1. Notes 2 Complex Vectors ECE 3317 Prof. Ji Chen Adapted from notes by Prof. Stuart A. Long Spring 2014 1

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

3. Complex Numbers Real part Magnitude Imaginary part Phase (always in radians) Euler's identity: Im Re 3

4. Complex Numbers Im Re Im Re Complex conjugate 4

5. Complex Algebra 5

6. Complex Algebra (cont.) 6

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

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

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

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

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

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

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

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. 14 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! 15

16. Time Average of Time-Harmonic Quantities 16 Hence

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

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

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

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

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

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