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λ
4 λ
λ
3λ
2
4
4
(b) ZL = 0 (short circuit)
λ/2 λ λ
λ
3λ
2
4
4
(c) ZL = (open circuit) 0 z ~
V(z)
2V0+ 0 z ~
V(z)
2V0+ 0 z Figure 13: Standing waves for matched load, SC and OC.
Figure 212 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 52 ˜
• The matched load ⇒ Γ = 0 ⇒ V (z ) = V0+ , i.e. without any
reﬂected waves there can be no interference ⇒ no standing waves .
• SC and OC cases have Γ = 1, or Γ = −1 for SC and Γ = 1
for OC.
• SC and OC have the same maximum value: 2V0+ , and minimum value of zero. Their patterns are shifted by λ/4.
• The ﬁrst voltage minimum is at z = 0 for SC, while OC has
ﬁrst maximum at z = 0; why? Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 53 What about a general expression for the position of the ﬁrst maximum (or minimum)? We’ve already seen that when
˜
2βdmax − θr = 2nπ ⇒ V max = V0+ [1 + Γ] (98) with n=0 or a positive integer. So, what’s dmax ?
dmax = θr λ nλ
θr + 2nπ
=
+
2β
4π
2 (99) where n = 1, 2, . . . if θr < 0, and n = 0, 1, 2 . . . if θr ≥ 0.
• θr is bounded by −π and π ,
• When θr ≥ 0 the ﬁrst dmax = θr λ/4π , otherwise it occurs at
dmax = (θr λ/4π ) + λ/2.
• Maximum of voltage standing wave is also where current standing wave has a minimum! Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 54 • Derivation of positions for minima is analogous and yields
˜
V min = V0+ [1 − Γ], for (2βdmin − θr ) = (2n + 1)π (100)
and the ﬁrst minimum occurs for n = 0.
• Spacing between dmax and dmin is λ/4 ⇒ no need to calculate
minima separately:
dmin = dmax + λ/4, if dmax < λ/4
dmin = dmax − λ/4, if dmax ≥ λ/4 (101) ˜
˜
• Finally, the ratio Vmax /Vmin  is the
voltage standing wave ratio S , aka SWR or VSWR:
S= ˜
Vmax 
1 + Γ
=
˜min 
1 − Γ
V Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (102) Electromagnetics I: Transmission lines 55
Sliding probe To detector
Probe tip ~
Vg + Slit Zg  40 cm 30 cm 20 cm 10 cm ZL Figure 14: Slotted coaxial line.
Figure 213 1.7. Input Impedance
We’ve learned a lot without actually solving our original equations
• We understand that the voltage and current magnitudes are
oscillatory with position on the line and are out of phase with
each other.
• Since impedance is the ratio of voltage to current, we can talk Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 56 about the input impedance which is,
Z (d) = ˜
V (d)
˜
I (d) (103) ˜
˜
• Substitute the solution we found for V (z ) and I (z )
Z (d) = ˜
V + ejβd + Γe−jβd
V (d)
1 + Γe−j 2βd
= 0+...
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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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