Notes 5 Complex Vectors ECE 3317 Prof. D. R. Wilton Adapted from notes by Prof. Stuart A. Long

References to equations and pages in your book will be written in green. Appendices A, B, C, and D in the text book list frequently used symbols and their units. V(t) is a time-varying function. V is a phasor (complex number). A bar underneath indicates a vector: V(t), V. Notation

Complex Numbers Real part Magnitude Imaginary part Phase Im Re Euler's identity: Hence

Complex Numbers Im Re Im Re complex conjugate

Complex Algebra Im Re Im Re

Complex Algebra (cont.)

Square Root where n is an integer Note: the complex square root will have two possible values. (principal branch) Principal square root (principal root)

Time-Harmonic Quantities Amplitude Angular Phase Frequency Charles Steinmetz

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

B B A V(t)V(t) t C Φ C A Graphical Illustration The complex number V

Time-Harmonic Quantities

Transform each component of a time-harmonic vector function into complex vector. Complex Vectors

Example 1.15 ωt = 3π/2 ωt = π ωt = π/2 y x ωt = 0, 2 π The vector rotates with time! Assume Find the corresponding time-domain vector

Example 1.15 (cont.) Practical application: A circular-polarized plane wave (discussed later). z For a fixed value of z, the electric field vector rotates with time. E (z,t) For a fixed value of t, the electric field vector appears to spiral in space as shown at left.

Example 1.16 We have to be careful about drawing conclusions from cross and dot products in the phasor domain!

Time Average of Time-Harmonic Quantities

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

Time Average of Time-Harmonic Quantities (cont.) (from previous slide) Hence Now consider

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

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

Time Average of Time-Harmonic Quantities (cont.) To illustrate, consider the time-average stored electric energy.