TELECOMMUNICATIONS

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TELECOMMUNICATIONS

• Dr. Hugh Blanton
• ENTC 4307/ENTC 5307

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Transmission Lines
3

• Transmission lines

• Transmission line parameters, equations
• Wave propagations
• Lossless line, standing wave and reflection
  coefficient
• Input impedence
• Special cases of lossless line
• Power flow
• Smith chart
• Impedence matching
• Transients on transmission lines

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• Transmission line parameters, equations

B
A
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)),
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• 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.
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• Transmission line parameters

• time delay

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

• Reflection the voltage has to be treated 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

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• 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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• Types of transmission lines

• Higher-order transmission lines

a) Optical fiber
b) Rectangular waveguide
c) Coplanar waveguide
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• 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
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• 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)
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• 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)
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• 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)
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• 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)
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• 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)
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• 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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• Telegraphers equation in phasor domain

Take d /dz on both sides of eq. (4)
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• 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)
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• 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)
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• 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)
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• Wave equations

d²V(z)/dz² - ?²V(z) 0, (10)
d² i(z)/dz² - ?²i(z) 0, (13)
? ? j?,
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• 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
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• Wave equations

-
V(z) V0 (14)

V0

-

i(z) I0 (15)
I0
where
and
are determined by boundary conditions.
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• Characteristic impedance Z0

recall
(17)
(18)
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• Characteristic impedance Z0

(17)
(18)
Define characteristic impedance Z0
recall

Z0 ?


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• Summary

(19)
(20)
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• 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
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• lossless transmission line

? ? j?,
(R j?L) (G j?C)
If Rltlt j ?L and G ltlt j?C,
?
? 0
lossless line
?
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• lossless transmission line

lossless line
? 0
?
? 2?/?
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• For TEM transmission line

LC ??
Vp
?

• summary

Vp
?
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• Voltage reflection coefficient

VL


-
iL


ZL


-
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• Voltage reflection coefficient

-
ZL
Z0
? ?


ZL
Z0

• Current reflection coefficient

• Notes

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

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• An example

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


ZL
Z0