9/21/2019
Electromagnetics:
Electromagnetic Field Theory
Transmission Line Equations
1
Lecture Outline
• Introduction
• Transmission Line Equations
• Transmission Line Wave Equations
Slide 2
2
1
9/21/2019
Introduction
Slide 3
3
Map of Waveguides (LI Media)
Waveguides
• Confines and transports waves.
• Supports higher‐order modes.
“Pipes”
Transmission Lines
• Contains two or more conductors.
• No low‐frequency cutoff.
• Thought of more as a circuit clement
Homogeneous
• Has TEM mode.
• Has TE and TM
modes.
Inhomogeneous
• Supports only
quasi‐(TEM, TE,
& TM) modes.
Metal Shell Pipes
Dielectric Pipes
• Enclosed by metal.
• Does not support TEM mode.
• Has a low frequency cutoff.
Single‐Ended
microstrip
stripline
coplanar
rectangular
circular
Differential
coplanar strips
shielded pair
slotline
• Supports TE and TM modes only
if one axis is uniform.
• Otherwise supports quasi‐TM
and quasi‐TE modes.
no uniform axis
(no TE or TM)
• Confinement along two axes.
• TE & TM modes only supported in
circularly symmetric guides.
dual‐ridge
Inhomogeneous
buried parallel
plate
• Composed of a core and a cladding.
• Symmetric waveguides have no low‐
frequency cutoff.
Channel Waveguides
Homogeneous
• Supports TE and TM modes
coaxial
• Has one or less conductors.
• Usually what is implied by the
label “waveguide.”
uniform axis
(has TE and TM)
optical Fiber photonic crystal
rib
Slab Waveguides
• Confinement only along one axis.
• Supports TE and TM modes.
• Interfaces can support surface waves.
dielectric Slab
large‐area
parallel plate
interface
Slide 4
4
2
9/21/2019
Signals in Transmission Lines: Coax
Slide 5
5
Signals in Transmission Lines: Microstrip
Slide 6
6
3
9/21/2019
Signals in Transmission Lines: Twisted Pair
Slide 7
7
Transmission Line Parameters RLGC
It can be useful to think of transmission lines as being composed of millions of tiny little
circuit elements that are distributed along the length of the line.
In fact, these circuit element are not discrete, but continuous along the length of the
transmission line.
Slide 8
8
4
9/21/2019
RLGC Circuit Model
It is not technically correct to represent a
transmission line with discrete circuit elements
like this.
However, if the size of the circuit z is very
small compared to the wavelength of the signal
on the transmission line, it becomes an
accurate and effective way to model the
transmission line.
z  0
Slide 9
9
L‐Type Equivalent Circuit Model
Distributed Circuit Parameters
There are many possible circuit models for transmission lines,
but most produce the same equations after analysis.
R (/m)
Resistance per unit length.
Arises due to resistivity in the
conductors.
L (H/m)
Inductance per unit length.
Arises due to stored magnetic
energy around the line.
R  z
G (1/m)
Conductance per unit length.
Arises due to conductivity in the
dielectric separating the
conductors.
G
L  z
G  z
C  z
1
R
C (F/m)
Capacitance per unit length.
Arises due to stored electric
energy between the conductors.
z
z  z
Slide 10
10
5
9/21/2019
L‐Type Equivalent Circuit Model
Distributed Circuit Parameters
There are many possible circuit models for transmission lines,
but most produce the same equations after analysis.
R (/m)
Resistance per unit length.
Arises due to resistivity in the
conductors.
L (H/m)
Inductance per unit length.
Arises due to stored magnetic
energy around the line.
R  z
G (1/m)
Conductance per unit length.
Arises due to conductivity in the
dielectric separating the
conductors.
G
L  z
G  z
C  z
1
R
C (F/m)
Capacitance per unit length.
Arises due to stored electric
energy between the conductors.
z
z  z
Slide 11
11
L‐Type Equivalent Circuit Model
Distributed Circuit Parameters
There are many possible circuit models for transmission lines,
but most produce the same equations after analysis.
R (/m)
Resistance per unit length.
Arises due to resistivity in the
conductors.
L (H/m)
Inductance per unit length.
Arises due to stored magnetic
energy around the line.
R  z
G (1/m)
Conductance per unit length.
Arises due to conductivity in the
dielectric separating the
conductors.
G
L  z
G  z
C  z
1
R
C (F/m)
Capacitance per unit length.
Arises due to stored electric
energy between the conductors.
z
z  z
Slide 12
12
6
9/21/2019
L‐Type Equivalent Circuit Model
Distributed Circuit Parameters
There are many possible circuit models for transmission lines,
but most produce the same equations after analysis.
R (/m)
Resistance per unit length.
Arises due to resistivity in the
conductors.
L (H/m)
Inductance per unit length.
Arises due to stored magnetic
energy around the line.
R  z
G (1/m)
Conductance per unit length.
Arises due to conductivity in the
dielectric separating the
conductors.
G
L  z
G  z
C  z
1
R
C (F/m)
Capacitance per unit length.
Arises due to stored electric
energy between the conductors.
z
z  z
Slide 13
13
Relation to Electromagnetic Parameters
,  , 
Every transmission line with
a homogeneous fill has:
LC  
G 

C 
Slide 14
14
7
9/21/2019
Example RLGC Parameters
RG‐59 Coax
CAT5 Twisted Pair
Microstrip
R  36 mΩ m
L  430 nH m
G  10   m
R  176 mΩ m
L  490 nH m
G  2  m
R  150 mΩ m
L  364 nH m
C  69 pF m
C  49 pF m
Z 0  75 
Z 0  100 
G  3  m
C  107 pF m
Z 0  50 
Surprisingly, almost all transmission lines have parameters very close to these same values.
Slide 15
15
Transmission Line
Equations
Slide 16
16
8
9/21/2019
E & H  V and I
Fundamentally, all circuit problems are electromagnetic problems and can be solved as
such.
All two‐conductor transmission lines either support a TEM wave or a wave very closely
approximated as TEM.
An important property of TEM waves is that E is uniquely related to V and H and uniquely
related to E.
 
V   E  d 
L


I   H  d 
L
This reduces analysis of transmission lines to just V and I. This makes analysis much simpler
because these are scalar quantities!
Slide 17
17
Transmission Line Equations
The transmission line equations do for transmission lines the same thing as Maxwell’s curl
equations do for waves.
Maxwell’s Equations


H
  E  
t


E
 H  
t
Transmission Line Equations


V
I
 RI  L
z
t
I
V
 GV  C
z
t
Like Maxwell’s equations, the transmission line equations are rarely directly useful.
Instead, we will derive all of the useful equations from them.
Slide 18
18
9
9/21/2019
Derivation of First TL Equation (1 of 2)
+
V  z, t 
R  z
L  z
2
3
I  z, t 
+
C  z
G  z
1
4
V  z  z , t 
‐
‐
z
z  z
Apply Kirchoff’s voltage law (KVL) to the outer loop of the equivalent circuit:
I  z , t 
 V  z  z, t   0
 V  z , t   I  z , t  Rz  Lz

  
t
2
4
1
3
Slide 19
19
Derivation of First TL Equation (2 of 2)
We rearrange the equation by bringing all of the voltage terms to the left‐hand side of the equation, bringing
all of the current terms to the right‐hand side of the equation, and then dividing both sides by z.
V  z , t   I  z , t  Rz  Lz
I  z , t 
t
 V  z  z , t   0


V  z  z , t   V  z , t 
I  z , t 
 RI  z , t   L
z
t
In the limit as z  0, the expression on the left‐hand side becomes a derivative with respect to z.

V  z , t 
I  z , t 
 RI  z , t   L
z
t
Slide 20
20
10
9/21/2019
Derivation of Second TL Equation (1 of 2)
R  z
L  z
2
1
I  z  z , t 
I  z, t 
+
4
3
V  z, t 
+
V  z  z , t 
C  z
G  z
‐
‐
z
z  z
Apply Kirchoff’s current law (KCL) to the main node the equivalent circuit:
V  z  z, t 
0
I  z , t   I  z  z , t   GzV  z  z , t   C z
   
t 

1
2
3
4
Slide 21
21
Derivation of Second TL Equation (2 of 2)
We rearrange the equation by bringing all of the current terms to the left‐hand side of the equation, bringing
all of the voltage terms to the right‐hand side of the equation, and then dividing both sides by z.
I  z , t   I  z  z , t   G zV  z  z , t   C z
V  z  z , t 
0
t


I  z  z , t   I  z , t 
V  z  z , t 
 GV  z  z , t   C
z
t
In the limit as z  0, the expression on the left‐hand side becomes a derivative with respect to z.

I  z , t 
V  z , t 
 GV  z , t   C
z
t
Slide 22
22
11
9/21/2019
Transmission Line
Wave Equations
Slide 23
23
Starting Point – Telegrapher Equations
Start with the transmission line equations derived in the previous section.

V  z , t 
I  z , t 
 RI  z , t   L
z
t

I  z , t 
V  z , t 
 GV  z , t   C
z
t
time‐domain
For time‐harmonic (i.e. frequency‐domain) analysis, Fourier transform the equations above.

dV  z 
  R  j L  I  z 
dz

dI  z 
  G  jC  V  z 
dz
frequency‐domain
Note: The derivative d/dz became an ordinary derivative
because z is the only independent variable left.
These last equations are commonly referred to as the
telegrapher equations.
Slide 24
24
12
9/21/2019
Wave Equation in Terms of V(z)

dV  z 
  R  j L  I  z 
dz
Eq. (1)

dI  z 
  G  jC  V  z 
dz
Eq. (2)
To derive a wave equation in terms of V(z), first differentiate Eq. (1) with respect to z.

d 2V  z 
dI  z 
  R  j L 
dz 2
dz
Eq. (3)
Second, substitute Eq. (2) into the right‐hand side of Eq. (3) to eliminate I(z) from the equation.
d 2V  z 
   R  j L  G  jC  V  z 
dz 2

Last, rearrange the terms to arrive at the final form of the wave equation.
d 2V  z 
  R  j L  G  jC  V  z   0
dz 2
Slide 25
25
Wave Equation in Terms of I(z)

dV  z 
  R  j L  I  z 
dz
Eq. (1)

dI  z 
  G  jC  V  z 
dz
Eq. (2)
To derive a wave equation in terms of I(z), first differentiate Eq. (2) with respect to z.
d 2I  z
dV  z 
  G  jC 
dz 2
dz

Eq. (3)
Second, substitute Eq. (1) into the right‐hand side of Eq. (3) to eliminate V(z).

d 2I  z
   G  jC  R  j L  I  z 
dz 2
Last, rearrange the terms to arrive at the final form of the wave equation.
d 2I  z
  G  jC  R  j L  I  z   0
dz 2
Slide 26
26
13
9/21/2019
Propagation Constant, 
In the wave equations, there is the common term 𝐺
𝑗𝜔𝐶 𝑅
𝑗𝜔𝐿 .
Define the propagation constant  to be
    j 
 G  jC  R  j L 
Given this definition, the transmission line equations are written as
d 2V  z 
  2V  z   0
2
dz
d 2I  z
  2I  z   0
2
dz
Slide 27
27
Solution to the Wave Equations
If the wave equations are handed off to a mathematician, they will return with the following
solutions.
d 2V  z 
  2V  z   0
dz 2
 V  z   V0 e  z  V0 e z
d 2I  z
  2I  z   0
dz 2
 I  z   I 0 e  z  I 0 e z
Forward wave
Backward wave
Both V(z) and I(z) have the same differential equation so it makes sense they
have the same solution.
Slide 28
28
14