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Unit 1 Transmission Line

V SEM ECE / JJ/JECRC/JOD

1

OVERVIEW

RF equivalent circuit representation of a Transmission Line

Primary constant R, L, G, C ; Secondary constants Z 0, P, , . Series impedance Z =R+jwL ;Shunt admittance Y=G+jwC Propagation constant P= +j (where neper/km and radian/ km)Where = attenuation constant and = phase constant1 neper = 8.686 db ; 1 radian = 57.3 o

Characteristic Impedance Z 0 = jwC)(G jwL)R(

+

+

Propagation constant P = jwC)(G jwL)R( ++ coshpx =(e px+e -px)/2 ; sinhpx=(e px-e -px )/2 ; tanhjx l = jtanx l , in term of primary and secondary constant

= )]CwG)(Lw(RLC)w-(1/2[(RG 2222222 +++

= )]CwG)(Lw(RRG)-LC(1/2[(w 2222222 ++ Relation between Series impedance and Sunt admittance with P, Z0 R + jwL = P Z 0 ; G + jwC= P /Z 0Transmission line equation (Voltage and Current ) along the lengthV= VS coshpx- I SZ0 sinhpx I= IS coshpx- V S / Z 0 sinhpx

Input impedance of transmission line terminated with any load impedanceZ R

Z s = tanhpl)Z(Z tanhpl)Z(ZZ

R0

0R0

+

+

For loss less =0, P=j ; tanhj l = jtan l ; = LC = 2

Z s = l)tan jZ(Z )ltanZ(ZZ

R0

0R0

+

+ j=

l)tan jZ(Z

)ltanZ(ZZ

R0

0R0

LC

LC j

+

+

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Unit 1 Transmission Line

V SEM ECE / JJ/JECRC/JOD

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Input Impedance in term of k Reflection co-efficient

Z s =)ke-(1 )e(1Z 2Pl-

-2P

0 k +

where k=)Z(Z)Z-(Z

R0

0R+

Prove: Input impedance of Infinite line is char impedance Z 0

Prove : Infinite line is equivalent to finite line with teminated any impedance Z 0

Distortion in line (Delay , Phase and Frequency distortion)Losses in line (Radiation loss, Copper loss, Dielectric loss, Sheath loss ) Dielectric loss= VI cos where =losses angleDistortion less line

Z 0 =C

L=

G

R;L=

GCR

value of L at which minimum attenuation

Transmission line at a high frequency

Skin effect :Skin depth =

w 2

where w =angular freq.(rad/sec);

=conductivity (siemens/m); =4 *10 -7permeability H/m

For copper = f

0664.0meters

Char. Impedance Z 0 =C L

=G R ; Because ( L>>R ) and ( C>> G)

, at High Frequency

=C LG

LC R

22+ ; = LC =

2

Velocity of Propagation VP =

= LC

1; V c = gP VV

Group Velocity Vg =12

12

; Vg =

P

P

V V

1

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Unit 1 Transmission Line

V SEM ECE / JJ/JECRC/JOD

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Reflection co-efficient k = (Reflected Voltage or Current )/ (Incident Voltage or Current)

Voltage transmission co-efficient k =V r / V i = k=)Z(Z

)Z-(Z

R0

0R

+

Current transmission co-efficient - k =I r / Iik=0 for impedance matching, k=1 open ckt, k= -1 short ckt at load impedanceStanding wave ratio VSWR and CSWR

VSWR =min

max

VV

; CSWR =min

max

II

VSWR=K1

K1

+Range (1 to infinite and 1 for matched condition )

Input Impedance of short circuited and open circuited transmission line

Zoc = Z o cothpl ; Zsc = Z otanhplZo =( Z oc . Zsc) ; tanhpl =

ZocZsc

Quarter wave transformer ( /4 line uses for impedance matching )Input impedance of quarter wave transformer Z in = Z o

2 / Z l If P=1/2L(log e(r,)) than =(log e r)/2L, = ( * /180) /2LEfficiency of transmission

= %100*)(P) (P

S

R ; P S = VS*IS cos

Reflection Loss f L = 10 log 10 (power to load /incident power ) = 20 log 100

0*2

Z Z

Z Z

R

R

+

For Open wire line ( suitable working range less than 100MHz ) Zo = 276 log e ohmsad

where d is the spacing between two wire from center and a is the radius of the wire

L = mhenrysad

er / 10log21.97

+ ;C = meter frad

a

d e

d / log

where r d =

Rdc and Rac is dc and ac loop resistance

Rdc = loopmeter a

/ 2

2

;R ac = Rdc loopmeter f a

Rc

dc / 2

For Co-axial cable ( suitable working range upto 1GHz )

L= m H d

De / )(log2

;C= mF

d D

e

/ )(log

2 ; R dc = m

D Dd e /

'11

41

2

2

+

Rac = md D

f /

11

+

; Z o = 138 log e ohmsd

D

D and d is the diameter of inner and outer conductor D is the inner diameter of outerconductor