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Unformatted text preview: 1 ECE 303 Fall 2007 Farhan Rana Cornell University Lecture 14 Time Harmonic Fields In this lecture you will learn: Complex mathematics for time-harmonic fields Maxwells equations for time-harmonic fields Complex Poynting vector ECE 303 Fall 2007 Farhan Rana Cornell University E and H-fields for a plane wave are (from last lecture): ( ) ( ) r k t E n t r E o r r r r . cos , = 2 2 2 2 . z y x z y x k k k k k k z k y k x k k + + = = + + = r r r . = n k r c k = ( ) ( ) ( ) r k t E n k t r H o o r r r r . cos , = Time-Harmonic Fields Fields for which the time variation is sinusoidal are called time-harmonic fields. Plane waves are just one example of time-harmonic fields In the rest of this course, 95% of the material will deal with time-harmonic fields 2 ECE 303 Fall 2007 Farhan Rana Cornell University Time-Harmonic Signals in Circuits Sinusoidal Steady State Consider an RC circuit driven by a sinusoidal voltage source: ~ +- R C ( ) ( ) t V t V o cos = ( ) t V R +- ( ) t V C ( ) [ ] t j o e V t V Re = ( ) ( ) [ ] t j R R e V t V Re = ( ) RC j RC j V V o R + = 1 ( ) t I ( ) ( ) [ ] t j e I t I Re = ( ) RC j C j V I o + = 1 Time-average power dissipation in the resistor: ( ) ( ) ( ) ( ) [ ] ( ) ( ) + = = 2 2 2 * 1 2 Re 2 1 C R C R R V I V t I t V o R R C j 1 Remember phasors from ECE210.. ECE 303 Fall 2007 Farhan Rana Cornell University Some useful trigonometric Identities to recall before we start: ( ) 2 cos j j e e + = ( ) j e e j j 2 sin = Expression for the E-field of a plane wave in complex notation: ( ) ( ) ( ) ( ) = + = r k t j o r k t j r k t j o e E n e e E n t r E r r r r r r r r . . . Re 2 , ( ) ( ) sin cos j e j + = ( ) ( ) sin cos j e j = Time-Harmonic Fields and Complex Notation Basic idea: 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 may...
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- Fall '06