Electronic 's Math
Complex
numbers
A
Complex
Real
I
number
we
they
'
-
,
number
complex
number is
Imaginary
2 Module 2 Lecture Notes
-7.8
=
44
number
operator
,
IT
,
or
imaginary
negative
number
number
.
e
represent
and
is
-
written
transforms the number to
an
.
It
is
prefix
as a
imaginary
called
that
number
.
Polar ( Complex )
Cartesian
plane
plane
fall
components
2
is
to
used
Real number t
any positive
,
has
-
AXIS
900
y
-
←
Imaginary
+900 →
j
<
~
§
L
goo
j
-
°
1804g
fo
[ Real axis
X axis
-
v
→
-
i
axis
goo
Polar ( Complex )
ja
Plane
•
L
a
E
j
j
.
'
j
j j
4
'
18%-1
y
.
.
12
j j g
2
1/+j = -j
Complex
a
-
a
Hoo
and 1/-j = +j
represent phasors
numbers
.
xtjy
j
C-
5tyi6)
n
6
(
4
-15
-13
4
-5
(-3-25)
>
"
7
(4-22)
-2
-
7tj4)
X
,
¥00
Complex
written
numbers
in
representing
RECTANGULAR
Form
phasors
or
Polar
can
be
Form
jn
¥4
T
r
y
← x→
xtjy
r
p
-
-
-
magnitude
hypotenuse
xatya
phaser
O
y=r Cos ( )
y
'
r
O
-
from
Otani
r
'
O
r
-
-
r
Sin ( O)
xtjy
tx axis
Y
O
Phasors have
Electronics : AC
changing signal
magnitude
and direction
signals represent
at
a
a
constantly
specified point
.
.
Trigonometric
Functions
and
in
I
.
'
1531
I -53
3tj4 5 53.10
-53.10
3tj4 5¢
.
-
.
.
"
-
g
g.
-
4=75453.10
3-24=7-5453.50
Notice that the
is
NOT the
angle
and
For
3tj4 5 53.10
-31-24=754-53.10
-
g
y
-
'd
angles
the
from the
calculator gives
positive
you
x-axis
must add 1800
54-53.101-180=54-126.9
5453.10=75453.101-1800--54233.10
3-24=754-53.50
.
Examples
14.3-4/6=21.5
r
-
-
Matu
E- tan
'
-
s)
&
48.20
¥
215
-
'
48.20
,
.
)
'
x
yr
4.6
21=4.6
+
-340=3.732-2.52
Cost-347=3.73
2=4,6×5
nc
-
347=2.52
¢
! -340
v
d
-19.41-27.5=20.8
75
1590
no
-
f- ✓ 19,421-7.52=20.8
O
-
tan
①
=
'
=
-19.4
In
In
j
2360=-403-25.98
calculator
yields
y
-21.80
-21.81-1800=1590
21
×
correct
×
answers
4--7.21×662369=-4.03
y=72lxSinC236°)=
-5.98
7.21
Realpart=Z
phaser
vertical
,
imaginary
tan (E)
'
790
the
on
r=IyT=y
your
axis
undefined
is
.
You cannot
use
calculator for this You have to
.
recognize
the
phaser
is
+900
0-jl60.si
-90
)
-
jv
-900
or
j
p
160
jl2
12h
Otjy
is
otjl2
'
x
0tj34
34
160
900
-
-
sat
x
x
L
Imaginary
part
-_
,
xtjo
phaser
zero
is
on
horizontal real
the
oo
1800
x
axis
r=IyT=y
O
-
-
tan ( 09=00
'
on
14 tj 0=14
14
1800
-
16120=-46
16
00
180
j
ji
l
14
X
l
-
16
X
Add
(subtract
Rectangular Form :
9-j6 t
9tj4V
Multiply
-
Rectangular
(3
complex
numbers
Addlsubtract
the real
Add /subtract the
pants
imaginary
parts
91-7
tj -61-8=161-22
3.6
-j4V=( 9-3,6 tj 41-4 5. ttj8V
7tj8
complex
numbers
form : FOIL method
jt )(2tj5)
-
(3×2)*3×25)tCj4x2)tCjtxj5)
j8
j
tj
G) 20
Gt (
)
1528
j
6
15
-
-
-
Gtj 71-20
261-27
220
( Recall ja
-
-
)
l
Polar
the
Multiply
magnitudes
Add the
form :
(4.6
angles
)
-16° )(3.7
87
(4.6×3.7) (-169-870)
17.02
Division of
Rectangular
71°
complex
form :
numbers
Multiply
numerator
the
denominator
by
the
and
complex
conjugate of the denominator
4 tjb
is
of 4
_j6
conjugate
.
Complex
6.
2tj8
is
atj
6.2
-j8
z.jo/3tj2--6tj4tj3tja2=6tj7-23tj29tj6-j6-j24
9+4
=
=
4+25--4-3 tj%
0.308
tj 0.538
Polar form: Divide
the
magnitude
Subtract the
1g ,
,
go
=
64 ) 670
-
=
6¥
angles
(150--670)
0.1875
82
°