16'360 Lecture 7

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16.360 Lecture 7

• Last lecture

• Standing wave
• Input impedance

j?z

e
V(z) V0 ( )

?
-
i(z)
)
?

V(z) V0
?


i(z) V0 /Z0

1/2
V0/Z0 1 ?² - 2?cos(2?z ?r)

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16.360 Lecture 7
Special cases

• ZL Z0, ? 0

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

1/2
V0 2 2cos(2?z ?)
V(z)
3. ZL ?, open circuit, ? 1

1/2
V(z)
V0 2 2cos(2?z )
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16.360 Lecture 7

• Voltage maximum and minimum

when 2?z ?r 2n?.
when 2?z ?r (2n1)?.

• Voltage standing-wave ratio VSWR or SWR

S 1, when ? 0,
S ?, when ? 1,
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16.360 Lecture 7

• Input impudence

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


Zin(-l)

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16.360 Lecture 7

• Today special cases

• short circuit line
• open circuit line
• quarter-wave transformer
• matched transmission line

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16.360 Lecture 7
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)
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16.360 Lecture 7
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

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16.360 Lecture 7
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 ?, ? ? 2?/? 2?/Vp 62.8 (rad/m)
tan (?l) - 1/?CeqZ0 -0.354,
-1
?l tan (-0.354) n?, -0.34 n?,
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16.360 Lecture 7
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)
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16.360 Lecture 7
Application for short-circuit and open-circuit

• Network analyzer

• Measure S paremeters

• Calculate Z0

• Calculate ?l

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16.360 Lecture 7
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.
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16.360 Lecture 7
Quarter-wave transformer l ?/4 n?/2
?l (2?/?)(?/4 n?/2) ?/2 ,
-j ?
e

?
)
(1
Zin(-l)


Z0

-j ?
e
-
?
(1
)
Z0²/ZL
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16.360 Lecture 7
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)
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16.360 Lecture 7
Matched transmission line

• ZL Z0
• ? 0
• All incident power is delivered to the load.

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16.360 Lecture 7
Next lecture

• Power flow on a lossless transmission line

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