notes 1 - transmission line theory.pptx

PowerPoint Presentation

Prof. David R. JacksonDept. of ECENotes 1ECE 5317-6351 Microwave EngineeringFall 2011

Transmission Line Theory

1A waveguiding structure is one that carries a signal (or power) from one point to another. There are three common types: Transmission lines Fiber-optic guides WaveguidesWaveguiding StructuresNote: An alternative to waveguiding structures is wireless transmission using antennas. (antenna are discussed in ECE 5318.) 2Transmission Line

Has two conductors running parallel Can propagate a signal at any frequency (in theory) Becomes lossy at high frequency Can handle low or moderate amounts of power Does not have signal distortion, unless there is loss May or may not be immune to interference Does not have Ez or Hz components of the fields (TEMz) PropertiesCoaxial cable (coax)Twin lead (shown connected to a 4:1 impedance-transforming balun)3Transmission Line (cont.)CAT 5 cable(twisted pair)

The two wires of the transmission line are twisted to reduce interference and radiation from discontinuities. 4Transmission Line (cont.)MicrostriphwererwStriplinehTransmission lines commonly met on printed-circuit boardsCoplanar stripsherwwCoplanar waveguide (CPW)herw5Transmission Line (cont.)Transmission lines are commonly met on printed-circuit boards.

A microwave integrated circuitMicrostrip line6Fiber-Optic GuideProperties

Uses a dielectric rod Can propagate a signal at any frequency (in theory) Can be made very low loss Has minimal signal distortion Very immune to interference Not suitable for high power Has both Ez and Hz components of the fields

7Fiber-Optic Guide (cont.)Two types of fiber-optic guides:1) Single-mode fiber2) Multi-mode fiberCarries a single mode, as with the mode on a transmission line or waveguide. Requires the fiber diameter to be small relative to a wavelength. Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect).8Fiber-Optic Guide (cont.)

http://en.wikipedia.org/wiki/Optical_fiber

Higher index core region9Waveguides Has a single hollow metal pipe Can propagate a signal only at high frequency: > c The width must be at least one-half of a wavelength Has signal distortion, even in the lossless case Immune to interference Can handle large amounts of power Has low loss (compared with a transmission line) Has either Ez or Hz component of the fields (TMz or TEz) Properties

http://en.wikipedia.org/wiki/Waveguide_(electromagnetism)

10 Lumped circuits: resistors, capacitors,inductors neglect time delays (phase)account for propagation and time delays (phase change)

Transmission-Line Theory Distributed circuit elements: transmission lines

We need transmission-line theory whenever the length of a line is significant compared with a wavelength.11Transmission Line2 conductors4 per-unit-length parameters:C = capacitance/length [F/m]L = inductance/length [H/m]R = resistance/length [/m]G = conductance/length [ /m or S/m]Dz1212Transmission Line (cont.)

+ + + + + + +- - - - - - - - - -

xxxB13RDzLDzGDzCDzzv(z+z,t)+-v(z,t)+-i(z,t)i(z+z,t)13

Transmission Line (cont.)14RDzLDzGDzCDzzv(z+z,t)+-v(z,t)+-i(z,t)i(z+z,t)14Hence

Now let Dz 0:

TelegraphersEquationsTEM Transmission Line (cont.)1515To combine these, take the derivative of the first one with respect to z:

Switch the order of the derivatives.TEM Transmission Line (cont.)1616

The same equation also holds for i.Hence, we have:

TEM Transmission Line (cont.)1717

TEM Transmission Line (cont.)Time-Harmonic Waves:1818Note that= series impedance/length

= parallel admittance/lengthThen we can write:

TEM Transmission Line (cont.)1919LetConvention:Solution:

ThenTEM Transmission Line (cont.) is called the "propagation constant."

2020TEM Transmission Line (cont.)

Forward travelling wave (a wave traveling in the positive z direction):

The wave repeats when:Hence:2121Phase VelocityTrack the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest.

vp(phase velocity)2222Phase Velocity (cont.)

SetHence

In expanded form:2323Characteristic Impedance Z0

so

+ V+(z)- I+ (z)zA wave is traveling in the positive z direction.(Z0 is a number, not a function of z.)2424Use Telegraphers Equation:

so

Hence

Characteristic Impedance Z0 (cont.) 2525From this we have:UsingWe have

Characteristic Impedance Z0 (cont.)

Note: The principal branch of the square root is chosen, so that Re (Z0) > 0. 2626

Note:wave in +z directionwave in -z direction

General Case (Waves in Both Directions)2727Backward-Traveling Wave

so+ V -(z)- I - (z)zA wave is traveling in the negative z direction.Note: The reference directions for voltage and current are the same as for the forward wave. 2828General Case

A general superposition of forward and backward traveling waves:Most general case:Note: The reference directions for voltage and current are the same for forward and backward waves. 29+ V (z)- I (z)z29

guided wavelength gphase velocity vpSummary of Basic TL formulas30Lossless Case

so

(indep. of freq.)(real and indep. of freq.)3131Lossless Case (cont.)

In the medium between the two conductors is homogeneous (uniform) and is characterized by (, ), then we have that

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). (proof given later)3232

Where do we assign z = 0?

The usual choice is at the load.

Terminating impedance (load)Ampl. of voltage wave propagating in negative z direction at z = 0.Ampl. of voltage wave propagating in positive z direction at z = 0.Terminated Transmission LineNote: The length l measures distance from the load:

33What if we know

Terminated Transmission Line (cont.)

HenceCan we use z = - l as a reference plane?

Terminating impedance (load)34


Compare:Note: This is simply a change of reference plane, from z = 0 to z = -l.


What is V(-l )?

propagating forwardspropagating backwardsTerminated Transmission Line (cont.)l distance away from loadThe current at z = - l is then


Total volt. at distance l from the loadAmpl. of volt. wave prop. towards load, at the load position (z = 0).Similarly,Ampl. of volt. wave prop. away from load, at the load position (z = 0).

L Load reflection coefficientTerminated Transmission Line (cont.)

l Reflection coefficient at z = - l37

Input impedance seen looking towards load at z = -l .


38At the load (l = 0):

Thus,


Recall39Simplifying, we have


Hence, we have40

Impedance is periodic with period g/2

Terminated Lossless Transmission Line

Note:

tan repeats when

41For the remainder of our transmission line discussion we will assume that the transmission line is lossless.

Terminated Lossless Transmission Line

42Matched load: (ZL=Z0)

For any l No reflection from the load AMatched Load

43Short circuit load: (ZL = 0)

Always imaginary!Note:B

S.C. can become an O.C. with a g/4 trans. line

Short-Circuit Load

44Using Transmission Lines to Synthesize Loads

A microwave filter constructed from microstrip.This is very useful is microwave engineering.45

ExampleFind the voltage at any point on the line.46Note:

At l = d :HenceExample (cont.)

47Some algebra:

Example (cont.) 48

where

Example (cont.) Therefore, we have the following alternative form for the result:Hence, we have49

Example (cont.)

Voltage wave that would exist if there were no reflections from the load (a semi-infinite transmission line or a matched load). 50

Example (cont.)

Wave-bounce method (illustrated for l = d ):51Example (cont.)

Geometric series:

52Example (cont.) or

This agrees with the previous result (setting l = d ).Note: This is a very tedious method not recommended. Hence53

At a distance l from the load:

If Z0 real (low-loss transmission line)Time- Average Power Flow

Note: 54Low-loss line

Lossless line ( = 0)Time- Average Power Flow

55

Quarter-Wave Transformer

so

HenceThis requires ZL to be real.ZLZ0Z0TZin56

Voltage Standing Wave Ratio

57Coaxial CableHere we present a case study of one particular transmission line, the coaxial cable. a b

Find C, L, G, RWe will assume no variation in the z direction, and take a length of one meter in the z direction in order top calculate the per-unit-length parameters. 58For a TEMz mode, the shape of the fields is independent of frequency, and hence we can perform the calculation using electrostatics and magnetostatics. Coaxial Cable (cont.)-l0l0ab

Find C (capacitance / length)Coaxial cableh = 1 [m]

From Gausss law:

59-l0l0ab

Coaxial cableh = 1 [m]

HenceWe then have

Coaxial Cable (cont.)60

Find L (inductance / length)From Amperes law: Coaxial cableh = 1 [m]

I

ShI I zcenter conductorMagnetic flux:Coaxial Cable (cont.)61Note: We ignore internal inductance here, and only look at the magnetic field between the two conductors (accurate for high frequency.Coaxial cableh = 1 [m]

I

HenceCoaxial Cable (cont.)62

Observation:

This result actually holds for any transmission line.Coaxial Cable (cont.)63

For a lossless cable:

Coaxial Cable (cont.)64-l0l0ab

Find G (conductance / length)Coaxial cableh = 1 [m]

From Gausss law:

Coaxial Cable (cont.)65-l0l0ab

We then have

orCoaxial Cable (cont.)66Observation:

This result actually holds for any transmission line.

Coaxial Cable (cont.)67

To be more general:

Note: It is the loss tangent that is usually (approximately) constant for a material, over a wide range of frequencies.Coaxial Cable (cont.)As just derived,The loss tangent actually arises from both conductivity loss and polarization loss (molecular friction loss), ingeneral.68This is the loss tangent that would arise from conductivity effects.General expression for loss tangent:

Effective permittivity that accounts for conductivityLoss due to molecular frictionLoss due to conductivityCoaxial Cable (cont.)69Find R (resistance / length)Coaxial cableh = 1 [m]Coaxial Cable (cont.)

ab

Rs = surface resistance of metal70General Transmission Line Formulas

(1)(2)(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.

(4)Equation (4) can be used to find R (discussed later).

71General Transmission Line Formulas (cont.)

Al four per-unit-length parameters can be found from

72Common Transmission Lines

CoaxTwin-lead

ab

h

aa73Common Transmission Lines (cont.) Microstrip

hwert74Common Transmission Lines (cont.) Microstrip

hwert

75At high frequency, discontinuity effects can become important. Limitations of Transmission-Line Theory

BendincidentreflectedtransmittedThe simple TL model does not account for the bend.ZTHZLZ0+-76At high frequency, radiation effects can become important. When will radiation occur?We want energy to travel from the generator to the load, without radiating. Limitations of Transmission-Line Theory (cont.)ZTHZLZ0+-77

abzThe 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 Transmission-Line Theory (cont.)78The 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.)+-Limitations of Transmission-Line Theory (cont.)Having fields that extend to infinity is not the same thing as having radiation, however. 79The infinite twin lead will not radiate by itself, regardless of how far apart the lines are.hincidentreflectedThe incident and reflected waves represent an exact solution to Maxwells equations on the infinite line, at any frequency.

S

Limitations of Transmission-Line Theory (cont.)No attenuation on an infinite lossless line80A discontinuity on the twin lead will cause radiation to occur.Note: Radiation effects increase as the frequency increases.Limitations of Transmission-Line Theory (cont.)hIncident wavepipeObstacleReflected waveBendhIncident wavebendReflected wave81To reduce radiation effects of the twin lead at discontinuities:hReduce the separation distance h (keep h

notes 1 - transmission line theory.pptx

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