lec.2 transmission lines theory
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RF & Microwave EngineeringBETE-Fall 2009
2 Basit Ali ZebDepartment of Electrical Engineering, AU
Topics of Discussion
• Why Transmission Lines?
• Lumped element circuit model
• Wave propagation on Transmission Line
• Calculation of:– Characteristics Impedance
– Propagation constant
– Standing wave ratio
• The lossless Line
• The terminated lossless transmission line
• Power Flow on lossless transmission line
RF & Microwave EngineeringBETE-Fall 2009
3 Basit Ali ZebDepartment of Electrical Engineering, AU
Guided Wave to Free Space
RF & Microwave EngineeringBETE-Fall 2009
4 Basit Ali ZebDepartment of Electrical Engineering, AU
Why Transmission Lines ?
• When voltage at A changes state, does the new voltage appear at B instantaneously?
If separation distance is electrically large, there will
be a propagation delay
RF & Microwave EngineeringBETE-Fall 2009
5 Basit Ali ZebDepartment of Electrical Engineering, AU
Why Transmission Lines ?
• In high frequency circuits, even smaller distances are “comparable” and hence propagation delay for a voltage signal becomes significant.
• We have to consider the propagation effects of voltage/current signals, which are modeled as a “Transmission line”.
• Both Voltage and Current can propagate along a Transmission line (TL)
RF & Microwave EngineeringBETE-Fall 2009
6 Basit Ali ZebDepartment of Electrical Engineering, AU
Transmission Lines
RF & Microwave EngineeringBETE-Fall 2009
7 Basit Ali ZebDepartment of Electrical Engineering, AU
Types of Transmission Lines
• Coaxial Cable
• Two-wire Twisted pair
• Microstrip, Stripline and coplanar waveguides, etc.
RF & Microwave EngineeringBETE-Fall 2009
8 Basit Ali ZebDepartment of Electrical Engineering, AU
Printed Circuit Transmission Lines
Integrated Circuit
Microstrip
Stripline
Via
Cross section view taken here
PCB substrate
T
W
Cross Section of Above PCB
T
Signal (microstrip)
Ground/Power
Signal (stripline)
Signal (stripline)
Ground/Power
Signal (microstrip)
Copper Trace
Copper Plane
FR4 Dielectric
W
MicrostripStripline
Frequency (f) is approaching 10 GHz Wavelength (λ) is 3 cm
RF & Microwave EngineeringBETE-Fall 2009
9 Basit Ali ZebDepartment of Electrical Engineering, AU
Transmission Lines
RF & Microwave EngineeringBETE-Fall 2009
10 Basit Ali ZebDepartment of Electrical Engineering, AU
Role of Wavelength
RF & Microwave EngineeringBETE-Fall 2009
11 Basit Ali ZebDepartment of Electrical Engineering, AU
Role of Wavelength
RF & Microwave EngineeringBETE-Fall 2009
12 Basit Ali ZebDepartment of Electrical Engineering, AU
Characterization of TLs
• Several types of transmission lines have been developed for various applications. They are characterized by their:
• Attenuation,
• Bandwidth,
• Dispersion
• Power-handling capability,
• Physical size, and applicability for integration..
Dispersion means the frequency dependence
characteristics of wave propagation
RF & Microwave EngineeringBETE-Fall 2009
13 Basit Ali ZebDepartment of Electrical Engineering, AU
Characterization of TLs
• All true transmission Lines share one common characteristics: the E, H fields and the direction of wave propagation are all mutually perpendicular
• What is the direction of propagation and what are they called?
The long axis of the geometry
TEM waves
TEM mode and Non-TEM mode Transmission Lines
RF & Microwave EngineeringBETE-Fall 2009
14 Basit Ali ZebDepartment of Electrical Engineering, AU
Lumped Element Circuit Model
3. Segmentation of the line
into small elements of
over which Kirchhoff’s law of constant voltage and current
can be applied.
1. Voltages and currents are
no longer spatially constant
over the geometric scale of interest to RF/Microwave
engineer
2. Kirchhoff’s law of constant voltage and current
cannot be applied over the
macroscopic dimension of
transmission line.
4. A finite length TL can be
viewed as cascade
connection of number of these
lumped element circuit models
RF & Microwave EngineeringBETE-Fall 2009
15 Basit Ali ZebDepartment of Electrical Engineering, AU
Lumped Element Circuit Model
R, L, C, G are frequency dependant distributed parameters expressed per unit length of line
RF & Microwave EngineeringBETE-Fall 2009
16 Basit Ali ZebDepartment of Electrical Engineering, AU
Equivalent Circuit Representation
• Provides a clear intuitive picture
• Lends itself to a 2-port network representation
• Permits the KCL & KVL analysis
ADVANTAGES
RF & Microwave EngineeringBETE-Fall 2009
17 Basit Ali ZebDepartment of Electrical Engineering, AU
Transmission Line Equations
• The terminal characteristics of TL model is determined from standard Kirchhoff’s laws for a short line segment .
• The equations so derived are commonly known as the Telegrapher Equations. In phasor form,
RF & Microwave EngineeringBETE-Fall 2009
18 Basit Ali ZebDepartment of Electrical Engineering, AU
Propagation Wave equations
The two first order differential equations can be solved to give wave equations for voltage & current along the Transmission line:
where
Attenuation constant Phase constant
Complex
Propagation
Constant
RF & Microwave EngineeringBETE-Fall 2009
19 Basit Ali ZebDepartment of Electrical Engineering, AU
Voltage & Current Waves
• Traveling wave solution to wave equations gives us the voltage and current along the line. It can be found as:
term represents wave propagation in + z direction
term represents wave propagation in – z direction
V0+, I0
+ are the wave amplitudes in +z direction
V0-, I0
- are the wave amplitudes in -z direction
RF & Microwave EngineeringBETE-Fall 2009
20 Basit Ali ZebDepartment of Electrical Engineering, AU
Characteristics Impedance
• We can easily relate the current wave amplitudes to the voltage wave amplitude by using the following two equations:
And solving for the value of current wave I(z):
Hence the characteristics impedance is calculated by
comparing the two current equations as:
COMPLEX
QUANTITY
RF & Microwave EngineeringBETE-Fall 2009
21 Basit Ali ZebDepartment of Electrical Engineering, AU
Transmission Line Parameters
A transmission line is characterized by two fundamental
parameters, its propagation constant γ and characteristics impedance Z0
All TEM transmission line share the following useful relations:
LC = µε G/C = σ/ε
RF & Microwave EngineeringBETE-Fall 2009
22 Basit Ali ZebDepartment of Electrical Engineering, AU
The Lossless TL
• The characteristic impedance, in general, is a complex quantity and hence losses must be taken into account in real transmission lines
• However, transmission lines can be designed to minimize ohmic losses and dielectric loss, by selecting conductors with high conductivities and dielectric materials (filling in between wires) with negligible conductivities.
• In such case, we can safely assume very small values of R and G (R << jωL and G << jωC.)
• Z0 is a purely real quantity as we let R = G = 0 given by the value:
RF & Microwave EngineeringBETE-Fall 2009
23 Basit Ali ZebDepartment of Electrical Engineering, AU
The Lossless TL- parameters
• Wavelength of the propagating signal:
• Phase velocity
What is the
characteristic impedance of
free space?
Recap: We expect that V(z) and I(z) are not constant along the RF &
microwave circuit interconnect. Rather they vary along the transmission line all the times.
RF & Microwave EngineeringBETE-Fall 2009
24 Basit Ali ZebDepartment of Electrical Engineering, AU
Voltage Reflection Coefficient
Transmission line of length ℓ connected on one end with a generator and on the other end to a load ZL . The load is
located at z = 0 and the generator terminals are at z = - ℓ.
RF & Microwave EngineeringBETE-Fall 2009
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Reflection Coefficient
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Reflection Coefficient
Given the reflection coefficient, total voltages and currents on TL can be found.
This is the general reflection coefficient …
For lossless TL, what would it be?
RF & Microwave EngineeringBETE-Fall 2009
27 Basit Ali ZebDepartment of Electrical Engineering, AU
Standing Wave Ratio (SWR)
Voltage and current
on the line consist of
a superposition of
incident and reflected
waves.
Standing waves do not occur when there is matched load or Γ = 0.
1 ≤ SWR ≤ ∞
Practical RF and
Microwave systems
should exhibit a value
close to 1
RF & Microwave EngineeringBETE-Fall 2009
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SWR
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SWR
RF & Microwave EngineeringBETE-Fall 2009
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SWR
SWR and Reflection Coefficient are the representation of same phenomenon, i.e., impedance mismatch
RF & Microwave EngineeringBETE-Fall 2009
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Return Loss
In practical microwave systems, we seek the highest
possible value of RL
A matched load has infinite return loss.
A load that reflects back all power has zero return loss.
RF & Microwave EngineeringBETE-Fall 2009
32 Basit Ali ZebDepartment of Electrical Engineering, AU
Input Impedance
What if we need to find the voltage at the input of the transmission line terminals?
RF & Microwave EngineeringBETE-Fall 2009
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Input Impedance
At a distance z = - l from the load
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Input Impedance of a TL
It takes into account the frequency
of operation through wave number β
It predicts how the load impedance
can be transformed along a TL of Z0
and length L
RF & Microwave EngineeringBETE-Fall 2009
35 Basit Ali ZebDepartment of Electrical Engineering, AU
Input Impedance Vs. Frequency
length = 10 cm
Practical
measurements with
network analyzer
permits the recording
of graphs as shown here.
If we fix the frequency and vary the line
length, we will get the
identical response
RF & Microwave EngineeringBETE-Fall 2009
36 Basit Ali ZebDepartment of Electrical Engineering, AU
Special Cases of Lossless Terminated Transmission Lines
RF & Microwave EngineeringBETE-Fall 2009
37 Basit Ali ZebDepartment of Electrical Engineering, AU
Termination of TLs
• We will now consider the termination of transmission lines that are excited by sinusoidal steady state sources
• Adding terminations produces reflection so that total voltage and current anywhere on the TL is the sum of forward and reverse propagating waves.
RF & Microwave EngineeringBETE-Fall 2009
38 Basit Ali ZebDepartment of Electrical Engineering, AU
Special Cases
RF & Microwave EngineeringBETE-Fall 2009
39 Basit Ali ZebDepartment of Electrical Engineering, AU
Short Circuit Termination
At the load z = 0, the voltage VL is minimum while current
IL is maximum
Voltage, Current and Input Impedance expressions!
RF & Microwave EngineeringBETE-Fall 2009
40 Basit Ali ZebDepartment of Electrical Engineering, AU
Short Circuit Termination
Observe the periodic transition of input
impedance as the distance from the
load increases
Periodic transition of Input impedance
with λ/2
RF & Microwave EngineeringBETE-Fall 2009
41 Basit Ali ZebDepartment of Electrical Engineering, AU
Open Circuit Termination
At the load z =0, the voltages VL is maximum
and current IL = 0
Voltage, Current and Input Impedance expressions!
RF & Microwave EngineeringBETE-Fall 2009
42 Basit Ali ZebDepartment of Electrical Engineering, AU
Open Circuit Termination
Periodic transition of Input impedance
with λ/2
RF & Microwave EngineeringBETE-Fall 2009
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Inductive & Capacitive Behavior
ininin jXRZ +=
In general, the input impedance may be complex which consists of a real part and an imaginary part:
In case of a short-circuited and open-circuited lossless line, the input impedance is purely reactive, i.e., have imaginary part
only. Through proper choice of the lengths of a short-circuited
line, desired inductance or capacitance can be achieved.
1. If tan βℓ ≥ 0: the line appears as an equivalent inductor Leq
whose impedance is equal to j Zo tan βℓ
2. If tan βℓ ≤ 0: the line appears as an equivalent capacitor Ceq
whose impedance is equal to j Zo tan βℓ
RF & Microwave EngineeringBETE-Fall 2009
44 Basit Ali ZebDepartment of Electrical Engineering, AU
Inductive and Capacitive Behavior
lβω tan0
jZLj eq =
= −
0
1tan
1
Z
Ll
eqω
β
For short circuit termination, the minimum line length ℓ that would result in an input impedance Zin equivalent to that of
inductance Leq is:
1.
lβω
tan1
0jZ
Cj eq
=
−= −
0
1 1tan
1
ZCl
eqωπ
β
The minimum line length ℓ that would result in an input impedance Zin equivalent to that of capacitance Ceq is:
2.
RF & Microwave EngineeringBETE-Fall 2009
45 Basit Ali ZebDepartment of Electrical Engineering, AU
Advantages
• Through proper choice of the length of a short-circuited line, we can make substitutes for capacitors and inductors with any desired reactance.
• This practice in indeed common and desirable in the design of microwave circuits and high-speed ICs where making an actual capacitor or inductor is much difficult.
RF & Microwave EngineeringBETE-Fall 2009
46 Basit Ali ZebDepartment of Electrical Engineering, AU
Half-Wavelength Line
How can we make the input impedance of a line equal to ZL?
0tantan
2
==
=
πβ
λ
nl
nl
Consequently, the input impedance expression reduces to:
Lin ZZ =Thus a generator connected to load through a half wavelength
lossless line would induce the same voltage across the load
and current through it as when the line is not there.
RF & Microwave EngineeringBETE-Fall 2009
47 Basit Ali ZebDepartment of Electrical Engineering, AU
Quarter Wave Transformer
It is used to match the real load impedance with the desired input impedance.
±∞==
ll
βλ
tan
4
RF & Microwave EngineeringBETE-Fall 2009
48 Basit Ali ZebDepartment of Electrical Engineering, AU
Quarter Wave Transformer
When a finite transmission line is terminated with its own
characteristic impedance, the voltage and current distributions are exactly the same as though the line had been extended to infinity.
Practically, this technique is easy to build but gives narrowband matching and is not suitable for wideband matching
RF & Microwave EngineeringBETE-Fall 2009
49 Basit Ali ZebDepartment of Electrical Engineering, AU
Matched Transmission Line
• For a matched lossless transmission line with:
0ZZL =
1. The input impedance becomes for all locations of z on the transmission line
2. Reflection coefficient is zero
3. All the incident power is delivered to the load, regardless of the line length.
LinZZ =
RF & Microwave EngineeringBETE-Fall 2009
50 Basit Ali ZebDepartment of Electrical Engineering, AU
Power Flow on a TL
A Hugely important part of electrical engineering is delivering
signal power to a load. Examples include efficiently delivering electromagnetic power from a source to an antenna, or
maximizing the power delivered from a filter to an amplifier.
So far our discussion is based on the voltage and current aspects of wave propagation on a transmission line.
The incident and reflected waves carry power with them. So
we look into the power flow on a lossless transmission line from source to load.
RF & Microwave EngineeringBETE-Fall 2009
51 Basit Ali ZebDepartment of Electrical Engineering, AU
Time Average Power Flow
Often the power we are ultimately concerned is the real time average power rather than instantaneous power:
This expression is similar to that used in circuit analysis
Recall the values of voltage and current along the lossless transmission line:
RF & Microwave EngineeringBETE-Fall 2009
52 Basit Ali ZebDepartment of Electrical Engineering, AU
Time Average Power Flow
Substituting these values in time average power flow equation, we get the net average power delivered to the load:
Watts
Since this power is not a function of z (true for lossless TL), a
z-dependence is no longer indicated.
RF & Microwave EngineeringBETE-Fall 2009
53 Basit Ali ZebDepartment of Electrical Engineering, AU
Time Average Power Flow
• The last equation shows that the total time averaged power delivered to the load is equal to
the incident time averaged power
minus the reflected time averaged power
RF & Microwave EngineeringBETE-Fall 2009
54 Basit Ali ZebDepartment of Electrical Engineering, AU
Return Loss
The time averaged power that is not delivered to the load is
considered as a loss. What is this loss called?
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