E.E. Essential Knowledge Sereies My Online Courses: https://www.byparams.com/courses 傳輸線理論

1. Department of Electronic Engineering
National Taipei University of Technology

2. •
•
•
•
Department of Electronic Engineering, NTUT2/47

3. Two-wire line
Coaxial ( )
Microstrip( )
Department of Electronic Engineering, NTUT3/47

4. •
•
•
( )
•
Department of Electronic Engineering, NTUT4/47

5. ( )
Department of Electronic Engineering, NTUT5/47

6. •
( ) :
L
L s
s L
Z
v v
Z Z
=
+
Source
Source
impedance Load
impedance
(Transmission line)
sv
sZ
LZ
l
•
? ?
Lv
Department of Electronic Engineering, NTUT6/47

7. (Distributed Circuit Model)
• l
∆x
• R: /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

8. •
• :
• :
• 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

9. (I)
• (KVL):
• (KCL):
( ) ( ) ( ) ( ) ( )
( ),
, , ,
i x t
v x t v x x t R x i x t L x
t
∂
− + ∆ = ∆ + ∆
∂
( ) ( ) ( ) ( ) ( )
( ),
, , ,
v x x t
i 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

10. (II)
( )
( )
( ), ,
,
v x t i x t
Ri x t L
x t
∂ ∂
= − −
∂ ∂
( )
( )
( ), ,
,
i x t v x t
Gv x t C
x t
∂ ∂
= − −
∂ ∂
( ) ( )
( )
( ), , ,
,
v x t v x x t i x t
Ri x t L
x t
− + ∆ ∂
= +
∆ ∂
( ) ( )
( )
( ), , ,
,
i x t i x x t v x x t
Gv x x t C
x t
− + ∆ ∂ + ∆
= + ∆ +
∆ ∂
•
• ∆x
(transmission-line equation)
(telegrapher equation)
Department of Electronic Engineering, NTUT10/47

11. (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

12. (IV) −
( )
( ) ( )
dV x
RI x j LI x
dx
ω= − −
( )
( ) ( )
dI x
GV x j CV x
dx
ω= − −
( ) ( )
( ) ( )
V x V x x
RI x j LI x
x
ω
− + ∆
= +
∆
( ) ( )
( ) ( )
I x I x x
GV x x j CV x x
x
ω
− + ∆
= + ∆ + + ∆
∆
•
• ∆x
(transmission-line equation)
(telegrapher equation)
Department of Electronic Engineering, NTUT12/47

13. ( ) ( ), ,v x t i x t
L
x t
∂ ∂
= −
∂ ∂
( ) ( ), ,i x t v x t
C
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 t
Ri x t L
x t
∂ ∂
= − −
∂ ∂
( )
( )
( ), ,
,
i x t v x t
Gv x t C
x t
∂ ∂
= − −
∂ ∂
Department of Electronic Engineering, NTUT13/47

14. ( ) ( ) ( )( ) ( ) ( )( )
{ } ( ) ( )
{ }, cos Re Re
j t x j x j t
v x t f x t x f x e f x e e
ω ϕ ϕ ω
ω ϕ
+
= + = =
( ) ( ) ( )( ) ( ) ( )( )
{ } ( ) ( )
{ }, cos Re Re
j t x j x j t
i x t g x t x g x e g x e e
ω η η ω
ω η
+
= + = =
( ) ( ) ( )j x
I x g x e
η
=
( ) ( ) ( )j x
V x f x e
ϕ
= ( ) ( ){ }, Re j t
v x t V x e ω
=
( ) ( ){ }, Re j t
i x t I x e ω
=
•
f(x) g(x) ( )xϕ ( )xη
•
time-domain
Department of Electronic Engineering, NTUT14/47

15. ( )
( )
V x
j LI x
x
ω
∂
= −
∂
( )
( )
I x
j CV x
x
ω
∂
= −
∂
( )
( )
dV x
j LI x
dx
ω= −
( )
( )
dI x
j CV x
dx
ω= −
( ) ( )V x dV x
x dx
∂
=
∂
( ) ( )
( )
2
2
2
d V x dI x
j L LCV x
dx dx
ω ω= − = −
•
( )
( )
( )
( )
2 2
2 2
2 2
0
d V x d V x
LCV x V x
dx dx
ω β+ = + =
LCβ ω=•
•
( ) j x j x
V x Ae Beβ β−
= + A B (complex constant)
• (Phasor)
(Think that if approaches zero?)ω
( )β
Phasor
Department of Electronic Engineering, NTUT15/47

16. ( )
( )
( ) ( )
1 1 j x j xdV x
I x A j e B j e
j L dx j L
β β
β β
ω ω
−
 = = − + − −
LCβ ω=
• 0
L L L
Z
CLC
ω ω
β ω
= = =
j x j x
A e B e
L L
β ββ β
ω ω
−  
= −  
 
( ) j x j x
V x Ae Beβ β−
= +
( )
( )
dV x
j LI x
dx
ω= −
Z0
Z0
( )
0 0
j x j xA B
I x e e
Z Z
β β−
= −
Department of Electronic Engineering, NTUT16/47

17. − (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

18. − (II)
•
2 0
1 1 0
2 0
2 0
1
1 1ab
Z Z
Z Z Z Z
Z Z
Z Z
= + = + =
++
2
1 1
0 1 2
2 4
Z Z
Z Z Z
 
= + + 
 
Z0
a-b
Z0 c-d
Z0( )
1Z
2Z 2Z
1Z
2Z
1Z 1Z
c
d
a
b
a
b
c
d
2Z
1Z
0Z 0Z
a
b
Department of Electronic Engineering, NTUT18/47

19. − (III)
•
2
1 1
0 1 2
2 4
Z Z
Z Z Z
 
= + + 
 
L
2 2
0
0
lim
4L
L L L
Z
C C
ω
→
= − =
2 2
0
4
L L
Z
C
ω
= −
4 LCω <
4 LCω >
0, 0
lim 4
L C
LCω
→ →
= = ∞
1Z j Lω=
2
1
Z
j Cω
=
L
C C
La
b
CCC
2L 2L 2L2L
a
b
a′
Department of Electronic Engineering, NTUT19/47

20. ( ) ( ){ } ( ) ( )
{ }, Re Re j x t j x tj t
v x t V x e Ae Beβ ω β ωω − − +
= = +
( ) ( ){ } ( ) ( )
0 0
, Re Re
j x t j x tj t A B
i x t I x e e e
Z Z
β ω β ωω − − + 
= = − 
 
( ) ( )cos cosA x t B x tβ ω β ω= − + +
( ) ( )
0 0
cos cos
A B
x t x t
Z Z
β ω β ω= − − +
( ) j x j x
V x Ae Beβ β−
= +
( )
0 0
j x j xA B
I x e e
Z Z
β β−
= −
• Phasor
Department of Electronic Engineering, NTUT20/47

21. (Wavelength)
• +x
x (rads)
x
j x
e β−
xβ
2 x
x λ
π β βλ=
= =
2π
λ
β
=
( ) { } ( )1 , Re cosj x j t
v x t Ae e A t xβ ω
ω β−
= = −
x λ=
( ) j x
V x Ae β−
=
0x = x λ=
0t = t T=
distance
time
phase 0x
xβ =
2x
x λ
β π=
=
For simplification, assume the
wave starts from x=0 and t=0.
Department of Electronic Engineering, NTUT21/47

22. (Wave Velocity)
( )7
0 4 10 Wb/A-mL µ π −
= ⋅≃
( )12
0 8.85419 10 F/mC ε −
⋅≃ ≃
( ) ( )8
,
0 0
1
light speed 3 10 /p vacuumv c m s
µ ε
= = = ⋅
( )0
0
0
377
L
Z
C
µ
ε
= = = Ω
,
0
p vaccumv
f
λ =
2 1
2 2
pv f
T LC
λ ω π ω ω
λ λ
π β π β
= = ⋅ = = = =
• : ( T )
( )
Department of Electronic Engineering, NTUT22/47

23. 0µ µ≃0,rε ε ε≃
( )
8
0 0
1 3 10
/p
r r r
c
v m s
µ ε ε ε ε
⋅
= = =
(non-magnetic material)
0p r
g
r
c
v
f f
ε λ
λ
ε
= = =
εr
81.5rε =
( )
8
7
,
3 10
3.32 10 /
81.5
p waterv m s
⋅
= ⋅≃
0
, 00.11
81.5
g water
λ
λ λ= ≅
Department of Electronic Engineering, NTUT23/47

24. @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

25. 2t T
tω π=
=
2
T
π
ω
=
( ) ( )1 , cosv x t A x tβ ω= −
0x =
( )1 0, cosv t A tω=
π 2π
tω
0
0t
tω =
=
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

26. 2x
x λ
β π=
=
2π
λ
β
=
( ) ( )1 , cosv x t A x tβ ω= −
0t =
( )1 ,0 cosv x A xβ=
π 2π
xβ
0
0x
xβ =
=
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

27. ( )
2λ λ
x
A
A−
t
t T=
2t T=
x
x
t
Department of Electronic Engineering, NTUT27/47

28. x =
2λ λ
x
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

29. (Terminated Transmission Line)
LZ LZ
0Z 0Z
j x
Ae β−
j x
Be β j x
Be β
j x
Ae β−
0x = x = l 0d =d = l
( ) j x j x
V x Ae Beβ β−
= +
( )
0 0
j x j xA B
I x e e
Z Z
β β−
= −
( )IN dΓ
( ) 1 1
j d j d
V d Ae B eβ β−
= +
( ) 1 1
0 0
j d j dA B
I d e e
Z Z
β β−
= −
1
j
A Ae β−
= l
1
j
B Be β
= l
incident wave
reflected wave
d x= −l
Department of Electronic Engineering, NTUT29/47

30. (Reflected Coefficient)
( ) 1 1
j d j d
V d Ae B eβ β−
= +
( ) 2 21 1
0
1 1
j d
j d j d
IN j d
B e B
d e e
Ae A
β
β β
β
−
− −
Γ = = = Γ
( ) 1
0
1
0IN
B
A
Γ = Γ =
• (at d = l ) (at d = 0)
d=0 :0Γ ( )0INΓ
Department of Electronic Engineering, NTUT30/47

31. (I)
( ) ( ) ( )2
1 0 1 01j d j d j d j d
V d A e e Ae eβ β β β− −
= + Γ = + Γ
( ) ( ) ( )21 1
0 0
0 0
1j d j d j d j dA A
I d e e e e
Z Z
β β β β− −
= − Γ = − Γ
( )
( )
( )
0
0
0
j d j d
IN j d j d
V d e e
Z d Z
I d e e
β β
β β
−
−
+ Γ
= =
− Γ
( ) 0
0
0
1
0
1
IN LZ Z Z
+ Γ
= =
− Γ
0
0
0
L
L
Z Z
Z Z
−
Γ =
+
( ) 1 1
j d j d
V d Ae B eβ β−
= +
( ) 1 1
0 0
j d j dA B
I d e e
Z Z
β β−
= −
d
(d = 0)
0 0Γ =0LZ Z=
LZ( )V d
0d =d = l
( )IN dΓ
( )INZ d
( )I d
+
−
1
0
1
B
A
Γ =
Department of Electronic Engineering, NTUT
0Z
31/47

32. (II)
( )
( ) ( )
( ) ( )
0 0
0
0 0
j d j d
L L
IN j d j d
L L
Z Z e Z Z e
Z d Z
Z Z e Z Z e
β β
β β
−
−
+ + −
=
+ − −
0 0
0 0
0 0
cos sin tan
cos sin tan
L L
L L
Z d jZ d Z jZ d
Z Z
Z d jZ d Z jZ d
β β β
β β β
+ +
= =
+ +
0
0
0
L
L
Z Z
Z Z
−
Γ =
+
( )
( )
( )
0
0
0
j d j d
IN j d j d
V d e e
Z d Z
I d e e
β β
β β
−
−
+ Γ
= =
− Γ
d = 0 ( )0IN LZ Z=
d = l ( ) 0
0
0
tan
tan
L
IN
L
Z jZ
Z Z
Z jZ
β
β
+
=
+
l
l
l
( )
•
Department of Electronic Engineering, NTUT32/47

33. (VSWR)
( ) 2
1 01 j d
V d A e β−
= + Γ
( )
( )
0max
0min
1
1
V d
VSWR
V d
+ Γ
= =
− Γ
( ) ( ) ( )2
1 0 1 01j d j d j d j d
V d A e e Ae eβ β β β− −
= + Γ = + Γ
( voltage standing-wave ratio, VSWR):
1/4
•
( ) ( )1 0max
1V d A= + Γ ( ) ( )1 0min
1V d A= − Γ
Department of Electronic Engineering, NTUT33/47

34. (Matched Line)
• (Matched line):
( )IN LZ d Z=
d
VSWR 1
0 0Γ = 1VSWR =
0Z
0LZ Z=
0d =d = l
( ) 0INZ Z=l
( ) 0INZ d Z=
0Z
( ) 0
0
0
tan
tan
L
IN
L
Z jZ d
Z d Z
Z jZ d
β
β
+
=
+
0LZ Z=
Department of Electronic Engineering, NTUT34/47

35. • 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=( ) 0
0
0
tan
tan
L
IN
L
Z jZ d
Z d Z
Z jZ d
β
β
+
=
+
0LZ =
Department of Electronic Engineering, NTUT35/47

36. 4
λ
=l ( ) 0ocZ =l
2
λ
=l ( )ocZ = ∞l
( )
VSWR
0 1Γ =
• dLZ = ∞
( ) 0 cotocZ d jZ dβ= −( ) 0
0
0
tan
tan
L
IN
L
Z jZ d
Z d Z
Z jZ d
β
β
+
=
+
LZ → ∞
LZ = ∞
0d =d = l
( ) 0 cotINZ jZ β= −l l
( ) 0 cotocZ d jZ dβ= −
0Z
Department of Electronic Engineering, NTUT36/47

37. ¼
• ¼ :
( )
2
0
4IN
L
Z
Z
Z
λ =
0 x LZ Z Z= ⋅
1/4
4d λ= =l
LZ xZ
: 1/4
75LZ = Ω 50 Ω
You can simply use a transmission line
with 61.2 Ohm characteristic impedance!
4λ
( )0 4 50 75 61.2IN LZ Z Zλ= ⋅ = ⋅ = Ω
( ) 0
0
0
tan
tan
L
IN
L
Z jZ d
Z d Z
Z jZ d
β
β
+
=
+
LZ
0d =4d λ=
2
0
4
IN
L
Z
Z
Z
λ 
= 
 
0Z
Department of Electronic Engineering, NTUT37/47

38. •
1/2
(
)
( )2IN LZ Zλ =
2d λ= =l
( ) 0
0
0
tan
tan
L
IN
L
Z jZ d
Z d Z
Z jZ d
β
β
+
=
+
LZ
0d =2d λ=
2
IN LZ Z
λ 
= 
 
0Z
Department of Electronic Engineering, NTUT38/47

39. ( ) ( )1 12 sinj d j d
V d A e e j A dβ β
β−
= − =
( ) ( ){ } 2
1, Re Re 2 sin
j t
j t
v d t V d e A d e
π
ω
ω
β
 
+ 
 
  
= = ⋅ 
  
( ) ( )1 14 2 sin 2 2V j A j Aλ π= =
( ) ( )12 2 sin 0V j Aλ π= =
4
d
λ
=
2
d
λ
=
•
Department of Electronic Engineering, NTUT39/47

40. ( ) 1, 2 sin cos
2
v d t A d t
π
β ω 
= ⋅ + 
 
π
2
π3
2
π2π
( )
( )
max
min
V d
VSWR
V d
= = ∞
( )V d
( )1 max
2A V d=
( )min
0 V d=dβ
d
2
λ
4
λ3
4
λλ
•
2
π3
2
π π2π
( ),v d t
12A
( )min
0 V d=dβ
d
2
λ
4
λ3
4
λλ
12A
3
2
t πω =5
4
t πω =
3,
4 4
t π πω =
2
t πω =
0,tω π=
Department of Electronic Engineering, NTUT40/47

41. (I)
100 50sZ j= + Ω
50 50LZ j= + Ω
10 0sv °
= ∠
( )
( ) ( )
50 50
10 0 3.92 11.31 (V)
50 50 100 50
L
L s
L s
jZ
V V
Z Z j j
+
= = ∠ = ∠
+ + + +
•
Department of Electronic Engineering, NTUT41/47

42. (II)
• 50 Ohm
(load reflection coefficient)
VSWR
10 0sv °
= ∠
100 50sZ j= + Ω
50 50LZ j= + Ω
( )
( )
0
0
0
50 50 50
0.447 63.44
50 50 50
L
L
jZ Z
Z Z j
+ −−
Γ = = = ∠
+ + +
( )
( )
( )
( )
50 50 50tan45
8 50 100 50
50 50 50 tan45
IN
j j
Z j
j j
λ
°
°
 + +
= = − Ω 
+ + 
0
0
1 1 0.447
2.62
1 1 0.447
VSWR
+ Γ +
= = =
− Γ −
8λ
0 50Z = Ω
l 8λ=
( )8INZ λ
LV
Department of Electronic Engineering, NTUT42/47

43. (II) –
( )
( )
( )
( )
( ) ( )
8 100 50
8 10 0 5.59 26.57
8 100 50 100 50
IN
s
IN s
Z j
V V
Z Z j j
λ
λ
λ
−
= = ∠ = ∠ −
+ − + +
( ) ( )2
1 01j d j d
V d Ae eβ β−
= + Γ
( ) 4 2
18 5.59 26.57 1 0.447 63.44
j j
V Ae 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
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A1
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44. LZ
+
−
sE
sZ
V
I
+
−
source
impedance
load
impedance
Phasor
•
( ) ( )
2 2
2 2
2 2
1 1 1
2 2 2
s s L
L rms L L L
s L s L s L
E E R
P I R I R R
Z Z R R X X
 
= = = =  + + + + 
• s LX X= −s LR R= s LZ Z∗
=
• ( ) ( ) ?
( )
2
,max
1
8
s
L
s
E
P
R
=
s
s L
E
I
Z Z
=
+
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45. (Power Waves)
•
0
L o
L o
Z Z
Z Z
−
Γ =
+
( )
s sV E Z I= −
LZ
+
−
sE
sZ
V
I
+
−
source
impedance
load
impedance
oZ
d = l
LZ
0Γ
0d =
( )IN dΓ
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46. LZ
+
−
sE
sZ
V
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+
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Z
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47. •
( )
• 1
• 20λ
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•
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