Yoonchan Jeong
School of Electrical Engineering, Seoul National University Tel: +82 (0)2 880 1623, Fax: +82 (0)2 873 9953
Email: [email protected]
Electromagnetics:
General Transmission-Line Equations
(9-3)
General Transmission Line Equations
d G =
σ
wc
f c
R w
σ µ π
= 2
w L =
µ
dd C =
ε
w→ Equivalent circuit
Faraday’s law:
0 ) , ) (
, ) (
, ( )
,
( − + ∆ =
∂
∆ ∂
−
⋅
∆
−
→ v z z t
t t z z i L t
z i z R t
z t v
t z L i t
z z Ri
t z v
∂ + ∂
∂ =
− ∂ ( , )
) , ) (
, (
Ampère’s law:
0 ) , ) (
, ) (
, (
) ,
( − + ∆ =
∂
∆ +
∆ ∂
−
∆ +
⋅
∆
−
→ i z z t
t t z z
z v C t
z z
v z G t
z i t
t z C v
t z z Gv
t z i
∂ + ∂
∂ =
− ∂ ( , )
) , ) (
, (
) ( ) ) (
( R j L I z
dz z
dV = + ω
−
[ V z e
j t] → t
z
v ( , ) = Re ( )
ωd
→ Kirchhoff’s voltage law
→ Kirchhoff’s current law
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
Wave Characteristics on an Infinite Transmission Line
) ( ) ) (
( R j L I z
dz z
dV = + ω
−
) ( ) ) (
( G j C V z
dz z
dI = + ω
−
Time-harmonic transmission-line equations:
) ) (
(
22 2
z dz V
z V
d = γ
) ) (
(
22 2
z dz I
z I
d = γ
β α
γ = + j
←
Propagation const.
(Attenuation const. & phase const.)
General solutions:
) ( )
( )
( z V z V z
V =
++
−= V
0+e
−γz+ V
0−e
+γz)
( )
( )
( z I z I z
I =
++
−= I
0+e
−γz+ I
0−e
+γz−
− +
+
= −
=
0 0 0
0
0
I
V I
Z V
Characteristic impedance:
γ ω L j R +
= G j ω C
γ
= +
C j G
L j R
ω ω +
= +
3
) )(
(R + j
ω
L G + jω
C=
Line Characteristics (1)
Lossless line:
Propagation constant:
γ = α + j β = j ω LC
Phase velocity:
β
= ω u
pLC
= 1
Characteristic impedance:
Z
0= R
0+ jX
0C
= L
Low-loss line:
Propagation constant:
γ = α + j β
2 / 1 2
/ 1
1
1
+
+
= j C
G L
j LC R
j ω ω ω
Phase velocity:
β
= ω u
pLC
≅ 1
Characteristic impedance:
Z
0= R
0+ jX
02 / 1 2
/ 1
1 1
−
+
+
= j C
G L
j R C
L
ω ω
LC C j
G L L
R C + ω
+
≅ 2 1
−
−
≅ C
G L R j C
L
ω 1 2
) )(
(R + j
ω
L G + jω
C= 0
,
0 =
=
→ R G
C G
L
R <<
ω
<<ω
→ ,
C j G
L j R
ω ω +
= +
Transmission-Line Parameters (1)
Propagation constant:
γ
=α
+ jβ
= (R+ jω
L)(G + jω
C)2 / 1
1
+
=
→ j C
LC G
j
ω ω
γ
LR <<
ω
2 / 1
1
+
=
ωε
µε σ ω
γ
j jPropagation constant for a TEM wave in a lossy dielectric medium:
ε
= σ
→ C G
µε
=
→ LC
Useful relations to determine transmission-line parameters if any single of those is known!
For low loss conductors:
5
Transmission-Line Parameters (2)
Two-wire transmission line:
( D a )
C = cosh
−1πε / 2
(See Eq. 4-47)
L = µε C
→
C G ε
= σ
→
R
sR = 2 f
cσ µ π π
= 1
Coaxial transmission line:
a L ln b
2 π
= µ
→
) / ln(
2 a C = b πε
) / ln(
2 a G = b πσ
→
+
= 1 f
c1 1 σ
µ π ) π
1 ( 1
R
sR = +
(See Eq. 3-139)
=
−
a D cosh
12 π
µ
) 2 / (
cosh
−1D a
= πσ
D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989. D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989.
Attenuation Constant from Power Relations
) )(
(R j L G j C
j
ω ω
β α
γ
+ +
= + Propagation constant: =
Attenuation constant:
α = Re[ ( R + j ω L )( G + j ω C ) ]
z
e
jV z
V ( ) =
0 −(α+ β)z
e
jI z
I ( ) =
0 −(α+ β) Time-averaged power loss:[ ( ) ( ) ]
2 Re ) 1
( z V z I
*z
P =
→ R e
zZ
V
2α2 0 0 2 0
2
=
− ze
jZ
V
( )0
0 −α+ β
=
) ) (
( P z
z z P
=
L∂
− ∂
→
Time-averaged power loss per unit length
) ( 2 α P z
= ) ( 2
) (
z P
z P
L=
→ α
[ I ( z )
2R V ( z )
2G ]
2
1 +
= R G Z e
zZ
V
2 2α2 0 0 2
0
( )
2
+
−=
) 2 (
1
20 0
Z G R R +
=
(
02)
2
01 R GR
R +
≅
→ α
+
= C
G L L R C 2 1
For a low-loss line:Z
0≅ R
0= L / C
7
0 0
0
R jX
Z = +
←
Loss by conducting walls Loss by dielectric
