16.360 Lecture 4

1
16.360 Lecture 4

• 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 4

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 4

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 4

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 4

• 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 4

• 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 4

• Types of transmission lines

• Higher-order transmission lines

a) Optical fiber
b) Rectangular waveguide
c) Coplanar waveguide
8
16.360 Lecture 4

• 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
Expressions will be derived in later chapters
10
Definitions of TL dimensions
TEM (Transverse Electromagnetic) Electric and
magnetic fields are orthogonal to one another,
and both are orthogonal to direction of
propagation
11
16.360 Lecture 4

• 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
12
16.360 Lecture 4

• 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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16.360 Lecture 4

• 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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16.360 Lecture 4

• 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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16.360 Lecture 4

• 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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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)
17
16.360 Lecture 4

• 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)