Electromagnetics Electromagnetics (ENGR 367)(ENGR 367)The Complete The Complete

Sourced & Loaded T-lineSourced & Loaded T-line

T-line Power: Big or SmallT-line Power: Big or Small• While T-lines carry from the high voltage of While T-lines carry from the high voltage of

Power Generators to the miniscule voltage Power Generators to the miniscule voltage detected by Communications Receivers, the detected by Communications Receivers, the load can only absorb as much signal power load can only absorb as much signal power as is transmitted, and that depends initially as is transmitted, and that depends initially on the source and input impedances. on the source and input impedances.

• To address this issue adequately, we must To address this issue adequately, we must consider the consider the fully sourced and loaded T-linefully sourced and loaded T-line..

Outline of LectureOutline of Lecture• Model the Fully Sourced & Loaded T-lineModel the Fully Sourced & Loaded T-line• Define Wave & Input Impedance of T-lineDefine Wave & Input Impedance of T-line• Form an Equivalent Circuit at the InputForm an Equivalent Circuit at the Input• Observe Special Input Impedance CasesObserve Special Input Impedance Cases

– For a half-wavelength lineFor a half-wavelength line– For a quarter-wavelength lineFor a quarter-wavelength line

• Work Out Examples of T-line ProblemsWork Out Examples of T-line Problems

Introduction: Practical T-lines Introduction: Practical T-lines Connect to Both Source and Connect to Both Source and

LoadLoad• Practical T-line may be mismatched at both endsPractical T-line may be mismatched at both ends

• If fully unmatched, reflections occur at If fully unmatched, reflections occur at bothboth endsends

Sourced & Loaded T-line Sourced & Loaded T-line (lossless)(lossless)(in case of lossy T-line, replace j(in case of lossy T-line, replace j with with ))• Express the Phasor line Voltage & CurrentExpress the Phasor line Voltage & Current

asas0 0

0 0

0 0 0 0

( )

( )

where , , and may be complex

j z j z

j z j z

V z V e V e

I z I e I e

V V I I

Introduce Wave ImpedanceIntroduce Wave Impedance• Define Wave Impedance as the Ratio Define Wave Impedance as the Ratio

of Total Voltage to Current Anywhere of Total Voltage to Current Anywhere on lineon line

• In terms of the voltage reflection In terms of the voltage reflection coeff. (coeff. ())

0 0

0 0

( )( )( )

j z j z

W j z j z

V e V eV zZ zI z I e I e

0 00 0 0 0

0 0

0

, and

( )j z j z

W j z j z

V VV V I IZ Z

e eZ z Ze e

Wave Impedance Wave Impedance & Input Impedance& Input Impedance

• In terms of line and load impedancesIn terms of line and load impedances

• Define Input Impedance as the Wave Define Input Impedance as the Wave Impedance at the front (source) end of Impedance at the front (source) end of lineline

00

0

0

0

cos( ) sin( )( )cos( ) sin( )

(by substituting and applying Euler's Identity)

LW

L

L

L

Z z jZ zZ z ZZ z jZ z

Z ZZ Z

0in 0

0

cos( ) sin( )@ : ( )

cos( ) sin( )L

WL

Z l jZ lz l Z Z l Z

Z l jZ l

Equivalent CircuitEquivalent Circuit• The Practical Sourced & Loaded T-line with The Practical Sourced & Loaded T-line with

ZZinin

• Acts like the equivalent circuit:Acts like the equivalent circuit:

Special Case ISpecial Case I• Half-Wavelength (Half-Wavelength (/2) Line/2) Line

– Also known as the Half-Wave Transformer (HWT)Also known as the Half-Wave Transformer (HWT)– Electrical LengthElectrical Length

– Input ImpedanceInput Impedance

• Notes on the HWTNotes on the HWT– Ideally, this T-line section is Ideally, this T-line section is transparent to the sourcetransparent to the source

and acts and acts as if it were absent from the circuitas if it were absent from the circuit– Actually, any Actually, any line loss results in less than ideal line loss results in less than ideal

performanceperformance, so , so the shortest possible section is the shortest possible section is usually preferred in applicationusually preferred in application

– Results Results depend on frequencydepend on frequency::

2 for 0,1,2,...2ml m m

in ( / 2) LZ l m Z

valid only at 2pmv

fl

HWT BehaviorHWT Behavior

• Due to the length of the HWT:Due to the length of the HWT:

• Provided:Provided:

in LZ Z

22 2

2 2p

p

vl l fv l

Special Case IISpecial Case II• Quarter-Wavelength (Quarter-Wavelength (/4) Line/4) Line

– Also known as the Quarter-Wave Transformer (QWT)Also known as the Quarter-Wave Transformer (QWT)– Electrical LengthElectrical Length– Input ImpedanceInput Impedance

• Notes on the QWTNotes on the QWT– Ideally, can be used to Ideally, can be used to match impedances at a match impedances at a

junctionjunction– Affected by line lossesAffected by line losses so shortest section preferred so shortest section preferred– Results Results depend on frequencydepend on frequency::

(2 1) for 0,1, 2,...2

ml m

2in 0( / 4) / LZ l Z Z

(2 1)valid only at

4pm v

fl

QWT MatchingQWT Matching

• Desire:Desire:

• Provided:Provided:

202

in 01 02 01 LL

ZZ Z Z Z ZZ

2 4p

p

vl l fv l

Examples of T-line ProblemsExamples of T-line Problems• Consider a lossless 2-wire 300-Consider a lossless 2-wire 300- line line

such as the lead-in wire from the such as the lead-in wire from the antenna to a TV or FM radio antenna to a TV or FM radio communications receivercommunications receiver

T-line Problem: Example AT-line Problem: Example A• GivenGiven: Z: ZGG = Z = Z00 = Z = ZLL = (R = (Rinin of Rx)= 300 of Rx)= 300

vvpp = 2.5 x 10 = 2.5 x 1088 m/s and m/s and ll = 2.0 m= 2.0 m

• FindFind: : , SWR, , SWR, , , , , , Z, Zinin, V, Vinin, I, Iinin, V, VLL, I, ILL, , PPLL

• SolutionSolution: line is matched at both ends!: line is matched at both ends!

8

6

o

2.5 10 m/sa) 0 c) 2.5m100 10 Hz

2 2 radb) 1 d) 0.8 rad/m; 02.5m

e) (0.8 rad/m)(2 m)

1.6 rad 288 0.8

pvf

SWR

l

T-line Problem: Example AT-line Problem: Example A• SolutionSolution: (continued): (continued)

1.6

1.6

f) 300 (due to matched conditions)6 V 3 V (by voltage division)

2 23 V 10 nA300

g) 3 V 3 V 1.6 rad

10 nA 10 nA 1.6 ra

in

s in sin

in G

inin

in

j jL in

j jL in

ZV Z VVZ ZVIZ

V V e e

I I e e

15

dh) (due to matched conditions)

1 1 (3 V)(10 nA) 15 fW( 15 10 W)2 2

in L

in in

P P

V I

T-line Problem: Example BT-line Problem: Example B

T-line Problem: Example BT-line Problem: Example B• GivenGiven:: a 2a 2ndnd identical Rx in parallel with 1 identical Rx in parallel with 1stst

so that so that ZZGG = Z = Z0 0 = 300 = 300 ; Z; ZLL = R = Rin1in1 R Rin2in2

• FindFind: : , SWR, , SWR, , , , , , Z, Zinin, V, Vinin, I, Iinin, V, VLL, I, ILL, P, PLL with with locations & values of Vlocations & values of Vminmin, V, Vmaxmax

• SolutionSolution: Z: ZLL = (300 = (300 )) (300 (300 )) =150 =150 0

0

1 411150 300 150 1 3 3a) b) 21 2150 300 450 3 1 13 3

c, d, e) , , all same as before, since no changes were made to the T-line or frequency

L

L

Z Z VSWRZ Z

T-line Problem: Example BT-line Problem: Example B• SolutionSolution: (continued): (continued)

• Good News: ZGood News: Zinin may be calculated or checked may be calculated or checked by MATLAB Zx on a lossless line to save complex by MATLAB Zx on a lossless line to save complex arithmetic labor and show a graphical result!arithmetic labor and show a graphical result!

00

0

o

cos( ) sin( )f) cos( ) sin( )

150cos(.8 ) 300sin(.8 )(300 )300cos(.8 ) 150sin(.8 )

510 23.8 (466 206) a cap. impedance

Lin

L

in

Z l jZ lZ ZZ l jZ l

jZj

j

T-line Problem: Example BT-line Problem: Example B• SolutionSolution: (continued): (continued)

22 9

1 2

2

6 Vf) 7.56 nA .263 rad(300 466 206)

1 1Re{ } 7.65 10 A 466 13.3 fW2 2

g) (due to lossless line)

but 6.67 fW each2

1h) since where 1 or 2

sin

G in

in in in

L in

LL L

LiLi

ini

VIZ Z j

P I Z

P PPP P

VP i

R

2 1 1

2 for either of the two Rx's

then 2.0 V each (vs. 3.0 V w/single Rx)L L in LV V R P

T-line Problem: Example BT-line Problem: Example B• SolutionSolution: (continued): (continued)

• Check with the graphical output of MATLAB ZxCheck with the graphical output of MATLAB Zx

max

max

min max

min

1 1i) ( 2 ) ( 0,1, 2,...) where 2 3

and 0.8 0.625 m and -1.875 m( .25 ) and ( .75 )

j) locations are 0.625 m from positions4

0 and 1.25 m (

j jz m m e e

z

z z

z

only due to 2-m length of line)

T-line Problem: Example BT-line Problem: Example B• SolutionSolution: (note that for all real, resistive Z: (note that for all real, resistive ZLL & Z & Z00))

– The voltage min. occurs @ load if ZThe voltage min. occurs @ load if ZLL<Z<Z0 0

– The voltage max. occurs @ load if ZThe voltage max. occurs @ load if ZLL>Z>Z00

• For the phase of the voltage at the input versus the load For the phase of the voltage at the input versus the load

min

max min

k) 2.0 V and ( ) (2)(2.0 V) 4.0 V

LV VV VSWR V

9

o

0

0

o0

(7.56 10 A .263 rad)(510 .415 rad)

3.85 V .152 rad 3.85 V 8.7

but anywhere on the line: ( ) ( )

so @ the input ( ): ( ) ( )

3.0 V 72

and @

in in in

j z j z

j l j lin

V I Z

V z e e V

z l V V l e e V

V

o

0

o o

the output ( 0) : (0) (1 ) 2.0 V 288

the reflected wave causes and to differ by 279 vs. 288L

L in

z V V V

V V

T-line Problem: Example CT-line Problem: Example C

T-line Problem: Example CT-line Problem: Example C(based on Ex. 11.8, H&B, 7/e, p. (based on Ex. 11.8, H&B, 7/e, p.

363)363)• GivenGiven: same T-line but add –j300: same T-line but add –j300 at load at load• FindFind: : , VSWR, Z, VSWR, Zinin, I, Iinin, P, PLL

• SolutionSolution: now Z: now ZLL = (150 = (150))(–j300(–j300) )

o0

0

o

(150 )( 300 ) (120 60) (150 300)

120 60 300a) .447 153.4120 60 300

1 1 .447b) 2.621 1 .447

c) (755 138.5) 767.5 10.4 (e.g., from MATLAB ZX)

L

L

L

in

jZ jj

Z Z jZ Z j

VSWR

Z j

T-line Problem: Example CT-line Problem: Example C• SolutionSolution: :

(continued)(continued)

o

2 2

1 2

6.0 Vd) [300 (755 138.5)]

5.64 nA 7.51 1e) Re{ } (5.64 nA) (755 )2 2

12 fW12 fWwhere and 6 fW

2 2

sin

G in

in in in

LL in L L

VIZ Z j

P I Z

PP P P P

T-line Problem: Example DT-line Problem: Example D

T-line Problem: Example DT-line Problem: Example D (based on Ex. 11.9, H&B, 7/e, pp. 363, (based on Ex. 11.9, H&B, 7/e, pp. 363,

364)364)• GivenGiven:: same T-line as before but now use same T-line as before but now use

the the purely capacitive load Zpurely capacitive load ZLL = - = -j300j300

• FindFind: : , VSWR, Z, VSWR, Zin in SolutionSolution::o0

0

300 300a) 1 1 90 (total reflection)300 300

1 1 1b) ( pure standing waves)1 1 1

c) 589 (purely inductive reactance) No real power can be delivered to this input

L

L

in

Z Z j jZ Z j

VSWR

Z j

impedance by the source, nor can any average power be delivered to the load!

T-line Problem: Example ET-line Problem: Example E (based on D11.4&5, H&B, 7/e, p. (based on D11.4&5, H&B, 7/e, p.

364)364)• GivenGiven: a 50-: a 50- lossless line, lossless line, l l = 0.4= 0.4 long, long,

operating at 300 MHz with Zoperating at 300 MHz with ZLL = 40+j30 = 40+j30 connected at z = 0, and a Thevenin Equiv. connected at z = 0, and a Thevenin Equiv. source of 12esource of 12ej0j0 V at z = - V at z = -l l in series with in series with ZZThTh== 50+j0 50+j0

• FindFind: : , VSWR, Z, VSWR, Zinin, V, Vinin, V, VLL, P, PLL • SolutionSolution::

o0

0

o

1(40 30) 50 1 1 1 1/ 3 4 / 3a) 90 b) 2(40 30) 50 3 3 1 1 1/ 3 2 / 3

2c) = 0.4 0.8 rad 144 (25.5 5.90) (e.g., from MATLAB Zx)

L

L

in

Z Z j j s VSWRZ Z j

l Z j

T-line Problem: Example ET-line Problem: Example E• Solution: Solution:

(continued)(continued) o

o

0

o0

o0

(12 V 0 )(25.5 5.90) d) (50 0) (25.5 5.90)

4.15 V 8.6

e) also ( ) ( )

6.00 144

@ the load ( 0) : (0) (1 ) 6.33 V 126

and di

ffer by +2.1

Th inin

Th in

j l j lin

L

L in

V Z jVZ Z j j

V V l e e V

V

z V V V

V V

o8 V in magnitude and +18 in phase

T-line Problem: Example ET-line Problem: Example E• Solution: Solution:

(continued)(continued)o o

o

o

2 2

f) to calculate average power at the load we can use

6.33 V 126 6.33 V 126(40 30) 50 36.9

127 mA 1631 1Re{ } (.127 A) (40 )2 2

.32 W 320 mW

LL

L

L L L

VIZ j

P I Z

Methods of Solving T-line Methods of Solving T-line ProblemsProblems

• Analytical Expressions (as illustrated here)Analytical Expressions (as illustrated here)+ No special tools needed other than calculator+ No special tools needed other than calculator– Complex number calc’s may become tediousComplex number calc’s may become tedious

• Software ToolsSoftware Tools1) MATLAB Zx for lossless lines (have access)1) MATLAB Zx for lossless lines (have access)2) Other commercial programs for calc. & sim.2) Other commercial programs for calc. & sim.

• Graphical Tools: Smith ChartGraphical Tools: Smith Chart

ConclusionsConclusions• Practical T-line problems must address Practical T-line problems must address

both source and load conditions; if fully both source and load conditions; if fully unmatched, reflections occur at both unmatched, reflections occur at both ends.ends.

• Input impedance (ZInput impedance (Zinin) is the complex wave ) is the complex wave impedance at the source end of the T-line impedance at the source end of the T-line evaluated by analytical formula, MATLAB, evaluated by analytical formula, MATLAB, other software tools or the Smith Chart.other software tools or the Smith Chart.

ConclusionsConclusions• The input impedance (ZThe input impedance (Zinin) reduces any sourced ) reduces any sourced

and loaded T-line to an equivalent circuit for and loaded T-line to an equivalent circuit for the purpose of voltage, current and power the purpose of voltage, current and power analysis. analysis.

• Certain T-line lengths (Certain T-line lengths (ll) transform the load ) transform the load impedance (Zimpedance (ZLL) at the input in a special way ) at the input in a special way – QWT: QWT: l l == /4 /4 Z Zinin = Z = Z00

22/Z/ZL L (good for matching)(good for matching)– HWT: HWT: l l == /2 /2 Z Zinin = Z = ZL L (good for transparency)(good for transparency)

ConclusionsConclusions• To find average power delivered to the To find average power delivered to the

load (Pload (PLL) of a sourced and loaded T-line ) of a sourced and loaded T-line – If one knows theIf one knows the

•Thevenin equivalent source (VThevenin equivalent source (Vthth, Z, Zthth))•Characteristic Line & Load impedances (ZCharacteristic Line & Load impedances (Z00,, ZZLL) ) •T-line length, frequency & prop. velocity (T-line length, frequency & prop. velocity (ll, f, v, f, vpp))

– then one can calculatethen one can calculate•T-line electrical length (T-line electrical length ( = = ll))• Input impedance (ZInput impedance (Zinin))•Power input (PPower input (Pinin) via equivalent circuit at the ) via equivalent circuit at the

source and real power absorbed in the load source and real power absorbed in the load

ConclusionsConclusions• If the If the loadload is not matched to a T-line, is not matched to a T-line,

some power reflects back to the source some power reflects back to the source and may damage a generator if and may damage a generator if excessiveexcessive

• If the If the sourcesource is not matched to a T-line is not matched to a T-line (not addressed in examples here!), (not addressed in examples here!), power may reflect at both ends and power may reflect at both ends and calculations become more involvedcalculations become more involved

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.