fundamental transmission line theory · fundamental transmission line theory (part 3) dr. josé...

Fundamental Transmission Line Theory (Part 3)Dr. José Ernesto Rayas-Sánchez

February 6, 2020

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Fundamental Transmission Line Theory(Part 3)

Dr. José Ernesto Rayas-Sánchez

2Dr. J. E. Rayas-Sánchez

Outline

Power along the transmission line

Return loss

Standing wave ratio

Summary on reflection parameters

Transmission coefficient

Insertion loss

Summary on transmission parameters

Analogies and illustrations


February 6, 2020

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Average Power along the Transmission Line

][)( 0zjzj eeVzV

][)(0

0 zjzj eeZV

zI

})()(Re{21 *zIzVPav

}||1Re{||

21 222*

0

20

zjzjav ee

ZV

P

)||1(||

21 2

0

20

ZV

Pav

The average power flow is constant along the line (if V0+ is constant)

The power delivered is the incident power minus the reflected power

If = 0, maximum power is delivered to the load

If || = 1, no power is delivered to the load

Z0,

0

ZL

z


Power Transfer Efficiency

RS

RL

Maximum power is transferred when RL = RS

RL / RS

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10

Lo

ad

Po

wer

(no

rmaliz

ed

)

(Hewlett-Packard's RF Design and Measurement Seminar, 2000)


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Power Transfer Efficiency (cont)

For complex impedances, maximum power transfer occurs when ZL = ZS* (conjugate match)

Zs = R + jX

ZL = Zs* = R - jX

Zo

Zo

Rs

RL

+jX

-jX

At high frequencies, maximum power transfer occurs when RS = RL = Zo



Return Loss, RL

)||1(||

21 2

0

20

ZV

Pav

It is a way to measure the ability of the TL to avoid energy reflection

It is the ratio of the incident power to the reflected power

dB)(||log20 RL

dB)0|(| RL dB0)1|(| RL

dB)(||log10 2RLZ0, ZL

l


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Standing Wave Ratio, SWR

It is a way to measure the mismatch of a line

A mismatch implies a “standing wave” along the line

It is the ratio of the maximum to the minimum wave amplitude along the line

][)( 0zjzj eeVzV

]1[)( 20

zjzj eVzVe

|1||||)(| 20

zjeVzV

je|| Since |||1||||)(| )2(

0ljeVlV

|1||||)(| 20

ljeVlV

Z0,

0l

ZL

z


Standing Wave Ratio, SWR (cont)

|||1||||)(| )2(0

ljeVlV

1 when maximum is |)(| )2( ljelV

1 when minimum is |)(| )2( ljelV

|)|1(|||)(| 0maxmax VVlV

|)|1(|||)(| 0minmin VVlV

||1||1

min

max

VV

SWR SWR1

The distance between two maxima or two minima is /2 The distance between a maximum and a minimum is /4

Z0,

0l

ZL


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The Slotted Line

(D. M. Pozar, Microwave Engineering, Wiley, 2005)

(X-band waveguide slotted line)

Used to measure:SWRlmin, lmax

ZL


Standing Wave Ratio – EM Simulation

Microstrip lineLaminate: Rogers RO3003

H = 5 mil

r = 3

tan() = 0.0013 at 10 GHz

Lossy metals: Cu = 5.8×107 S/m

t = 0.6 mil (half-once copper)

W = 12.5 mil

L = 187.5 mil


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Standing Wave Ratio – EM Simulation (cont)

Using Zport1 = Zport2 = 50

f = 0.1 GHz f = 30 GHz


Standing Wave Ratio – EM Simulation (cont)

Using Zport1 = 50 , Zport2 = 1 K

f = 0.1 GHz f = 30 GHz


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Transmission Line Terminated with Zo

For reflection, a transmission line terminated in Zo behaves like an infinitely long transmission line

Zs = Zo

Zo

Vrefl = 0 (all the incident power is absorbed in the load)

Vinc

Zo = characteristic impedance of transmission line



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Transmission Line Terminated with Short, Open

Zs = Zo

Vrefl

V inc

For reflection, a transmission line terminated in a short or open reflects all power back to source

In phase (0°) for openOut of phase (180°) for short



Transmission Line Terminated with 25

Zs = Zo

ZL = 25

Vrefl

V inc

Standing wave pattern does not go to zero as with short or open



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Reflection Parameters

dB

No reflection(ZL = Z0)

RL

VSWR

0 1

Full reflection(ZL = open, short)

0 dB

1

=Z L - Z 0

ZL + 0Z

ReflectionCoefficient

=Vreflected

Vincident=

= Return loss = -20 log(),

VSWR = Emax

Emin

=1 + 1

Voltage Standing Wave RatioEmax

Emin



Reflection Parameters – Summary

Reflection coefficient, l

Return Loss, RL

Standing Wave Ratio, SWR


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Transmission Coefficient, T

It is a measure of the capacity of a TL to transmit an incoming wave

0for )()( 0 zeeVzV zjzj 01

01

ZZZZ

infinite) TL2 (assuming 0for )( 0 zTeVzV zj

TL2TL1

00 ]1[)0( TVVV 1T


Transmission Coefficient, T (cont)

T is the ratio of the total transmitted voltage to the incident wave at a transition

T can be greater than 1 (no amplification implied)

TL2TL1

00 ]1[)0( TVVV 1TzjeV

zVT

0

)(


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Insertion Loss, IL

It is a measure of the reflections caused by an inserted TL

TL2TL1

(dB) ||log20 TIL

dB)0|(| TIL dB0)1|(| TIL


Transmission Parameters – Summary

Transmission coefficient, T

Insertion Loss, IL


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Lightwave Analogy to RF Energy

RF

Incident

Reflected

Transmitted

Lightwave


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