Complex Mathematics

1
Complex Mathematics

• Chapter 14
• Polar and Rectangular Coordinates

2
Complex Numbers

• For R, L, C components, what are the two
  significant parts of the voltage and current
  relationships.
• The angular frequency is the same for
  combination.
• Need to keep track of
• amplitude
• phase angle

3
ac Amplitude

• The most common ac measurement device is a
  multimeter, not an oscilloscope.
• Therefore ac amplitudes are (nearly) always
  given as the effective value such as measured by
  a multimeter.
• ac amplitudes will be assumed in rms unless
  otherwise noted.

4
Amplitude, Phase Angle Graph
Every sinusoid of the same frequency can be
represented by a phasor with length
proportional to amplitude and an angle from the
horizontal equal to the phase angle.
amplitude
q
5
Polar Form
Consistent with text, the phasor C represents a
sinusoid with amplitude C, and phase angle q.
CCÐq
6
Example Phasors
Draw v, i, on a phasor diagram.
2
1
120
30
7
Retangular Form

• A point on an plane can be represented by
  Cartesian, x-y, coordinates.

C(A,B)
y
B
A
x
8
Complex Rectangular Form

• We will define the
• x axis as the real axis
• y axis as the imaginary axis
• there is a correspondence to power
• voltage and current on the real axis result in
  power consumed as heat
• voltage and current on the imaginary axis result
  in power stored, not used.
• You pay real money for real power that is
  consumed.

9
j operator

• In mathematics, the symbol i is used to denote
  the imaginary coordinate.
• In electronics, j, is used as i is reserved for
  current.
• An algebraic solution may involve taking the
  square root of a negative number.
• Cant do that, so define j operator.

10
Define j operator
11
Why j operator?

• Consider

This complex solution is common in electricity.
12
Rectangular Form

• Now the phasor is written

To locate the phasor, move A units along the
horizontal axis then B units along the vertical
axis. Draw a line from the origin (0,0) to the
point. This is the phasor.
13
Rectangular to Polar

• The phasor is the same whether in polar or
  rectangular form.

14
Polar to Rectangular
15
Lab Practice

• Laboratory will have conversion practice.
• Bring calculator and learn to use it efficiently.

16
Complex Conjugate

• mirror image about horizontal axis

17
Complex Addition

• Must be in rectangular form.
• exception if angle is 0 or 180.
• Add real and imaginary parts separately.
• Subtraction is addition with a sign change.

18
Graphical Addition

• A B C
• move origin of B to end of A

A
C
B
19
Graphical Subtraction

• A - B A (-B) C
• move origin of -B to end of A

-B
C
-B
A
B
20
Multiplication Rectangular

• Rectangular

21
Multiplication Polar
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Division Rectangular

• Easy way
• convert to polar.

23
Division Polar
24
Calculator Operations

• learn your own

25
ac Sinusoid

• standard equation
• caution with amplitude gt effective value

26
Phasors

• Draw a phasor diagram for the following

Answer It cant be done. The frequencies must
be the same to use phasor analysis.