Transmission Lines

Transmission Transmission LinesLines

Dr. Sandra Cruz-PolECE Dept. UPRM

Exercise 11.3 A 40-m long TL has Vg=15 Vrms, Zo=30+j60 , and VL=5e-j48o Vrms. If the line is

matched to the load and the generator, find: the input impedance Zin, the sending-end current Iin and Voltage Vin, the propagation constant .

Answers:

ZL

Zg

Vg

+

Vin

-

Iin

+

VL

-

Zo=30+j60+j

40 m

We’ll solve this problem later, but look at Vin and VL

At high frequencies,We cannot

apply regular circuit

theory to electric circuits!

Transmission LinesI. TL parameters

II. TL Equations

III. Input Impedance, SWR, power

IV. Smith Chart

V. Applications Quarter-wave

transformer Slotted line Single stub

VI. Microstrips

Transmission Lines (TL)

TL have two conductors in parallel with a dielectric separating them

They transmit TEM waves inside the lines

Common Transmission LinesTwo-wire (ribbon)

Coaxial

Microstrip

Stripline (Triplate)

Other TL (higher order)[Chapter 12]

Fields inside the TL V proportional to E, I proportional to H

dlHI

dlEV

Distributed parameters

The parameters that characterize the TL are given in terms of per length.

R = ohms/meter L = Henries/ m C = Farads/m G = mhos/m

cmc

kmc

GHz

Hz

15000,000,2000/

000,560/

2

60

Common Transmission LinesR, L, G, and C depend on the particular transmission line structure and the material properties. R, L, G, and C can be calculated using fundamental EMAG techniques.

Parameter Two-Wire Line Coaxial Line Parallel-Plate Line

Unit

R

L

G

C

1

a cond

1

2 cond1

a

1

b

2

w cond

/m

acoshD

2a

2

lnb

a

d

w

H /m

diel

acosh D / 2a

2 diel

ln b /a

diel

w

d

S /m

acosh D / 2a

2ln b /a

w

d

F /m

TL representation

Distributed line parametersUsing KVL:

Distributed parameters Taking the limit as z tends to 0 leads to

Similarly, applying KCL to the main node gives

t

tzILtzRI

z

tzV

),(),(

),(

t

tzVCtzGV

z

tzI

),(),(

),(

ss

ss

VCjGz

I

ILjRz

V

0 22

2

ss V

z

V

Wave equation

Using phasors

The two expressions reduce to

Wave Equation for voltage

sss

sss

CVjGVz

I

LIjRIz

V

])(Re[),(

])(Re[),(tj

s

tjs

ezItzI

ezVtzV

CjGLjR 2

TL Equations Note that these are the wave eq. for

voltage and current inside the lines.

The propagation constant is and the wavelength and velocity are

fu

CjGLjRj

Idz

IdV

dz

Vds

ss

s

2

))((

00 22

22

2

2

Transmission LinesI. TL parameters (R’,L’, G’, C’)

II. TL Equations ( Zo, …)

III. Input Impedance, SWR, power

IV. Smith Chart

V. ApplicationsI. Quarter-wave transformer

II. Slotted line

III. Single stub

VI. Microstrips

Waves moves through line The general solution is

In time domain is

Similarly for current, I)cos()cos(),(

)cos()cos(

])(Re[),(

zteIzteItzI

zteVzteV

ezVtzV

eVeVV

zz

zz

tjs

zzs

z

Characteristic Impedance of a Line, Zo

Is the ratio of positively traveling voltage wave to current wave at any point on the line

I

VjXR

CjG

LjRLjR

I

VZ

eILjReV

eIzI

eVzV

zILjRdz

zdV

ooo

zz

z

z

)()(

)(

)(

)()()(

z

Example: An air filled planar line with w=30cm, d=1.2cm,

t=3mm, c=7x107 S/m. Find R, L, C, G for 500MHz Answer See next

0

/2213.0

/24.50

/036.02

d

wG

mpFd

w

d

wC

mnHw

dL

mw

R

o

c

w

d

Common Transmission LinesR, L, G, and C depend on the particular transmission line structure and the material properties. R, L, G, and C can be calculated using fundamental EMAG techniques.

Parameter Two-Wire Line Coaxial Line Parallel-Plate Line

Unit

R

L

G

C

1

a cond

1

2 cond1

a

1

b

2

w cond

/m

acoshD

2a

2

lnb

a

d

w

H /m

diel

acosh D / 2a

2 diel

ln b /a

diel

w

d

S /m

acosh D / 2a

2ln b /a

w

d

F /m

Exercise 11.1

A transmission line operating at 500MHz has Zo=80 , =0.04Np/m, =1.5rad/m. Find the line parameters R,L,G, and C.

Answer: 3.2 /m, 38.2nH/m, 0.0005 S/m, 5.97 pF/m

03.00005.

1202.3

jCjGZ

jLjRZ

o

o

Different cases of TL

Lossless

Distortionless

Lossy

Transmission line

Transmission line

Transmission line

Lossless Lines (R=0=G)

Has perfect conductors and perfect dielectric medium between them.

Propagation:

Velocity:

Impedance C

LRZX

fLC

u

LCj

ooo

0

2,

1

,,0

Distortionless line (R/L = G/C)

Is one in which the attenuation is independent on frequency.

Propagation:

Velocity:

Impedance G

R

C

LRZX

fLC

u

LCRG

j

ooo

0

1

Summaryj Zo

General

Lossless

(R=0=G)

Distortionless

RC = GLLCjRG

LCj 0

CjG

LjRZo

G

R

C

LRZ oo

))(( CjGLjR

C

LRZ oo

Excersice 11.2 A telephone line has R=30 /km, L=100 mH/km, G=0, and C=

20F/km. At 1kHz, obtain: the characteristic impedance of the line, the propagation constant, the phase velocity.

Is this a distortionless line? Solution:

Transmission LinesI. TL parameters

II. TL Equations

III. Input Impedance, SWR, power (Zin , s, Pave)

IV. Smith Chart

V. Applications Quarter-wave transformer Slotted line Single stub

VI. Microstrips

Define reflection coefficient at the load, L

V

V

eVeVzV

L

zzs

)(

Terminated TL

zL

zs eeVzV )(

zL

z

os ee

Z

VzI

)(

z

Lz

zL

z

os

s

ee

eeZ

zI

zVzZ

)(

)()(

Then,

Similarly,

The impedance anywhere along the line is given by

The impedance at the load end, ZL, is given by

L

LoL ZZZ

1

1)0(

Terminated, Lossless TL

oL

oLL ZZ

ZZ

LCjj 0

Then,

Conclusion: The reflection coefficient is a function of the load impedance and the characteristic impedance.

Recall for the lossless case,

Then zjL

zjs eeVzV )(

zjL

zj

os ee

Z

VzI

)(

Terminated, Lossless TL

djL

djdj

o

djL

djdj

eeeZ

VdI

eeeVdV

)(

)(

It is customary to change to a new coordinate system, z = - l , at this point.

Rewriting the expressions for voltage and current, we have

Rearranging,

ljL

lj

o

ljL

lj

eeZ

VlI

eeVlV

)(

)(

ljL

lj

o

ljL

lj

eeZ

VlI

eeVlV

2

2

1)(

1)(

-z

z = - l

The impedance anywhere along the line is given by

The reflection coefficient can be modified as follows

Then, the impedance can be written as

After some algebra, an alternative expression for the impedance is given by

Conclusion: The load impedance is “transformed” as we move away from the load.

Impedance (Lossless line)

lj

L

ljL

o e

eZ

lI

lVlZ

2

2

1

1

)(

)()(

)(1

)(1)(

l

lZlZ o

ljZZ

ljZZZZlZ

Lo

oLoin

tan

tan)(

ljjL

ljL eeel 22)(

ljL

lj

o

ljL

lj

eeZ

VlI

eeVlV

)(

)(

The impedance anywhere along the line is given by

The reflection coefficient can be modified as follows

Then, the impedance can be written as

After some algebra, an alternative expression for the impedance is given by

Conclusion: For Lossy TL we use hyperbolic tangent

Impedance (Lossy line)

lL

lL

o e

eZ

lI

lVlZ

2

2

1

1

)(

)()(

)(1

)(1)(

l

lZlZ o

lZZ

lZZZZlZ

Lo

oLoin

tanh

tanh)(

)()( 222 ljlL

lL eeel

zL

zs eeVzV )(

Exercise : using formulas A 2cm lossless TL has Vg=10 Vrms, Zg=60 , ZL=100+j80 and Zo=40,

=10cm . Find: the input impedance Zin, the sending-end Voltage Vin,

ZL

Zg

Vg

+

Vin

-

Iin

+

VL

-

Zo

j

2 cm

52tan)80100(40

52

tan40)80100(40)2(

jj

jjcmZZ in

17.212.12 jZ in

Voltage Divider:

o

oL

oLL ZZ

ZZ4.2362.0

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Example A generator with 10Vrms and Rg=50, is connected

to a 75load thru a 0.8 50-lossless line. Find VL

Transmission LinesI. TL parameters

II. TL Equations

III. Input Impedance, SWR, power

IV. Smith Chart

V. Applications Quarter-wave transformer Slotted line Single stub

VI. Microstrips

SWR or VSWR or sWhenever there is a reflected wave, a standing wave will form out of the combination of incident and reflected waves.

(Voltage) Standing Wave Ratio – SWR=VSWR= s is defined as:

L

Ls

I

I

V

VSWRs

1

1

min

max

min

max

Transmission LinesI. TL parameters

II. TL Equations

III. Input Impedance, SWR, power

IV. Smith Chart

V. Applications Quarter-wave transformer Slotted line Single stub

VI. Microstrips

Power

The average input power at a distance l from the load is given by

which can be reduced to

The first term is the incident power and the second is the reflected power. Maximum power is delivered to load if =0

)()(Re2

1 * lIlVPave

2

2

12

o

o

ave Z

VP

Three Common Cases of line-load combinations: Shorted Line (ZL=0)

Open-circuited Line (ZL=∞)

Matched Line (ZL = Zo)

ljZZ oin cot

oin ZZ

sjbljZZ Loin ,1 tan0

sL ,1

1,0 sL

Standing Waves -Short

Shorted Line (ZL=0), we had

So substituting in V(z)

sljZZ Loin ,1 , tan

-z /4/2

|V(z)|

])1([)( ljlj eeVzV

)sin2()( ljVzV

lVzV sin2)(

lVzV2

sin2)(

Voltage maxima

*Voltage minima occurs at same place that impedance has a

minimum on the line

Standing Waves -Open

Open Line (ZL=∞) ,we had

So substituting in V(z) |V(z)|

lVzV

lVzV

lVzV

eeVzV

sljZZ

ljlj

Loin

2cos2)(

cos2)(

)cos2()(

])1([)(

,1 , cot

-z /4/2

Voltage minima

Standing Waves -MatchedMatched Line (ZL = Zo), we had

So substituting in V(z)|V(z)|

VzV

eVzV

eVzV

eeVzV

sZZ

lj

lj

ljlj

Loin

)(

)(

)(

])0([)(

1,0 ,

-z /4/2

Java applets

http://www.amanogawa.com/transmission.html

http://physics.usask.ca/~hirose/ep225/ http://www.home.agilent.com/agilent/applic

ation.jspx?nid=-34943.0.00&cc=PR&lc=eng

Transmission LinesI. TL parameters

II. TL Equations

III. Input Impedance, SWR, power

IV. Smith Chart

V. Applications Slotted line Quarter-wave transformer Single stub

VI. Microstrips

The Smith Chart

Smith Chart

Commonly used as graphical representation of a TL.

Used in hi-tech equipment for design and testing of microwave circuits

One turn (360o) around the SC = to /2

What can be seen on the

screen?

Network Analyzer

Smith Chart Suppose you use as coordinates the reflection

coefficient real and imaginary parts.

and define the normalized ZL:

oL

oLir ZZ

ZZj

i

r

Now relating to z=r+jx

After some algebra, we obtain two eqs.

Similar to general Equation of a Circle of radius a, center at (x,y)= h,k)

Circles of r

Circles of x

Examples of Circles of r and x

2

2

1Radius

1,1 Center

1

1Radius 0,

1 Center

xx

rr

r

rr

r

1

1Radius 0,

1 Center

Circles of r Circles of x

xx

1Radius

1,1 Center

Examples of circles of r and x

Circles of r Circles of x

rr

r

1

1Radius 0,

1 Center

xx

1Radius

1,1 Center

r

i

Fun facts about the Smith Chart A lossless TL is represented as

a circle of constant radius, ||, or constant s

Moving along the line from the load toward the generator, the phase decrease, therefore, in the SC equals to moves clockwisely.

ljjL

ljL eeel 22)( To

generator

The joy of the SC

Numerically s = r on the +axis of r in the SC

Proof:

1s

1-s0 but then

1

1)when (

1

1

j

r

rrz

z

z

r

L

L

Fun facts about the Smith Chart

One turn (360o) around the SC = to /2 because in the formula below, if you substitute length for half-wavelength, the phase changes by 2, which is one turn.

Find the point in the SC where =+1,-1, j, -j, 0, 0.5What is r and x for each case?

ljLel 2)(

Fun facts : Admittance in the SC

The admittance, y=YL/Yo where Yo=1/Zo, can be found by moving ½ turn (/4) on the TL circle

1

1

1

1

1

11)0(

1

1

)1(1

)1(1

1

1)4/(

1

1

1

1

1

1

1

11)0(

4

222

0

0

2

2

02

02

2

2

2

2

j

j

lj

lj

o

o

j

j

j

j

lj

lj

lj

o

lj

oo

L

e

e

eV

eZV

Yly

e

elz

e

e

e

e

eZV

eV

ZZ

Zlz

l

Fun facts about the Smith Chart

The r +axis, where r > 0 corresponds to Vmax

The r -axis, where r < 0 corresponds to Vmin

Vmax (Maximum

impedance)

Vmin

Exercise: using S.C. A 2cm lossless TL has Vg=10 Vrms, Zg=60 , ZL=100+j80 and Zo=40, =10cm . find: the input impedance Zin, the sending-end Voltage Vin,

Load is at .2179 @ S.C. Move .2 and arrive to .4179 Read

ZL

Zg

Vg

+

Vin

-

Iin

+

VL

-

Zo

j

2 cm

55.3. jzin

Voltage Divider:

radZZ

ZVV

gin

ingin 775.032.3

25.240

80100j

jzL

2.l

2212 jZ in

oL 5.23622.0

ocm 120622.0)2(

4.176. jyin

zL

zin 0.2towards generator

.2179

.4179

VminVmax

Exercise: cont….using S.C. A 2cm lossless TL has Vg=10 Vrms, Zg=60 , ZL=100+j80 and Zo=40, =10cm . find: the input impedance Zin, the sending-end Voltage Vin,

Distance from the load (.2179to the nearest minimum & max Move to horizontal axis toward the generator and arrive to .5Vmax and to .25 for the Vmin. .

Distance to 1st max=.25-.2179=.0321 Distance to 1st min=.5-.2179=.282 Distance to 2st voltage maximum is .282See drawing

ZL

Zg

Vg

+

Vin

-

Iin

+

VL

-

Zo

j

2 cm

25.2 jzL 2.lo

L 5.23622.0

Exercise : using formulas A 2cm lossless TL has Vg=10 Vrms, Zg=60 , ZL=100+j80 and Zo=40,

=10cm . find: the input impedance Zin, the sending-end Voltage Vin,

ZL

Zg

Vg

+

Vin

-

Iin

+

VL

-

Zo

j

2 cm

52tan)80100(40

52

tan40)80100(40)2(

jj

jjcmZZ in

17.212.12 jZ inVoltage Divider:

radZZ

ZVV

gin

ingin 766.030.3

o

oL

oLL ZZ

ZZ4.2362.0

ooL

ljLecm

6.12062.0144

)2( 2

Another example: A 26cm lossless TL is connected to load ZL=36-j44 and Zo=100, =10cm . find: the input impedance Zin

Load is at .427 @ S.C. Move .1 and arrive to .527 Read

ZL

Zg

Vg

+

Vin

-

Iin

+

VL

-

Zo

j

26cm

16.31. jzin

Distance to first Vmax:

44.36. jzL 1.)5(.56.2 l

1631 jZ in

oL 12754.0

Exercise 11.4 A 70 lossless line has s =1.6 and =300o. If the line is 0.6 long, obtain , ZL, Zin and the distance of the first minimum voltage from the

load. Answer

The load is located at: Move to .4338and draw line from center to this place, then read where it crosses you TL circle. Distance to Vmin in this case, lmin =.5-.3338=

oL 30023.0

1s

1-s

6.335.80

48.15.1

jzZZ

jz

LoL

L

6/

5.176.47

25.68.0

jZ

jz

in

in3338.

Java Applet : Smith Chart

http://education.tm.agilent.com/index.cgi?CONTENT_ID=5

Transmission LinesI. TL parameters

II. TL Equations

III. Input Impedance, SWR, power

IV. Smith Chart

V. Applications Slotted line Quarter-wave transformer Single stub

VI. Microstrips

Slotted Line

Used to measure frequency and load impedance

HP Network Analyzer in Standing Wave Display http://www.ee.olemiss.edu/software/naswave/Stdwave.pdf

Slotted line exampleGiven s, the distance between adjacent minima, and lmin for an

“air” 100 transmission line, Find f and ZL

s=2.4, lmin=1.5 cm, lmin-min=1.75 cm

Solution: =8.6GHz

Draw a circle on r=2.4, that’s your T.L.

move from Vmin to zL

3850

38.5.

jzZZ

jz

LoL

L

cm

cf

5.3

103 8

429./5.3

5.1min l

Transmission LinesI. TL parameters

II. TL Equations

III. Input Impedance, SWR, power

IV. Smith Chart

V. Applications Slotted line Quarter-wave transformer Single stub

VI. Microstrips

Quarter-wave transformer …for impedance matching

2/tan

tan

42

tan

ljZZ

jZZ

ZZLo

oL

oin

ZLZin Zo ,

l= /4

L

oin Z

ZZ

2

Conclusion: **A piece of line of /4 can be used to change the impedance to a desired value (e.g. for impedance matching)

Applications Slotted line as a frequency meter

Impedance MatchingIf ZL is Real: Quarter-wave Transformer (/4 Xmer)

If ZL is complex: Single-stub tuning (use admittance Y)

Transmission LinesI. TL parameters

II. TL Equations

III. Input Impedance, SWR, power

IV. Smith Chart

V. Applications Quarter-wave transformer Slotted line Single stub

VI. Microstrips

Single Stub Tuning …for impedance matching A stub is connected in parallel to sum the

admittances Use a reactance (Y) from a short-circuited

stub or open-circuited stub to cancel reactive part

When matched: Zin=Zo therefore z =1 or y=1 (this is our goal!)

Single Stub Basics

We work with Y, because in parallel connections, they add.

YL (=1/ZL) is to be matched to a TL having characteristic admittance Yo by means of a "stub" consisting of a shorted (or open) section of line having the same characteristic admittance Yo

http://web.mit.edu/6.013_book/www/chapter14/14.6.html

Single Stub Steps

First, the length l is adjusted so that the real part of the admittance at the position where the stub is connected is equal to Yo or yline = 1+jb

Then the length of the shorted stub is adjusted so that it's susceptance cancels that of the line, or ystub= -jb

Example: Single StubA 75 lossless line is to be matched to

a 100-j80 load with a shorted stub. Calculate the distance from the load, the stub length, and the necessary stub admittance.

Answer: Change zL to admitance: Find d=distance to circle with real=1 as:

d=.4338- .3393=0.094or

(both yield same d)

Read(1+jbj [or next intersection i.e. 1-

jb :d=0.272,]

Short stub:.25-.124=0.126 Or 0.376.25= 0.126(both yield same

distance)

With ystub= -j.96/75 *j mhos

*[Note que para denormalizar admitancia se divide entre Zo]

.4338

.3393

.0662

.1607

.124

.376

Transmission LinesI. TL parameters

II. TL Equations

III. Input Impedance, SWR, power

IV. Smith Chart

V. Applications Quarter-wave transformer Slotted line Single stub

VI. Microstrips

Microstrips

Microstripsanalysisequations &Pattern of EM fields

1/8

ln60

hw

h

w

w

hZ

eff

o

]1/for[)444.1/ln(667.0393.1

1201

hw

hwhw

Zeff

o

MicrostripDesign Equations

Falta un radical en eff

eff

cu

2/2

8/

2

hw

e

ehw

A

A

2/61.039.)1ln(

2

1

)12ln(12

/

hwB

BB

hw

rr

r

rr

rroZA

11.0

23.01

1

2

1

60

roZB

160 2

Microstrip Design Curves

ExampleA microstrip with fused quartz (r=3.8) as a

substrate, and ratio of line width to substrate thickness is w/h=0.8, find:

Effective relative permittivity of substrate Characteristic impedance of line Wavelength of the line at 10GHz

Answer:eff=2.75, Zo=86.03 , =18.09 mm

75.28.0/1212

8.2

2

8.4/1212

1

2

1

wh

rreff

Diseño de microcinta:

Dado (r=4) para el substrato, y h=1mm halla w para Zo=30 y cuánto es eff?

Solución: Suponga que como Z es pequeña w/h>2