16.360 Lecture 2

1
16.360 Lecture 2

• Transmission lines

1. Transmission line parameters, equations
2. Wave propagations
3. Lossless line, standing wave and reflection
   coefficient
4. Input impedence
5. Special cases of lossless line
6. Power flow
7. Smith chart
8. Impedence matching
9. Transients on transmission lines

2
16.360 Lecture 2

1. Transmission line parameters, equations

B
A
VBB(t) VAA(t)
VBB(t)
Vg(t)
VAA(t)
L
A
B
VAA(t) Vg(t) V0cos(?t),
Low frequency circuits
VBB(t) VAA(t)
Approximate result
VBB(t) VAA(t-td) VAA(t-L/c)
V0cos(?(t-L/c)),
3
16.360 Lecture 2

1. Transmission line parameters, equations

Recall ??c, and ? 2??
VBB(t) VAA(t-td) VAA(t-L/c)
V0cos(?(t-L/c)) V0cos(?t- 2?L/?),
If ?gtgtL, VBB(t) ? V0cos(?t) VAA(t),
If ?lt L, VBB(t) ?VAA(t), the circuit theory
has to be replaced.
4
16.360 Lecture 2

1. Transmission line parameters, equations

e. g ? 1GHz, L 1cm
Time delay
?t L/c 1cm /3x1010 cm/s 30 ps
?? 2?f?t 0.06 ?
Phase shift
VBB(t) VAA(t)
? 10GHz, L 1cm
Time delay
?t L/c 1cm /3x1010 cm/s 30 ps
?? 2?f?t 0.6 ?
Phase shift
VBB(t) ?VAA(t)
5
16.360 Lecture 2

• Transmission line parameters

• time delay

VBB(t) VAA(t-td) VAA(t-L/vp),

• Reflection the voltage has to be treat as wave,
  some bounce back

• power loss due to reflection and some other loss
  mechanism,

• Dispersion in material, Vp could be different
  for different wavelength

6
16.360 Lecture 2

• Types of transmission lines

• Transverse electromagnetic (TEM) transmission
  lines

B
E
a) Coaxial line
b) Two-wire line
c) Parallel-plate line
d) Strip line
e) Microstrip line
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16.360 Lecture 2

• Types of transmission lines

• Higher-order transmission lines

a) Optical fiber
b) Rectangular waveguide
c) Coplanar waveguide
8
16.360 Lecture 2

• Lumped-element Model

• Represent transmission lines as parallel-wire
  configuration

A
B
Vg(t)
VBB(t)
VAA(t)
B
A
?z
?z
?z
R?z
L?z
L?z
R?z
L?z
R?z
Vg(t)
G?z
C?z
C?z
C?z
G?z
G?z
9
16.360 Lecture 2

• Transmission line equations

• Represent transmission lines as parallel-wire
  configuration

i(z,t)
i(z?z,t)
L?z
R?z
V(z,t)
V(z ?z,t)
G?z
C?z
V(z,t) R?z i(z,t) L?z ? i(z,t)/ ?t V(z
?z,t), (1)
i(z,t) G?z V(z ?z,t) C?z ?V(z ?z,t)/?t
i(z?z,t), (2)
10
16.360 Lecture 2

• Transmission line equations

V(z,t) R?z i(z,t) L?z ? i(z,t)/ ?t V(z
?z,t), (1)
-V(z ?z,t) V(z,t) R?z i(z,t) L?z ?
i(z,t)/ ?t
- ?V(z,t)/?z R i(z,t) L ? i(z,t)/ ?t,
(3)
Rewrite V(z,t) and i(z,t) as phasors, for
sinusoidal V(z,t) and i(z,t)
11
16.360 Lecture 2

• Transmission line equations

Recall
j?t
di(t)/dt
Re(d i e
)/dt
- ?V(z,t)/?z R i(z,t) L ? i(z,t)/ ?t,
(3)
12
16.360 Lecture 2

• Transmission line equations

• Represent transmission lines as parallel-wire
  configuration

i(z,t)
i(z?z,t)
L?z
R?z
V(z,t)
V(z ?z,t)
G?z
C?z
V(z,t) R?z i(z,t) L?z ? i(z,t)/ ?t V(z
?z,t), (1)
i(z,t) G?z V(z ?z,t) C?z ?V(z ?z,t)/?t
i(z?z,t), (2)
13
16.360 Lecture 4

• Transmission line equations

i(z,t) G?z V(z ?z,t) C?z ?V(z ?z,t)/?t
i(z?z,t), (2)
- i (z ?z,t) i (z,t) G?z V(z ?z ,t)
C?z ? V(z ?z,t)/ ?t
- ? i(z,t)/?z G V(z,t) C ? V(z,t)/ ?t,
(5)
Rewrite V(z,t) and i(z,t) as phasors, for
sinusoidal V(z,t) and i(z,t)
14
16.360 Lecture 2

• Transmission line equations

Recall
j?t
dV(t)/dt
Re(d V e
)/dt
- ?i(z,t)/?z G V(z,t) C ? V(z,t)/ ?t,
(6)
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16.360 Lecture 2

• Telegraphers equation in phasor domain

Take d /dz on both sides of eq. (4)
16
16.360 Lecture 2

• Telegraphers equation in phasor domain

- d i(z)/dz G V(z) j?C V(z), (7)
substitute (7) to (8)
d²V(z)/dz² (R j?L) (G j?C)V(z),
or
d²V(z)/dz² - (R j?L) (G j?C)V(z) 0,
(9)
d²V(z)/dz² - ?²V(z) 0, (10)
?² (R j?L) (G j?C), (11)
17
16.360 Lecture 2

• Telegraphers equation in phasor domain

Take d /dz on both sides of eq. (7)
- d² i(z)/dz² G dV(z)/dz j?C dV(z)/dz,
(12)
18
16.360 Lecture 2

• Telegraphers equation in phasor domain

- d i(z)/dz G V(z) j?C V(z), (7)
- d² i(z)/dz² G dV(z)/dz j?C dV(z)/dz,
(12)
substitute (4) to (12)
d² i(z)/dz² (R j?L) (G j?C)i(z),
or
d² i(z)/dz² - (R j?L) (G j?C) i(z) 0,
(9)
d² i(z)/dz² - ?²i(z) 0, (13)
?² (R j?L) (G j?C), (11)
19
16.360 Lecture 2

• Wave equations

d²V(z)/dz² - ?²V(z) 0, (10)
d² i(z)/dz² - ?²i(z) 0, (13)
? ? j?,
20
16.360 Lecture 2

• Wave equations

d²V(z)/dz² - ?²V(z) 0, (10)
d² i(z)/dz² - ?²i(z) 0, (13)
? ? j?,
Solving the second order differential equation

-
V(z) V0 (14)

V0

-

i(z) I0 (15)
I0
21
16.360 Lecture 2

• Wave equations

-
V(z) V0 (14)

V0

-

i(z) I0 (15)
I0
where
and
are determined by boundary conditions.
22
16.360 Lecture 2

• Characteristic impedance Z0

recall
(17)
(18)
23
16.360 Lecture 5

• Characteristic impedance Z0

(17)
(18)
Define characteristic impedance Z0
recall

Z0 ?


24
16.360 Lecture 5

• Summary

(19)
(20)
25
16.360 Lecture 5

• Example, an air line

R 0 ?, G 0 /?, Z0 50?, ? 20 rad/m, f
700 MHz
L ? and C ?
solution
50?
? ? j?,
? ?
20 rad/m
26
16.360 Lecture 5

• lossless transmission line

? ? j?,
(R j?L) (G j?C)
If Rltlt j ?L and G ltlt j?C,
?
? 0
lossless line
?
27
16.360 Lecture 2

• lossless transmission line

lossless line
? 0
?
? 2?/?
28
16.360 Lecture 2

• For TEM transmission line

LC ??
Vp
?

• summary

Vp
?
29
16.360 Lecture 5

• Voltage reflection coefficient

VL


-
iL


ZL


-
30
16.360 Lecture 5

• Voltage reflection coefficient

-
ZL
Z0
? ?


ZL
Z0

• Current reflection coefficient

• Notes

1. ?? 1, how to prove it?
2. If ZL Z0, ? 0. Impedance match, no reflection
   from the load ZL.

31
16.360 Lecture 2

• An example

A
RL 50?
Z0 100?
f 100MHz
A
CL 10pF
z 0
ZL RL j/?CL 50 j159
-
ZL
Z0
?


ZL
Z0
32
16.360 Lecture 2

• Standing wave
• Input impedance

j?z

e
V(z) V0 ( )

?
-
i(z)
)
?

V(z) V0
?

33
16.360 Lecture 2

• Standing wave

j?z

e
V(z) V0 ( )

?
-
i(z)
)
?

V(z) V0
?

34
16.360 Lecture 2

• Standing wave

V(z) V0 ( )
-
i(z)
)
?

i(z) V0 /Z0


1/2
V0/Z0 1 ?² - 2?cos(2?z ?r)
V(z)
35
16.360 Lecture 2
Special cases

1. ZL Z0, ? 0

V(z)
V0
2. ZL 0, short circuit, ? -1

1/2
V0 2 2cos(2?z ?)
V(z)
36
16.360 Lecture 2
Special cases
3. ZL ?, open circuit, ? 1

1/2
V0 2 2cos(2?z )
V(z)
37
16.360 Lecture 2

• Voltage maximum

when 2?z ?r 2n?.
z ??r/4? n?/2
n 1, 2, 3, , if ?r lt0
n 0, 1, 2, 3, , if ?r gt 0
38
16.360 Lecture 2

• Voltage minimum

when 2?z ?r (2n1)?.
z ??r/4? n?/2 ?/4
Note
voltage minimums occur ?/4 away from voltage
maximum, because of the 2?z, the special
frequency doubled.
39
16.360 Lecture 2

• Voltage standing-wave ratio VSWR or SWR

S 1, when ? 0,
S ?, when ? 1,
40
16.360 Lecture 2

• An example

Voltage probe
S 3, Z0 50?, ?lmin 30cm, lmin 12cm, ZL?
Solution
?lmin 30cm,? ? 0.6m,
S 3, ? ? 0.5,
-2?lmin ?r -?, ? ?r -36º,
? ?, and ZL.
41
16.360 Lecture 2

• Input impudence

B
Ii
A
Zg
Vg(t)
VL
Z0
ZL
Vi
l
z 0
z - l
Zin(z)
Z0


Zin(-l)

42
16.360 Lecture 2
An example
A 1.05-GHz generator circuit with series
impedance Zg 10-? and voltage source given by
Vg(t) 10 sin(?t 30º) is connected to a load ZL
100 j5-? through a 50-?, 67-cm long lossless
transmission line. The phase velocity is
0.7c. Find V(z,t) and i(z,t) on the line.
Solution
Since, Vp ƒ?, ? Vp/f 0.7c/1.05GHz 0.2m.
? 2?/?, ? 10 ?.
? (ZL-Z0)/(ZLZ0), ? 0.45exp(j26.6º)
Zin(-l)

21.9 j17.4 ?
Zin(-l)

V0exp(-j?l) ?exp(j?l)
Vg

Zin(-l) Zg
43
16.360 Lecture 2
short circuit line
B
Ii
A
Zg
Vg(t)
sc
VL
Z0
Zin
ZL 0
l
z 0
z - l
ZL 0, ? -1, S ?

V(z) V0 )
-
-2jV0sin(?z)


i(z)
)
2V0cos(?z)/Z0
V(-l)
Zin

jZ0tan(?l)
i(-l)
44
16.360 Lecture 2
short circuit line
V(-l)
Zin
jZ0tan(?l)

i(-l)

• If tan(?l) gt 0, the line appears inductive,

j?Leq jZ0tan(?l),

• If tan(?l) lt 0, the line appears capacitive,

1/j?Ceq jZ0tan(?l),

• The minimum length results in transmission line
  as a capacitor

45
16.360 Lecture 2
An example
Choose the length of a shorted 50-? lossless line
such that its input impedance at 2.25 GHz is
equivalent to the reactance of a capacitor with
capacitance Ceq 4pF. The wave phase velocity
on the line is 0.75c.
Solution
Vp ?ƒ, ? ? 2?/? 2?ƒ/Vp 62.8 (rad/m)
tan (?l) - 1/?CeqZ0 -0.354,
-1
?l tan (-0.354) n?, -0.34 n?,
46
16.360 Lecture 2
open circuit line
B
Ii
A
Zg
Vg(t)
oc
VL
Z0
Zin
ZL ?
l
z 0
z - l
ZL 0, ? 1, S ?

V(z) V0 )

-
i(z)
)
2jV0sin(?z)/Z0
V(-l)
oc
Zin

-jZ0cot(?l)
i(-l)
47
16.360 Lecture 2
Application for short-circuit and open-circuit

• Network analyzer

• Measure S paremeters

• Calculate Z0

• Calculate ?l

48
16.360 Lecture 2
Line of length l n?/2
tan(?l) tan((2?/?)(n?/2)) 0,
Zin(-l)

ZL
Any multiple of half-wavelength line doesnt
modify the load impedance.
49
16.360 Lecture 2
Quarter-wave transformer l ?/4 n?/2
?l (2?/?)(?/4 n?/2) ?/2 ,
-j ?
e

?
)
(1
Zin(-l)


Z0

-j ?
e
-
?
(1
)
Z0²/ZL
50
16.360 Lecture 2
An example
A 50-? lossless tarnsmission is to be matched to
a resistive load impedance with ZL 100 ? via a
quarter-wave section, thereby eliminating
reflections along the feed line. Find the
characteristic impedance of the quarter-wave
tarnsformer.
ZL 100 ?
Z01 50 ?
?/4
Zin Z0²/ZL 50 ?
½
½
Z0 (ZinZL) (50100)
51
16.360 Lecture 2
Matched transmission line

1. ZL Z0
2. ? 0
3. All incident power is delivered to the load.

52
16.360 Lecture 8

• Instantaneous power
• Time-average power

j?z

e
V(z) V0 ( )

?
-
i(z)
)
?
At load z 0, the incident and reflected
voltages and currents
i
i

V V0
i
r
-
r
V V0
i
53
16.360 Lecture 2

• Instantaneous power

i
i
i
P(t) v(t) i(t) ReV exp(j?t) Re i
exp(j?t)




ReV0exp(j? )exp(j?t) ReV0/Z0 exp(j?
)exp(j?t)


(V0²/Z0) cos²(?t ? )
r
r
r
P(t) v(t) i(t) ReV exp(j?t) Re i
exp(j?t)
-

-

ReV0exp(j? )exp(j?t) ReV0/Z0 exp(j?
)exp(j?t)


- ?²(V0²/Z0) cos²(?t ? ?r)
54
16.360 Lecture 2

• Time-average

Time-domain approach
i
?
T
T
i
Pav
P (t)dt

2?
0
0

(V0²/2Z0)
r

Pav
-?² (V0²/2Z0)
Net average power
i
r
Pav
Pav
Pav

(1-?²) (V0²/2Z0)
55
16.360 Lecture 2

• Time-average

Phasor-domain approach
Pav
(½)ReV i
i


Pav (1/2) ReV0 V0 /Z0
(V0²/2Z0)
r

Pav
-?² (V0²/2Z0)

Pav
(1-?²) (V0²/2Z0)