Transmission Lines

Transmission Line Theory In an electronic system, the delivery of power requires the connection of two wires between the source and the load. At low frequencies, power is considered to be delivered to the load through the wire. In the microwave frequency region, power is considered to be in electric and magnetic fields that are guided from lace to place by some physical structure. Any physical structure that will guide an electromagnetic wave place to place is called a Transmission Line. 1

Types of Transmission Lines 1. 2. 3.

Two wire line Coaxial cable Waveguide  

1.

Planar Transmission Lines      

2

Rectangular Circular Strip line Microstrip line Slot line Fin line Coplanar Waveguide Coplanar slot line

Analysis of differences between Low and High Frequency At low frequencies, the circuit elements are lumped since voltage and current waves affect the entire circuit at the same time.  At microwave frequencies, such treatment of circuit elements is not possible since voltag and current waves do not affect the entire circuit at the same time.  The circuit must be broken down into unit sections within which the circuit elements are considered to be lumped.  This is because the dimensions of the circuit are comparable to the wavelength of the waves according to the formula:     c/f where, c = velocity of light f = frequency of voltage/current 

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Metallic Cable Transmission Media

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Metallic Cable Transmission Media • Metallic transmission lines • Balanced and Unbalanced Transmission Lines • Metallic Transmission Line Equivalent Circuit • Wave Propagation on a Metallic Transmission Line • Transmission Line Losses • Phasor Current and Voltages • Single section of transmission line • Characteristic Impedance and Propagation Constant • Standing waves, reflection 5

Types of Transmission Lines Coaxial Twisted-Pair Open-Wire Twin-Lead

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Metallic transmission lines Open-wire

7

Twin lead

Metallic transmission lines Unshielded twisted-pair

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Metallic transmission lines Coaxial cable

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Coaxial components 

Connectors: Microwave coaxial connectors required to connect two



Coaxial sections: Coaxial line sections slip inside each other while



Attenuators: The function of an attenuator is to reduce the power of

coaxial lines are als called connector pairs (male and female). They must match the characteristic impedance of the attached lines and be designed to have minimum reflection coefficients and not radiate power through the connector. E.g. APC-3.5, BNC, SMA, SMC still making electrical contact. These sections are useful for matching loads and making slotted line measurements. Double and triple stub tuning configurations are available as coaxial stub tuning sections.

the signal through it by a fixed or adjustable amount. The different types of attenuators are: 1. Fixed attenuators 2. Step attenuators 3. Variable attenuators

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Coaxial components (contd.)  Coaxial cavities: Coaxial cavities are concentric

lines or coaxial lines with an air dielectric and closed ends. Propagation of EM waves is in TEM mode.

 Coaxial wave meters: Wave meters

use a cavity to allow the transmission or absorption of a wave at a frequency equal to the resonant frequency of the cavity. Coaxial cavities are used as wave meters. 11

Attenuators Attenuators are components that reduce the amount of power 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 of interfering with the electric field or magnetic field or both. 

Slide vane attenuators: They work on the principle that a resistive material placed in parallel with the E-lines of a field current will induce a current in the material that will result in I2R power loss.



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



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

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Isolators Mismatch or discontinuities cause energy to be reflected back down the line. Reflected energy is undesirable. Thus, to prevent reflected energy from reaching the source, isolators are used.  Faraday Rotational Isolator: It combines ferrite material to shift the phase of an electromagnetic wave in its vicinity and attenuation vanes to attenuate an electric field that is parallel to the resistive plane.  Resonant absorption isolator: A device that can be used for higher powers. It consists of a section of rectangular waveguide with ferrite material placed half way to the center of the waveguide, along the axis of the guide.

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Metallic transmission lines Differential, or balanced, transmission system

Balanced lines have equal impedances from the two conductors to ground Twisted-pair and parallel lines are usually balanced

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Metallic transmission lines Differential, or balanced, transmission system

signal voltages

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noise voltages

Metallic transmission lines Single-ended, or unbalanced, transmission system

Unbalanced lines usually have one conductor grounded Coaxial lines usually have outer conductor grounded

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Ideal Transmission Line  No losses  conductors

have zero resistance  dielectric has zero conductance  possible only with superconductors  approximated by a short line  No capacitance or inductance  possible with a real line only at dc  with low frequencies and short lines this can be approximated 17

Two-wire parallel transmission line electrical equivalent circuit

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Traveling wave The input voltage can be described as

v ( t ) = Vc cos ( ω t ) ω is the angular frequency (rad/sec)

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Traveling wave The input voltage can be described as

v ( t ) = Vc cos ( ω t ) ω is the angular frequency (rad/sec) The traveling wave can be described as

v ( z , t ) = Vc cos ( ω t − β z ) β is the propagation constant (rad/m)

i ( z , t ) = I c cos ( ω t − β z ) current and voltage are in phase?!?!? 20

Phase velocity and wavelength

v ( t ) = Vc cos ( ω t − β z )

βλ = 2π 2π β= λ ω = 2π f

distance λ ω vp = = = fλ = 1 time β f dω vg = The energy travels with the group velocity dβ 21

Attenuation v ( z , t ) = Vc e

−α z

cos ( ωt − β z )

α is the attenuation coefficient (nepers/meter) What is the attenuation in dB per meter?

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Attenuation

v ( z , t ) = Vc e

−α z

( )

log a x g = g log a ( x )

cos ( ωt − β z )

α is the attenuation coefficient (nepers/meter) What is the attenuation in dB per meter?

( )

AttdB = 20 log eα = 20α log ( e ) = 8.686α m

(One neper is 8.686 dB) 23

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

v ( t ) = Ve

−α z

cos ( ωt − β z ) = Re ( Ve

−α z − j β z

e

e

jωt

)

has a phasor V which can be represented in amplitude-angle form as V∠ φ , or in component form a+jb where a=Vcosφ and b=Vsinφ or in φ complex-exponential Ve jform 24

Phasor currents and voltages The phasor of the driving voltage is V0 The phasor of the voltage at distance x from the driving point is

Vx = V0 e

−α x − jβ x

e

= V0e

− ( α + jβ ) x

= V0e

where γ is the propagation constant γ = α + jβ

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−γ x

Phasor currents and voltages

Vx = V0 e

−γ z

γ = α + jβ

I x = I 0 e −γ z Remember, I0 and V0 are themselves phasors, and their angles are not necessarily the same.

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Transmission Line Model

 At low frequencies only resistance has to

be considered  At higher frequencies capacitance and inductance must be included  All of these are distributed along the line 27

Single section of transmission line Iz

I z +δ I

δ V

Vz = V0 e −γ z I z = I 0 e −γ z 1 jω C Z L = jω L ZC =

Vz

Cδ z

Iz

Gδ z

δ I

Vz + δ V

Total series resistance Rδ z Iz+δ Total series inductance Lδ z

I

δ V = − Rδ zI z − jω Lδ zI z dVz = − ( R + jω L ) I z γ Vz = − ( R + jω L ) I z dz δ I = −Gδ zVz − jω Cδ zVz dI z γ I z = − ( G + jω C ) Vz = − ( G + jωC ) Vz dz 28

Characteristic Impedance  Ratio between voltage and current on line  Depends only on line geometry and

dielectric  Not a function of length  Has units of ohms but not the same as the resistance of the wire in the line

Vz Z0 = Iz 29

Characteristic Impedance γ Vz = − ( R + jω L ) I z Vz R + jω L I z = I z G + jωC Vz

γ I z = − ( G + jω C ) Vz Vz R + jω L Z0 = = Iz G + jω C

Z0 is the characteristic impedance R = conductor resistance in Ω /unit length L = inductance in H/unit length G = dielectric conductance in S/unit length C = capacitance in F/unit length For an RF line R and G are zero (valid for high RF frequencies)

Vz L Z0 = = Iz C 30

Current and voltage are in phase

Velocity Factor  Step moves down line at a finite speed  Velocity cannot be greater than speed

of light and is usually lower  Velocity factor is ratio between actual propagation velocity and speed of light  Velocity factor depends only on line dielectric 31

c=

Velocity Factor

vf =  vp

vp c

= propagation velocity on the line

 c = speed of light in vacuum

= 300 × 106 m/s

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1 µ 0ε 0

Propagation Constant

γ Vz = − ( R + jω L ) I z

γ = α + jβ ω vp = β

γ I z = − ( G + jω C ) Vz

γ 2 Vz I z = ( R + jω L ) ( G + jω C ) Vz Iz γ=

( R + jω L ) ( G + jω C )

For an ideal line R and G are zero

γ = jω LC α =0

purely imaginary and no attenuation

β = ω LC 33

vp =

1 LC

Metallic transmission lines Two-wire parallel transmission line

D Z 0 = 276 log r 1 vp = µ oε

ε = ε rε 0 c=

Z0 = the characteristic impedance (ohms) D = the distance between the centers r = the radius of the conductor ε 0 = the permittivity of free space (F/m) ε r = the relative permittivity or dielectric constant of the medium (unitless) µ 0 = the permeability of free space (H/m) 34

1 µ oε 0

Metallic transmission lines Coaxial cable

138 D Z0 = log d εr 1 vp = µ oε

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Z0 = the characteristic impedance (ohms) D = the diameter of the outer conductor d = the diameter of the inner conductor ε = the permittivity of the material ε r = the relative permittivity or dielectric constant of the medium µ 0 = the permeability of free space

ε = ε rε 0 c=

1 µ oε 0

Transmission Lines Losses • Conductor Losses

•Increases with frequency due to skin effect

• Dielectric Heating Losses •Also increases with frequency

• Radiation Losses

• Not significant with good quality coax properly installed • Can be a problem with open-wire cable

• Coupling Losses • Corona 36

Skin effect

Transmission Lines Losses

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Step Applied to Infinite Line  Voltage step will propagate down line  Energy is stored in line capacitance and

inductance  Energy from source appears to be dissipated by line but is really stored  If line is infinitely long the step never reaches the end  Voltage and current have definite, finite values

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Reflection of Voltage Step  Infinite line: no reflection  Finite line with load impedance ZL

= Z0

 no

reflection  the load looks to the source like an extension of the line  Voltage and currents are compatible 

Z = √(L/C)

≠ Z0  Some or all of the step will reflect from the load end of the line

 Finite line with load impedance ZL

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Reflection of Pulses Transmission Line R0 Short circuit

Reflection hyperlink

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

equal and opposite  Incident and reflected currents are equal with same polarity  Time for surge to reach end of line is T = L/vp

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Reflection of Pulses Transmission Line R0

Open Line

Reflection hyperlink

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Open-Circuited Line  Total current at open end = 0  Incident and reflected currents must be

equal and opposite  Incident and reflected voltages are equal with same polarity  Time for surge to reach end of line is T = L/vp

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Reflection Coefficient  More complex situation: Load has an arbitrary

impedance  not equal to Z0  not

shorted or open  impedance may be complex (either capacitive 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 Coefficient The amount of voltage reflected back is called voltage reflection coefficient.

Vr Ir Γ = or Vi Ii

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Γ = reflection coefficient Vi = incident voltage Vr = reflected voltage Ii = incident current Ir = reflected current

Reflection of Pulses Transmission Line Z0

Vi Vr = = Z0 Ii I r

Vr Ir Γ @ or Vi Ii 47

ZL

total voltage = Vi + Vr total current = I i − I r Vi + Vr = ZL Ii − I r ZL − Z0 Γ= Z L + Z0

Wave Propagation on Lines  Start by assuming a matched line  Waves move down the line at propagation

velocity  Waves are the same at all points except for phase  Phase changes 360 degrees in the distance a wave travels in one period  This distance is called the wavelength

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Standing Waves  When an incident wave reflects from a mismatched

load, an interference pattern develops

 Both incident and reflected waves move at the

propagation velocity, but the interference pattern is stationary

 The interference pattern is called a set of standing

waves

 It is formed by the addition of incident and reflected

waves and has nodal points that remain stationary with time

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Incident and Reflected Waves

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Standing Waves

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Standing-Wave Ratio  When line is mismatched but neither

open nor shorted, voltage varies along line without ever falling to zero  Greater mismatch leads to greater variation  Voltage standing-wave ratio (VSWR or SWR) is defined:

Vmax SWR = Vmin 52

( ≥ 1)

Standing waves Vmax SWR = Vmin

Z L − Z0 Γ= Z L + Z0

( ≥ 1)

Vmax = Vi + Vr = Vi + Γ Vi Vmin = Vi − Vr = Vi − Γ Vi 1+ Γ

Z0 ZL SWR = = or 1− Γ ZL Z0

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SWR − 1 Γ= SWR + 1

SWR and Reflection Coefficient  SWR is a positive real number Γ

may be positive, negative or complex  SWR ≥ 1  Magnitude of Γ ≤ 1

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Standing waves on an Open Line

This is only the amplitude!!!

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Standing waves on an Shorted Line

This is only the amplitude!!!

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Effects of High SWR  High SWR causes voltage peaks on

the line that can damage the line or connected equipment such as a transmitter  Current peaks due to high SWR cause losses to increase

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Reflected Power

 When a signal travels down a mismatched line, some of the

power reflects from the load  This power is dissipated in the source, if the source matches the line  A high SWR causes the load power to be reduced

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Reflected Power

Pr = Γ Pi 2

PL = Pi (1 − Γ ) 2

4 SWR PL = Pi 2 (1 + SWR )

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Pr = reflected power Pi = incident power PL = power delivered to load

SWR − 1 Γ= SWR + 1

Time-Domain Reflectometry Reflecto meter

transmission line reflection

v ⋅t d= 2 60

General Input Impedance Equation  Input impedance of a transmission

line at a distance L from the load impedance ZL with a characteristic Zo is Zinput = Zo [(ZL + j Zo BL) (Zo + j ZL BL)] where B is called phase constant or wavelength constant and is defined by the equation B = 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 waves 

The effect of resistance in a transmission line is to continuously reduce the amplitude of both incident and reflected voltage and current waves.

Skin Effect: As frequency increases, depth of penetration into adjacent conductive surfaces decreases for boundary currents associated with electromagnetic waves, that results in the confinement of the voltage and current waves at the boundary of the transmission line, thus making the transmission more lossy. Skin depth (m) = 1  √ f where f = frequency, Hz  = permeability, H/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 Chart  Horizontal line: The horizontal line running through

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

 Circles of constant resistance and conductance:

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

 Lines of constant reactance: Lines of constant

reactance are shown on the Smith chart with curves that start from a given reactance value on the outer circle and end at the right-hand side of the center line. 65

Type of Microwave problems that Smith chart can be used 1. 2. 3. 4. 5. 6. 7. 8. 9.

Plotting a complex impedance on a Smith chart Finding VSWR for a given load Finding the admittance for a given impedance Finding the input impedance of a transmission line terminated in a short or open. Finding the input impedance at any distance from a load ZL. Locating the first maximum and minimum from any load Matching a transmission line to a load with a single series stub. Matching a transmission line with a single parallel stub Matching a transmission line to a load with two parallel stubs.

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Plotting a Complex Impedance on a Smith Chart  To locate a complex impedance, Z = R+-jX or

admittance Y = G +- jB on a Smith chart, normalize the real and imaginary part of the complex impedance. Locating the value of the normalized real term on the horizontal line scale locates the resistance circle. Locating the normalized value of the imaginary term on the outer circle locates the curve of constant reactance. The intersection of the circle and the curve locates the complex impedance on the Smith chart. 67

Finding the VSWR for a given load  Normalize the load and plot its location on

the Smith chart.  Draw a circle with a radius equal to the distance between the 1.0 point and the location of the normalized load and the center of the Smith chart as the center.  The intersection of the right-hand side of the circle with the horizontal resistance line locates the value of the VSWR. 68

Finding the Input Impedance at any Distance from the Load  The load impedance is first normalized and is

located on the Smith chart.  The VSWR circle is drawn for the load.  A line is drawn from the 1.0 point through the load to the outer wavelength scale.  To locate the input impedance on a Smith chart of the transmission line at any given distance from the load, advance in clockwise direction from the located point, a distance in wavelength equal to the distance to the new location on the transmission line. 69

Power Loss  Return Power Loss: When an electromagnetic

wave travels down a transmission line and encounters a mismatched load or a discontinuity in the line, part of the incident power is reflected back down the line. The return loss is defined as: Preturn = 10 log10 Pi/Pr Preturn = 20 log10 1/  Mismatch Power Loss: The term mismatch loss is used to describe the loss caused by the reflection due to a mismatched line. It is defined as Pmismatch = 10 log10 Pi/(Pi - Pr) 70

Notes:  Metallic circuit current – currents that flow in

opposite directions in a balanced wire pair  Longitudinal current – currents that flow in the same direction  Common Mode Rejection (CMR) – cancellation of common mode signals or noise interference induced equally on both wires producing longitudinal currents that cancel in the load CMRR = 40 to 70 dB

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

Constant  For maximum power transfer, Z

L

= Zo, thus no

reflection  Characteristic impedance = Surge impedance  Transmission line stores energy in its distributed inductance and capacitance

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Notes:  Transmission lines: 



  





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The input impedance of an infinitely long line at radio frequencies is resistive and equal to Zo Nonresonant – when electromagnetic waves travel the line without reflections Ratio of voltage to current at any point is equal to Zo Incident voltage and current at any point are in phase Line losses on a non-resonant line are minimum per unit length Any transmission line that is terminated in a load equals to Zo acts as if it were an infinite line. Prop. Cons. = attenuation coeff. + phase shift coeff. γ = α + jβ

Notes:

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Material

Velocity Factor

air rubber polyethylene teflon teflon foam teflon pins teflon spiral

0.95 – 0.975 0.56 – 0.65 0.66 0.70 0.82 0.81 0.81

Notes: Material Vacuum Air Teflon polyethylene polystyrene paper, paraffined rubber PVC Mica Glass 75

Dielectric Constant 1 1.0006 2.1 2.27 2.5 2.5 3.0 3.3 5.0 7.5

Notes: 





Velocity factor (Velocity constant) = actual vel. Of prop. vel. In free space Vf = Vp / c Electrical length of transmission line  Long – length exceeds λ/16  Short – length less than or equal λ/16 Delay lines – transmission lines designed to intentionally introduce a time delay in the path of an electromagnetic wave td = LC (seconds) td = 1.016 Є

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Notes: 

The disadvantages of not having a matched line:  100 percent of the source incident power does not reach the load  The dielectric separating the two conductors can break down and cause corona due to high VSWR  Reflections and rereflections cause more power loss  Reflections cause ghost images  Mismatches cause noise interference

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Notes: 

Characteristics of transmission line terminated at open  voltage incident wave is reflected back (no phase reversal)  current incident wave is reflected back 180 degrees from how it would have continued  sum of the incident and reflected current waveforms is minimum  sum of the incident and reflected voltage waveforms is maximum

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Notes: 

Characteristics of transmission line terminated at short  voltage standing wave is reflected back 180 degrees reversed from how it would have continued  current standing wave is reflected back the same as if it had continued  sum of the incident and reflected current waveforms is maximum  sum of the incident and reflected voltage waveforms is zero at the short

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Input Impedance Zin = resistive, max

short

Parallel LC circuit, resistive and maximum

Zin = resistive, min

open

Series LC circuit, resistive and minimum

Zin = inductive

short

Zin = capacitive

open

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capacitor

Zin = capacitive

short

capacitor

Zin = inductive

open

inductor

λ/4 Input end

inductor

Output end

Notes: 



The impedance transformation for a quarter wavelength transmission line is:  R = Zo: quarter λ line acts 1:1 turns ratio transformer L 

RL > Zo: quarter λ line acts as a step down transformer



RL < Zo: quarter λ line acts as a step up transformer

Characteristic Impedance of quarter wavelength X’former Zo’ = √(ZoZL)



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

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Notes: 

Stub Matching  Stubs are used to eliminate the reactive component to match the transmission line to the load  It is just a piece of additional transmission line that is placed across the primary line as close to the load as possible  Susceptance of stub is used to tune out the susceptance of the load  Shorted stubs are preferred because open stubs have the tendency to radiate at higher frequencies

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Notes: 

Process of Stub Matching  locate a point as close to the load as possible where the conductive component of the input admittance is equal to the characteristic admittance of transmission line Yin = G – jB, G = 1 / Zo  Attach the shorted stub to the point on the transmission line  Depending whether the reactive component at the point is inductive or capacitive, the stub length is adjusted Yin = Go – jB + jBstub Yin = Go 83

Notes: 



 

Time Domain Reflectometry (TDR)  technique used to locate an impairment in the metallic cable  How much of the transmitted signal returns depends on the type and magnitude of the impairment  Impairment represents a discontinuity in the signal For higher frequency applications (300 MHz – 3000 MHz), microstrip and stripline is constructed to interconnect components on PC boards When the distance between source and load ends is a few inches or less, coaxial cable is impractical Microstrip and Stripline use the tracks on the PC board. 84

Notes: 





Microstrip and Stripline are used to construct transmission lines, inductors, capacitors, tuned circuits, filters, phase shifters, and impedance matching devices. Microstrip – when the lines are etched in the middle layer of the multilayer PC board Zo = 87 ln 5.98h__ Є fiberglass = 4.5 √(Є + 1.41) 0.8w + t Є teflon = 3 w = width of Cu trace t = thickness of Cu trace h = thickness of dielectric Stripline – if the lines are etched onto the surface of the PC board only Zo = 60 ln 4d __ d = dielectric thick Є 0.67πw(0.8 + t/h)

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86

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Problems: 1.

2.

3.

Determine the characteristic impedance for an air dielectric two-wire parallel transmission line with a D/r ratio = 13.5 (311.97 ohms) Determine the characteristic impedance for an RG-59A coaxial cable with parameters: L=0.121 μH/ft, C=30 pF/ft, d=0.042 in., D=0.22 in, and Є=2.15 (63.509 ohms, 67.685 ohms) For a given length of RG8A/U coaxial cable with parameters: C=98.4 pF/m, L=262.45 nH/m, Єr=2.15. Find Vp and Vf (1.968x108 m/s, 0.656 or 0.682) 88

Problems: 4.

5.

For a transmission line with incident voltage of 5.2V and reflected voltage of 3.8V, find reflection coefficient and SWR (0.731, 6.429) Determine the physical length and Zo for a quarter wavelength transformer that is used to match a section of RG8A/U (Zo=50 ohms) to a 175 ohm resistive load. The frequency of operation is 220 MHz and the velocity factor is 1 (0.341 m, 93.54 ohms)

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Stub Matching 

   

Use to remove the reactive component of the complex impedance of the load to match the transmission line to the load It is a piece of additional transmission line that is placed across the primary line as close to the load as possible The susceptance of the stub is used to tune out the susceptance of the load Either a shorted or open stub is used with greater preference on the shorted stub A transmission line that is one-half wavelength or shorter is used to tune out the reactive component of the load

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Stub Matching Process Locate a point as close as possible to the load where the conductive component of the Zin = Zo Yin = G – jB where G = 1 / Zo 2. Attach the shorted stub on the identified point 3. Depending on whether the reactive component at that point is inductive or capacitive, the stub length is adjusted accordingly Yin = Go – jB + jBstub ~ Yin = Go 1.

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