lecture 2_transmission line theory(part1)

7/29/2019 Lecture 2_Transmission Line Theory(Part1)

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Lecture 2:

Transmission Line TheoryDr. Osama M. H. AminElectrical Engineering Department

Assiut University

Power Line Communications

November 18, 2012 1


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What is this Lecture about?

1. Introduction2. Examples of Multiconductor Transmission Line (MTL)

Structures

3. Maxwells Equations4. Properties of the TEM mode of propagation5. Transmission-Line Equations for MTL6. Per Unit Length parameters for MTL

Reference:Analysis of Multiconductor Transmission Lines, Second Edition, by

Clayton R. Paul, 2008 John Wiley & Sons, Inc. Chapters: 1 to 5.

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1- Introduction

Transmission Line Theory objectives for MulticonductorTransmission Lines (MTL):

u Per-unit-length parametersu Frequency domain analysisu Time-domain analysisu Incident field excitationu Transmission line networks

The analysis of MTL is somewhat more difficult than the analysis oftwo conductor lines. (For example the matching impedancerequirement).

In the case of an MTL consisting ofn+ 1 conductors parallel to thez axis, we have 2nfirst-order matrixpartial differential equations

relating the nline voltages Vi(z, t) and nline currents Ii(z, t).

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The analysis of an MTL for the resultingnline voltages Vi(z, t) and nline currents Ii(z, t) is in general, a three-step process (direct method

not numerical):

1. Step 1: Determine the per-unit-length parameters of inductance,capacitance, conductance, and resistance for the given line.

2. Step 2: Determine the general solution of the resulting MTL equations.For an MTL consisting of n+ 1 conductors, the general solution

consists of the sum ofnforward- and nbackward-traveling waves. In

the case where the sources driving the line are general excitationwaveforms, these waves are represented by 2n unknownfunctionsthat are

functions of position along the line zand time t. In the case ofsinusoidal steady-state excitation of the line, there are 2n complex-valued

undetermined constants.

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Step 3: Incorporate the terminal conditions to determine theunknown functions or unknown coefficients in the general form

of the solution. A transmission line will have terminations at the

left and right ends consisting of independent voltage and/or

current sources and lumped elements such as resistors,capacitors, inductors, diodes, transistors, and so on. Theseterminal constraintsprovide the additional 2nequations (nfor the

left termination and nfor the right termination), which can be

used to explicitly determine the 2nundetermined functions orthe 2ncoefficients in the general form of the MTL equation

solution that was obtained in step 2.

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The excitation sources for the MTL will have several forms:1. Independent lumped sources2. Crosstalk (unintentional): the electromagnetic fields associated

with the current and voltage on that line interact with

neighboring lines inducing signals at those endpoints.

3. Interference (unintentional): with an incident electromagneticfield such as radio, radar, or TV signals, or a lightning pulse

In order to obtain the complete solution for the line voltages andcurrents via the direct solution method, each of the above three stepsmust be performedand in the above order.

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Question?

When should we use the lumped-circuit theory or follow thedirect solution method of the transmission line equations?

If a structures largest dimension is electrically small, that is,much less than a wavelength, we can use the lumped-circuit

theory, otherwise we need to find direct solution of thetransmission-line equations.

The electrical small size means also that the cross-sectionaldimensions, (such as conductor separations), must be electrically

small in order for the analysis to yield valid results. Thefundamental assumption for MTL is that the electromagnetic

field surrounding the conductors has a transverseelectromagnetic (TEM) structure.

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Examples of MTL Structures

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November 18, 2012 9

coupled microstrip

common on PCBs

coupled stripline


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Notes

The surrounding medium is said to be homogeneouswhen thepermittivityand permeabilityof the surrounding dielectricmedium are constants and are independent of the position.

The free space with parameters ofpermittivity0 andpermeability0. For the homogeneous dielectric, the parameters are definedas=r0 and=0. The permeability of all dielectricsis that of free space, whereas the permittivity is characterized bya relative permittivity (relative to that of free space) ofr .

Thus dielectrics affect electric fields and do not affect magnetic

fields. Nonuniform lines in which the conductors either are not of

uniform cross section along their length or are not parallel arisefrom either nonintentional or intentional reasons.

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For example, the conductors of a high-voltage powerdistribution line, because of their weight, sag and are not parallel

to the ground. Tapered lines are intentionally designed to give

certain desirable characteristics in microwave filters.

The velocity of propagation of the waves on those lines is equalto that of the medium in which they are immersed or

whereis thepermeabilityof the surrounding medium andisthepermittivityof the surrounding medium. For free space, these

become0 = 4

107

H/m and0

1/36

109

F/m.The velocity of propagation in free space is2.99792458 108 m/s.


= 1

= 1 00=


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Maxwells Equations

Describe exactly how electric (E-fields) and magnetic (H-fields) behave. Divergence operator Divergence at a point (x,y, z) is the measure of the vector flow out of a

surface surrounding that point.

Curl operatorThe curl is a measure of the rotation of a vector field


. =x

ax+

y

ay+

z

az

1 Gauss Law

2 GaussMagnetism Law

3 Faradays Law

4 Amperes Law

.

D = v

.B = 0

E =

t

H =

D

t+

J

D =

E electric flux density C/m2

B=

H

magnetic flux density TJ=

E density of free current

v volume density of free charge

H magnetic field density A/m

E electric field density V/m


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Gauss Law

The electric flux density flow out of a surface surrounding a certainpoint (x,y, z) equal to the volume density of the enclosed charge.

Total outward flux of the electric displacement (or simply, totaloutward electric flux) over any closed surface is equal to the total

free charge enclosed in the surface. Gauss Law for magnetism

The divergence of the B or H fields (magnetic flux densityflowing out of a certain surface is always zero through any

volume.

Magnetic monopoles do not exist, magnetic fields flow in aclosed loop. This is true even for plane waves, which just so

happen to have an infinite radius loop.


.D = v

D.d

s

S =Qenc

.B = 0

B.d

s

S = 0


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Faradays Law

A changing E-field in space gives rise to a changing B-field intime.

A circulating E-field in time gives rise to a Magnetic FieldChanging in time.

A Magnetic Field Changing in Time gives rise to an E-fieldcirculating around it.

Electric Current gives rise to magnetic fields. Magnetic Fieldsaround a circuit gives rise to electric current.


E =

t

E.d

l

C =

t.d

s

S


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Amperes Law

A flowing electric current (J) gives rise to a Magnetic Field thatcircles the current.

A time-changing Electric Flux Density (D) gives rise to aMagnetic Field that circles the D field.

Ampere's Law with the contribution of Maxwell nailed down thebasis for Electromagnetics as we currently understand it. And so

we know that a time varyingD gives rise to an H field, but from

Farday's Law we know that a varyingH field gives rise to an E

field.... and so on and so forth and the electromagnetic wavespropagate


H =

D

t+

J

H.d

l

C =

J +

D

t

.ds

S


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Properties of the TEM mode of propagation

The electric field intensity vector E(x, y, z, t) and the magneticfield intensity vector H (x, y, z, t) satisfy the TEM field structure,

that is, they lie in a plane (the xyplane) transverse or

perpendicular to the line axis (the zaxis).

Maxwell equations:


Et=

Ht

t

Ht

=

Et

+

Et

t

=

x

ax+

y

ay

+

z

az=

t+

z


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Et= Ex x,y,z,t( )

ax +Ey x,y,z,t( )

ay

Ex

x,y,z,t( ) = Emx+

ez

cos t z +x+( )+Emx

ez

cos t+ z +x( )

Ht= H

xx,y,z,t( )

ax+ H

yx,y,z,t( )

ay

Hy

x,y,z,t( ) =1

E

mx

+

ez

cos t z +x+

( )

1

E

mx

ez

cos t+ z +x

( )

=

j

+ j=

e

j Intrinsic Impedance

= j + j( ) =+ j Propagation constantAttenuation constant


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Definition of Voltage and Current for the TEMMode of Propagation


V z,t( ) = Et .d

l

c

= Et .d

l

P0

P1

I(z,t) =Ht

c .d

l


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Defining the Per-Unit-Length Parameters


A transverse conduction currentJt =Et,is induced by this transverse electric

field to flow in the lossy medium due toits conductivityfrom the top

conductor to the bottom conductor inthis transverse plane.Aper-unit-lengthconductance g

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The transverse current flowing from the top conductor to the bottomconductor is related to the voltage between the two conductors by

Theper-unit-length capacitanceThe (displacement) current flowing in the transverse plane from the top

conductor to the bottom conductor is

The electric flux density vector on the surface of the perfect conductoris Dt =Et and, according to the boundary conditions on the surface

of this perfect conductor, is normal to the surface. In order todetermine the charge on the conductor surface, we surround it with a

closed surface sthat is just off the surface of the conductor anddetermine the total electric flux through that surface. The total

capacitance is the ratio of this total charge to the voltage between thetwo conductors:


cz =

E

t.ds

a

ns

V z,t( )

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The current flowing along the top conductor and returning along thebottom conductor will generate a magnetic field intensity in the

transverse plane, Ht. The transverse magnetic flux density is Bt =Ht

This produces a magnetic fluxthrough the surface that lies betweenthe two conductors.

Hence if a section of line of lengthzhas a total inductance L, then aper-unit-length inductance lwhose units are H/m, is given by

The total inductance is the ratio of the magnetic flux through thissurface to the current that caused it:

This will produce a longitudinal voltage drop around the loop containedby the two conductors of:


lz = =

H.ds

an

sI z,t( )

=

Bt.d

ss

= 0

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The per-unit-length parameters satisfy the following relations:

Taking the ratio of above relations:

Hence for a homogeneousmedium surrounding the conductors thatis characterized by the parameters,, and, we only needto determine one of the three parameters.

For example, if we determine the per-unit-length capacitance c,then the other two parameters are obtained in terms of cas l=

(/c) andg= (/)c.


lc =

gl =

g

c=

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Transmission-Line Equations for Two Conductors


V z,t( ) = E

t.d

l

c

= E

t.d

l

P0

P1

I(z,t) =Ht

c .d

l

First, they will be derivedfrom the integral form of

Maxwells equationsSecond they will be derived

from the per-unit lengthdistributed parameterequivalent circuit

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Using Maxwells Equations


To allow for imperfect conductors, we define the per-unit-length conductorresistance of each conductor as r1/m and r0/m. Thus,

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Taking the limit asz0 and substituting, yields the first

transmission line equation

The total per-unit-length resistance of azsection of the line isthe sum of the per unit length resistances of each conductor, r1 and

r0, and we denote this total resistance as r= r1 + r0.

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the continuity equation or equationof conservation of charge

A portion of the left hand side of (2.10) contains the transverseconduction currentflowing between the conductors

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This can again be considered by defining aper-unit-length conductance, gS/m, between the two conductors as the ratio of conduction current

flowing between the two conductors in the transverse plane to the

voltage between the two conductors (see Figure 2.4(b)). Therefore,

Similarly, the charge enclosed by the surface (residing on the conductorsurface) is, by Gauss law,

The charge per unit of line length can be defined in terms of theper-unit-length capacitance cbetween the conductors as

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Taking the limit asz0 and substituting, yields the secondtransmission line equation

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From the per unit length Equivalent Circuit




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Incorporating Frequency Dependent Losses


The per-unit-length parameters will, in general, befrequency dependentand willdepend to varying degrees on the frequency of excitation of the line,=

2f , which are denoted as l(), c(),g(), and r().For perfect conductors, the line currents will reside on the surfaces of those

conductors and therefore r= 0. For imperfect conductors, the currents willbe uniformly distributed over the conductor cross sections at low frequencies,

but at higher frequencies will, because of skin effect, migrate toward the surfaces

of the conductors lying in a thickness on the order of a skin depth,= 1/fc.

Similarly, some of the magnetic flux will lie internal to the conductors

giving a per-unit-length internal inductance that is also frequency dependent, li(). As the frequency of excitation increases and the currents migrate to the

conductor surfaces, this internal inductance due to magnetic flux internal to theconductors will decrease at a rate off eventually going to zero as the

frequency increases without bound. Hence, the total per-unit-length inductance

will be the sum of this frequency dependentinternal inductance and the external inductance due to the magnetic flux

external to the conductors, le, as l= le + li ().



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November 8, 0 31

Incorporating Frequency Dependent Losses



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The Transmission Line Equations for MTL

, 32



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,



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In the case of a reference conductor being a large ground plane,each current returning in the ground plane will be concentrated

beneath the going down conductor giving the followingresistance matrix:

,



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The second equation is:



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Prosperities of the Per Unit Length ParametersMatrices L, C, G For the case of an MTL consisting ofn+ 1 conductors immersed

in a homogeneous medium characterized by permeability,

conductivity, and permittivity, the per-unit-length parameter

matrices are similarly related by



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We will assume that any medium surrounding the line conductors(homogeneous or inhomogeneous) is not ferromagnetic and

therefore has a permeability of free space,=0. Designate thecapacitance matrix with the surrounding medium (homogeneous or

inhomogeneous) removed and replaced by free space havingpermeability0 and permeability0 as C0. Since inductancedepends on the permeability of the surrounding medium and doesnot depend on the permittivity of the medium, and the

permeability of dielectrics is that of free space,0, the inductance

matrix L can be obtained from C0 using the relations for ahomogeneous medium



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Per Unit Length parameters Definitions

The per-unit-length inductance lis the ratio of the magnetic fluxpenetrating the cross-sectional surface between the two

conductors per unit of line length,, and the current along the

conductors that produced it:



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The per-unit length capacitance cis the ratio of the charge per unitof line length, q, and the voltage between the two conductors

The per-unit-length conductancegis the ratio of the per-unit-lengthconduction current It flowing in the transverse, xy, plane from

the positive conductor to the reference conductor through a lossy

dielectric surrounding the conductors and the voltage between thetwo conductors



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Circular conductors with large separation

Consider the case of two perfectly conducting wires of radii rw1and rw2 that are separated by a distance sin a homogeneous

medium,

l = 2

ln

s

rw1( )

s

rw2( )r

w1rw2

2

lns2

rw1rw2

H/m

C=2

lns r

w1( ) s rw2( )rw1

rw2

2

lns2

rw1rw2

F/m



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Circular conductors with small separation

Dont forget, the inductance is calculatedbased on free space parameters.



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one wire parallel to an infinite, perfectlyconducting plane



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Coaxial Cable



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Per-Unit-Length Conductance and Resistancefor Wire-Type Lines

In the preceding section, we have represented the losses in ahomogeneous mediumby conductivity.

The conductivityrepresents losses in the medium that are dueto free charge in the dielectric.

There is another loss mechanism that usually dominates theresistive losses in a dielectric. This is due to the polarization ofthe dielectric

that is due to bound chargein the dielectric. Within the dielectricare dipoles of charge consisting of equal but opposite-sign

charges that are tightly bound together. Some materials such aswater consist of permanent dipoles that are randomly orientedso that the material has no net polarization. Dipoles are createdin other materials when an external electric field slightly deformsthe atoms.



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Polarization loss is represented by a complex permittivity of thedielectric as

The effective conductivity, representing conductive andpolarization losses, isAn alternative way of representing both these losses is with the

loss tangent

The polarization losses will dominate the resistive losses intypical dielectrics of practical interest. Handbooks tabulate this

loss tangent for various dielectric materials at various frequenciesTherefore, the effective conductivity of a dielectric is related to

the loss tangent as



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Hence, in a homogeneous medium, the per-unit-length conductanceis related to the per-unit-length capacitance as

The loss tangent, tan, is not frequency independent, andhence the per-unit-length conductance does not increase linearly

with frequency as it seems to imply. For typical dielectricmaterials used to construct transmission lines, the variation of

the loss tangent with frequency is relatively constant over certain

frequency ranges.

For sinusoidal excitation, the total per-unit-length admittancebetween the two conductors is the sum of the per-unit-length

conductance and the per-unit-length capacitive reactance as



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InhomogeneousSurrounding Medium

Determining the capacitance by using a complex permittivity:This gives a complex capacitance as

See the reference for coaxial example.



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Per Unit Length Resistance of Wires

The exact per-unit-length high-frequencyresistance for a two-wireline in a homogeneous medium consisting of two identical wires

of radii rwseparated bysas

The surface resistance

The skin depth is defined as:

The DC resistance



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Exact results



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Per Unit Length parameters for MTL

Main assumption:u The wires must be widely separatedu The dielectric medium surrounding the wires must be homogeneous; that is,

circular dielectric insulations are ignored.

Computing the capacitance matrix amounts to several two-conductor capacitance calculations



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In order to numerically determine the entries in L,C, and G, weneed only a capacitance solver. First, we solve for the scalar

capacitance matrix with the dielectric(s) surrounding theconductors removedand replaced with free space that is denoted

as C0. The per-unit-length inductance matrix can be obtainedfrom

Use the capacitance solver with each dielectric replaced by itscomplex permittivity:



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The Generalized Capacitance Matrix C

We describe a technique for computing a certain per-unit-lengthparameter matrix, thegeneralized capacitance matrix C, without regard to

the choice of the reference conductor. The dimensions of thisgeneralized capacitance matrix are (n+ 1) X(n+ 1).



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The nMTL voltages Viare defined to be between eachconductor and the chosen reference conductor. We may also

define the potentialsiof each of the n+ 1 conductors withrespect to some reference point or line that is parallel to the zaxis. The total charge per unit of line length, qi, of each of the n

+ 1 conductors can be related to their potentialsifor i= 0, 1,2, , nwith the (n+ 1) X(n+ 1) generalized capacitance

matrix Cas



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Numerical Methods for the General Case

If the wires are closely spaced, proximity effect will cause thecharge distributions to be nonuniform around the wire

peripheries. In the case of wires that are closely spaced, the

charge distributions will tend to concentrate on the adjacent

surfaces (proximity effect) In order to model this effect, we will assume aformof the charge

distribution around the ith wire periphery in the form of aFourier series as a function of the peripheralangle

ias

Number of unknowns



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Then voltage between two points



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Applications to Inhomogeneous Dielectric MediaAt the interface between two dielectric surfaces, the boundary

condition is that the components of the electric flux density vector Dthatare normal to the interface must be continuous, that is, D1n=1E1n=2E2n= D2n.

A simple way of handling inhomogeneous dielectric media is toreplace the dielectrics with free space having bound charge at theinterface.

At places where the dielectric is adjacent to a perfect conductor, wehave both free charge and bound charge, and the free charge densityon the surface of the conductor is equal to the component of the

electric flux density vector that is normal to the conductor surface, Dn(C/m2).

Of course, the component of the electric field intensity vector that istangent to a boundary is continuous across the boundary for aninterface between two dielectrics, Et1 = Et2, and is zero at the surfaceof a perfect conductor.



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Thank you!

First Report Duplicate the results of sec. 5.2.3 Computed Results: Ribbon

Cables in page 187

lecture 2_transmission line theory(part1)

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