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Ali FotowatAhmady
Office hours: Sat Mon 1:30pm to 2pm
Sat Mon 6:00pm 6:30pm
Telephone contact: 09121117364 (=RFMi)
Calls OK in reasonable hours!
Technical Q’s on SMS OK 24 hours!
Please DO NOT KNOCK ON MY DOOR BEFORE MY CLASSES!
Do the HW’s every week and U will do well.
Lecture 2: Transmission Lines
Dr. FotowatAhmady
Sharif University of Technology
Fall1391
Prepared by: Siavash Kananian & AlirezaZabetian
A wave guiding structure is one that carries a signal (or power) from one point to another.
There are three common types:
Properties
Twin lead
(shown connected to a 4:1 impedancetransforming balun)
Coaxial cable (coax)
CAT 5 cable
(twisted pair)
The two wires of the transmission line are twisted to reduce interference and radiation from discontinuities.
Transmission lines commonly met on printedcircuit boards
w
er
h
er
h
w
Microstrip
Stripline
w
w
w
er
h
er
h
Coplanar waveguide (CPW)
Coplanar strips
Transmission lines are commonly met on printedcircuit boards.
Microstrip line
A microwave integrated circuit
Properties
Two types of fiberoptic guides:
1) Singlemode fiber
Carries a single mode, as with the mode on a transmission line or waveguide. Requires the fiber diameter to be small relative to a wavelength.
2) Multimode fiber
Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect).
Properties
http://en.wikipedia.org/wiki/Waveguide_(electromagnetism)
neglect time delays (phase)
account for propagation and time delays (phase change)
We need transmissionline theory whenever the length of a line is significant compared with a wavelength.
Dz
Transmission Line
2 conductors
4 perunitlength parameters:
C= capacitance/length [F/m]
L= inductance/length [H/m]
R= resistance/length [/m]
G= conductance/length [ /m or S/m]
B
         
x
x
x
Transmission Line (cont.)
RDz
LDz
i(z,t)
i(z+z,t)
+
+
v(z+z,t)
CDz
v(z,t)
GDz


z
To combine these, take the derivative of the first one with respect to z:
Switch the order of the derivatives.
TimeHarmonic Waves:
Note that
= series impedance/length
= parallel admittance/length
Then we can write:
Let
Then
Solution:
is called the "propagation constant."
Convention:
Forward travelling wave (a wave traveling in the positive z direction):
The wave “repeats” when:
Hence:
Track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest.
vp
(phase velocity)
I+(z)
+
V+(z)

z
Characteristic Impedance Z0
A wave is traveling in the positive z direction.
so
(Z0is a number, not a function of z.)
Characteristic Impedance Z0 (cont.)
From this we have:
Using
We have
Note: The principal branch of the square root is chosen, so that Re (Z0) > 0.
I (z)
+
V (z)

z
BackwardTraveling Wave
A wave is traveling in the negative z direction.
so
Note: The reference directions for voltage and current are the same as for the forward wave.
I(z)
+
V(z)

z
General Case
Most general case:
A general superposition of forward and backward traveling waves:
Note: The reference directions for voltage and current are the same for forward and backward waves.
In the medium between the two conductors is homogeneous (uniform) and is characterized by (, ), then we have that
(proof given later)
The speed of light in a dielectric medium is
Hence, we have that
The phase velocity does not depend on the frequency, and it is always the speed of light (in the material).
Terminating impedance (load)
Ampl. of voltage wave propagating in positive z direction at z = 0.
Ampl. of voltage wave propagating in negative z direction at z = 0.
Where do we assign z = 0?
The usual choice is at the load.
Note: The length l measures distance from the load:
Terminated Transmission Line (cont.)
Terminating impedance (load)
What if we know
Can we use z =  las a reference plane?
Hence
Terminated Transmission Line (cont.)
Terminating impedance (load)
Compare:
Note: This is simply a change of reference plane, from z = 0 to z = l.
Terminated Transmission Line (cont.)
Terminating impedance (load)
What is V(l )?
propagating backwards
propagating forwards
The current at z =  lis then
l distance away from load
Terminated Transmission Line (cont.)
Total volt. at distance l from the load
Ampl. of volt. wave prop. towards load, at the load position (z = 0).
Ampl. of volt. wave prop. away from load, at the load position (z = 0).
L Load reflection coefficient
l Reflection coefficient at z =  l
Similarly,
Terminated Transmission Line (cont.)
Input impedance seen “looking” towards load at z = l .
Terminated Lossless Transmission Line
For the remainder of our transmission line discussion we will assume that the transmission line is lossless.
Short circuit load: (ZL = 0)
B
Note:
Always imaginary!
S.C. can become an O.C. with a g/4 trans. line
Using Transmission Lines to Synthesize Loads
This is very useful is microwave engineering.
A microwave filter constructed from microstrip.
Find the voltage at any point on the line.
Some algebra:
Hence, we have
where
Therefore, we have the following alternative form for the result:
Voltage wave that would exist if there were no reflections from the load (a semiinfinite transmission line or a matched load).
Wavebounce method (illustrated for l = d):
Geometric series:
Hence
or
This agrees with the previous result (setting l = d).
Note: This is a very tedious method – not recommended.
At a distance l from the load:
Note:
If Z0 real (lowloss transmission line)
Here we present a “case study” of one particular transmission line, the coaxial cable.
a
b
Find C, L, G, R
For a TEMz mode, the shape of the fields is independent of frequency, and hence we can perform the calculation using electrostatics and magnetostatics.
We will assume no variation in the z direction, and take a length of one meter in the z direction in order top calculate the perunitlength parameters.
h = 1[m]
Coaxial cable
a
l0
l0
b
Coaxial Cable (cont.)
Find C (capacitance / length)
From Gauss’s law:
Find L(inductance / length)
h = 1[m]
From Ampere’s law:
I
Coaxial cable
Note: We ignore “internal inductance” here, and only look at the magnetic field between the two conductors (accurate for high frequency.
I
z
I
Magnetic flux:
center conductor
S
h
For a lossless cable:
h = 1[m]
Coaxial cable
a
l0
l0
b
Coaxial Cable (cont.)
Find G (conductance / length)
From Gauss’s law:
As just derived,
To be more general:
This is the loss tangent that would arise from conductivity effects.
The loss tangent actually arises from both conductivity loss and polarization loss (molecular friction loss), ingeneral.
Note: It is the loss tangent that is usually (approximately) constant for a material, over a wide range of frequencies.
General expression for loss tangent:
Effective permittivity that accounts for conductivity
Loss due to molecular friction
Loss due to conductivity
h = 1[m]
Coaxial cable
a
b
Coaxial Cable (cont.)
Find R (resistance / length)
Rs = surface resistance of metal
General Transmission Line Formulas
(1)
(2)
(4)
(3)
Equations (1) and (2) can be used to find L and C if we know the material properties and the characteristic impedance of the lossless line.
Equation (3) can be used to find G if we know the material loss tangent.
Equation (4) can be used to find R (discussed later).
General Transmission Line Formulas (cont.)
Al four perunitlength parameters can be found from
ZTH
+
Z0
ZL

Limitations of TransmissionLine Theory
At high frequency, discontinuity effects can become important.
transmitted
incident
Bend
reflected
The simple TL model does not account for the bend.
Limitations of TransmissionLine Theory (cont.)
At high frequency, radiation effects can become important.
We want energy to travel from the generator to the load, without radiating.
When will radiation occur?
ZTH
+
Z0
ZL

b
z
Limitations of TransmissionLine Theory (cont.)
The coaxial cable is a perfectly shielded system – there is never any radiation at any frequency, or under any circumstances.
The fields are confined to the region between the two conductors.
Limitations of TransmissionLine Theory (cont.)
The twin lead is an open type of transmission line – the fields extend out to infinity.
The extended fields may cause interference with nearby objects. (This may be improved by using “twisted pair.”)
+

Having fields that extend to infinity is not the same thing as having radiation, however.
Limitations of TransmissionLine Theory (cont.)
The infinite twin lead will not radiate by itself, regardless of how far apart the lines are.
S
reflected
incident
h
No attenuation on an infinite lossless line
The incident and reflected waves represent an exact solution to Maxwell’s equations on the infinite line, at any frequency.
Limitations of TransmissionLine Theory (cont.)
A discontinuity on the twin lead will cause radiation to occur.
Incident wave
pipe
obstacle
h
Reflected wave
Note: Radiation effects increase as the frequency increases.
Incident wave
h
bend
bend
Reflected wave
Limitations of TransmissionLine Theory (cont.)
To reduce radiation effects of the twin lead at discontinuities:
h
CAT 5 cable
(twisted pair)