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Department of Electronic EngineeringNational Taipei University of Technology
••••
Department of Electronic Engineering, NTUT2/47
Two-wire line
Coaxial ( )
Microstrip( )
Department of Electronic Engineering, NTUT3/47
•
•
•
( )
•
Department of Electronic Engineering, NTUT4/47
( )
Department of Electronic Engineering, NTUT5/47
•( ) :
LL s
s L
Zv v
Z Z=
+
Source
Source impedance Load
impedance
(Transmission line)
sv
sZLZ
l
•? ?
Lv
Department of Electronic Engineering, NTUT6/47
(Distributed Circuit Model)
• l
∆x
• R: Ω/m, L: H/m, C: F/m, G: S/m
G x∆
L x∆R x∆
C x∆sZ
sv LZ
llll
dx dx dx
Department of Electronic Engineering, NTUT7/47
•• :
• :
• R: G:
( ) ( ), , ,v x t i x t
( ) ( ), , ,v x x t i x x t+ ∆ + ∆
( ),v x x t+ ∆
( ),i x x t+ ∆R x∆ L x∆
G x∆ C x∆( ),v x t
( ) ,i x t
Department of Electronic Engineering, NTUT8/47
(I)
• (KVL):
• (KCL):
( ) ( ) ( ) ( ) ( ) ( ),, , ,
i x tv x t v x x t R x i x t L x
t
∂− + ∆ = ∆ + ∆
∂
( ) ( ) ( ) ( ) ( ) ( ),, , ,
v x x ti x t i x x t G x v x x t C x
t
∂ + ∆− + ∆ = ∆ + ∆ + ∆
∂
( ),v x x t+ ∆
( ),i x x t+ ∆R x∆ L x∆
G x∆ C x∆( ),v x t
( ) ,i x t
Department of Electronic Engineering, NTUT9/47
(II)
( ) ( ) ( ), ,,
v x t i x tRi x t L
x t
∂ ∂= − −
∂ ∂
( ) ( ) ( ), ,,
i x t v x tGv x t C
x t
∂ ∂= − −
∂ ∂
( ) ( ) ( ) ( ), , ,,
v x t v x x t i x tRi x t L
x t
− + ∆ ∂= +
∆ ∂
( ) ( ) ( ) ( ), , ,,
i x t i x x t v x x tGv x x t C
x t
− + ∆ ∂ + ∆= + ∆ +
∆ ∂
•
• ∆x
(transmission-line equation)(telegrapher equation)
Department of Electronic Engineering, NTUT10/47
(III) −
• (KVL):
• (KCL):
( ) ( ) ( ) ( ) ( ) ( )V x V x x R x I x j L x I xω− + ∆ = ∆ + ∆
( ) ( ) ( ) ( ) ( ) ( )I x I x x G x V x x j C x V x xω− + ∆ = ∆ + ∆ + ∆ + ∆
( )V x x+ ∆
( )I x x+ ∆R x∆ j L xω ∆
G x∆1
j C xω ∆( )V x
( )I x
( ) ( ), cosv x t V x tω=
Department of Electronic Engineering, NTUT11/47
(IV) −
( ) ( ) ( )dV xRI x j LI x
dxω= − −
( ) ( ) ( )dI xGV x j CV x
dxω= − −
( ) ( ) ( ) ( )V x V x xRI x j LI x
xω
− + ∆= +
∆
( ) ( ) ( ) ( )I x I x xGV x x j CV x x
xω
− + ∆= + ∆ + + ∆
∆
•
• ∆x
(transmission-line equation)(telegrapher equation)
Department of Electronic Engineering, NTUT12/47
( ) ( ), ,v x t i x tL
x t
∂ ∂= −
∂ ∂
( ) ( ), ,i x t v x tC
x t
∂ ∂= −
∂ ∂
• R = G = 0
( ),v x x t+ ∆
( ),i x x t+ ∆L x∆
C x∆( ),v x t
( ) ,i x t
( ) ( ) ( ), ,,
v x t i x tRi x t L
x t
∂ ∂= − −
∂ ∂
( ) ( ) ( ), ,,
i x t v x tGv x t C
x t
∂ ∂= − −
∂ ∂
Department of Electronic Engineering, NTUT13/47
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) , cos Re Rej t x j x j tv x t f x t x f x e f x e eω ϕ ϕ ωω ϕ += + = =
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) , cos Re Rej t x j x j ti x t g x t x g x e g x e eω η η ωω η += + = =
( ) ( ) ( )j xI x g x e η=
( ) ( ) ( )j xV x f x e ϕ= ( ) ( ) , Re j tv x t V x e ω=
( ) ( ) , Re j ti x t I x e ω=
• :
f(x) g(x) ( )xϕ ( )xη
•
time-domain
Department of Electronic Engineering, NTUT14/47
( ) ( )V xj LI x
xω
∂= −
∂
( ) ( )I xj CV x
xω
∂= −
∂
( ) ( )dV xj LI x
dxω= −
( ) ( )dI xj CV x
dxω= −
( ) ( )V x dV x
x dx
∂=
∂
( ) ( ) ( )2
22
d V x dI xj L LCV x
dx dxω ω= − = −
•
( ) ( ) ( ) ( )2 2
2 22 2
0d V x d V x
LCV x V xdx dx
ω β+ = + =
LCβ ω=•
•
( ) j x j xV x Ae Beβ β−= + A B (complex constant)
• (Phasor)
(Think that if approaches zero?)ω
( )β
Phasor
Department of Electronic Engineering, NTUT15/47
( ) ( ) ( ) ( )1 1 j x j xdV xI x A j e B j e
j L dx j Lβ ββ β
ω ω− = = − + − −
LCβ ω=
• 0
L L LZ
CLC
ω ωβ ω
= = =
j x j xA e B eL L
β ββ βω ω
− = −
( ) j x j xV x Ae Beβ β−= + ( ) ( )dV xj LI x
dxω= −
Z0
Z0
( )0 0
j x j xA BI x e e
Z Zβ β−= −
Department of Electronic Engineering, NTUT16/47
− (I)
•
1Z
2Z 3Z 3 1 2Z Z Z= +
1Z
2Z 2Z
1Z
2Z
1Z
3Z
1Z
4Z 5Z
4 2 3||Z Z Z=
Department of Electronic Engineering, NTUT17/47
− (II)
•
2 01 1 0
2 0
2 0
11 1ab
Z ZZ Z Z Z
Z ZZ Z
= + = + =++
21 1
0 1 22 4
Z ZZ Z Z
= + +
Z0
a-bZ0 c-d
Z0( )
1Z
2Z 2Z
1Z
2Z
1Z 1Zc
d
a
b
a
b
c
d
2Z
1Z
0Z 0Z
a
b
Department of Electronic Engineering, NTUT18/47
− (III)
•2
1 10 1 22 4
Z ZZ Z Z
= + +
L
2 2
0 0lim
4L
L L LZ
C C
ω→
= − =
2 2
0 4
L LZ
C
ω= −4 LCω <
4 LCω >
0, 0lim 4
L CLCω
→ →= = ∞
1Z j Lω=
2
1Z
j Cω=
L
C C
La
b
CCC
2L 2L 2L2La
b
a′
Department of Electronic Engineering, NTUT19/47
( ) ( ) ( ) ( ) , Re Re j x t j x tj tv x t V x e Ae Beβ ω β ωω − − += = +
( ) ( ) ( ) ( )
0 0
, Re Re j x t j x tj t A Bi x t I x e e e
Z Zβ ω β ωω − − +
= = −
( ) ( )cos cosA x t B x tβ ω β ω= − + +
( ) ( )0 0
cos cosA B
x t x tZ Z
β ω β ω= − − +
( ) j x j xV x Ae Beβ β−= +
( )0 0
j x j xA BI x e e
Z Zβ β−= −
• Phasor
Department of Electronic Engineering, NTUT20/47
(Wavelength)
• +x
x (rads)x
j xe β− xβ
2x
x λπ β βλ=
= = 2πλβ
=
( ) ( )1 , Re cosj x j tv x t Ae e A t xβ ω ω β−= = −
x λ=
( ) j xV x Ae β−=
0x = x λ=
0t = t T=
distance
time
phase 0xxβ
=2
xx λβ π
==
For simplification, assume thewave starts from x=0 and t=0.
Department of Electronic Engineering, NTUT21/47
(Wave Velocity)
( )70 4 10 Wb/A-mL µ π −= ⋅≃
( )120 8.85419 10 F/mC ε −⋅≃ ≃
( ) ( )8,
0 0
1 light speed 3 10 /p vacuumv c m s
µ ε= = = ⋅
( )00
0
377 L
ZC
µε
= = = Ω
,0
p vaccumv
fλ =
2 1
2 2pv fT LC
λ ω π ω ωλ λπ β π β
= = ⋅ = = = =
• : ( T )
( )
Department of Electronic Engineering, NTUT22/47
0µ µ≃0,rε ε ε≃
( )8
0 0
1 3 10/p
r r r
cv m s
µ ε ε ε ε⋅= = =
(non-magnetic material)
0p rg
r
cv
f f
ε λλε
= = =
ε r
81.5rε =
( )8
7,
3 103.32 10 /
81.5p waterv m s
⋅= ⋅≃
0, 00.11
81.5g water
λλ λ= ≅
Department of Electronic Engineering, NTUT23/47
@1 GHz * @10 GHz
Air ( ) 1 30 cm − 3 cm −
Alumina( ,96%) 9.6 9.68 cm 0.48 cm 0.968 cm 0.48 mm
Sapphire ( ) 9.4 9.78 cm 0.49 cm 0.978 cm 0.49 mm
Glass ( ) 5 13.42 cm 0.67 cm 1.342 cm 0.67 mm
Polyimide ( ) 3.2 16.77 cm 0.84 cm 1.677 cm 0.84 mm
Quartz ( ) 3.8 15.39 cm 0.77 cm 1.539 cm 0.77 mm
FR4 ( ) 4.5 14.14 cm 0.71 cm 1.414 cm 0.71 mm
RT-duroid 5880 ( ) 2.16 20.41 cm 1.02 cm 2.041 cm 1.02 mm
RT-duroid 6010 ( ) 10.2 9.39 cm 0.47 cm 0.939 cm 0.47 mm
Si ( ) 11.9 8.70 cm 0.44 cm 0.870 cm 0.44 mm
GaAs ( ) 12.85 8.37 cm 0.42 cm 0.837 cm 0.42 mm
LTCC ( ) 7.8 10.74 cm 0.54 cm 1.074 cm 0.54 mm
* : 1/20
Department of Electronic Engineering, NTUT24/47
2t T
tω π=
= 2T
πω
=
( ) ( )1 , cosv x t A x tβ ω= −
0x =
( )1 0, cosv t A tω=
π 2πtω
00
ttω
==
A
A−
( )1 0, cosv t A tω=
0x = x = l
0x =
x = l•
We only pay attention to this point
Department of Electronic Engineering, NTUT25/47
2x
x λβ π=
= 2πλβ
=
( ) ( )1 , cosv x t A x tβ ω= −
0t =
( )1 ,0 cosv x A xβ=
π 2πxβ
00
xxβ
==
A
A−
( )1 ,0 cosv x A xβ=
0x = x = l
0x =
x = l•
We now pay attention to the whole line at any time instant (here, t=0)
Department of Electronic Engineering, NTUT26/47
( )
2λ λx
A
A−
t
t T=
2t T=
x
x
t
Department of Electronic Engineering, NTUT27/47
x =
2λ λx
A
A−
t
t T=
2t T=
t
λ
0x = x = l
We only pay attention to this point
x λ=
Department of Electronic Engineering, NTUT28/47
(Terminated Transmission Line)
LZ LZ0Z 0Z
j xAe β−
j xBe β j xBe β
j xAe β−
0x = x = l 0d =d = l
( ) j x j xV x Ae Beβ β−= +
( )0 0
j x j xA BI x e e
Z Zβ β−= −
( )IN dΓ
( ) 1 1j d j dV d A e B eβ β−= +
( ) 1 1
0 0
j d j dA BI d e e
Z Zβ β−= −
1jA Ae β−= l
1jB Be β= l
incident wave
reflected wave
d x= −l
Department of Electronic Engineering, NTUT29/47
(Reflected Coefficient)
( ) 1 1j d j dV d A e B eβ β−= +
( ) 2 21 10
1 1
j dj d j d
IN j d
B e Bd e e
A e A
ββ β
β
−− −Γ = = = Γ
( ) 10
1
0IN
B
AΓ = Γ =
• (at d = l ) (at d = 0)
d=0 :0Γ ( )0INΓ
Department of Electronic Engineering, NTUT30/47
(I)
( ) ( ) ( )21 0 1 01j d j d j d j dV d A e e A e eβ β β β− −= + Γ = + Γ
( ) ( ) ( )21 10 0
0 0
1j d j d j d j dA AI d e e e e
Z Zβ β β β− −= − Γ = − Γ
( ) ( )( )
00
0
j d j d
IN j d j d
V d e eZ d Z
I d e e
β β
β β
−
−
+ Γ= =− Γ
( ) 00
0
10
1IN LZ Z Z+ Γ= =− Γ
00
0
L
L
Z Z
Z Z
−Γ =+
( ) 1 1j d j dV d A e B eβ β−= +
( ) 1 1
0 0
j d j dA BI d e e
Z Zβ β−= −
d
(d = 0)
0 0Γ =0LZ Z=
LZ( )V d
0d =d = l
( )IN dΓ
( )INZ d
( )I d
+
−
10
1
B
AΓ =
Department of Electronic Engineering, NTUT
0Z
31/47
(II)
( ) ( ) ( )( ) ( )
0 00
0 0
j d j dL L
IN j d j dL L
Z Z e Z Z eZ d Z
Z Z e Z Z e
β β
β β
−
−
+ + −=
+ − −
0 00 0
0 0
cos sin tan
cos sin tanL L
L L
Z d jZ d Z jZ dZ Z
Z d jZ d Z jZ d
β β ββ β β
+ += =+ +
00
0
L
L
Z Z
Z Z
−Γ =+
( ) ( )( )
00
0
j d j d
IN j d j d
V d e eZ d Z
I d e e
β β
β β
−
−
+ Γ= =− Γ
d = 0 ( )0IN LZ Z=
d = l ( ) 00
0
tan
tanL
INL
Z jZZ Z
Z jZ
ββ
+=+
ll
l
( )
•
Department of Electronic Engineering, NTUT32/47
(VSWR)
( ) 21 01 j dV d A e β−= + Γ
( )( )
0max
0min
1
1
V dVSWR
V d
+ Γ= =
− Γ
( ) ( ) ( )21 0 1 01j d j d j d j dV d A e e A e eβ β β β− −= + Γ = + Γ
( voltage standing-wave ratio, VSWR):
1/4
•
( ) ( )1 0max1V d A= + Γ ( ) ( )1 0min
1V d A= − Γ
Department of Electronic Engineering, NTUT33/47
(Matched Line)
• (Matched line):
( )IN LZ d Z=
dVSWR 1
0 0Γ = 1VSWR =
0Z
0LZ Z=
0d =d = l
( ) 0INZ Z=l
( ) 0INZ d Z=
0Z
( ) 00
0
tan
tanL
INL
Z jZ dZ d Z
Z jZ d
ββ
+=+
0LZ Z=
Department of Electronic Engineering, NTUT34/47
• d
( )VSWR
0LZ =
0 1Γ = −
4
λ=l ( )scZ = ∞l
2
λ=l ( ) 0scZ =l
0LZ =
0d =d = l
( ) 0 tanINZ jZ β=l l
( ) 0 tanscZ d jZ dβ=
0Z
( ) 0 tanscZ d jZ dβ=( ) 00
0
tan
tanL
INL
Z jZ dZ d Z
Z jZ d
ββ
+=+
0LZ =
Department of Electronic Engineering, NTUT35/47
4
λ=l ( ) 0ocZ =l
2
λ=l ( )ocZ = ∞l
( )VSWR
0 1Γ =
• dLZ = ∞
( ) 0 cotocZ d jZ dβ= −( ) 00
0
tan
tanL
INL
Z jZ dZ d Z
Z jZ d
ββ
+=+
LZ → ∞
LZ = ∞
0d =d = l
( ) 0 cotINZ jZ β= −l l
( ) 0 cotocZ d jZ dβ= −
0Z
Department of Electronic Engineering, NTUT36/47
¼
• ¼ :
( )204INL
ZZ
Zλ =
0 x LZ Z Z= ⋅
1/4
4d λ= =l
LZ xZ
: 1/475 LZ = Ω 50 Ω
You can simply use a transmission line with 61.2 Ohm characteristic impedance!
4λ
( )0 4 50 75 61.2 IN LZ Z Zλ= ⋅ = ⋅ = Ω
( ) 00
0
tan
tanL
INL
Z jZ dZ d Z
Z jZ d
ββ
+=+
LZ
0d =4d λ=
20
4INL
ZZ
Z
λ =
0Z
Department of Electronic Engineering, NTUT37/47
•
1/2(
)
( )2IN LZ Zλ =2d λ= =l( ) 00
0
tan
tanL
INL
Z jZ dZ d Z
Z jZ d
ββ
+=+
LZ
0d =2d λ=
2IN LZ Zλ =
0Z
Department of Electronic Engineering, NTUT38/47
( ) ( )1 12 sinj d j dV d A e e j A dβ β β−= − =
( ) ( ) 21, Re Re 2 sin
j tj tv d t V d e A d e
πωω β
+
= = ⋅
( ) ( )1 14 2 sin 2 2V j A j Aλ π= =
( ) ( )12 2 sin 0V j Aλ π= =
4d
λ=
2
dλ=
•
Department of Electronic Engineering, NTUT39/47
( ) 1, 2 sin cos2
v d t A d tπβ ω = ⋅ +
π2
π3
2
π2π
( )( )
max
min
V dVSWR
V d= = ∞
( )V d
( )1 max2A V d=
( )min
0 V d=dβ
d
2
λ4
λ3
4
λλ
•
2
π3
2
π π2π
( ),v d t
12A
( )min
0 V d=dβ
d
2
λ4
λ3
4
λλ
12A
32t πω =5
4t πω =
3,4 4t π πω = 2t πω =
0,tω π=
Department of Electronic Engineering, NTUT40/47
(I)
100 50sZ j= + Ω50 50LZ j= + Ω
10 0sv °= ∠
( )( ) ( )
50 5010 0 3.92 11.31 (V)
50 50 100 50L
L sL s
jZV V
Z Z j j
+= = ∠ = ∠
+ + + +
•
Department of Electronic Engineering, NTUT41/47
(II)
• 50 Ohm(load reflection coefficient)
VSWR
10 0sv °= ∠
100 50sZ j= + Ω50 50LZ j= + Ω
( )( )
00
0
50 50 500.447 63.44
50 50 50L
L
jZ Z
Z Z j
+ −−Γ = = = ∠+ + +
( ) ( )( ) ( )50 50 50 tan 45
8 50 100 50 50 50 50 tan 45IN
j jZ j
j jλ
°
°
+ += = − Ω + +
0
0
1 1 0.4472.62
1 1 0.447VSWR
+ Γ += = =− Γ −
8λ
0 50Z = Ω
l 8λ=
( )8INZ λ
LV
Department of Electronic Engineering, NTUT42/47
(II) –
( ) ( )( )
( )( ) ( )
8 100 508 10 0 5.59 26.57
8 100 50 100 50IN
sIN s
Z jV V
Z Z j j
λλ
λ−
= = ∠ = ∠ −+ − + +
( ) ( )21 01j d j dV d A e eβ β−= + Γ
( ) 4 218 5.59 26.57 1 0.447 63.44
j jV A e e
π π
λ−°
= ∠ − = + ∠ ⋅
1 3.95 63.44A °= ∠ −
( )0 3.95 63.44 1.77 5 45 (V)LV V ° °= = ∠ − + = ∠
100 50sZ j= + Ω
10 0sv °= ∠( )8 100 50INZ jλ = − Ω
Equivalent circuit
Department of Electronic Engineering, NTUT
A1
41
43/47
LZ
+
−sE
sZV
I
+
−
sourceimpedance
loadimpedance
Phasor
•( ) ( )
2 22 2
2 2
1 1 1
2 2 2s s L
L rms L L Ls L s L s L
E E RP I R I R R
Z Z R R X X
= = = = + + + +
•s LX X= −s LR R= s LZ Z ∗=
• ( ) ( ) ?
( )
2
,max
1
8s
Ls
EP
R=
s
s L
EI
Z Z=
+
Department of Electronic Engineering, NTUT44/47
(Power Waves)
•
0L o
L o
Z Z
Z Z
−Γ =+
( )
s sV E Z I= −
LZ
+
−sE
sZV
I
+
−
sourceimpedance
loadimpedance
oZ
d = l
LZ
0Γ
0d =
( )IN dΓ
Department of Electronic Engineering, NTUT45/47
LZ
+
−sE
sZV
I
+
−
LR+
−sE
sR
V
+
−
sjX LjX
( )( )
( )( )
L s s LL s L ss L s
s L s L s L s L s s L
R R j X XR j X X RZ R Z Z
Z R R j X X R Z Z R R j X X
∗ − − + + + −− − Γ = = = =+ + + + + + + +
L s
L s
Z Z
Z Z
∗−Γ =+
• ( )*L sZ Z= Γ = 0
• (Z0)Rs
LZ+
−sE
sR
V
+
−
sjX
L LR jX+
Department of Electronic Engineering, NTUT
Z
46/47
•
( )
• 1
• 20λ
Department of Electronic Engineering, NTUT
•
47/47