COMPLEX NUMBERS TUTORIAL
Table of Contents
· Section 5: Phasor Notation and Complex
Multiplication and Division Revisited
· Motivation: “Sinusoidal steady-state analysis is greatly facilitated if the currents and voltages are represented as vectors in the complex-number plane known as phasors.” (Boylestad, USNA EE text.)
·
Objective: Introduce
phasor
notation and show resulting easier form of complex
products/divisions. Below we see a description of phasors taken from a course
introduction at the University of Dayton.
Definition: A phasor is a vector quantity (or more typically a complex number),
representing a sinusoidally varying current or voltage. The length of the
vector (magnitude of the complex number) is proportional to the magnitude of
the current or voltage. The direction or angle of the phasor (or angle of the
complex number) is equal to the phase of the current or voltage relative to
some arbitrary or common reference. The phase is negative if the current or
voltage lags the reference and is positive if the current or voltage leads
the reference.
|
5.1
Phasor Notation:
|
Phasor
notation provides two other ways of writing the polar form of a complex number: |
|
r cos q [means
r (cosq + j sin q)] |
|
r Ð q [means r (cosq + j sin q)] |
· 5.2 Complex Products with Phasors
|
Example: Find (3e4j)(2e1.7j)
(3e4j)(2e1.7j) = (3)(2)e4j+1.7j =
6e5.7j In general,
From this, we can multiply using the polar form:
or in phasor notation
|
|
·
Verbal summary of complex
products using phasors: Magnitudes
are multiplied and
the angles are added together. |
· 5.3 Complex Division with Phasors:
|
Example in Exponential Form:
From this, we can conclude the following:
or
Example : Find
Answer:
Example : Find Answer:
|
|
·
Verbal summary of complex
divisions using phasors: Division is accomplished by 1) simply dividing the magnitude of the
numerator by the magnitude of the denominator and 2) subtracting the angle of
the denominator from that of the numerator. Given, C1 |
, 