Ransmission Line Parameters & Theory

UNIT-1

RANSMISSION LINE PARAMETERS & THEORY

CONTENTS

Transmission line Parameters
Characteristic impedance
Transmission lines as a cascade of T Sections
Definition of Propagation Constant
General Solution of the transmission line
The two standard forms for voltage and current of a line terminated by an impedance
Physical significance of the equation and the infinite line
The two standard forms for the input impedance of a transmission line terminated by an impedance
Meaning of reflection coefficient – wavelength and velocity of propagation
Waveform distortion – distortion less transmission line
Input impedance of lossless lines – reflection on a line not terminated by Zo
Transfer impedance – reflection factor and reflection loss
T and Π Section equivalent to lines.

LEARNING OBJECTIVES OF UNIT-1

Fundamentals of transmission lines
State what sound waves are and define a propagating medium
Define light waves and list their characteristics
State the difference between sound waves and light waves
State the electromagnetic wave theory and list the components of the electromagnetic wave
Load impedance and its effect on current flow through transmission lines

THEORETICAL BACKGROUND

A TRANSMISSION LINE is a device designedto guide electrical energy from one point to another.It is used, for example, to transfer the output rf energyof a transmitter to an antenna. This energy will nottravel through normal electrical wire without greatlosses. Although the antenna can be connecteddirectly to the transmitter, the antenna is usuallylocated some distance away from the transmitter. Onboard ship, the transmitter is located inside a radioroom, and its associated antenna is mounted on a mast.A transmission line is used to connect the transmitterand the antenna.The transmission line has a single purpose for boththe transmitter and the antenna. This purpose is totransfer the energy output of the transmitter to theantenna with the least possible power loss. How wellthis is done depends on the special physical andelectrical characteristics (impedance and resistance)of the transmission line.

1.1 CHARACTERISTIC IMPEDANCE

You can describe a transmission line in terms ofits impedance. The ratio of voltage to current (ein/iin)at the input end is known as the input impedance(zin). This is the impedance presented to the transmitterby the transmission line and its load, the antenna.The ratio of voltage to current at the output (eout/iout)end is known as the output impedance (zout).This is the impedance presented to the load by thetransmission line and its source. If an infinitely longtransmission line could be used, the ratio of voltageto current at any point on that transmission line wouldbe some particular value of impedance. This impedanceis known as the characteristic impedance.

The maximum (and most efficient) transfer ofelectrical energy takes place when the source impedanceis matched to the load impedance. This fact isvery important in the study of transmission lines andantennas. If the characteristic impedance of thetransmission line and the load impedance are equal,energy from the transmitter will travel down thetransmission line to the antenna with no power losscaused by reflection.

1.2 THE PROPAGATION CONSTANT

The propagation constant is separated into two components that have very different effects on signals:

The real part of the propagation constant is the attenuation constant and is denoted by Greek lowercase letter (alpha). It causes a signal amplitude to decrease along a transmission line. The natural units of the attenuation constant are Nepers/meter, but we often convert to dB/meter in microwave engineering. To get loss in dB/length, multiply Nepers/length by 8.686.

The phase constant is denoted by Greek lowercase letter (beta) adds the imaginary component to the propagation constant. It determines the sinusoidal amplitude/phase of the signal along a transmission line, at a constant time. The phase constant's "natural" units are radians/meter, but we often convert to degrees/meter. A transmission line of length "l" will have an electrical phase of l, in radians or degrees. To convert from radians to degrees, multiply by 180/.

The two parts of the propagation constant have radically different effects on a wave. The amplitude of a wave (frozen in time) goes as cosine(l). In a lossless transmission line, the wave would propagate as a perfect sine wave. In real life there is some loss to the transmission line, and that is where the attenuation constant comes in. The amplitude of the signal decays as Exp(-l). The composite behavior of the propagation constant is observed when you multiply the effects of and.

The graph below represents wave propagation in a fairly lossy line, we made it lossy so you could observe the familiar exponential curve of amplitude decay. In this graph, =1 and =0.05.

1.3 REFLECTION COEFFICIENT

A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave. The reflection coefficient is closely related to the transmission coefficient.

it is the complex ratio of the electric field strength of the reflected wave (E− ) to that of the incident wave (E+ ). This is typically represented with a Γ (capital gamma) and can be written as;

The reflection coefficient may also be established using other field or circuit quantities.The reflection coefficient can be given by the equations below, where ZS is the impedance toward the source, ZL is the impedance toward the load:

Simple circuit configuration showing measurement location of reflection coefficient.

Notice that a negative reflection coefficient means that the reflected wave receives a 180°, or π, phase shift.The absolute magnitude (designated by vertical bars) of the reflection coefficient can be calculated from the standing wave ratio, SWR:

The reflection coefficient is displayed graphically using a Smith chart.

1.4 VELOCITY OF PROPAGATION

This is a measure of how fast a signal travels along a line. A radio signal travels in free space atthe speed of light, approximately 3x108 m/sec. A signal travels in a transmission line at much lessthan this. In twisted pair cable the velocity of propagation may be between 40% and 75% of thevelocity in free space.

There is a direct relationship between Velocity of Propagation (V) and Wavelength, l:

V =l f

At high frequencies, V can also be simplified as it becomes a constant for that line:

Velocity of propagation is often stated either as a percentage of the speed of light or as time-todistance.When the time-to-distance figure is used, it mat be known as Propagation Delay, andwill be expressed as ns/100m or ms/km.

1.5 DISTORTIONLESS TRANSMISSIONLINE

A distortionless line does not distort the signal phase, but does introduce a signal loss since they are not super conductors. This is also known as the Heaviside condition. Phase distortion does not occur if the phase velocity Vp is constant at all frequencies.

By definition, a phase shift of 2π radians occurs over one wavelength λ.

Since

then

This tells us that in order for phase velocity Vp to be constant, the phase shift coefficient b, must vary directly with frequency w .

Recall

The problem now is to find b. This can be done as follows:

The 2nd and 3rd roots can be expanded by means of the Binomial Expansion. Recall:

In this instance n = 1/2. Since the contribution of successive terms diminishes rapidly, g is expanded to only 3 terms:

Since g = a + jb, equate the imaginary terms to find b.

Note that if then:

From this we observe that b is directly proportional to w. This means that the requirement for distortionless transmission is:

If we equate the real terms, we obtain:

1.6 INPUT IMPEDANCE

The input impedance of a line of length l with a load ZL at xi=i0 is

and using equation 16 to substitute for K

There are three interesting special cases:

(1) When the load is a short-circuit

and since a lossless line has a real Z0 the input impedance is purely reactive.

(2) When the load is an open-circuit

which is the same as for a short-circuit at a position shifted by a quarter of a wavelength.

(3) When the line is a quarter of a wavelength long, i.e. kl = ! / 2 and

So, the input impedance is real when a quarter-wavelength line is terminated by any real impedance. This has the important application that two transmission lines of different characteristic impedance can be joined without causing reflections if a suitable quarter-wave transformer is used between the two lines. This technique obviously only works for a fairly narrow band of frequencies specific to a given transformer and can also be used for matching inconvenient loads.

1.7 REFLECTION LOSS

Return loss or Reflection loss is the reflection of signal power resulting from the insertion of a device in a transmission line or optical fiber. It is usually expressed as a ratio in dB relative to the transmitted signal power.

If the power transmitted by the source is PT and the power reflected is PR, then the return

loss in dB is given by

Optical Return Loss is a positive number, historically ORL has also been referred to as a negative number. Within the industry expect to see ORL referred to variably as a positive or negative number.

This ORL sign ambiguity can lead to confusion when referring to a circuit as having high or low return loss; so remember:- High Return Loss = lower reflected power = large ORL

number = generally good. Low Return Loss = higher reflected power = small ORL number = generally bad.

In metallic conductor systems, reflections of a signal traveling down a conductor can occur at a discontinuity or impedance mismatch. The ratio of the amplitude of the reflected wave Vr to the amplitude of the incident wave Vi is known as the reflection coefficient Ã.

When the source and load impedances are known values, the reflection coefficient is given by

where ZS is the impedance toward the source and ZL is the impedance toward the load.

1.8 TRANSMISSION LINE EQUATIONS

A typical engineering problem involves the transmission of a signal from a generator to a load. A transmission line is the part of the circuit that provides the direct link between generator and load. Transmission lines can be realized in a number of ways. Common examples are the parallel-wire line and the coaxial cable. For simplicity, we use in most diagrams the parallel-wire line to represent circuit connections, but the theory applies to all types of transmission lines.

If you are only familiar with low frequency circuits, you are used to treat all lines connecting the various circuit elements as perfect wires, with no voltage drop and no impedance associated to them (lumped impedance circuits). This is a reasonable procedure as long as the length of the wires is much smaller than the wavelength of the signal. At any given time, the measured voltage and current are the same for each location on the same wire.

Let’s look at some examples. The electricity supplied to households consists of high power sinusoidal signals, with frequency of 60Hz or 50Hz, depending on the country. Assuming that the insulator between wires is air (ε ≈ ε0), the wavelength for 60Hz is:

Let’s compare to a frequency in the microwave range, for instance 60 GHz. The wavelength is given by

For sufficiently high frequencies the wavelength is comparable with the length of conductors in a transmission line. The signal propagates as a wave of voltage and current along the line, because it cannot change instantaneously at all locations. Therefore, we cannot neglect the impedance properties of the wires (distributed impedance circuits).

Note that the equivalent circuit of a generator consists of an ideal alternating voltage generator in series with its actual internal impedance. When the generator is open (ZR → ∞) we have:

Iin = 0 and Vin = VG

If the generator is connected to a load ZR

The simplest circuit problem that we can study consists of a voltage generator connected to a load through a uniform transmission line. In general, the impedance seen by the generator is not the same as the impedance of the load, because of the presence of the transmission line, except for some very particular cases:

Our first goal is to determine the equivalent impedance seen by the generator, that is, the input impedance of a line terminated by the load. Once that is known, standard circuit theory can be used.

A uniform transmission line is a “distributed circuit” that we can describe as a cascade of identical cells with infinitesimal length. The conductors used to realize the line possess a certain series inductance and resistance. In addition, there is a shunt capacitance between the conductors, and even a shunt conductance if the medium insulating the wires is not perfect. We use the concept of shunt conductance, rather than resistance, because it is more convenient for adding the parallel elements of the shunt. We can represent the uniform transmission line with the distributed circuit below (general lossy line)

The impedance parameters L, R, C, and G represent:

L = series inductance per unit length

R = series resistance per unit length

C = shunt capacitance per unit length

G = shunt conductance per unit length.

Each cell of the distributed circuit will have impedance elements with values: Ldz, Rdz, Cdz, and Gdz, where dz is the infinitesimal length of the cells. If we can determine the differential behavior of an elementary cell of the distributed circuit, in terms of voltage and current, we can find a global differential equation that describes the entire transmission line. We can do so, because we assume the line to be uniform along its length. So, all we need to do is to study how voltage and current vary in a single elementary cell of the distributed circuit.

Loss-less Transmission Line

In many cases, it is possible to neglect resistive effects in the line. In this approximation there is no Joule effect loss because only reactive elements are present. The equivalent circuit for the elementary cell of a loss-less transmission line is shown in the figure below.

The series inductance determines the variation of the voltage from input to output of the cell, according to the sub-circuit below

The corresponding circuit equation is

(V + dV) −V = −jωL dz I

which gives a first order differential equation for the voltage

The current flowing through the shunt capacitance determines the variation of the current from input to output of the cell.

The circuit equation for the sub-circuit above is

dI = −jωCdz(V + dV)= −jωCVdz –jωCdVdz

The second term (including dV dz) tends to zero very rapidly in the limit of infinitesimal length dz leaving a first order differential equation for the current

We have obtained a system of two coupled first order differential equations that describe the behavior of voltage and current on the uniform loss-less transmission line. The equations must be solved simultaneously.

One can easily obtain a set of uncoupled equations by differentiating with respect to the space coordinate. The first order differential terms are eliminated by using the corresponding telegraphers’ equation

We have now two uncoupled second order differential equations for voltage and current, which give an equivalent description of the loss-less transmission line. Mathematically, these are wave equations and can be solved independently. The general solution for the voltage equation is

where the wave propagation constant is

β = ωLC

Note that the complex exponential terms including βhave unitarymagnitude and purely “imaginary” argument, therefore they onlyaffect the “phase” of the wave in space.

We have the following useful relations:

Here, λ = vp f is the wavelength of the dielectric medium surrounding the conductors of the transmission line and

Vp is the phase velocity of an electromagnetic wave in the dielectric. As you can see, the propagation constant βcan be written in many different, equivalent ways.

1 | Page Prepared by V.NAVANEETHAKRISHNAN AP/ECE