transmission lines

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Transmission Line TheoryTransmission Line TheoryIn an electronic system, the delivery of In an electronic system, the delivery of

power requires the connection of two wires power requires the connection of two wires between the source and the load. At low between the source and the load. At low frequencies, power is considered to be delivered frequencies, power is considered to be delivered to the load through the wire. to the load through the wire.

In the microwave frequency region, power In the microwave frequency region, power is considered to be in electric and magnetic is considered to be in electric and magnetic fields that are guided from lace to place by some fields that are guided from lace to place by some physical structure. Any physical structure that physical structure. Any physical structure that will guide an electromagnetic wave place to will guide an electromagnetic wave place to place is called a place is called a Transmission LineTransmission Line..

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Types of Transmission LinesTypes of Transmission Lines

1.1. Two wire lineTwo wire line2.2. Coaxial cableCoaxial cable3.3. WaveguideWaveguide

RectangularRectangular CircularCircular

4.4. Planar Transmission LinesPlanar Transmission Lines Strip lineStrip line Microstrip lineMicrostrip line Slot lineSlot line Fin lineFin line Coplanar WaveguideCoplanar Waveguide Coplanar slot lineCoplanar slot line

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Analysis of differences between Low and High Analysis of differences between Low and High FrequencyFrequency

At low frequencies, the circuit elements are lumped since At low frequencies, the circuit elements are lumped since voltage and current waves affect the entire circuit at the voltage and current waves affect the entire circuit at the same time.same time.

At microwave frequencies, such treatment of circuit At microwave frequencies, such treatment of circuit elements is not possible since voltag and current waves elements is not possible since voltag and current waves do not affect the entire circuit at the same time.do not affect the entire circuit at the same time.

The circuit must be broken down into unit sections within The circuit must be broken down into unit sections within which the circuit elements are considered to be lumped.which the circuit elements are considered to be lumped.

This is because the dimensions of the circuit are This is because the dimensions of the circuit are comparable to the wavelength of the waves according to comparable to the wavelength of the waves according to the formula:the formula:

c/fc/fwhere,where,c = velocity of light c = velocity of light f = frequency of voltage/currentf = frequency of voltage/current

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Metallic Cable Metallic Cable Transmission MediaTransmission Media

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Metallic Cable Transmission MediaMetallic Cable Transmission Media• Metallic transmission linesMetallic transmission lines

• Balanced and Unbalanced Transmission LinesBalanced and Unbalanced Transmission Lines

• Metallic Transmission Line Equivalent CircuitMetallic Transmission Line Equivalent Circuit

• Wave Propagation on a Metallic Transmission LineWave Propagation on a Metallic Transmission Line

• Transmission Line LossesTransmission Line Losses

• Phasor Current and VoltagesPhasor Current and Voltages

• Single section of transmission lineSingle section of transmission line

• Characteristic Impedance and Propagation Characteristic Impedance and Propagation ConstantConstant

• Standing waves, reflectionStanding waves, reflection

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Types of Transmission LinesTypes of Transmission Lines

CoaxialCoaxialTwisted-PairTwisted-PairOpen-WireOpen-WireTwin-LeadTwin-Lead

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Metallic transmission linesMetallic transmission lines

Open-wireOpen-wire Twin leadTwin lead

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Unshielded twisted-pairUnshielded twisted-pair

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Coaxial cableCoaxial cable

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Coaxial componentsCoaxial components ConnectorsConnectors: Microwave coaxial connectors required to connect : Microwave coaxial connectors required to connect

two coaxial lines are als called two coaxial lines are als called connector pairs (male connector pairs (male andand female). female). They must match the characteristic impedance of the attached lines They must match the characteristic impedance of the attached lines and be designed to have minimum reflection coefficients and not and be designed to have minimum reflection coefficients and not radiate power through the connector. radiate power through the connector. E.g. APC-3.5, BNC, SMA, E.g. APC-3.5, BNC, SMA, SMCSMC

Coaxial sectionsCoaxial sections: Coaxial line sections slip inside each other while : Coaxial line sections slip inside each other while still making electrical contact. These sections are useful for matching still making electrical contact. These sections are useful for matching loads and making slotted line measurements. Double and triple stub loads and making slotted line measurements. Double and triple stub tuning configurations are available as coaxial stub tuning sections.tuning configurations are available as coaxial stub tuning sections.

AttenuatorsAttenuators: The function of an attenuator is to reduce the power of : The function of an attenuator is to reduce the power of the signal through it by a fixed or adjustable amount. The different the signal through it by a fixed or adjustable amount. The different types of attenuators are:types of attenuators are:1.1. Fixed attenuatorsFixed attenuators2.2. Step attenuatorsStep attenuators3.3. Variable attenuatorsVariable attenuators

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Coaxial components (contd.)Coaxial components (contd.)

Coaxial cavities: Coaxial cavities are concentric Coaxial cavities: Coaxial cavities are concentric lines or coaxial lines with an air dielectric and lines or coaxial lines with an air dielectric and closed ends. Propagation of EM waves is in TEM closed ends. Propagation of EM waves is in TEM mode.mode.

Coaxial wave meters: Wave meters use a cavity Coaxial wave meters: Wave meters use a cavity to allow the transmission or absorption of a to allow the transmission or absorption of a wave at a frequency equal to the resonant wave at a frequency equal to the resonant frequency of the cavity. Coaxial cavities are frequency of the cavity. Coaxial cavities are used as wave meters.used as wave meters.

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AttenuatorsAttenuatorsAttenuators are components that reduce the amount of power Attenuators are components that reduce the amount of power

a fixed amount, a variable amount or in a series of fixed steps from a fixed amount, a variable amount or in a series of fixed steps from the input to the output of the device. They operate on the principle the input to the output of the device. They operate on the principle of interfering with the electric field or magnetic field or both.of interfering with the electric field or magnetic field or both.

Slide vane attenuatorsSlide vane attenuators: They work on the principle that a resistive : They work on the principle that a resistive material placed in parallel with the E-lines of a field current will material placed in parallel with the E-lines of a field current will induce a current in the material that will result in induce a current in the material that will result in II22RR power loss. power loss.

Flap attenuatorFlap attenuator: A flap attenuator has a vane that is dropped into the : A flap attenuator has a vane that is dropped into the waveguide through a slot in the top of the guide. The further the waveguide through a slot in the top of the guide. The further the vane is inserted into the waveguide, the greater the attenuation. vane is inserted into the waveguide, the greater the attenuation.

Rotary vane attenuatorRotary vane attenuator: It is a precision waveguide attenuator in : It is a precision waveguide attenuator in which attenuation follows a mathematical law. In this device, which attenuation follows a mathematical law. In this device, attenuation is independent on frequency.attenuation is independent on frequency.

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IsolatorsIsolatorsMismatch or discontinuities cause energy to be Mismatch or discontinuities cause energy to be

reflected back down the line. Reflected energy is reflected back down the line. Reflected energy is undesirable. Thus, to prevent reflected energy from reaching undesirable. Thus, to prevent reflected energy from reaching the source, isolators are used.the source, isolators are used.

Faraday Rotational IsolatorFaraday Rotational Isolator: It combines ferrite material to : It combines ferrite material to shift the phase of an electromagnetic wave in its vicinity and shift the phase of an electromagnetic wave in its vicinity and attenuation vanes to attenuate an electric field that is parallel attenuation vanes to attenuate an electric field that is parallel to the resistive plane. to the resistive plane.

Resonant absorption isolatorResonant absorption isolator: A device that can be used for : A device that can be used for higher powers. It consists of a section of rectangular higher powers. It consists of a section of rectangular waveguide with ferrite material placed half way to the center waveguide with ferrite material placed half way to the center of the waveguide, along the axis of the guide.of the waveguide, along the axis of the guide.

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Balanced lines have equal impedances from the two Balanced lines have equal impedances from the two conductors to groundconductors to ground

Twisted-pair and parallel lines are usually balancedTwisted-pair and parallel lines are usually balanced

Differential, or balanced, transmission systemDifferential, or balanced, transmission system

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Metallic transmission linesMetallic transmission linesDifferential, or balanced, transmission systemDifferential, or balanced, transmission system

signal voltagessignal voltages noise voltagesnoise voltages

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Unbalanced lines usually have one conductor groundedUnbalanced lines usually have one conductor groundedCoaxial lines usually have outer conductor groundedCoaxial lines usually have outer conductor grounded

Single-ended, or unbalanced, transmission systemSingle-ended, or unbalanced, transmission system

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Ideal Transmission LineIdeal Transmission Line

No lossesNo lossesconductors have zero resistanceconductors have zero resistancedielectric has zero conductancedielectric has zero conductancepossible only with superconductorspossible only with superconductorsapproximated by a short lineapproximated by a short line

No capacitance or inductanceNo capacitance or inductancepossible with a real line only at dcpossible with a real line only at dcwith low frequencies and short lines this with low frequencies and short lines this

can be approximated can be approximated

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Two-wire parallel transmission lineTwo-wire parallel transmission lineelectrical equivalent circuitelectrical equivalent circuit

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Traveling waveTraveling wave

coscv t V t is the angular frequency (rad/sec)is the angular frequency (rad/sec)

The input voltage can be described asThe input voltage can be described as

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Traveling waveTraveling wave

coscv t V t

, coscv z t V t z

is the angular frequency (rad/sec)is the angular frequency (rad/sec)

The input voltage can be described asThe input voltage can be described as

is the propagation constant (rad/m)is the propagation constant (rad/m)

, cosci z t I t z

current and voltage are in phase?!?!?current and voltage are in phase?!?!?

The traveling wave can be described asThe traveling wave can be described as

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Phase velocity and wavelengthPhase velocity and wavelength

coscv t V t z 2 2

distance1timepv f

f

2 f

The energy travels with the group velocityThe energy travels with the group velocityg

dv

d

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AttenuationAttenuation

, coszcv z t V e t z

is the attenuation coefficient is the attenuation coefficient (nepers/meter)(nepers/meter)

What is the attenuation in dB per What is the attenuation in dB per meter?meter?

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AttenuationAttenuation

, coszcv z t V e t z

is the attenuation coefficient (nepers/meter)is the attenuation coefficient (nepers/meter)

What is the attenuation in dB per meter?What is the attenuation in dB per meter?

20log 20 log 8.686dBm

Att e e

(One neper is 8.686 dB)(One neper is 8.686 dB)

log logga ax g x

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Phasor currents and voltagesPhasor currents and voltages

A phasor can be used to represent the amplitude of a A phasor can be used to represent the amplitude of a sinusoidal voltage or current and is phase difference from sinusoidal voltage or current and is phase difference from a reference sinusoid of the same frequency. A phasor a reference sinusoid of the same frequency. A phasor does not include any representation of the frequency.does not include any representation of the frequency.

cos Rez z j z j tv t Ve t z Ve e e

has a phasor has a phasor VV which can be represented in which can be represented in amplitude-angle form as amplitude-angle form as VV, , or in component or in component form form a+a+jjb b where where a=Va=Vcoscos and and b=Vb=Vsinsin or in complex- or in complex-exponential formexponential form jVe

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The phasor of the driving voltage is The phasor of the driving voltage is VV00

The phasor of the voltage at distance The phasor of the voltage at distance xx from the driving point isfrom the driving point is

jj xx x xe e e e x 0 0 0V V V V

where where is the propagation constant is the propagation constant


j

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z

z

e

e

x 0

x 0

V V

I I

Remember, Remember, II0 0 and and VV0 0 are themselves phasors, are themselves phasors,

and their angles are not necessarily the same.and their angles are not necessarily the same.


j

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Transmission Line ModelTransmission Line Model

At low frequencies only resistance has to At low frequencies only resistance has to be consideredbe considered

At higher frequencies capacitance and At higher frequencies capacitance and inductance must be includedinductance must be included

All of these are distributed along the lineAll of these are distributed along the line

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Iz

GzCz

Iz

Iz+I

Iz+I

Vz Vz+V

Total series resistance Total series resistance RRzzTotal series inductance Total series inductance LLzz

jR z L z z zV I I

jd

R Ldz

zz

VI

jd

G Cdz

zz

IV

jR L z zV I

jG C z zI V

Single section of transmission lineSingle section of transmission line

C

L

1

j

j

zz

zz

e

e

ZC

Z L

0

0

V V

I I

I

jI G z C z z zV V

V

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Characteristic ImpedanceCharacteristic Impedance

Ratio between voltage and current on lineRatio between voltage and current on lineDepends only on line geometry and Depends only on line geometry and

dielectricdielectricNot a function of lengthNot a function of lengthHas units of ohms but not the same as Has units of ohms but not the same as

the resistance of the wire in the linethe resistance of the wire in the line

0Z z

z

V

I

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jR L z zV I jG C z zI V

0

j

j

R LZ

G C

z

z

V

I

Characteristic ImpedanceCharacteristic Impedance

j

j

R L

G C

z z

z z

V I

I V

ZZ00 is the characteristic impedance is the characteristic impedance

For an RF line For an RF line RR and and GG are zero (valid for high RF frequencies) are zero (valid for high RF frequencies)

0

LZ

C z

z

V

I Current and voltage are in phaseCurrent and voltage are in phase

RR = conductor resistance in = conductor resistance in /unit length/unit length

LL = inductance in H/unit length = inductance in H/unit length

G G = dielectric conductance in S/unit length= dielectric conductance in S/unit length

CC = capacitance in F/unit length = capacitance in F/unit length

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Velocity FactorVelocity Factor

Step moves down line at a finite speedStep moves down line at a finite speedVelocity cannot be greater than speed Velocity cannot be greater than speed

of light and is usually lowerof light and is usually lowerVelocity factor is ratio between actual Velocity factor is ratio between actual

propagation velocity and speed of lightpropagation velocity and speed of lightVelocity factor depends only on line Velocity factor depends only on line

dielectricdielectric

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Velocity FactorVelocity Factor

vvpp = propagation velocity on the line = propagation velocity on the line

cc = speed of light in vacuum = speed of light in vacuum

= = 300 300 10 1066 m/s m/s

c

vv pf

0 0

1c

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Propagation ConstantPropagation Constant

jR L z zV I

jG C z zI V

2 j jR L G C z z z zV I V I

j jR L G C

For an For an ideal lineideal line RR and and GG are zero are zero

j LC purely imaginary and no attenuationpurely imaginary and no attenuation

j

0

LC 1

pvLC

pv

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Metallic transmission linesMetallic transmission linesTwo-wire parallel transmission lineTwo-wire parallel transmission line

0 276logD

Zr

ZZ00 = the characteristic impedance (ohms) = the characteristic impedance (ohms)

D D = the distance between the centers = the distance between the centersr r = the radius of the conductor = the radius of the conductor00 = the permittivity of free space (F/m) = the permittivity of free space (F/m)

rr = the relative permittivity or dielectric constant of = the relative permittivity or dielectric constant of

the medium (unitless)the medium (unitless)00 = the permeability of free space (H/m) = the permeability of free space (H/m)

1p

o

v

0r

0

1

o

c

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Metallic transmission linesMetallic transmission linesCoaxial cableCoaxial cable

0

138log

r

DZ

d

ZZ00 = the characteristic impedance (ohms) = the characteristic impedance (ohms)

DD = the diameter of the outer conductor = the diameter of the outer conductord d = the diameter of the inner conductor = the diameter of the inner conductor = the permittivity of the material= the permittivity of the materialrr = the relative permittivity or dielectric constant = the relative permittivity or dielectric constant

of the mediumof the medium00 = the permeability of free space = the permeability of free space

1p

o

v

0r

0

1

o

c

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Transmission Lines LossesTransmission Lines Losses• Conductor LossesConductor Losses

•Increases with frequency Increases with frequency due to skin effectdue to skin effect

• Dielectric Heating LossesDielectric Heating Losses•Also increases with Also increases with frequencyfrequency

• Radiation LossesRadiation Losses• Not significant with good Not significant with good quality coax properly quality coax properly installedinstalled• Can be a problem with Can be a problem with open-wire cableopen-wire cable

• Coupling LossesCoupling Losses• CoronaCorona Skin effect

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Transmission Lines LossesTransmission Lines Losses

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Step Applied to Infinite LineStep Applied to Infinite Line

Voltage step will propagate down lineVoltage step will propagate down lineEnergy is stored in line capacitance and Energy is stored in line capacitance and

inductanceinductanceEnergy from source appears to be dissipated Energy from source appears to be dissipated

by line but is really storedby line but is really stored If line is infinitely long the step never reaches If line is infinitely long the step never reaches

the endthe endVoltage and current have definite, finite Voltage and current have definite, finite

valuesvalues

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Reflection of Voltage StepReflection of Voltage Step

Infinite line: no reflectionInfinite line: no reflectionFinite line with load impedance Finite line with load impedance ZZLL = = ZZ00

no reflectionno reflection the load looks to the source like an extension the load looks to the source like an extension

of the lineof the lineVoltage and currents are compatibleVoltage and currents are compatible Z = Z = √√(L/C)(L/C)

Finite line with load impedance Finite line with load impedance ZZLL ZZ00

Some or all of the step will reflect from the Some or all of the step will reflect from the load end of the lineload end of the line

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Reflection of PulsesReflection of PulsesTransmission LineTransmission Line

Short circuitShort circuit

R0

ReflectionReflectionhyperlinkhyperlink

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Shorted LineShorted Line

Total voltage at shorted end = 0Total voltage at shorted end = 0 Incident and reflected voltages must be Incident and reflected voltages must be

equal and oppositeequal and opposite Incident and reflected currents are equal Incident and reflected currents are equal

with same polaritywith same polarity Time for surge to reach end of line isTime for surge to reach end of line is

T = L/vT = L/vpp

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Open LineOpen Line

Transmission LineTransmission LineR0

Reflection of PulsesReflection of Pulses

ReflectionReflectionhyperlinkhyperlink

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Open-Circuited LineOpen-Circuited Line

Total current at open end = 0Total current at open end = 0 Incident and reflected currents must be Incident and reflected currents must be

equal and oppositeequal and opposite Incident and reflected voltages are equal Incident and reflected voltages are equal

with same polaritywith same polarityTime for surge to reach end of line isTime for surge to reach end of line is

T = L/vT = L/vpp

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Reflection CoefficientReflection CoefficientMore complex situation: Load has an arbitrary More complex situation: Load has an arbitrary

impedanceimpedancenot equal to not equal to ZZ00

not shorted or opennot shorted or open impedance may be complex (either capacitive impedance may be complex (either capacitive

or inductive as well as resistive)or inductive as well as resistive)When the ZL ≠ Z0, part of the power is reflected

back and the remainder is absorbed by the load.

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Reflection CoefficientReflection Coefficient

r r

i i

V Ior

V I

= reflection coefficient= reflection coefficientVVii = incident voltage= incident voltage

VVrr = reflected voltage= reflected voltage

IIii = incident current= incident current

IIrr = reflected current= reflected current

The amount of voltage reflected back is The amount of voltage reflected back is called called voltage reflection coefficient.voltage reflection coefficient.

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ZZLL

Transmission LineTransmission Line

ZZ00

Reflection of PulsesReflection of Pulses

r r

i i

V Ior

V I

total voltage i rV V

total current i rI I

i rL

i r

V VZ

I I

0i r

i r

V VZ

I I

0

0

L

L

Z Z

Z Z

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Wave Propagation on LinesWave Propagation on Lines

Start by assuming a matched lineStart by assuming a matched line Waves move down the line at propagation Waves move down the line at propagation

velocityvelocity Waves are the same at all points except Waves are the same at all points except

for phasefor phase Phase changes 360 degrees in the Phase changes 360 degrees in the

distance a wave travels in one perioddistance a wave travels in one period This distance is called the wavelengthThis distance is called the wavelength

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Standing WavesStanding Waves When an incident wave reflects from a When an incident wave reflects from a

mismatched load, an interference pattern developsmismatched load, an interference pattern develops Both incident and reflected waves move at the Both incident and reflected waves move at the

propagation velocity, but the interference pattern propagation velocity, but the interference pattern is stationaryis stationary

The interference pattern is called a set of standing The interference pattern is called a set of standing waveswaves

It is formed by the addition of incident and It is formed by the addition of incident and reflected waves and has nodal points that remain reflected waves and has nodal points that remain stationary with timestationary with time

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Incident and Reflected WavesIncident and Reflected Waves

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Standing WavesStanding Waves

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Standing-Wave RatioStanding-Wave Ratio

When line is mismatched but neither When line is mismatched but neither open nor shorted, voltage varies open nor shorted, voltage varies along line without ever falling to zeroalong line without ever falling to zero

Greater mismatch leads to greater Greater mismatch leads to greater variationvariation

Voltage standing-wave ratio (VSWR or Voltage standing-wave ratio (VSWR or SWR) is defined:SWR) is defined:

min

max

V

VSWR ( ( 1) 1)

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Standing wavesStanding waves

max

min

VSWR

V

max i r i iV V V V V

min i r i iV V V V V

0

0

1 or

1L

L

Z ZSWR

Z Z

1

1

SWR

SWR

0

0

L

L

Z Z

Z Z

( 1)

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SWR and Reflection CoefficientSWR and Reflection Coefficient

SWR is a positive real numberSWR is a positive real number may be positive, negative or complexmay be positive, negative or complexSWR SWR 1 1Magnitude of Magnitude of 1 1

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Standing waves on an Open LineStanding waves on an Open Line

This is only the amplitude!!!This is only the amplitude!!!

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Standing waves on an Shorted LineStanding waves on an Shorted Line

This is only the amplitude!!!This is only the amplitude!!!

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Effects of High SWREffects of High SWR

High SWR causes voltage peaks on High SWR causes voltage peaks on the line that can damage the line or the line that can damage the line or connected equipment such as a connected equipment such as a transmittertransmitter

Current peaks due to high SWR cause Current peaks due to high SWR cause losses to increaselosses to increase

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Reflected PowerReflected Power

When a signal travels down a mismatched When a signal travels down a mismatched line, some of the power reflects from the loadline, some of the power reflects from the load

This power is dissipated in the source, if the This power is dissipated in the source, if the source matches the linesource matches the line

A high SWR causes the load power to be A high SWR causes the load power to be reducedreduced

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Reflected PowerReflected Power

PPr r = reflected power= reflected power

PPii = incident power = incident power

PPLL = power delivered to load = power delivered to load

1

1

SWR

SWR

iL

iL

ir

PSWR

SWRP

PP

PP

2

2

2

)1(

4

)1(

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Time-Domain ReflectometryTime-Domain Reflectometry

2

v td

ReflectoReflectometermeter transmission linetransmission line

reflectionreflection

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General Input Impedance EquationGeneral Input Impedance Equation

Input impedance of a transmission line at a Input impedance of a transmission line at a distance distance L L from the load impedance Zfrom the load impedance ZL L with with

a characteristic Za characteristic Zoo is is

Zinput = Zinput = ZZo o [(Z[(ZLL + j Z + j Zoo BL) BL)

(Z(Zoo + j Z + j ZLL BL)] BL)]

where B is called phase constant or where B is called phase constant or wavelength constant and is defined by the wavelength constant and is defined by the equation equation

B = 2B = 2

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Half and Quarter wave transmission lines

The relationship of the input impedance at the input of the half-wave transmission line with its terminating impedance is got by letting L = in the impedance equation.

Zinput = ZL

The relationship of the input impedance at the input of the quarter-wave transmission line with its terminating impedance is got by letting L = in the impedance equation.

Zinput = √(Zinput Zoutput)

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Effect of Lossy line on V and I wavesEffect of Lossy line on V and I waves The effect of resistance in a transmission line is to The effect of resistance in a transmission line is to

continuously reduce the amplitude of both incident and continuously reduce the amplitude of both incident and reflected voltage and current waves.reflected voltage and current waves.

Skin Effect: As frequency increases, depth of Skin Effect: As frequency increases, depth of penetration into adjacent conductive surfaces decreases penetration into adjacent conductive surfaces decreases for boundary currents associated with electromagnetic for boundary currents associated with electromagnetic waves, that results in the confinement of the voltage and waves, that results in the confinement of the voltage and current waves at the boundary of the transmission line, current waves at the boundary of the transmission line, thus making the transmission more lossy.thus making the transmission more lossy.

Skin depth (m) = 1 Skin depth (m) = 1 √√ffwhere f = frequency, Hzwhere f = frequency, Hz = permeability, H/m= permeability, H/m

= conductivity, S/m= conductivity, S/m

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Smith chart For complex transmission line problems, the use

of the formulae becomes increasingly difficult and inconvenient. An indispensable graphical method of solution is the use of Smith Chart.

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Components of a Smith ChartComponents of a Smith Chart HHorizontal lineorizontal line: The horizontal line running through : The horizontal line running through

the center of the Smith chart represents either the the center of the Smith chart represents either the resistive or the conductive component. Zero resistive or the conductive component. Zero resistance is located on the left end and infinite resistance is located on the left end and infinite resistance is located on the right end of the line.resistance is located on the right end of the line.

CCircles of constant resistance and conductanceircles of constant resistance and conductance: : Circles of constant resistance are drawn on the Smith Circles of constant resistance are drawn on the Smith chart tangent to the right-hand side of the chart and chart tangent to the right-hand side of the chart and its intersection with the centerline. These circles of its intersection with the centerline. These circles of constant resistance are used to locate complex constant resistance are used to locate complex impedances.impedances.

LLines of constant reactanceines of constant reactance: Lines of constant : Lines of constant reactance are shown on the Smith chart with curves reactance are shown on the Smith chart with curves that start from a given reactance value on the outer that start from a given reactance value on the outer circle and end at the right-hand side of the center circle and end at the right-hand side of the center line.line.

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Type of Microwave problems that Smith Type of Microwave problems that Smith chart can be usedchart can be used

1.1. Plotting a complex impedance on a Smith chartPlotting a complex impedance on a Smith chart2.2. Finding VSWR for a given loadFinding VSWR for a given load3.3. Finding the admittance for a given impedanceFinding the admittance for a given impedance4.4. Finding the input impedance of a transmission line Finding the input impedance of a transmission line

terminated in a short or open.terminated in a short or open.

5.5. Finding the input impedance at any distance from a load ZFinding the input impedance at any distance from a load ZLL..6.6. Locating the first maximum and minimum from any loadLocating the first maximum and minimum from any load7.7. Matching a transmission line to a load with a single series Matching a transmission line to a load with a single series

stub.stub.8.8. Matching a transmission line with a single parallel stubMatching a transmission line with a single parallel stub9.9. Matching a transmission line to a load with two parallel Matching a transmission line to a load with two parallel

stubs.stubs.

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Plotting a Complex Impedance on a Plotting a Complex Impedance on a Smith ChartSmith Chart

To locate a complex impedance, Z = R+-jX or To locate a complex impedance, Z = R+-jX or admittance Y = G +- jB on a Smith chart, admittance Y = G +- jB on a Smith chart, normalize the real and imaginary part of the normalize the real and imaginary part of the complex impedance. Locating the value of the complex impedance. Locating the value of the normalized real term on the horizontal line normalized real term on the horizontal line scale locates the resistance circle. Locating scale locates the resistance circle. Locating the normalized value of the imaginary term on the normalized value of the imaginary term on the outer circle locates the curve of constant the outer circle locates the curve of constant reactance. The intersection of the circle and reactance. The intersection of the circle and the curve locates the complex impedance on the curve locates the complex impedance on the Smith chart.the Smith chart.

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Finding the VSWR for a given Finding the VSWR for a given loadload

Normalize the load and plot its location on Normalize the load and plot its location on the Smith chart.the Smith chart.

Draw a circle with a radius equal to the Draw a circle with a radius equal to the distance between the 1.0 point and the distance between the 1.0 point and the location of the normalized load and the location of the normalized load and the center of the Smith chart as the center.center of the Smith chart as the center.

The intersection of the right-hand side of The intersection of the right-hand side of the circle with the horizontal resistance the circle with the horizontal resistance line locates the value of the VSWR.line locates the value of the VSWR.

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Finding the Input Impedance at any Finding the Input Impedance at any Distance from the LoadDistance from the Load

The load impedance is first normalized and is The load impedance is first normalized and is located on the Smith chart.located on the Smith chart.

The VSWR circle is drawn for the load.The VSWR circle is drawn for the load. A line is drawn from the 1.0 point through the A line is drawn from the 1.0 point through the

load to the outer wavelength scale.load to the outer wavelength scale. To locate the input impedance on a Smith To locate the input impedance on a Smith

chart of the transmission line at any given chart of the transmission line at any given distance from the load, advance in clockwise distance from the load, advance in clockwise direction from the located point, a distance in direction from the located point, a distance in wavelength equal to the distance to the new wavelength equal to the distance to the new location on the transmission line.location on the transmission line.

70

Power LossPower Loss Return Power LossReturn Power Loss: When an electromagnetic : When an electromagnetic

wave travels down a transmission line and wave travels down a transmission line and encounters a mismatched load or a encounters a mismatched load or a discontinuity in the line, part of the incident discontinuity in the line, part of the incident power is reflected back down the line. The power is reflected back down the line. The return loss is defined as:return loss is defined as:

PPreturnreturn = 10 log = 10 log1010 P Pii/P/Prr

PPreturnreturn = 20 log = 20 log1010 1/ 1/ Mismatch Power LossMismatch Power Loss: The term mismatch loss : The term mismatch loss

is used to describe the loss caused by the is used to describe the loss caused by the reflection due to a mismatched line. It is defined reflection due to a mismatched line. It is defined asas

PPmismatch mismatch = 10 log= 10 log10 10 PPii/(P/(Pii - P - Prr))

71

Notes:Notes: Metallic circuit currentMetallic circuit current – currents that flow in – currents that flow in

opposite directions in a balanced wire pairopposite directions in a balanced wire pair Longitudinal currentLongitudinal current – currents that flow in the – currents that flow in the

same directionsame direction Common Mode Rejection (CMR)Common Mode Rejection (CMR) – cancellation – cancellation

of common mode signals or noise interference of common mode signals or noise interference induced equally on both wires producing induced equally on both wires producing longitudinal currents that cancel in the load longitudinal currents that cancel in the load

CMRR = 40 to 70 dBCMRR = 40 to 70 dB

72

Notes:Notes: Primary electrical constantsPrimary electrical constants – R, L, C, G – R, L, C, G Secondary constantsSecondary constants – Zo, Propagation – Zo, Propagation

ConstantConstant For maximum power transfer, ZFor maximum power transfer, ZLL = Zo, thus no = Zo, thus no

reflectionreflection Characteristic impedance = Surge impedanceCharacteristic impedance = Surge impedance Transmission line stores energy in its Transmission line stores energy in its

distributed inductance and capacitancedistributed inductance and capacitance

73

Notes:Notes: Transmission lines:Transmission lines:

The input impedance of an infinitely long line at radio The input impedance of an infinitely long line at radio

frequencies is resistive and equal to Zofrequencies is resistive and equal to Zo Nonresonant – when electromagnetic waves travel Nonresonant – when electromagnetic waves travel

the line without reflectionsthe line without reflections Ratio of voltage to current at any point is equal to ZoRatio of voltage to current at any point is equal to Zo Incident voltage and current at any point are in phase Incident voltage and current at any point are in phase Line losses on a non-resonant line are minimum per Line losses on a non-resonant line are minimum per

unit lengthunit length Any transmission line that is terminated in a load Any transmission line that is terminated in a load

equals to Zo acts as if it were an infinite line. equals to Zo acts as if it were an infinite line. Prop. Cons. = attenuation coeff. + phase shift coeff.Prop. Cons. = attenuation coeff. + phase shift coeff.

γγ = = αα + j + jββ

74

Notes:Notes: Material Material Velocity FactorVelocity Factor

airair 0.95 – 0.9750.95 – 0.975

rubberrubber 0.56 – 0.65 0.56 – 0.65

polyethylene polyethylene 0.660.66

teflonteflon 0.700.70

teflon foamteflon foam 0.820.82

teflon pinsteflon pins 0.810.81

teflon spiralteflon spiral 0.810.81

75

Notes:Notes: Material Material Dielectric ConstantDielectric Constant

VacuumVacuum 11

AirAir 1.00061.0006

TeflonTeflon 2.12.1

polyethylenepolyethylene 2.272.27

polystyrenepolystyrene 2.52.5

paper, paraffinedpaper, paraffined 2.52.5

rubberrubber 3.03.0

PVCPVC 3.33.3

MicaMica 5.05.0

GlassGlass 7.57.5

76

Notes:Notes: Velocity factor (Velocity constant) = Velocity factor (Velocity constant) = actual vel. Of prop.actual vel. Of prop.

vel. In free spacevel. In free space

Vf = Vp / cVf = Vp / c Electrical length of transmission line Electrical length of transmission line

Long – length exceeds Long – length exceeds λλ/16/16 Short – length less than or equal Short – length less than or equal λλ/16 /16

Delay lines – transmission lines designed to intentionally Delay lines – transmission lines designed to intentionally introduce a time delay in the path of an electromagnetic introduce a time delay in the path of an electromagnetic wavewave

td = LC (seconds)td = LC (seconds)

td = 1.016 td = 1.016 ЄЄ

77

Notes:Notes:

The disadvantages of not having a matched line:The disadvantages of not having a matched line: 100 percent of the source incident power does not 100 percent of the source incident power does not

reach the loadreach the load The dielectric separating the two conductors can The dielectric separating the two conductors can

break down and cause corona due to high VSWRbreak down and cause corona due to high VSWR Reflections and rereflections cause more power lossReflections and rereflections cause more power loss Reflections cause ghost imagesReflections cause ghost images Mismatches cause noise interferenceMismatches cause noise interference

78

Notes:Notes:

Characteristics of transmission line terminated at openCharacteristics of transmission line terminated at open voltage incident wave is reflected back (no phase voltage incident wave is reflected back (no phase

reversal)reversal) current incident wave is reflected back 180 degrees current incident wave is reflected back 180 degrees

from how it would have continuedfrom how it would have continued sum of the incident and reflected current waveforms sum of the incident and reflected current waveforms

is minimumis minimum sum of the incident and reflected voltage waveforms sum of the incident and reflected voltage waveforms

is maximumis maximum

79

Notes:Notes:

Characteristics of transmission line terminated at shortCharacteristics of transmission line terminated at short voltage standing wave is reflected back 180 degrees voltage standing wave is reflected back 180 degrees

reversed from how it would have continued reversed from how it would have continued current standing wave is reflected back the same as current standing wave is reflected back the same as

if it had continuedif it had continued sum of the incident and reflected current waveforms sum of the incident and reflected current waveforms

is maximumis maximum sum of the incident and reflected voltage waveforms sum of the incident and reflected voltage waveforms

is zero at the shortis zero at the short

80

InputInputendend

OutputOutputendend

λλ/4/4

Zin = resistive, maxZin = resistive, max

Zin = resistive, minZin = resistive, min

Zin = inductiveZin = inductive

Zin = capacitiveZin = capacitive

Zin = capacitiveZin = capacitive

Zin = inductiveZin = inductive

shortshort

openopen

shortshort

openopen

shortshort

openopen

Input ImpedanceInput Impedance

Parallel LC circuit, Parallel LC circuit, resistive and maximum resistive and maximum

Series LC circuit, Series LC circuit, resistive and minimumresistive and minimum

inductor inductor

capacitorcapacitor

capacitorcapacitor

inductor inductor

81

Notes:Notes:

The impedance transformation for a quarter wavelength The impedance transformation for a quarter wavelength transmission line is:transmission line is: RRLL = Zo: quarter = Zo: quarter λλ line acts 1:1 turns ratio transformer line acts 1:1 turns ratio transformer

RRLL > Zo: quarter > Zo: quarter λλ line line acts as a step down transformer acts as a step down transformer

RRLL < Zo: quarter < Zo: quarter λλ line acts as a step up transformer line acts as a step up transformer

Characteristic Impedance of quarter wavelength X’formerCharacteristic Impedance of quarter wavelength X’former

Zo’ = Zo’ = √(Zo√(ZoZZLL))

When a load is purely inductive oir purely capacitive, no When a load is purely inductive oir purely capacitive, no energy is absorbed, thus, energy is absorbed, thus, ГГ = 1 and SWR = inf. = 1 and SWR = inf.

82

Notes:Notes:

Stub Matching Stub Matching Stubs are used to eliminate the reactive component Stubs are used to eliminate the reactive component

to match the transmission line to the load to match the transmission line to the load It is just a piece of additional transmission line that is It is just a piece of additional transmission line that is

placed across the primary line as close to the load as placed across the primary line as close to the load as possiblepossible

Susceptance of stub is used to tune out the Susceptance of stub is used to tune out the susceptance of the loadsusceptance of the load

Shorted stubs are preferred because open stubs Shorted stubs are preferred because open stubs have the tendency to radiate at higher frequencies have the tendency to radiate at higher frequencies

83

Notes:Notes:

Process of Stub Matching Process of Stub Matching locate a point as close to the load as possible where locate a point as close to the load as possible where

the conductive component of the input admittance is the conductive component of the input admittance is equal to the characteristic admittance of transmission equal to the characteristic admittance of transmission line Yin = G – jB, G = 1 / Zoline Yin = G – jB, G = 1 / Zo

Attach the shorted stub to the point on the Attach the shorted stub to the point on the transmission linetransmission line

Depending whether the reactive component at the Depending whether the reactive component at the point is inductive or capacitive, the stub length is point is inductive or capacitive, the stub length is adjustedadjusted

Yin = Go – jB + jBstubYin = Go – jB + jBstub

Yin = Go Yin = Go

84

Notes:Notes:

Time Domain Reflectometry (TDR)Time Domain Reflectometry (TDR) technique used to locate an impairment in the technique used to locate an impairment in the

metallic cablemetallic cable How much of the transmitted signal returns depends How much of the transmitted signal returns depends

on the type and magnitude of the impairmenton the type and magnitude of the impairment Impairment represents a discontinuity in the signalImpairment represents a discontinuity in the signal

For higher frequency applications (300 MHz – 3000 MHz), For higher frequency applications (300 MHz – 3000 MHz), microstrip and stripline is constructed to interconnect microstrip and stripline is constructed to interconnect components on PC boardscomponents on PC boards

When the distance between source and load ends is a When the distance between source and load ends is a few inches or less, coaxial cable is impracticalfew inches or less, coaxial cable is impractical

Microstrip and Stripline use the tracks on the PC board.Microstrip and Stripline use the tracks on the PC board.

85

Notes:Notes: Microstrip and Stripline are used to construct transmission Microstrip and Stripline are used to construct transmission

lines, inductors, capacitors, tuned circuits, filters, phase lines, inductors, capacitors, tuned circuits, filters, phase shifters, and impedance matching devices.shifters, and impedance matching devices.

Microstrip – when the lines are etched in the middle layer of Microstrip – when the lines are etched in the middle layer of the multilayer PC boardthe multilayer PC board

Zo = Zo = 87 87 ln ln 5.98h__ 5.98h__ ЄЄ fiberglass = 4.5 fiberglass = 4.5

√√((ЄЄ + 1.41) + 1.41) 0.8w + t 0.8w + t ЄЄ teflon = 3 teflon = 3

w = width of Cu tracew = width of Cu trace t = thickness of Cu trace t = thickness of Cu trace

h = thickness of dielectric h = thickness of dielectric Stripline – if the lines are etched onto the surface of the PC Stripline – if the lines are etched onto the surface of the PC

board only board only

Zo = Zo = 60 60 ln ln 4d __ 4d __ d = dielectric thick d = dielectric thick

ЄЄ 0.67 0.67ππww(0.8 + t/h)(0.8 + t/h)

88

Problems:Problems:1.1. Determine the characteristic impedance for an Determine the characteristic impedance for an

air dielectric two-wire parallel transmission air dielectric two-wire parallel transmission line with a D/r ratio = 13.5 line with a D/r ratio = 13.5 (311.97 ohms)(311.97 ohms)

2.2. Determine the characteristic impedance for an Determine the characteristic impedance for an RG-59A coaxial cable with parameters: RG-59A coaxial cable with parameters: L=0.121 L=0.121 μμH/ft, C=30 pF/ft, d=0.042 in., D=0.22 H/ft, C=30 pF/ft, d=0.042 in., D=0.22 in, and in, and ЄЄ=2.15 =2.15 (63.509 ohms, 67.685 ohms)(63.509 ohms, 67.685 ohms)

3.3. For a given length of RG8A/U coaxial cable For a given length of RG8A/U coaxial cable with parameters: C=98.4 pF/m, L=262.45 nH/m, with parameters: C=98.4 pF/m, L=262.45 nH/m, ЄЄr=2.15. Find Vp and Vf r=2.15. Find Vp and Vf (1.968x10(1.968x1088 m/s, 0.656 m/s, 0.656 or 0.682)or 0.682)

89

Problems:Problems:4.4. For a transmission line with incident voltage of For a transmission line with incident voltage of

5.2V and reflected voltage of 3.8V, find 5.2V and reflected voltage of 3.8V, find reflection coefficient and SWR reflection coefficient and SWR (0.731, 6.429)(0.731, 6.429)

5.5. Determine the physical length and Zo for a Determine the physical length and Zo for a quarter wavelength transformer that is used to quarter wavelength transformer that is used to match a section of RG8A/U (Zo=50 ohms) to a match a section of RG8A/U (Zo=50 ohms) to a 175 ohm resistive load. The frequency of 175 ohm resistive load. The frequency of operation is 220 MHz and the velocity factor is operation is 220 MHz and the velocity factor is 1 1 (0.341 m, 93.54 ohms)(0.341 m, 93.54 ohms)

90

Stub MatchingStub Matching Use to remove the reactive component of the complex Use to remove the reactive component of the complex

impedance of the load to match the transmission line to impedance of the load to match the transmission line to the loadthe load

It is a piece of additional transmission line that is placed It is a piece of additional transmission line that is placed across the primary line as close to the load as possibleacross the primary line as close to the load as possible

The susceptance of the stub is used to tune out the The susceptance of the stub is used to tune out the susceptance of the loadsusceptance of the load

Either a shorted or open stub is used with greater Either a shorted or open stub is used with greater preference on the shorted stubpreference on the shorted stub

A transmission line that is one-half wavelength or A transmission line that is one-half wavelength or shorter is used to tune out the reactive component of shorter is used to tune out the reactive component of the load the load

91

Stub Matching ProcessStub Matching Process1.1. Locate a point as close as possible to the load Locate a point as close as possible to the load

where the conductive component of the Zwhere the conductive component of the Zinin = Z = Zoo

YYinin = G – jB where G = 1 / Z = G – jB where G = 1 / Zoo

2.2. Attach the shorted stub on the identified pointAttach the shorted stub on the identified point

3.3. Depending on whether the reactive component at Depending on whether the reactive component at that point is inductive or capacitive, the stub length that point is inductive or capacitive, the stub length is adjusted accordinglyis adjusted accordingly

YYinin = G = Goo – jB + jB – jB + jBstubstub

~ Y~ Yinin = G = Goo

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