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Unformatted text preview: circuit (OC) load ⇒ Γ = 1, V0− = V0+ .
• Shortcircuit (SC) load ⇒ Γ = −1, V0− = −V0+ Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 47 • Standing waves
We are making progress, but we still don’t have a full solution! We’ve
made it this far:
V+
˜
˜
V (z ) = V0+ (e−jβz + Γejβz ), I (z ) = 0 (e−jβz − Γejβz )
Z0 (94) (What is the unknown?)
• Let’s do a little bit more examination by looking at the magni˜
tude of V (z ).
• After some manipulation
˜
V (z ) = V0+  1 + Γ2 + 2Γ cos(2βz + Θr ) 1/2 (95) ˜
and similarly for I (z ).
• Fig. 12 shows these magnitudes vs. position z , given the circuit
parameters.
Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 48
~
V(z) ~
V max 1.4 V
1.2
1.0
0.8
0.6
0.4
0.2 ~
V min λ 3λ
λ lmin λ
4
4
2
~
(a) V(z) versus z lmax 0 z ~ I(z)
30 mA
25
20
15
10
5 ~ I max
~ I min λ λ
3λ
λ
2
4
4
~
(b) I(z) versus z 0 z Figure 211 ˜
˜
Figure 12: Standing voltage (V ) and current (I ) waves. Z0 = 50Ω,
◦
Γ = 0.3ej 30 , V0+  = 1 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 49 • If we substitute in a position −d on our transmission line we
have,
˜
V (z ) = V0+  1 + Γ2 + 2Γ cos(−2βd + Θr )
˜
V (z ) = V0+  1 + Γ2 + 2Γ cos(2βd − Θr ) 1/2 1/2 (96)
(97) Some observations:
• The magnitudes show a sinusoidal pattern, which is caused by
interference of two waves,
• This pattern is called standing wave ,
• The maximum of the standing wave pattern happens when incident and reﬂected waves are in phase, i.e. the argument of the
cosine term, 2βd − Θr = 2nπ . In this case, magnitude of total
voltage is (1 + Γ)V0+ . Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 50 • Conversely, if the traveling waves are out of phase, we have a
minimum. This happens at 2βd − Θr = (2n + 1)π , and magnitude is (1 − Γ)V0+ .
• Standing wave pattern repeats every λ/2 , where λ is associated with the traveling waves.
• Note that ﬁg. 12 is vs. coordinate z , i.e. there is no time
dependence, which is OK since we are looking at magnitudes
(or amplitudes) only.
• What happens if we ﬁx the position and look at time? Voltage
has a cos ωt variation.
• Interestingly, current and voltage are in opposition; when voltage peaks, current has a minimum and vice versa. This is a
consequence of a minus sign in eq. 94.
There are three other special cases: matched load, OC and SC,
which are shown in ﬁg. 13.
Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 51
~
V(z) Matched line
λ λ 3λ
4 λ
2
(a) ZL = Z0
λ/2 V...
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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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