16.360 Lecture 6

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16.360 Lecture 6

• Last lecture

• Wave propagation on a Transmission line
• Characteristic impedance
• Standing wave and traveling wave
• Lossless transmission line
• Reflection coefficient

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16.360 Lecture 6

• Wave equations

d²V(z)/dz² - ?²V(z) 0, (10)
d² i(z)/dz² - ?²i(z) 0, (13)
? ? 0,
??0,
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16.360 Lecture 6

• lossless transmission line

? ? j?,
? 0,
LC ??
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16.360 Lecture 6

• For TEM transmission line

LC ??
Vp
?

• summary

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16.360 Lecture 6

• Voltage reflection coefficient

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

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16.360 Lecture 6

• Today

• Standing wave
• Input impedance

j?z

e
V(z) V0 ( )

?
-
i(z)
)
?

V(z) V0
?

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16.360 Lecture 6

• Standing wave

j?z

e
V(z) V0 ( )

?
-
i(z)
)
?

V(z) V0
?

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16.360 Lecture 6

• Standing wave

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

i(z) V0 /Z0


1/2
V0/Z0 1 ?² - 2?cos(2?z ?r)
V(z)
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16.360 Lecture 6
Special cases

1. ZL Z0, ? 0

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

1/2
V0 2 2cos(2?z ?)
V(z)
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16.360 Lecture 6
Special cases
3. ZL ?, open circuit, ? 1

1/2
V0 2 2cos(2?z )
V(z)
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16.360 Lecture 6

• 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
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16.360 Lecture 6

• 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.
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16.360 Lecture 6

• Voltage standing-wave ratio VSWR or SWR

S 1, when ? 0,
S ?, when ? 1,
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16.360 Lecture 6

• 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.
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16.360 Lecture 6

• 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 6
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
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16.360 Lecture 6
Next lecture

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

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