ECE 3317. Prof. D. R. Wilton. Note 2 Transmission Lines (Time Domain). Note about Notes 2. Disclaimer:
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ECE 3317
Prof. D. R. Wilton
Note 2 Transmission Lines(Time Domain)
Note about Notes 2
Disclaimer:
Transmission lines is the subject of Chapter 6 in the book. However, the subject of wave propagation on transmission lines in the time domain is not treated very thoroughly there, appearing only in the latter half of section 6.5.
Therefore, the material of this Note is roughly independent of the book.
Approach:
Transmission line theory can be developed starting from either circuit theory or from Maxwell’s equations directly. We’ll use the former approach because it is simpler, though it doesn’t reveal the approximations or limitations of the approach.
Transmission Lines
a
z
b
A transmission line is a twoconductor system that is used to transmit a signal from one point to another point.
Two common examples:
twin line
coaxial cable
A transmission line is normally used in the balanced mode, meaning equal and opposite currents (and charges) on the two conductors.
Transmission Lines (cont.)
Here’s what they look like in reallife.
coax to twin line matching section
coaxial cable
twin line
Transmission Lines (cont.)
Another common example (for printed circuit boards):
w
r
h
microstrip line
Transmission Lines (cont.)
microstrip line
Transmission Lines (cont.)
E

l


+
+
+
+
+

+


l
Some practical notes:
E
twin line
coax
Transmission Lines (cont.)
Load
Transmission Lines (cont.)
symbols:
z
4 parameters
Note: We use this schematic to represent a general transmission line, no matter what the actual shape of the conductors.
Transmission Lines (cont.)
z
Capacitance/m between the two conductors
Inductance/m due to stored magnetic energy
Resistance/m due to the conductors
Conductance/m due to the filling material between the conductors
Four fundamental parameters characterize any transmission line:
These are “per unit length” parameters.
4 parameters
C= capacitance/length [F/m]
L= inductance/length [H/m]
R= resistance/length [/m]
G= conductance/length [S/m]
Circuit Model
Dz
z
RDz
LDz
CDz
GDz
Circuit Model:
Dz
z
Circuit Model (cont.)
z
CDz
CDz
CDz
GDz
GDz
GDz
Dz
Dz
Dz
Dz
Circuit Model:
Dz
RDz
RDz
LDZ
RDz
LDZ
RDz
LDZ
LDZ
CDz
GDz
z
Coaxial Cable
a
z
b
Example: coaxial cable
d = conductivity of dielectric [S/m].
m = conductivity of metal [S/m].
(skin depth of metal)
Coaxial Cable (cont.)
E

l


+
+
+
+
+

+


l
Overview of derivation: capacitance per unit length
Coaxial Cable (cont.)
y
x
E
Js
Overview of derivation: inductance per unit length
Coaxial Cable (cont.)
Overview of derivation: conductance per unit length
RC Analogy:
Coaxial Cable (cont.)
Relation between L and C:
Speed of light in dielectric medium:
This is true for ALL twoconductor
transmission lines.
Hence:
Telegrapher’s Equations
RDz
LDz
I(z+Dz,t)
I(z,t)
+
V(z+Dz,t)

+
V(z,t)

CDz
GDz
z
z+Dz
z
Apply KVL and KCL laws to a small slice of line:
Telegrapher’s Equations (cont.)
Hence
Now let Dz 0:
“Telegrapher’s Equations (TE)”
Telegrapher’s Equations (cont.)
To combine these, take the derivative of the first one with respect to z:
To obtain an equation in V alone, eliminate I between eqs.:
Telegrapher’s Equations (cont.)
Hence, we have:
There is no exact solution to this differential equation, except for the lossless case.
The same equation also holds for i.
Telegrapher’s Equations (cont.)
Lossless case:
Note: The current satisfies the same differential equation:
The same equation also holds for i.
Solution to Telegrapher's Equations
Hence we have
Solution:
This is called the D’Alembert solution to the Telegrapher's Equations (the solution is in the form of traveling waves).
The same equation also holds for i.
Traveling Waves
Proof of solution:
General solution:
It is seen that the differential equation is satisfied by the general solution.
Traveling Waves
Example:
z
z0
t = t2 > t1
t = 0
t = t1 > 0
V(z,t)
…
…
z
z0
z0 + cdt1
z0 + cdt2
Traveling Waves
Example:
z
z0
t = 0
t = t2 > t1
t = t1 > 0
V(z,t)
…
…
z
z0  cdt1
z0  cdt2
z0
Traveling Waves (cont.)
Loss causes an attenuation in the signal level, and it also causes distortion (the pulse changes shape and usually becomes broader).
t = 0
V(z,t)
t = t1 > 0
t = t2 > t1
z
z0
z0 + cdt1
z0 + cdt2
(These effects can be studied numerically.)
Current
(first TE)
lossless
Our goal is to now solve for the current on the line.
Assume the following forms:
The derivatives are:
Current (cont.)
This becomes
Equating terms with the same space and time variation, we have
Hence we have
Constants C1, C2 represent time and
spaceindependent DC voltages or
currents on the line. Assuming no
initial line voltage or current we
conclude C1, C2=0
Current (cont.)
Observation about term:
Define (real) characteristic impedance Z0:
The units of Z0are Ohms.
Then
or
Current (cont.)
General solution:
OR
For a forward wave, the current waveform is the same as the voltage, but reduced in amplitude by a factor of Z0.
For a backward traveling wave, there is a minus sign as well.
Note that without this minus sign, the ratio of voltage to current would be constant rather than varying from pointtopoint and over time along the line as is generally the case!
Current (cont.)
Picture for a forwardtraveling wave:
forwardtraveling wave
z
+

Current (cont.)
Physical interpretation of minus sign for the backwardtraveling wave:
backwardtraveling wave
z
+

The minus sign arises from the reference direction for the current.
Coaxial Cable
a
z
b
Example: Find the characteristic impedance of a coax.
Coaxial Cable (cont.)
a
z
b
(intrinsic impedance of free space)
Twin Line
d
a = radius of wires
Twin Line (cont.)
These are the common values used for TV.
75300 [] transformer
75 [] coax
300 [] twin line
twin line
coaxial cable
Microstrip Line
w
r
h
parallelplate formulas:
Microstrip Line (cont.)
t = strip thickness
More accurate CAD formulas:
Note: the effective relative permittivity accounts for the fact that some of the field exists outside the substrate, in the air region. The effective widthw' accounts for the strip thickness.
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