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 D
d
! = permativity of material
µ = permeability of material
h
Typically derived in your basic physics text. From Wikipedia, the free encyclopedia SchemaMc RepresentaMon Coaxial cable From Wikipedia, the free encyclopedia L=total inductance C= total capacitance R= total resistance (small –> zero) G= total leakage resistance (very large –> infinite) L
C
Simplified SchemaMc This is not a very good model of a transmission line. Reminders from Basic Physics Courses For AC circuits: Vdrop = RI
ZR = R
Q
Vdrop =
C
dI
Vdrop = L
dt
V=Z I
1
ZC =
j! C
Z L = j! L
I
V
Z V
Z=
I
Derive the Capacitance of a Cylinder r
d
D
Q = charge on the inner conductor
!
h
Used Gauss’ Law and an imaginary surface to calculate the E field in the cylinder. Voltage difference is the integral of the E field. Q = CV
µ
! ! Q
! E i da = "
r2
Q
E2! rh =
"
E=
D/2
Q
! 2" rh
Q
1
Q
Q
% D(
V = ! E dr =
dr
=
ln(D
/
2)
$
ln(d
/
2)
=
ln
[
]
'& *)
!
"
2
#
h
r
"
2
#
h
"
2
#
h
d
r1
d/2
C=
Q
2!" h
=
# D&
V
ln % (
$d'
C
2!"
=
h ln(D / d)
Derive the Inductance of a Cylinder !
B
I = current on the inner conductor
r
D
d
!
µ
h
•  Used Ampere’s Law µI
! !
B
=
and an imaginary path B2! r = µ I
B i d " = µI
#
!
2! r
to calculate the B field r2
in the cylinder. ! ! D/2 µ I
µI
µI
% D(
!
=
B
i
d
a
=
h
dr
=
h
ln(D
/
2)
$
ln(d
/
2)
=
h
ln
[
]
'& *)
•  Determine the flux "r
" 2# r
2
#
2
#
d
d/2
through the imaginary 1
loop. d"
dI * µ $ D ' - dI
Vdrop = !EMF =
= L = ,h
ln & ) /
% d ( . dt
dt
dt
2
#
+
•  Inductance is proporMonal µ " D%
L µ " D%
to the induced EMF for a L=h
ln $ '
=
ln $ '
change in current. 2! # d &
h 2! # d &
CharacterisMc Impedance of a Coaxial Cable L h
L
Z0 =
=
=
C h
C
µ
ln(D / d)
1
2!
=
2!"
2!
ln(D / d)
1
v=
µ!
c=
Speed of EM wave in a medium. Index of RefracMon c
=
v
1
µ0! 0
=
1
µ!
µ
ln(D / d)
"
1
µ0! 0
Speed of EM wave in a vacuum. µ !
µ0 ! 0
Common Types What it is the characterisMc impedance of a transmission line? L /h
C /h
We need a be`er model! L /h
C /h L /h
C /h
L /h
C /h
L /h
C /h
…
…
x
I1
V1
V2
I2
dQ
dt
L
dI
"x
h
dt
!V = V (x2 ) " V (x1 )
!V
L dI1
="
!x
h dt
V =0
!x
V1 ! V2 =
!V #V
L dI
=
=$
!x"0 !x
#x
h dt
lim
I1 ! I 2 =
dQ C
dV
= "x 2
dt
h
dt
!I = I (x2 ) " I (x1 )
!I
C dV2
="
!x
h dt
!I #I
C dV
=
=$
!x"0 !x
#x
h dt
lim
Two Coupled Diff. Eq.s Solve for voltage and current. !V
L dI
="
!x
h dt
!I
C dV
="
!x
h dt
! !V
L d !I
L C d dV
="
=""
!x !x
h dt !x
h h dt dt
Wave EquaMon ! 2 V # L C & d 2V
"%
=0
(
!x 2 $ h h ' dt 2
Find the current for a posiMve going wave. I=!
Traveling wave soluMon for V V = Aei(kx!" t )
k2 = ! 2
LC
h h
k = ±!
LC
h h
!
=v=
k
dI
1 "V
1
=!
=!
ikAei(kx!# t )
dt
L / h "x
L /h
dI
1
1 "ik i(kx"# t )
1
dt = "
ikA ! ei(kx"# t ) dt =
Ae
=
dt
L /h
L / h "i#
L /h
I (x,t) =
C / h i(kx!" t )
Ae
L /h
L C i(kx"# t )
Ae
h h
1
LC
h h
What voltage and current do we use to calculate the characterisMc impedance? The voltage and current at any parMcular point along the transmission line. I (x,t)
V (x,t)
Assume V = 0 on the outside conductor.
V (x,t)
Z0 =
=
I(x,t)
Aei(kx!" t )
L /h
L
=
=
C /h
C
C / h i(kx!" t )
Ae
L /h
v=
1
1
=
LC
µ!
h h
DemonstraMon