ElectromagneticsElectromagnetics(ENGR 367)(ENGR 367)

T-line Power, Reflection & T-line Power, Reflection & SWRSWR

T-line Theory: T-line Theory: Something New or Not?!Something New or Not?!

• Power, Reflection and Standing Waves in Power, Reflection and Standing Waves in T-lines act just like Uniform Plane Waves T-lines act just like Uniform Plane Waves (UPW) in unbounded and layered media!(UPW) in unbounded and layered media!

• Once you understand UPWs, you can also Once you understand UPWs, you can also see by analogy how waves behave in see by analogy how waves behave in T-lines with a few simplifications! T-lines with a few simplifications!

Traveling Waves on T-linesTraveling Waves on T-lines• Space-time phenomena may be Space-time phenomena may be

described by phasor functions described by phasor functions representing eitherrepresenting either– Voltage and current disturbancesVoltage and current disturbances

– Electromagnetic wave disturbancesElectromagnetic wave disturbances

( , ) and ( , )V V z t I I z t

( , ) and ( , )E E z t H H z t

T-line Traveling WavesT-line Traveling Waves• Analagous to waves on a string or Analagous to waves on a string or

sound waves in a tube since all these wavessound waves in a tube since all these waves– carry real powercarry real power– reflect at boundaries and discontinuitiesreflect at boundaries and discontinuities– exhibit impedance at each point in the medium exhibit impedance at each point in the medium

• Unique from waveguides since on T-lines they Unique from waveguides since on T-lines they propagate in the (quasi-) Transverse propagate in the (quasi-) Transverse

Electromagnetic (TEM) mode: ~plane wavesElectromagnetic (TEM) mode: ~plane waves

Power in T-lines via Circuit Power in T-lines via Circuit Model during time harmonic Model during time harmonic

oscillationoscillation• Instantaneous Instantaneous Power over a Power over a fixed line length zfixed line length z– Express the real Express the real

parts of V, I in the parts of V, I in the (+) direction only(+) direction only

– Apply Euler’s Apply Euler’s IdentityIdentity

– ThusThus

( )

( )

( , ) Re{ } Re{ }

( , ) Re{ } Re{ }

z j t z

z j t z

V z t V V e e

I z t I I e e

0

0

0 0

( , ) cos( )

( , ) cos( )

where Re{ }, cos Re{ }

z

z

V z t V e t z

I z t I e t z

V V I I

20 0( , ) cos( ) cos( )zP z t V I e t z t z

( , ) ( , ) ( , )P z t V z t I z t

Power in T-lines via Circuit Power in T-lines via Circuit Model during time harmonic Model during time harmonic

oscillationoscillation• Time Averaged Time Averaged

PowerPower0

*

22 20

0 00

1 2( , ) (where for 1 cycle)

1[or alternatively Re{ ( ) }]2

1 1 cos cos [W]2 2

[circuit power] [attenuation] [power factor]

T

z z

P P z t dt TT

P V I

VP V I e eZ

Power Loss due to Power Loss due to AttenuationAttenuation

• ExplicitlyExplicitly

• In decibel (dB) In decibel (dB) unitsunits

2( ) (0)

power drops at twice the exponential rate with distance as either voltage or current

zP z P e

2 .869 2 .869

10

( )10 since 10

(0)

( )Power Loss (dB) 10log 8.69

(0)

z zP ze e

P

P zz

P

Power Loss due to Power Loss due to AttenuationAttenuation

• In terms of VoltageIn terms of Voltage

10 10

2

( ) ( )Power Loss (dB) 10log 20log

(0) (0)

8.69

since ( ) ( ) and ( ) (0) z

P z V zP V

z

P z V z V z V e

Example of Calculating Example of Calculating T-line Power LossT-line Power Loss

• Exercise 1 (based on D11.2, H&B, 7/e, p. 350) Exercise 1 (based on D11.2, H&B, 7/e, p. 350) GivenGiven:: two T-lines joined end-to-end by an adaptor. two T-lines joined end-to-end by an adaptor.

Line 1 is 30 m long and is rated at 0.1 dB/m, Line 1 is 30 m long and is rated at 0.1 dB/m, whereas line 2 is 45 m long and is rated at 0.15 whereas line 2 is 45 m long and is rated at 0.15 dB/m. Due to a poor adaptor, the joint imparts dB/m. Due to a poor adaptor, the joint imparts another 3 dB loss.another 3 dB loss.

FindFind: the percentage (%) of the input power that : the percentage (%) of the input power that reaches the output of this combinationreaches the output of this combination

SolutionSolution:: Total Power Loss from input to output is: (30 m)(0.1 dB/m) (3 dB) (45 m)(0.15 dB/m)

3 dB 3 dB 6.75 dB 12.75 dB

Example of Calculating Example of Calculating T-line Power LossT-line Power Loss

• Exercise 1 Exercise 1 (continued)(continued)SolutionSolution::

10

dB 12.751.27510 10

( )since Power Loss (dB) 10log

(0)

( )10 10 10 0.053

(0)

5.3%

P zP

P zP

Wave Reflection at Wave Reflection at T-line DiscontinuityT-line Discontinuity

• T-line discontinuity may consist ofT-line discontinuity may consist of– an actual load termination: device with an actual load termination: device with

complex complex input impedance (e.g., antenna input impedance (e.g., antenna or display)or display)

– a junction between lines: connector and/or a junction between lines: connector and/or line line mismatchmismatch

• Schematic modelSchematic model

Wave Reflection at Wave Reflection at T-line DiscontinuityT-line Discontinuity

• Energized T-line with discontinuityEnergized T-line with discontinuity

– Incident Voltage phasor Incident Voltage phasor – Reflected Voltage phasorReflected Voltage phasor(where the time dependence e(where the time dependence ejjtt has been supressed) has been supressed)

0 0

0 0

( )

( )

z z j zi i i

z z j zr r r

V z V e V e e

V z V e V e e

Wave Reflection at Wave Reflection at T-line DiscontinuityT-line Discontinuity

• Consider the situation at the load junction (z=0):Consider the situation at the load junction (z=0):– Voltages of opposite going waves Voltages of opposite going waves addadd

– Currents of opposite going waves Currents of opposite going waves addadd

where the – sign arises due to neg. z-going current where the – sign arises due to neg. z-going current wavewave

0 0L i rV V V

0 0 0 00

1 [ ] LL i r i r

L

VI I I V VZ Z

Wave Reflection at Wave Reflection at T-line DiscontinuityT-line Discontinuity

• Define Voltage Reflection Coefficient Define Voltage Reflection Coefficient (())

• Solving for Solving for in terms of impedances in terms of impedances onlyonly

0

0

reflected voltageincident voltage

r

i

VV

0 0 0 0 0

0 0 0 0 0

0 0

0 0

( ) ( ) 2( ) ( ) 2

i r i r r

i r i r i

jL L L L

L L L L

V V V V VV V V V VI Z I Z Z Z

eI Z I Z Z Z

Wave Transmission at Wave Transmission at T-line DiscontinuityT-line Discontinuity

• Define Voltage Transmission Define Voltage Transmission Coefficient (Coefficient ())

• Solving in terms of impedances onlySolving in terms of impedances only

0

transmitted voltageincident voltage

L

i

VV

0 0 0

0 0

0

1 1

2

i r L

i L

jL

L

V V Z ZV Z ZZ e

Z Z

Matching Condition Matching Condition at a T-line Junction at a T-line Junction

• An An impedance matchimpedance match becomes a desired becomes a desired design condition for most practical T-line design condition for most practical T-line systems because itsystems because it– Maximizes power transferred to the loadMaximizes power transferred to the load– Minimizes power reflected back to generatorMinimizes power reflected back to generator

• In terms of ZIn terms of ZLL and Z and Z00

00

0 0

2 1 or = 0LLL

L L

Z ZZ Z ZZ Z Z Z

Power Reflected and Power Reflected and Transmitted at a T-line JunctionTransmitted at a T-line Junction• Ratio of Reflected to Incident PowerRatio of Reflected to Incident Power

• Ratio of Transmitted to Incident PowerRatio of Transmitted to Incident Power

22 2

2 (since )r r

i i

P VP V

P V

2 21 ( ) (since )tt i r

i

PP P P

P

Calculating Power Calculating Power In Case of a Line-Load In Case of a Line-Load

MismatchMismatch• Exercise 2 (Ex. 11.5, H&B, 7/e, p. 352)Exercise 2 (Ex. 11.5, H&B, 7/e, p. 352)

GGiveniven:: a 50 a 50 lossless T-line terminated by lossless T-line terminated by a load impedance, Za load impedance, ZLL=50-j75 =50-j75 . Power . Power incident from the T-line to the load is 100 incident from the T-line to the load is 100 mW.mW.

FindFind: the power dissipated by the load: the power dissipated by the load

SolutionSolution: first calculate the reflection : first calculate the reflection coefficientcoefficient 0.930

0

(50 75) 50 0.36 0.48 0.60(50 75) 50

jL

L

Z Z j j eZ Z j

Calculating Power Calculating Power In Case of a Line-Load In Case of a Line-Load

MismatchMismatch• Exercise 2 (continued)Exercise 2 (continued)

SolutionSolution: next calculate the transmitted : next calculate the transmitted power in terms of incident power and power in terms of incident power and

2

2

(1 )

[1 (0.60) ](100 mW) 64 mWt iP P

Calculating Power In Case of Calculating Power In Case of Both Line Loss and Line-Load Both Line Loss and Line-Load MismatchMismatch• Exercise 3 (Ex. 11.6, H&B, 7/e, pp. 352, Exercise 3 (Ex. 11.6, H&B, 7/e, pp. 352,

353)353)GivenGiven: two lossy lines joined end-to-end. : two lossy lines joined end-to-end.

Line 1 is 10 m long and has a 0.20 dB/m loss. Line 1 is 10 m long and has a 0.20 dB/m loss. Line 2 is 15 m long and has a 0.10 dB/m loss. Line 2 is 15 m long and has a 0.10 dB/m loss. At the junction of these two lines At the junction of these two lines = 0.30. = 0.30. Power input to line 1 is PPower input to line 1 is Pi1i1 = 100 mW = 100 mW

FindFind: a) the total loss of the line combination in dB.: a) the total loss of the line combination in dB. b) the power transmitted to the output of line b) the power transmitted to the output of line 2.2.

Calculating Power In Case of Calculating Power In Case of Both Line Loss and Line-Load Both Line Loss and Line-Load MismatchMismatch• Exercise 3 (Ex. 11.6, H&B, 7/e, pp. 352, 353)Exercise 3 (Ex. 11.6, H&B, 7/e, pp. 352, 353)

SolutionSolution:: a) first calculate the dB loss of the a) first calculate the dB loss of the joint from joint from

then calculate the total loss of the linkthen calculate the total loss of the link

b) now calculate the output power as b) now calculate the output power as

210 10

10

(dB) 10log ( ) 10log (1 )

10log (1 0.09) 0.41 dB

tj

i

PL

P

(dB) (0.20)(10) 0.41 (0.10)(15) 3.91 dBtL

0.391out (100 mW) 10 41 mWP

Voltage Standing Wave Ratio Voltage Standing Wave Ratio (VSWR) for Terminated T-lines(VSWR) for Terminated T-lines

• The status of waves on a T-line depends on The status of waves on a T-line depends on the termination: 3 possibilities existthe termination: 3 possibilities exist1) Matched termination (Z1) Matched termination (ZLL = Z = Z0 0 = 0) = 0)

•All waves travel from source to loadAll waves travel from source to load•No waves reflect back to the source No waves reflect back to the source •No standing waves exist, only No standing waves exist, only pure traveling wavespure traveling waves

2) Perfectly reflective termination (2) Perfectly reflective termination ( = 1) = 1)•All waves travel from source to load and back againAll waves travel from source to load and back again•All waves completely reflectAll waves completely reflect•A A pure standing wavepure standing wave pattern exists with fixed null pattern exists with fixed null

and maximum voltage locations along the lineand maximum voltage locations along the line

Voltage Standing Wave Ratio Voltage Standing Wave Ratio (VSWR) for Terminated T-lines(VSWR) for Terminated T-lines

• The status of waves on a T-line depends on The status of waves on a T-line depends on the termination: 3 possibilities existthe termination: 3 possibilities exist3) A partially reflective termination (0<3) A partially reflective termination (0<<1)<1)

•Some waves travel from source to load and backSome waves travel from source to load and back•Some waves reflect; others pass to the loadSome waves reflect; others pass to the load•A A partial standing wavepartial standing wave pattern exists with fixed pattern exists with fixed

minimum and maximum locations along the line minimum and maximum locations along the line mixed with traveling waves!mixed with traveling waves!

(animated partial standing wave pattern)(animated partial standing wave pattern)

Terminated Lossless T-lineTerminated Lossless T-line• Total voltage wave phasor (w/load @ z=0)Total voltage wave phasor (w/load @ z=0)

• Complete space-time voltage wave functionComplete space-time voltage wave function

0 0

0

0

( )

where

j z j z

jL

L

V z V e V eZ Z eZ Z

( , ) Re{ ( ) }j tV z t V z e

Terminated Lossless T-lineTerminated Lossless T-line• After applying Euler’s Identity and taking After applying Euler’s Identity and taking

the real the real part the total voltage wave part the total voltage wave function becomesfunction becomes

0 0

max 0 0 0

min 0

( , ) (1 )cos( ) 2 cos( )cos( )2 2

[traveling wave part] [standing wave part]where (1 ) 2 (1 )

and (1 ) (where the standing wave part has a null)

V z t V t z V z t

V V V V

V V

Terminated Lossless T-lineTerminated Lossless T-line• Where are maximum and minimum voltages located?Where are maximum and minimum voltages located?

• In terms of wavelengths (In terms of wavelengths () between successive) between successive– VVmaxmax locations locations

– VVminmin locations locations

– VVmaxmax to V to Vminmin locations locations

max max max

min min min

1( , ) where ( 2 ) and 0,1, 2,...21( , ) where ( (2 1) ) where 0,1,2,...

2

V V z t z m m

V V z t z m m

max

min

max min

2

2

4

z

z

z z z

Graphical Standing Wave Graphical Standing Wave Patterns Patterns • Voltage Standing Wave Patterns for Voltage Standing Wave Patterns for

RealReal Reflection Coefficient Reflection Coefficient ComplexComplex

-2.5 -2 -1.5 -1 -0.5 00

5

10

15Voltage Standing Wave Pattern for V0 = 10v, Gamma = 0.5, Phi = 0

z-Position in Wavelengths

Mag

nitu

de o

f AC

Vol

tage

Am

plitu

de

-2.5 -2 -1.5 -1 -0.5 00

5

10

15

z-Position in Wavelengths

Mag

nitu

de o

f AC

Vol

tage

Am

plitu

de

Voltage Standing Wave Pattern for V0 = 10v, Gamma = 0.3, Phi = 45 deg.

40.3je

00.5 je

VSWR: Terminated Lossless T-VSWR: Terminated Lossless T-lineline

• Now define asNow define as

• Note special casesNote special cases– Matched termination:Matched termination:– Perfectly reflective termination:Perfectly reflective termination:

• Range: Range: • Significance: indicates the degree of standing Significance: indicates the degree of standing

waves vs. traveling waves present on the T-waves vs. traveling waves present on the T-lineline

max max

min min

1( )VSWR( ) 1

V V zV V z

0 VSWR 11 VSWR

1 VSWR

VSWR CalculationsVSWR Calculationsfor a Lossless Terminated T-for a Lossless Terminated T-

lineline• Exercise 4Exercise 4

GivenGiven:: = 3/5 = 3/5FindFind: VSWR = ?: VSWR = ?SolutionSolution::

• Exercise 5Exercise 5– GivenGiven: for a good match, we desire VSWR < 2.5: for a good match, we desire VSWR < 2.5– FindFind: the condition on : the condition on – SolutionSolution::

31 85VSWR 43 215

VSWR 1 2.5 1 3 0.43VSWR 1 2.5 1 7

must have 0.43

ConclusionsConclusions• Traveling waves on T-lines carry power Traveling waves on T-lines carry power

subject to the losses of attenuation over subject to the losses of attenuation over distance and any mismatch of distance and any mismatch of impedance at junctionsimpedance at junctions

• The power output expected from a T-The power output expected from a T-line may be computed from the input line may be computed from the input power by taking into account any dB power by taking into account any dB loss factorsloss factors

ConclusionsConclusions• The reflection (or transmission) The reflection (or transmission)

coefficient (coefficient ( or or ) at any T-line ) at any T-line discontinuity discontinuity – Indicates how much voltage and power will Indicates how much voltage and power will

be reflected (or transmitted) at the junctionbe reflected (or transmitted) at the junction– May be computed from the line impedance May be computed from the line impedance

(Z(Z00) on the source side and the effective ) on the source side and the effective input impedance (Zinput impedance (ZLL = Z = Zinin) on the load side ) on the load side

ConclusionsConclusions• The Voltage Standing Wave Ratio The Voltage Standing Wave Ratio

(VSWR) for a terminated T-line(VSWR) for a terminated T-line– Indicates the degreeIndicates the degree of standing waves of standing waves

versus traveling waves present on the lineversus traveling waves present on the line– Serves as a figure of merit for the quality Serves as a figure of merit for the quality

of impedance match at a junctionof impedance match at a junction– Represents the max. to min. voltage ratio Represents the max. to min. voltage ratio

along the line, but may be calculated along the line, but may be calculated directly from the reflection coefficient at a directly from the reflection coefficient at a junction junction

ReferencesReferences• Hayt & Buck, Hayt & Buck, Engineering Engineering

ElectromagneticsElectromagnetics, 7/e, McGraw Hill: , 7/e, McGraw Hill: New York, 2006.New York, 2006.

• Kraus & Fleisch, Kraus & Fleisch, Electromagnetics Electromagnetics with Applicationswith Applications, 5/e, McGraw Hill: , 5/e, McGraw Hill: New York, 1999.New York, 1999.