Lect13Phasors

# Lect13Phasors - ECE 3030 Electromagnetic Fields and Waves...

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1 ECE 3030 Electromagnetic Fields and Waves Fall 2009 Lecture 13 2009/9/25 Time Harmonic Fields Complex Mathematics and Phasors Phasor Notation for Maxwell’s Equations 1 Rana and Swartz 06/5/31 Electromagnetic Fields and Waves – Fall 2009 Lecture 13: Instructor: Dr. Wesley E. Swartz Complex Poynting Vectors λ c c z x y E H c Time-Harmonic Fields E and H-fields for a plane wave are: () ( ) r k t E n t r E o = ω cos ˆ , 2 2 2 2 . ˆ ˆ ˆ z y x z y x k k k k k k z k y k x k k + + = = + + = 0 ˆ . = n k c k = r k t E n k t r H o o × = η cos ˆ ˆ , 2 Rana and Swartz 06/5/31 Electromagnetic Fields and Waves – Fall 2009 Lecture 13: Fields for which the time variation is sinusoidal are called time- harmonic fields. Plane waves are just one example of time-harmonic fields. 95% of the remaining material in this course will deal with time-harmonic fields. Time-Harmonic Signals in Circuits: The Sinusoidal Steady State Consider an RC circuit driven by a sinusoidal voltage source: Remember phasors from ECE 210 ~ + - R C ( ) t V t V o cos = t V R + - t V C t I C j 1 3 Rana and Swartz 06/5/31 Electromagnetic Fields and Waves – Fall 2009 Lecture 13: Remember phasors from ECE 210 … Time-averaged power dissipation in the resistor is: [] t j o e V t V Re = t j R R e V t V Re = t j C C e V t V Re = RC j RC j V V o R + = 1 t j e I t I Re = RC j C j V I o + = 1 ()() + = = 2 2 2 * 1 2 Re 2 1 C R C R R V I V t I t V o R R Time-Harmonic Fields: Complex Notation If the time-variation of fields is known a-priori to be sinusoidal (i.e. the fields are known to be time-harmonic) then, in order to simplify the math, one need not carry around the time dependence in calculations. Lets look at plane waves as an example to see how the complex notation can be used to factor out the sinusoidal time dependence. Some useful trigonometric Identities: θ j j 4 Rana and Swartz 06/5/31 Electromagnetic Fields and Waves – Fall 2009 Lecture 13: Expression for the E-field of a plane wave in complex notation: 2 cos e e + = j e e j j 2 sin = () () r k t j o r k t j r k t j o e E n e e E n t r E = + = ˆ ˆ , Re 2 sin cos j e j + = sin cos j e j = ( ) r k t E n t r E o = cos ˆ , Plane Waves in Complex Notation For the E-field of a plane wave we had… Do a little more manipulation … ( ) = r k t

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Lect13Phasors - ECE 3030 Electromagnetic Fields and Waves...

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