Characteristics and
Electrical Circuit Model
June 2008
© 2006 Fabian Kung Wai Lee
100
50
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
101
Distributed Electrical Circuit Model for
Transmission Line (2)
z
5.8GHz, 3.0V
x
y
Magnitude
of Ez in
YZ plane
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. June 2008
It
Vt
© 2006 Fabian Kung Wai Lee
102
51
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.
I
L ∆z
∆z
V1
C ∆z
V2
Electric
field linkage
Magnetic field
linkage
H
E
June 2008
© 2006 Fabian Kung Wai Lee
103
Distributed Parameters (2)
•
•
•
•
When the conductor has small conductive loss a series resistance R∆ z
can 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.
Conductor loss
Under lossy condition, R, L and G
are usually function of frequency,
hence the Tline is dispersive.
L ∆z
I
V1
C ∆z
R ∆z
G ∆z
V2
Dielectric loss
June 2008
© 2006 Fabian Kung Wai Lee
104
52
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.
It
∆z→0
Vt
Distributed
RLCG circuit
It
Vt
June 2008
© 2006 Fabian Kung Wai Lee
105
Finding the RLCG Parameters
L=
µ
It
R=
r 2
H
2 ∫∫∫ t
V
C=
This indicates the
volume enclosing
the conductors
r 2
1
σ cδ s I t
dv
2
∫ Ht
dl
2
ωσ c µ
• These formulas are
derived from energy
consideration.
• Note that conductor loss
results in R, while
dielectric loss results
in G.
June 2008
Vt
E
2 ∫∫∫ t
dv
V
(4.1a)
r 2
ωε "
G=
Et dv
∫∫∫
2
Vt V
C1 + C 2
δ s = skin depth =
r 2
ε'
(4.1b) ε ' = ε ε
r o
Skin depth
and G depend
on frequency
• 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.
ε " = ε r ε o tan δ
Loss tangent of
the dielectric
σ c = conductivity of metalic object
1m
C1
Conductor 1
© 2006 Fabian Kung Wai Lee
Conductor 2
S
C2
106
53
Finding RLCG Parameters From
Energy Consideration
The instantaneous power absorbed by an inductor L is:
i(t)
Pind (t ) = v (t )i (t )
v(t)
L
io
Assuming i(t) increases from 0 at t = 0 to Io at t = to, total energy stored
by inductor is:
t
t
t
i(t)
Eind = ∫ o Pind (τ )dτ = ∫ o v(τ )i(τ )dτ = ∫ o L di (τ )i (τ )dτ
0
0
0
dτ
I
⇒ Eind = ∫ o Li ⋅ di = 1 LI o2
t
0
2
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
r
field is given by: E = µ
H dxdydz
H
∫∫∫
2
Io
V
E
= EH
Both energy are the same, hence: ind
r
0
to
2
µ
⇒ 1 LI o2 = ∫∫∫ H dxdydz
H
2
2
⇒L=
June 2008
© 2006 Fabian Kung Wai Lee
µ
∫∫∫
I o2 V
V
r2
H dxdydz
107
Multi-Conductor Transmission Line
Symbol and Circuit
Long interconnect:
l > 0.1λ
TEM or
quasi-TEM mode
Parameters:
Per unit length
R, L, C, G matrices.
Usually as a function
Frequency.
 L11

 L12
L12 

L22 
 R11

 R21
R12   G11 G12 

 
R22  G21 G22 
June 2008
 C11

− C12
Distributed
RLCG circuit
L11
C12
R11
L22
L12
C1G
− C12 

C22 
Electrical Symbol
C2G
R22
C1G = C11 − C12
C2G = C22 − C12
© 2006 Fabian Kung Wai Lee
108
54
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-domain
∂V
∂I
= − RI − L
∂z
∂t
(4.2a)
∂I
∂V
= −GV − C
∂z
∂t
Fourier
Transforms
In time-harmonic form
Inverse Fourier
Transforms
∂V
= −(R + jωL )I = − ZI
∂z
(4.2b)
∂I
= −(G + jωC )V = −YV
∂z
I
Distributed
RLCG circuit
V
June 2008
© 2006 Fabian Kung Wai Lee
109
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:
Wave travelling in
-z direction
Wave travelling in
+z direction
+ −γz
− γz
I ( z ) = I o+ e −γz + I o− eγz (4.3a) V ( z ) = Vo e + Vo e
(4.3b)
Attenuation factor
Phase factor
γ = α (ω ) + jβ (ω ) =
(R + jωL )(G + jωC )
(4.3c)
Propagation coefficient
June 2008
© 2006 Fabian Kung Wai Lee
110
55
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.
The boundary of the tline circuit
Zs
Vs
June 2008
It
ZL
Vt
© 2006 Fabian Kung Wai Lee
111
Signal Propagation on Transmission
Line
•
Considering sinusoidal sources, the expression for V(z) and I(z) can be
written in time domain as:
+
+ jφ
+
+ jθ
Vo = Vo e
+
Io = Io e
+
v(z , t ) = Vo+ cos(ωt − βz + φ+ )e −αz + Vo− cos(ωt + βz + φ− )eαz (4.4a)
i ( z , t ) = I o+ cos(ωt − β z + θ + )e −αz + I o− cos(ωt + β z + θ − )eαz
•
(4.4b)
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.
June 2008
© 2006 Fabian Kung Wai Lee
112
56
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.
•
From Telegraphic Equations
∂V
= − ZI
∂z
⇒ −γVo+ e −γz = − ZI o+ e −γz
⇒ Vo+ = Z I o+
γ
V + e −γz Vo+ Z
R + jωL
Zc = o
=
= =
+
−
γ
z
+
γ
G
+ jωC
Io e
Io
Or
June 2008
V − eγz
R + jω L
Zc = − o
=
−
γ
z
G
+ jω C
Io e
(4.5)
A function of frequency
© 2006 Fabian Kung Wai Lee
113
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:
V (z , t ) = Vo+ cos(ωt − βz + φ+ )e −αz + Vo− cos(ωt + β z + φ− )eαz
Compare
general function describing propagating
f (ωt − β z ) Awave
in +z direction
•
And recognizing that both v(z,t) and i(z,t) are propagating waves, the
phase velocity is given by:
vp =
June 2008
ω
β
© 2006 Fabian Kung Wai Lee
(4.6)
114
57
Attenuation (α
α)
•
•
The attenuation factor α decreases the amplitude of the voltage and
current wave along the Tline.
For +ve traveling wave:
Vo+ cos(ωt − βz + φ + )e −αz
•
y
z
For -ve traveling wave:
Z=0
y
Vo− cos(ωt + βz + φ − )eαz
z
Z=0
June 2008
© 2006 Fabian Kung Wai Lee
115
Dispersion (1)
Since γ = γ (ω ) = α (ω ) + jβ (ω )
vp =
ω
β (ω )
Video
Cause of
dispersion
• We observe that the propagation
velocity is a function of the
Components
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. Low dispersion
Transmission Line vin
vin
June 2008
vout
vout
© 2006 Fabian Kung Wai Lee
116
58
Dispersion (2)
•
•
Video
Dispersion causes
distortion of the signal
propagating through a
transmission line.
This is particularly evident
in a long line.
High dispersion
Transmission Line
vin
vin
vout
vout
June 2008
© 2006 Fabian Kung Wai Lee
117
Dispersion (3)
•
At the output, the sinusoidal components overlap at the wrong ‘timing’,
causing distortion of the pulse.
1
1
0.5
0.5
Vout( i⋅ ∆t , 1)
Vin( i⋅ ∆t , 1)
Vin( i⋅ ∆t , 3)
Vout( i⋅ ∆t , 3)
0
0
Vout( i⋅ ∆t , 5)
Vin( i⋅ ∆t , 5)
0.5
0.5
1
1
0
10
20
30
40
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.
1.5
1
Vint( i⋅ ∆t ) 0.5
0
0.5
vout
vin
June 2008
10
20
i
30
40
10
20
30
40
30
40
i
1.5
1
Voutt ( i⋅ ∆t ) 0.5
0
0.5
0
0
0
10
20
i
© 2006 Fabian Kung Wai Lee
118
59
The Implications
Tline supporting TEM or quasi-TEM mode
June 2008
© 2006 Fabian Kung Wai Lee
119
The Lossless Transmission Line
•
•
When the tline is lossless, R= 0 and G = 0.
We have:
γ = jβ = jω LC
•
•
Zc =
L
C
vp =
1
=
LC
1
µε
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).
June 2008
© 2006 Fabian Kung Wai Lee
120
60
Appendix 2
Advanced Concepts –
Distributed RLCG Model for
Transmission Line and
Telegraphic Equations
June 2008
© 2006 Fabian Kung Wai Lee
121
Distributed Parameters Model (1)
•
For TEM or quasi-TEM mode propagation along z direction:
Conductor
y
Loop C
s3
s2
V1
C
V2
r
r ∆z
− ∫ E ⋅ dl = − ∫∫ ∇ × E ⋅ ds
C
r
= − ∫ E ⋅ dl −
s1
S
r
r
r
∫ E ⋅ dl − ∫ E ⋅ dl − ∫ E ⋅ dl
s2
s3
s4


r
r
r


⇒ − ∫∫ ∇ × E ⋅ ds = − ∫ E ⋅ dl −  − ∫ E ⋅ dl 


S
s2
 − s4

r
⇒ − ∫∫ − jµωH ⋅ ds = V1 − V2
(
SJune 2008
)
s1
z
Stoke’s
Theorem
Flux linkage
(Definition for
Inductance)
s4
r


− V1 + V2 = − jω  µ ∫∫ H ⋅ ds 


 S
 When ∆z
is small
⇒ V2 − V1 = − jω (L∆zI )
as compared
This means the
relation between
V1 and V2 is as if
an inductor is
between them
L is the
inductance
per meter
© 2006 Fabian Kung Wai Lee
V1
to wavelength
r
r
H + ≅ ht (x, y ) + hz (x , y )zˆ
L∆z
Can be represented
in circuit as:
V2
122
61
Distributed Parameters Model (2)
S
This means the
relation between
I1 and I2 is as if
a capacitor is
between them
W
r
∂ρ
1
∫∫ E ⋅ d S = ε ∫∫∫ ∂t dW
S
W
r
⇒ ∂ ∫∫ E ⋅ d S = 1 ∫∫∫ − ∇ ⋅ J dW
∂t
ε
S
W
r
r r
r r
⇒ ε ∂ ∫∫ E ⋅ d S = − ∫∫ J ⋅ dS = − ∫∫ J ⋅ dS −
⇒ ∂
∂t
Volume W
C2
Using Divergence Theorem
W
r
ρ
Surface S ⇒ ∫∫ E ⋅ d S = ∫∫∫ ε dW
A1
I1
ρ
∫∫∫ ∇ ⋅ EdW = ∫∫∫ ε dW
W
A2
C1
r
I2
∆z
See the following
slide for more
proof.
I1
∂t
I2
⇒ε ∂
∂t
V
∂t
S
A1
r
∫∫ E ⋅ d S = − I1 + I 2
r
A2
When ∆z
is small
as compared
to wavelength
S
⇒ ∂ ((C∆z )V ) = (C∆z ) ∂V = I 2 − I1
∂t
∂t
C∆z
C∆z ∂V
S
r
∫∫ J ⋅ dS
Can be represented in circuit as
•
C is the per unit length capacitance between the 2 conductors of the Tline.
June 2008
© 2006 Fabian Kung Wai Lee
123
Distributed Parameters Model (3)
Volume W
∆z
r
Consider the ratio:
I2
ε ∫∫ E ⋅ ds
Q
S
=
Br
Br
− ∫ E ⋅ dl − ∫ A E ⋅dl
A
B
Surface S
I1
C1
r1
(x,y,z)
r2
A
C2
Hence the ratio becomes:
r
ε ∫∫ E ⋅ ds
For static or quasi-static condition, the E field
is given by:
r
σ s1( x, y , z ) rˆ1
σ s 2 (x, y , z ) rˆ2
E=
∫∫
surface on C1
⋅ ds +
r1
∫∫
surface on C 2
4πε
⋅
r2
ds =


f s1 ( x, y, z ) rˆ1
f s 2 ( x, y, z ) rˆ2 
= Q
⋅ ds +
⋅ ds
∫∫
∫∫
πε
r
πε
r
4
4

1
2 
surface on C 2
surface on C1

Q is the total charge on
conductors C1 or C2, σs is the surface charge density, while fs
is the normalized surface charge density with respect to Q.
Q
S
=
Br

− ∫ E ⋅ dl
f s1 ( x, y, z ) rˆ1
f s 2 ( x, y, z ) rˆ2 
B

A
−Q ∫ A
⋅ ds +
⋅ ds ⋅dl
∫∫
∫∫
4πε
r1
4πε
r2 
surface on C
surface
on
C
1
2


1
=
=C

f s1 (x, y, z ) rˆ1
f s 2 ( x, y , z ) rˆ2 
B
− ∫A
⋅ ds +
⋅ ds ⋅dl
∫∫
∫∫
4πε
r1
4πε
r2 
surface on C
surface on C 2

1

June 2008
4πε
Potential difference between A and B
© 2006 Fabian Kung Wai Lee
C does not depend on
the total charge on the
conductors, we called
r
this constant the
‘Capacitance’. ε ∫∫ E ⋅ ds = CV
S
124
62
Distributed Parameters (4)
•
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:
∆z
L ∆z
I
C ∆z
V1
•
V2
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.
June 2008
© 2006 Fabian Kung Wai Lee
125
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
I
*Under lossy condition, R, L and G
are usually function of frequency,
hence the Tline is dispersive.
June 2008
V1
© 2006 Fabian Kung Wai Lee
C ∆z
R ∆z
G ∆z
V2
126
63
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.
*We should also include
Loss current density
r r
r
r
r
r
hysterisis loss in
∇ × H = J + jωε r ε o E ⇒ ∇ × H = σ d E + jωε r ε o E
ferromagnetic material.
r
r
σd 
⇒ ∇ × H = jωε r ε o 1 +
E
j
ωε
rε o 

r
r
r
r
∇ × H = jω ε ' − jε " E
⇒ ∇ × H = jωε r ε o (1 − j tan δ )E
ε ' = ε r ε o ε " = ε r ε o tan δ
σd
tan δ =
ωε r ε o
(
This is called Loss Tangent
June 2008
)
Note that σd is a function of frequency
© 2006 Fabian Kung Wai Lee
127
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.
June 2008
© 2006 Fabian Kung Wai Lee
128
64
Derivation of Telegraphic Equations (1)
•
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:
Use Kirchoff’s Voltage Law (KVL):
I1
L ∆z
C ∆z
V1
jωL∆z
I1=I
I2
V1
V2
V2
V2 = V1 − jωL∆zI
V −V
⇒ 2 1 = − jωLI
∆z
Observing that:
V −V
lim ∆z →0  2 1  = ∂V
 ∆z  ∂z
June 2008
I2
⇒ ∂V = − jωLI
∂z
© 2006 Fabian Kung Wai Lee
129
Derivation of Telegraphic Equations (2)
•
Now considering the current on both ends, and using Kirchoff’s Current
Law (KCL):
Using KCL here:
I1
V1
L ∆z
C ∆z
I2
I1
V2
jωL ∆z
I1
I2
(jωC ∆z)-1
V2=V
I1 − ( jωC∆z )V = I 2
I −I
⇒ 2 1 = − jωCV
∆z
Again observing that:
I 2 − I1
∂V
= −C 2
∆z
∂t
June 2008
∂I = − jωCV
∂z
© 2006 Fabian Kung Wai Lee
Lossless
Telegraphic
Equations
130
65
Relationship Between Field Solutions
and Telegraphic Equations
Interconnect
‘Short’ interconnect
‘Long’ interconnect
Lumped RLCG circuit
Waveguide
Maxwell’s Equations
Wave Equations
TEM
TE
TM
Hybrid
E & H wave
KVL &
KCL
Transmission line
Vt, It and
Distributed
RLCG
circuit
Quasi-TEM
Only for stripline
structures
June 2008
Telegraphic
Equations
Zc, vp, attenuation factor,
dispersion, power handling.
© 2006 Fabian Kung Wai Lee
131
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):
ε = ε '− jε "
•
Use the expressions for Et and Ht as derived in Example A1.
L=
R=
June 2008
µd
W
C=
H/m
2
σ cδ sW
Ω/m
G=
© 2006 Fabian Kung Wai Lee
ε 'W
d
F/m
σ W
ωε "
C = d Ω −1/m
ε'
d
132
66
5.0 - Transmission Line
Synthesis On Printed Circuit
Board (PCB) And Related
Structures
June 2008
© 2006 Fabian Kung Wai Lee
133
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.
June 2008
© 2006 Fabian Kung Wai Lee
134
67
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
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135
The Substrate or Laminate for Stripline
Technology
Copper thickness
Standard
material consist
of epoxy with glass
fiber reinforcement
Dielectric thickness
Dielectric
Copper (Usually gold plated
to protect against oxidation)
• 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.
June 2008
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136
68
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
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137
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 Not suitable for coupled
structures
line structures
Need through holes to
connect to ground
Need through holes
to connect to ground
June 2008
© 2006 Fabian Kung Wai Lee
No through hole required
to connect to ground
138
69
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 quasiTEM. 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 Conformal
Mapping 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
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139
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.
June 2008
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70
Typical Iterative Flow for Transmission
Line Design
Start
Draw Tline physical
cross section
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 β
at the frequency of
interest
No
Design
equations
for Tline
Criteria met?
Yes
End
June 2008
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141
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.
•
•
263.177
µo , εo
W
300
Zc ( x, 1)
Zc ( x, 2)
200
εr=1
Zc
Zc ( x, 3)
Zc ( x, 4)
d
µ,ε
εr=2
εr=3
Zc ( x, 5) 100
Zc ( x, 6)
εr=6
15.313
0
0
0.1
June 2008
© 2006 Fabian Kung Wai Lee
2
4
x
W/d
6
8
8
142
71
Design Equations for Microstrip Line
Microstrip Line (see reference [3], Chapter 8)
Effective dielectric
constant (See Appendix 3)
ε eff = 1 +

ε r −1 
2
1
1 +
10d

1+

w
µo , εo
W





(5.1a)
t
µ,ε
d
−1
0.172 
377  w
 w
 + 1.98 
 (5.1b)
Zc =
ε eff  d
d 

1
vp =
(5.1c)
µε eff ε o
Only valid when quasi-TEM approximation and very low loss condition applies.
June 2008
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143
Design Equations for Stripline
Stripline (see reference [3], Chapter 8):
Z
K (k )
Zc ≅ o ⋅
4 K 1− k 2 




π
K ( x) =
2
∫
0
dφ
1 − x 2 sin 2 φ
d
d

 πw  
k = cosh  
 4d  

Complete elliptic integral Z o =
of the 2nd kind
vp =
µ,ε
(5.2a)
−1
(5.2b)
w
µ
ε
1
µε
(5.2c)
Only valid when TEM, quasi-TEM approximation and very low loss condition applies.
June 2008
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72
Design Equations for Co-planar Line
Co-planar Line ([3], assume d is large compare to s):
a
w
Effective dielectric
constant (See Appendix 3)
ε eff =
Zc ≅
εr +1
(5.3a)
2
Zo
π ε eff
 a
w
 for 0 < < 0.173
ln 2

w
a


 
w 
−1
πZ o   1 +
a 
Zc ≅
ln 2

4 ε eff   1 − w 
 
a 
Zo =
µo
εo
d
vp =
w
for 0.173 < < 1
a
(5.3b)
s
1
µε eff ε o
(5.3c)
Only valid when quasi-TEM approximation and very low loss condition applies.
June 2008
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145
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.
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73
Dispersive Property of Microstrip and
Co-planar Line (2)
εeff
Microstrip line
Stripline
Region where (5.1a)
and (5.1b) applies.
εeff
εr
εr
1
1
f
f
fcritical
Limit for quasi-TEM approximation, see Example A1 and on v =
p
how to estimate fcritical
For microstrip line, the EM
field is partly in the air and
dielectric. Hence the effective
dielectric εeff constant is between
those of air and dielectric.
June 2008
1
µε eff ε o
Note:
Beyond fcritical the concept of characteristic impedance
becomes meaningless.
© 2006 Fabian Kung Wai Lee
147
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
June 2008
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74
Example 5.1 - Microstrip Line Design
•
(a) Design a 50Ω
Ω microstrip line, given that d = 1.57 mm and
dielectric constant = 4.6 (Here it means find w).
160
•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
150
140
130
120
110
100
Zo S
4
i
90
80
70
3.75
60
εe S
i
3.5
50
3
40
50Ω
3.25
0
1
2
3
4
5
30
20
6
W/dS i
June 2008
© 2006 Fabian Kung Wai Lee
0
1
1.8
2
3
4
5
S
i
W/d
149
Microstrip Line Design Example Cont...
•
•
(b) If the length of the Tline is 6.5 cm, find the propagation delay.
(c) Using the l < 0.05λ
λ 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:
vp =
1
ε o µ o ⋅ 3.51
= 1.601 × 108 ms −1 ⇒ t delay =
0.065
≅ 406 ps
vp
To be represented as lumped, the wavelength λ must be > 20 x Length:
λ > 20 ⋅ l = 1.30 m
⇒
vp
f
⇒ f <
June 2008
> 1.30
vp
1.30
= 123.2 MHz
f lumped = 123.2 MHz
© 2006 Fabian Kung Wai Lee
150
75
Microstrip Line Design Example Cont...
•
(d) When the low loss microstrip line is considered short, derived
its equivalent LC network.
L
1
1
L = Z c 2C = 313.27 nH/m
×
=
C
C
LC
1
1
C=
=
= 124.6pF/m
Z c v p 50 × 1.601 × 108
Zcv p =
C ⋅ 0.065 = 8.096 pF
L ⋅ 0.065 = 20.36nH
For shot interconnect we could model the Tline as:
1 L∆z = 10.18nH
2
C∆z = 8.096 pF
June 2008
L∆z = 20.36nH
10.18nH
or
1 C∆z = 4.048 pF
2
4.048 pF
© 2006 Fabian Kung Wai Lee
151
Microstrip Line Design Example Cont...
•
(e) Finally estimate fcritical , the limit where quasi-TEM
approximation begins to break down.
Again using the criteria that wavelength > 20d for quasi-TEM mode
to propagate:
λ=
vp
vp
> 20d = 0.031 ⇒ f critical <
≅ 5.10GHz
f critical
0.031
We see that to increase fcritical, smaller thickness d
should be used.
June 2008
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152
76
Microstrip Line Design Example Cont...
µo , εo
65.0 mm
2.82 mm
1.57 mm
µ = µo , ε = 4.6εo
Zc = 50Ω
vp = 1.601x108 m/s
tdelay = 406psec
Maximum usable frequency = fcritical = 5.10 GHz.
Short interconnect limit = 123.2 MHz
June 2008
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153
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.
TL1
TL2
TL3
TL4
June 2008
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77
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 twodimensional (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
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155
Example 5.3 - Screen Shot
June 2008
© 2006 Fabian Kung Wai Lee
156
78
Example 5.3 - Stripline Cross Section
•
Draw the cross section of the model and assign material.
FR4 Substrate
“substrate”
Signal conductor
(PEC) “Trace1”
“GND2”
GND planes
(PEC)
0.6mm
“GND1”
8.0mm
• Set drawing units to micron.
• Set drawing size to 10000um for x
and 4000um for y.
• Set grid to dU = dV = 100um.
• Draw model, use direct entry mode
for conducting structures like GND
planes and signal.
• Name signal conductor “Trace1” and
GND planes “GND1” and “GND2”.
• Name FR4 as “substrate”, then group
both “GND1” and “GND2” as one object “GND”.
June 2008
0.036mm
0.3mm
0.4mm
0.3mm
0.036mm
© 2006 Fabian Kung Wai Lee
157
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
1V
0V
0V
0V
Balloon Boundaries
June 2008
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158
79
Example 5.3 - Electrostatics: Setup
Executive Parameters
•
Under ‘Setup Executive Parameters’ tab, select ‘Matrix…’ and proceed
to perform the capacitance matrix setup as shown.
June 2008
© 2006 Fabian Kung Wai Lee
159
Example 5.3 - Electrostatics: Setup
Solver and Solve for Scalar Potential φ
•
Setup the solver and
solve for the approximate
potential solution. Use
the ‘Suggested Values’ if
you are not sure.
June 2008
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160
80
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
y
object.
x
•
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.
r
r
ρ
∇ 2V = ε
∇ 2 A = µJ
•
•
Potential value inside the triangle can be estimated via interpolation.
For 2D problem the PDE can be written as:
r
)
)
ρ
∇ t 2V t =
∇t = ∂ x + ∂ y
∇ t 2 Az = µJ z
ε
∂x
∂y
V t = V t (x , y ) ρ = ρ ( x , y )
June 2008
Az = Az (x, y ) J z = J z (x, y )
© 2006 Fabian Kung Wai Lee
161
Example 5.3 - Finite Element Method (2)
•
2D quasi-static E
r field can then be obtained by:
Et (x, y ) = −∇ tVt (x, y )
•
Similarly magnetic flux intensity H can be obtained from:
r
)
H t (x, y ) = − 1 ∇ t × [Az (x, y )z ]
µ
•
For more information, refer to
– T. Itoh (editor), “Numerical techniques for microwave and milimeterwave passive structures”, John-Wiley & Sons, 1989.
– P. P. Silvester, R. L. Ferrari, “Finite elements for electrical
engineers”, Cambridge University Press, 1990.
– Other newer books.
June 2008
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81
Example 5.3 - Electrostatics: The Triangular
Mesh
June 2008
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163
Example 5.3 - Electrostatics: Plot of E
field Magnitude
June 2008
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164
82
Example 5.3 - Electrostatics: Plot of Voltage
Contour
June 2008
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165
Example 5.3 - Electrostatics: Capacitance
•
The estimated distributed capacitance is then computed using:
r 2
ε'
C=
Et dv
∫∫∫
2
Vt V
•
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.
June 2008
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166
83
Example 5.3 - Magnetostatics: Setup
Boundary Conditions
Solid source +1A
Solid source –1A (total for 2 planes)
Balloon Boundaries
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167
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.
June 2008
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168
84
Example 5.3 - Magnetostatics: Plot of B
field Magnitude
June 2008
© 2006 Fabian Kung Wai Lee
169
Example 5.3 - Magnetostatics: Inductance
•
The estimated distributed capacitance is then computed using:
L=
µ
It
•
2
r 2
H
∫∫∫ t dv
V
L ≈ 2.5093×10-7 H/m or 250.93 nH/m.
June 2008
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85
Example 5.3 - Derivation of Parameters for
Stripline
Zc =
vp =
June 2008
L = 2.5093×10 − 7 = 35.819Ω
C
1.9558×10 −10
1
LC
= 1.427 × 108 ms −1
© 2006 Fabian Kung Wai Lee
171
Example 5.4 – Tline Design Using
Agilent’s AppCAD V3.02
June 2008
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172
86
Example 5.4 - Derivation of LC Parameters
from AppCAD V3.02 Results
Z c = 36.0
v p = 0.466v p (vacumn )
v p (vacumn ) = 2.998 × 108
C=
1 = 1.988 × 10 −10 F / m
Zcv p
L = Z c 2C = 2.577 × 10 − 7 H / m
• 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
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173
Appendix 3
June 2008
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174
87
The Origin of Effective Dielectric Constant
(εεeff)
The approach can be traced to a paper by Bryant and Weiss1.
Assuming low loss and quasi-TEM mode:
vp = 1
Zc = L = 1
C v pC
µo , εo
LC
•
W
µ,ε
d
C
Define
ε eff = C
Then
vp =
With dielectric, C is the capacitance
per meter between the conductors
⇒ Zc =
Without dielectric, C1 is the capacitance
per meter between the conductors
June 2008
c
ε eff
C1
1
= c
ε eff
LC1 ⋅ ε eff
1
Zc =
µo , εo
µo , εo
C1
=
⋅C
1
c ε eff ⋅C1
c
ε eff
Speed of
light in vacumn
1
⋅C1ε eff
C1 is computed via numerical
methods for various W and d
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.
© 2006 Fabian Kung Wai Lee
175
When Quasi-TEM Approximation is not
Valid - Full Wave Analysis
•
•
•
•
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.
June 2008
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88
Full-Wave Analysis
•
•
Full-wave analysis is usually carried out using numerical methods such
as Method of Moments (MoM), Finite Element Methods (FEM), FiniteDifference 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
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177
THE END
June 2008
© 2006 Fabian Kung Wai Lee
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89
Example A1 – Solution for TE Mode
The EM fields for TE mode are shown below:
E t (x, y ) = jk(onZπo)Bn sin ( ndπ y )e − jβz xˆ
d
H t (x, y ) = j(βnπBn) sin ( ndπ y )e − jβz yˆ
d
Since n is an arbitrary integer,
the TE mode is usually called
the TEn mode.
H z (x, y ) = Bn cos( ndπ y )e − jβz
ko = ω µε
Zo =
June 2008
µo
εo
© 2006 Fabian Kung Wai Lee
179
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
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180
90
Exercise Cont…
•
Assuming this transmission line is used in the following system.
Estimate the maximum working distance R.
Omni-directional
Antenna
Parallelplate Tline
Transmission
Tower
Distance = R
Antenna gain G = 10
Receiver
June 2008
© 2006 Fabian Kung Wai Lee
If the Receiver can detect
the data when received power
is 2 µWatt.
181
91