Chapter2_Notes

# Applied electromagnetics ulaby et al for ece331 psu

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Unformatted text preview: ber (on the tan curve) is when βl = π/2 and continues negative until βl = π . To ﬁnd the minimum l for capacitive short-circuit transmission line, −1 = tan βl Ceq Z0 ω Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (131) Electromagnetics I: Transmission lines βl = − tan−1 68 1 ωCeq Z0 (132) since tan−1 (−θ) = − tan−1 (θ). But, this would give us a negative number. Remember though that the tan curve repeats itself every π so the minimum value for l would be the above plus π : βl = π − tan−1 l= 1 π − tan−1 β 1 ωCeq Z0 1 ωCeq Z0 (m) (m) (133) (134) • This means that depending on our choice for l the short circuit line can be a substitute for capacitors or inductors. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 69 l Z0 short circuit ⇒ Zc = 1 jωCeq Figure 17: Short-circuit transmission line as equivalent capacitor. Figure 2-16 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 70 • Open-circuited line Situation is analogous to the short-circuit case. For Z = ∞, we have Γ = 1 and voltage and current, ˜ V (z ) = V0+ [e−jβz + Γejβz ] (135) V+ ˜ I (z ) = 0 [e−jβz − Γejβz ] Z0 (136) ˜ Voc (d) = V0+ [ejβd + e−jβd ] = 2V0+ cos βd (137) ˜ Ioc (d) = V0+ Z0 [ejβd − e−jβd ] = 2jV0+ Z0 sin βd (138) Input impedance is again the ratio (when d = l), oc Zin = ˜ Voc (l) = −jZ0 cot βl ˜ Ioc (l) Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (139) Electromagnetics I: Transmission lines (a) oc 71 Z0 Zin ~ Voc(z) 2V0+ 1 Voltage (b) -λ -3λ 4 -λ 2 z -λ 4 -1 ~ Current (c) -λ -3λ 4 -λ 2 Ioc(z)Z0 2jV0+ 1 z -λ 4 -1 oc Impedance (d) -λ -3λ 4 -λ 2 -λ 4 Zin jZ0 z Figure 2-17 Figure 18: Input impedance of a O-C transmission line. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 72 • Application of short-circuit and open-circuit measurements If we measure a transmission line (we can use a network analyzer) that is short circuited and get Zsc and then measure again open circuit we get Zoc , we can combine to determine the characteristic impedance Z0 and β . • The product of impedances is, oc sc 2 Zin Zin = −jZ0 cot βljZ0 tan βl = Z0 (140) so, Z0 = oc sc Zin Zin (141) Similarly, take the ratio, sc Zin jZ0 tan βl 2 oc = −jZ cot βl = − tan βl Zin 0 tan βl = − sc Zin oc Zin Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (142) (143) Electromagnetics I: Transmission lines 73 • One-half wavelength lines The main point about this line is that multiples of λ/2 lengths do nothing in terms of input impedance! • If l = nλ/2 then tan βl = tan[(2π/λ)(nλ/2)] = tan nπ = 0 (144) • Recall, the input impedance, Zin = Z0 ZL + jZ0 tan...
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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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