Complex Numbers and Phasors

(Time-dependent Waves) 

A complex number, Z, is the sum of an imaginary and a real number.  Z = a + ib where a = Re{Z}, b = Im{Z} and i is the square root of -1.  Because of the two component nature of complex numbers, it is convenient to represent complex numbers as vectors in a two dimensional plane called the complex plane.  By convention, the horizontal axis is the real axis and the vertical axis is the imaginary one.  The cartesian coordinates of Z in the complex plane are {a,b}. 

Another set of valid coordinates specifying the location of Z are the polar coordinates {r,a} where r is the length of a vector from the origin to Z and a is the phase angle measured in a clockwise manner from the real axis to Z.  In this phasor representation, Z = |Z| exp( i * alpha).  A purely real number has alpha = 0 or alpha = pi and a purely imaginary number has alpha = pi/ 2 or alpha = -pi/ 2.

The animation below traces a circular path in the complex plane.  Z begins as a purely real number.  The radius of the circle, the length of Z, remains constant and alpha linearly increases with time.  The graphs to the left and below the circle trace the real and imaginary projections of Z.

Section 1a Exercises:

  1. What is the period of rotation of the vector in the above applet?  What is the angular frequency of the vector?  If the initial coordinates of Z are {2.5,0}, write the equation that expresses Z as a function of time.  (Do not use cursor to measure coordinates.)

The real and imaginary components of Z can be found using the relation:

exp( i alpha} = cos(alpha) + i sin(alpha) .

  1. At t = 1.7, what are the real and imaginary components of Z?  Is your answer in qualitative agreement with the graphs in the applet?
  2. At t = 3.5, what are the real and imaginary components of Z?
  3. Using your answers to #2 and #3 and the Pythagorean theorem, show that the length of Z is what you expect.
  4. What is the complex conjugate of the complex number represented by the cartesian coordinates {1.3, 2.4}?  What is the phase angle between a number and its complex conjugate?
  5. Given your answer to #1, write the equation for the complex conjugate of Z ,or Z*.  As time increases, how is alpha changing? In what direction is the vector rotating?

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