Chapter2_Notes

# i0 v0 92 note that is a complex number

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Unformatted text preview: -circuit (O-C) load ⇒ Γ = 1, V0− = V0+ . • Short-circuit (S-C) 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 + |Γ|2 + 2|Γ| 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 2-11 ˜ ˜ 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 + |Γ|2 + 2|Γ| cos(−2βd + Θr ) ˜ |V (z )| = |V0+ | 1 + |Γ|2 + 2|Γ| 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, O-C and S-C, 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.

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