16.360 Lecture 8

1
16.360 Lecture 8

• Last lecture

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

2
16.360 Lecture 8
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 8
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 8
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 8
Application for short-circuit and open-circuit

• Network analyzer

• Measure S paremeters

• Calculate Z0

• Calculate ?l

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16.360 Lecture 8
Transmission 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 8
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 8
Matched transmission line

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

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16.360 Lecture 8
Today Power flow on a lossless transmission line

• 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
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16.360 Lecture 8

• 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)
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16.360 Lecture 8

• 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)
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16.360 Lecture 8

• 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)
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16.360 Lecture 8
Next lecture

• The Smith Chart

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