Chapter1_Notes_v0

# Ulaby et al for ece331 psu electromagnetics i

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## 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.

Ask a homework question - tutors are online