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ω LC
LC • Using the relationship, up = √1
µ (83) , √
β = ω µ (rad/m)
Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (84) Electromagnetics I: Transmission lines 42 • Typical materials for transmission lines will have permeability
µ = µ0 = 4π × 10−7 H/m, while permittivity is given via
relative permittivity r = / 0 . Free space permittivity 0 =
8.854 × 10−12 F/m.
• Some simple manipulations leads to
c
up = √ ⇒λ=
r up
c1
λ0
= √ =√
f
f
r
r (85) One more property of transmission lines: dispersive or not? If the
phase velocity is independent of frequency ⇒ medium is nondispersive. Lossless TEM lines are of this type. Why do we care? It gets to
the signal integrity issues and how faithfully is the shape of the signal
preserved (ﬁg. 2).
Summary of several cases of transmission lines is given in table
2.2. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 43 • Voltage reﬂection coeﬃcient
After all this, we still don’t know the voltage! Actually, to solve eqs.
87 we need a full circuit, as presented in ﬁg. 10.
˜
V (z ) = V0+ e−γz + V0− eγz
˜
I (z ) = I + e−γz + I − eγz
0 0 (86)
(87) Remember that for lossless lines γ = jβ . Set up coordinate system
so that z = 0 at the load end and z = −l at the generator end.
• We know that at the load
˜˜
ZL = VL /IL (88) This has nothing to do with the waves!
• Now we make a connection with the wave picture:
V−
V+
˜
˜
˜
˜
VL = V (z = 0) = V0+ + V0− , IL = I (z = 0) = 0 − 0 (89)
Z0
Z0
Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines
Zg
~
Vg + ~
Ii 44 Transmission line
+ +
~
Vi Z0 ~
VL ~ IL
ZL   Generator Load
z = l z=0 Figure 29 Figure 10: Transmission line connected to a general load and generator.
• Plug this into eq. 88
⇒ ZL = V0+ + V0−
V0+ − V0− Z0 ⇒ V0− = ZL − Z0
ZL + Z0 V0+ (90) • Very interesting result: the ratio of incident and reﬂected wave
at the load depends only on the load impedance ZL and characteristic impedance Z0 !
Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 45 • This we call voltage reﬂection coeﬃcient
Γ= ZL /Z0 − 1
V0−
ZL − Z0
=
=
ZL + Z0
ZL /Z0 + 1
V0+ (91) • Similarly, for current waves
−
I0
V0−
+ = − + = −Γ
I0
V0 (92) • Note that Γ is a complex number! Z0 may be real (for lossless
lines), but ZL is generally complex.
⇒ Γ = Γej Θr (93) • For passive loads Γ ≤ 1
• The load is considered matched if ZL = Z0 ⇒ no reﬂection at
the load ⇒ V0− = 0
Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines
Transmission line 46
A
RL 50 W CL Z0 = 100 W 10 pF A' Figure 210
Figure 11: Transmission line for example 23. • Open...
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This note was uploaded on 09/25/2013 for the course ECE 331 taught by Professor Martinsiderious during the Fall '12 term at Portland State.
 Fall '12
 MartinSiderious
 Electromagnet

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