Coaxial  Cable:  It’s  not  just  a  piece  of  wire.  

Prof.  Carol  Tanner  For  PHYS40441  

Coaxial  Cable  �Coaxial  cable�  From  Wikipedia,  the  free  encyclopedia  

Capacitance  and  Inductance    per  Unit  Length  

! = permativity of materialµ = permeability of material

h

Dd

Typically  derived  in  your  basic  physics  text.  

From  Wikipedia,  the  free  encyclopedia  

SchemaMc  RepresentaMon  

L=total  inductance  C=  total  capacitance  R=  total  resistance  (small  –>  zero)  G=  total  leakage  resistance                                        (very  large  –>  infinite)  

Simplified  SchemaMc  

L

C

Coaxial  cable�  From  Wikipedia,  the  free  encyclopedia  

This  is  not  a  very  good  model  of  a  transmission  line.  

Reminders  from  Basic  Physics  Courses  

Vdrop = LdIdt

Vdrop =QC

Vdrop = RI

ZC = 1j!C

ZR = R

ZL = j!L

V = Z I Z = VI

I

V Z  

For  AC  circuits:  

Derive  the  Capacitance  of  a  Cylinder  

! µ

Dd

!E i d!a! = Q

"E2!rh = Q

"E = Q

!2"rh

V = Edrr1

r2

! = Q"2#h

1rdr =

d /2

D/2

!Q

"2#hln(D / 2)$ ln(d / 2)[ ] = Q

"2#hln D

d%&'

()*

C = QV

= 2!"h

ln Dd

#$%

&'(

Q = CV

Used  Gauss’  Law  and  an  imaginary  surface  to  calculate  the  E  field  in  the  cylinder.  

r

Q = charge on the inner conductor

Ch= 2!"ln(D / d)

Voltage  difference  is  the  integral  of  the  E  field.  

h

Derive  the  Inductance  of  a  Cylinder  

! µ

h

Dd

!B i d!"#! = µI B2!r = µI B = µI

2!r

! =

!B i d!a

r1

r2

" = µI2#r

hdr =d /2

D/2

"µI2#

h ln(D / 2)$ ln(d / 2)[ ] = µI2#

h ln Dd

%&'

()*

Vdrop = !EMF = d"dt

= L dIdt

= h µ2#ln D

d$%&

'()

*+,

-./dIdt

•  Used  Ampere’s  Law  and  an  imaginary  path  to  calculate  the  B  field  in  the  cylinder.  

•  Determine  the  flux  through  the  imaginary  loop.  

rI = current on the inner conductor !B

•  Inductance  is  proporMonal    to  the  induced  EMF  for  a  change  in  current.  

L = h µ2!ln D

d"#$

%&'

Lh= µ2!ln D

d"#$

%&'

CharacterisMc  Impedance  of  a  Coaxial  Cable  

Z0 =L hC h

= LC

=

µ2!ln(D / d)

2!"ln(D / d)

= 12!

µ"ln(D / d)

v = 1µ!

c = 1µ0!0

Speed  of  EM  wave  in  a  medium.   Speed  of  EM  wave  in  a  vacuum.  

cv=

1µ0!01µ!

= µµ0

!!0

Index  of  RefracMon  

Common  Types  

What  it  is  the  characterisMc  impedance  of  a  transmission  line?  

L / h L / h L / h L / h L / hC / h C / h C / h C / h C / h

… …

We  need  a  be`er  model!  

!x

V1I1 I2dQdt

V2

V = 0

V1 !V2 =Lh"x dI

dtI1 ! I2 =

dQdt

= Ch"x dV2

dt

!V!x

= " LhdI1dt

!V =V (x2 )"V (x1)

x

!I = I(x2 )" I(x1)

!I!x

= "ChdV2dt

lim!x"0

!V!x

= #V#x

= $ LhdIdt

lim!x"0

!I!x

= #I#x

= $ChdVdt

Two  Coupled  Diff.  Eq.s  !V!x

= " LhdIdt

!I!x

= "ChdVdt

!!x

!V!x

= " Lhddt

!I!x

= " " LhChddtdVdt

!2V!x2

" LhCh

#$%

&'(d 2Vdt 2

= 0k2 =! 2 L

hCh

dIdt

= ! 1L / h

"V"x

= ! 1L / h

ikAei(kx!#t )

I = dIdtdt! = " 1

L / hikA ei(kx"#t ) dt! = 1

L / h"ik"i#

Aei(kx"#t ) = 1L / h

LhChAei(kx"#t )

V = Aei(kx!"t )!k

= v = 1LhCh

Wave  EquaMon  

Find  the  current  for  a  posiMve  going  wave.  

k = ±! LhCh

I(x,t) = C / hL / h

Aei(kx!"t )

Solve  for  voltage  and  current.  

Traveling  wave  soluMon  for  V  

What  voltage  and  current  do  we  use  to  calculate  the  characterisMc  impedance?  

I(x,t)

V (x,t)

Assume V = 0 on the outside conductor.

Z0 =V (x,t)I(x,t)

= Aei(kx!"t )

C / hL / h

Aei(kx!"t )= L / h

C / h= L

C

v = 1LhCh

= 1µ!

The  voltage  and  current  at  any  parMcular  point  along  the  transmission  line.  

DemonstraMon