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Notes 2 Complex Vectors

ECE 3317. Prof. David R. Jackson. Spring 2013. Notes 2 Complex Vectors. Adapted from notes by Prof. Stuart A. Long. V ( t ) is a time-varying function. V is a phasor (complex number). A bar underneath indicates a vector: V ( t ), V .

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Notes 2 Complex Vectors

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  1. ECE 3317 Prof. David R. Jackson Spring 2013 Notes 2 Complex Vectors Adapted from notes by Prof. Stuart A. Long

  2. V(t) is a time-varying function. • V is a phasor (complex number). • A bar underneath indicates a vector: V(t), V. Appendices A, B, C, and D in the Shen & Kong text book list frequently used symbols and their units.

  3. Euler's identity: Im Re

  4. Im Im Re Re Complex conjugate

  5. Complex Algebra

  6. Complex Algebra (cont.)

  7. Square Root Principle square root (principal branch) where nis an integer The complex square root will have two possible values. (The principal branch is unique.)

  8. Time-Harmonic Quantities Amplitude Angular Phase Frequency From Euler’s identity: We then have

  9. Time-Harmonic Quantities (cont.) going from time domain to phasor domain going from phasor domain to time domain Time-domain Phasor domain

  10. Graphical Illustration The complex number V V(t) C A C A  t B B

  11. Time-Harmonic Quantities (cont.) This assumes that the two sinusoidal signals are at the same frequency. There are no time derivatives in the phasor domain! All phasors are complex numbers, but not all complex numbers are phasors!

  12. Complex Vectors Transform each component of a time-harmonic vector function into a complex vector. To see this: Hence

  13. Example 1.15 (Shen & Kong) Assume Find the corresponding time-domain vector y ωt = 3π/2 ωt = π ωt = 0 x ωt = π/2 The vector rotates with time!

  14. Example 1.15 (cont.) Practical application: A circular-polarized plane wave (discussed later). For a fixed value of z, the electric field vector rotates with time. E (z,t) z

  15. Example 1.16 (Shen & Kong) We have to be careful about drawing conclusions from cross and dot products in the phasor domain!

  16. Time Average of Time-Harmonic Quantities Hence

  17. Time Average of Time-Harmonic Quantities (cont.) Next, consider the time average of a productof sinusoids: Sinusoidal (time ave = 0)

  18. Time Average of Time-Harmonic Quantities (cont.) Next, consider Hence, Recall that (from previous slide) Hence

  19. Time Average of Time-Harmonic Quantities (cont.) The results directly extend to vectors that vary sinusoidally in time. Consider: Hence

  20. Time Average of Time-Harmonic Quantities (cont.) The result holds for both dot product and cross products.

  21. Time Average of Time-Harmonic Quantities (cont.) To illustrate, consider the time-average stored electric energy for a sinusoidal field. (from ECE 2317)

  22. Time Average of Time-Harmonic Quantities (cont.) Similarly,

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