Lec.2 Transmission Lines Theory

Department of Electrical EngineeringAir University

Transmission Lines Analysis

Lecture No. 2

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



Guided Wave to Free Space



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



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)



Transmission Lines



Types of Transmission Lines

• Coaxial Cable

• Two-wire Twisted pair

• Microstrip, Stripline and coplanar waveguides, etc.



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



Transmission Lines



Role of Wavelength



Role of Wavelength



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



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



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



Lumped Element Circuit Model

R, L, C, G are frequency dependant distributed parameters expressed per unit length of line



Equivalent Circuit Representation

• Provides a clear intuitive picture

• Lends itself to a 2-port network representation

• Permits the KCL & KVL analysis

ADVANTAGES



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,



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



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



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



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



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:



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.



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.



Reflection Coefficient



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?



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



SWR



SWR



SWR

SWR and Reflection Coefficient are the representation of same phenomenon, i.e., impedance mismatch



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.



Input Impedance

What if we need to find the voltage at the input of the transmission line terminals?



Input Impedance

At a distance z = - l from the load



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



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



Special Cases of Lossless Terminated Transmission Lines



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.



Special Cases



Short Circuit Termination

At the load z = 0, the voltage VL is minimum while current

IL is maximum

Voltage, Current and Input Impedance expressions!



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



Open Circuit Termination

At the load z =0, the voltages VL is maximum

and current IL = 0

Voltage, Current and Input Impedance expressions!



Open Circuit Termination

Periodic transition of Input impedance

with λ/2



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



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.



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.



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.



Quarter Wave Transformer

It is used to match the real load impedance with the desired input impedance.

±∞==

ll

βλ

tan

4



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



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 =



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.



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:




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.




• 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



Return Loss

The time averaged power that is not delivered to the load is

considered as a loss. What is this loss called?



Study

• Article 2.1, 2.2, 2.3, 2.5 and 2.6 from the text book

• Next lecture on “Generator and Load Mismatch”

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Lec.2 Transmission Lines Theory