Chapter1_Notes_v0

Ulaby et al for ece331 psu electromagnetics i

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Unformatted text preview: 58 Solution in phasor domain: We can solve for the current phasor, ˜ Vs ˜ I= R+ 1 jωC (41) ˜ Substitute in for the source voltage phasor, Vs = V0 ej (φ0 −π/2) , j (φ0 −π/2) ˜ V0 e I= 1 R + jωC (42) Multiply out denominator term, j (φ0 −π/2) jωC V0 ωCejφ0 ˜ V0 e I= = (1 + jωRC ) (1 + jωRC ) (43) We can write the denominator term as a magnitude and phase, 1 + jωRC = 1 + ω 2 R2 C 2 ejφ1 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (44) Electromagnetics I: Introduction: Waves and Phasors 59 Where the phase is φ1 = tan−1 (ωRC ). We can then write the current phasor as, V0 ωC ˜ ej (φ0 −φ1 ) I=√ 1 + ω 2 R2 C 2 (45) Solution in time domain: The last step is that we need to go back to a time domain solution. To find i(t), we multiply the phasor by ejωt and take the real part. ˜ i(t) = [Iejωt ] V0 ωC ej (φ0 −φ1 ) ejωt ] i(t) = [ √ 1 + ω 2 R2 C 2 V0 ωC i(t) = √ cos (ωt + φ0 − φ1 ) 1 + ω 2 R2 C 2 (46) (47) (48) Table 1-5 in the book gives various time domain expressions and their corresponding phasor domain expression. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Introduction: Waves and Phasors 60 Traveling waves in the phasor domain • A traveling wave has a dependency that is something of the form ωt ± βx. • For a time harmonic signal, A cos(ωt + βx) ⇔ Aejβx (49) • For a wave traveling in the positive x direction, the sign between ωt and βx are opposite (e.g., ωt − βx). • When the sign is the same the wave is moving in the negative direction. • So, a wave traveling in the negative x direction has a phasor of the form Aejβx and moving in the positive x direction is, Ae−jβx . • Therefore, the sign of the exponent is opposite to the direction of travel. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU....
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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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