Download Complex Numbers & Functions: Polar Form, Multiplication, Division & Properties and more Assignments Electrical Circuit Analysis in PDF only on Docsity!
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008 Complex Numbers and Complex Functions
So far, we have used real numbers to determine the unknown circuit variables. While it is possible to obtain solutions to circuit problems utilizing only the real numbers and their properties, it is sometimes very difficult or cumbersome.
Complex numbers often alleviate these problems. While, at first, the complex numbers seem somewhat esoteric, one should keep in mind that in circuit analysis they are used as an intermediate step in obtaining the solution. Any circuit problem can be described in the domain of real numbers, then manipulated in the domain of either real or complex numbers, and its solution be presented again in the domain of real numbers.
1 Basic Definitions
Definition 1 (Complex Number)
A complex number z is a number that can be expressed as
z = x + jy (1)
where x ,y ∈ R and j^2 =− 1.
∆
- x is called a real part of z , and we write x = Re (z) ,
- y is called an imaginary part of z, and we write y = Im (z) ,
- j is called an imaginary unit , sometimes expressed as j = − 1.
The representation (1) is called the rectangular form of a complex number z.
Example 1
z = 4 − 3 j ; Re ( ) z = 4 , Im( ) z =− 3 ,
z 1 = − 2 + j ln 6 ; Re ( z 1 ) =− 2 , Im( z 1 ) =ln 6
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008 The following two notations are equivalent
z = x + jy = x + yj
Note : The terminology used to describe these new numbers is somewhat unfortunate. There is nothing complex or imaginary about them. They could just as well be called two- part numbers with the real part called the first part and the imaginary part called the second part. The names complex or imaginary carry with them a connotation of being difficult to comprehend and thus potentially create a degree of apprehension in some students. As we will soon see, there is nothing magical or unreal about the complex numbers, and with time they will be easily assimilated into our repertoire of circuit analysis tools.
When dealing with the complex numbers, it is often expedient to represent them graphically. Since these numbers have two parts, a plane seems to be the most natural setting for such a representation.
Definition 2 (Complex Plane)
A complex plane is a plane in Cartesian coordinate system in which the x -axis is the real axis and the y -axis is the imaginary axis.
In the complex plane, a complex number z = x + jy can be represented as a point P with
the coordinates x and y as shown in Fig. 1.
Im z
P(x,y) y
x Re z
Figure 1 Complex plane
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008
Note: r and θ are polar coordinates of P defined by
θ θ
θ θ y rsin z sin
x rcos zcos = =
- r is called the absolute value or modulus of z , denoted z and
z = r = x^2 + y^2 (3)
Geometrically, z is the distance of the point P from the origin.
- θ is called the argument of z or angle of z and
x
y
θ = arg z = tan −^1 (4)
Geometrically θ is the directed angle from the positive x -axis measured in radians and
positive in the counterclockwise sense.
By substituting Eq. (2) into the rectangular form of z expressed by Eq. (1) we obtain another form of a complex number z
rcos jsin z cos j sin
z x jy rcos jrsin = + = +
or
z = z ( cos θ + jsin θ) (6)
This form is often denoted as
z = z ∠ θ (7)
The representation (6) or (7) is called the polar form of a complex number z.
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008 Note : Some authors call the representation (4) the rectangular form or trigonometric form.
♦
The rectangular and polar forms are equivalent.
z = x + jy = z cosθ + jz sinθ= z ( cosθ+ j sinθ)= z ∠ θ (8)
Example 3
Let (^) z = 1 + j. Obtain polar form of z.
Solution :
To determine the polar form we need to obtain the magnitude and the argument (angle) of the complex number z.
z = 1 + j = x + jy ⇒ x = 1 , y = 1
The magnitude of z is:
z = x^2 + y^2 = 1 + 1 = 2
The argument of z is:
2 , ( 0 , 1 K)
= tan −^1 =tan−^11 = ± n n = x
y
is called the principal argument. It is the value of θ : − π < θ< π.
Note : In circuit analysis, when determining the argument of a complex number, we will, by default, solve for the principal argument, and we will simply call it the argument or the angle of a complex number.
The polar form of complex number z is, therefore:
sin 4
cos sin 2 cos
π π π θ θ ⎟= ∠ ⎠
z = z + j = + j
♦
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008
- Determine the polar form of:
a) w = 3 + j 4
b) v (^) 1 = 7 j
c) u =− 5 + j 0
d) − 4 − j 5
- Determine the rectangular form of:
a) 6
z = ∠ −
b) w = 1 ∠ 0
c) u = 3 ∠ π
d) s = 8 ∠ 60 °
Note : The last problem has an angle expressed in degrees instead of radians. It is not uncommon to see that, even though we defined the polar form as the one in which the angle is expressed in radians.
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008 2 Complex Conjugate
Given a complex number
z = x + jy
one can create other related complex numbers:
z x jy
z x jy
z x j y
There are infinitely many such related numbers. Among them, one such related number is especially useful in analysis and has been given its own name.
Definition 3 (Complex Cojugate)
Let
x + jy (9)
be any complex number. Then the number
x − jy (10)
is called the complex conjugate of the number defined by (9).
♦
If the number in (9) is denoted by, say, z :
z = x + jy (11)
then its complex conjugate, say w , is often denoted by z* :(to emphasize the relation to the original number z )
w = z *^ = x − jy (12)
Note that the real part of z*^ is the same as the real part of z, and the imaginary part of z*^ is the negative of the imaginary part of z.
♦
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008
Im z
y
z = r
- θ
-y
z = r
P(x,y)
x Re z
P*^ (x,-y)
Figure 4 Complex number and its conjugate
3 Operations on Complex Numbers
Definition 4 (Equality of Complex Numbers in Rectangular Form)
Two complex numbers
z = x + jy, w = u + jv
are equal if and only if their real parts are equal and their imaginary parts are equal.
Thus
z = w iff x = u and y = v (16)
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008 Definition 5 (Equality of Complex Numbers in Polar Form)
Two complex numbers
z 1 = z 1 ∠θ 1 , z 2 = z 2 ∠ θ 2
are equal if and only if their magnitudes are equal and their arguments are equal.
Thus
z 1 = z 2 iff z 1 = z 2 and θ 1 = θ 2 (17)
Definition 6 (Addition of Complex Numbers)
Let
z = x + jy, w = u + jv.
Then
z+w
is called the sum of z and w and is defined as
z + w = ( x + jy ) +( u + jv ) =( x + u ) + j ( y + v ) (18)
Therefore,
Two complex numbers in rectangular form are added by adding the real parts and the imaginary parts separately.
Addition in polar form cannot be performed (except for the trivial cases) and therefore is Not defined.
♦
Example 5
z w 5 j 1 j 3 ( 5 1 ) j ( 1 3 ) 6 j 4
z 5 j, w 1 j 3
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008 Definition 9 (The negative in Rectangular Coordinates)
Let
z = x + jy, a =− 1.
Then the number
az = − 1 ( x + jy ) =− x − jy (21)
is called the negative of z.
♦
Definition 10 (The Negative in Polar Coordinates)
Let
z = z ∠ θ , a =− 1.
Then the number
az = − 1 ( z ∠θ ) = z ∠(θ+π) (^) (22)
is called the negative of z.
♦
Note : We often use the notation
− z ∠ θ (23)
when we mean
z ∠ (θ +π) (24)
Even worse, we often call the form (23) the polar form, even though, by definition, the magnitude of a polar number is non-negative.
Again, we permit that deviation, for it does not contradict the definition and it is obvious what we mean.
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008 Definition 11 (Subtraction)
Let
z = x + jy, w = u + jv.
Then the subtraction of z-w can be considered as addition of z and the negative of w
( x jy ) ( u jv ) ( x u ) j ( y v )
z w x jy u jv x jy 1 u jv = + + − − = − + −
Let’s restate the above result:
z − w = ( x − u ) + j ( y − v ) (26)
Therefore,
Two complex numbers in rectangular form are subtracted by subtracting the real parts and the imaginary parts separately.
Subtraction in polar form cannot be performed (except for the trivial cases) and therefore is not defined.
♦
Example 7
z w 5 j ( 1 j 3 ) ( 5 1 ) j ( 1 3 ) 4 j 2
z 5 j, w 1 j 3 − = + − + = − + − = −
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008 Example 8 (Product of a Complex Number and its Complex Conjugate)
Let
z = x + jy , z *= x − jy.
Then
z z *^ = ( x + jy )( x − jy ) = x^2 − jxy + jyx − j^2 y^2 = x^2 + y^2 (31)
Since this result is of extreme importance we will state it again
z z *^ = x^2 + y^2 (32)
Note that the product of a complex number and its complex conjugate is a real number.
Example 9 (Multiplication by j in Rectangular Form)
In rectangular form, let
z = x + jy.
Then
jz = j ( x + jy ) = jx + j^2 y =− y + jx (33)
Example 10 (Multiplication by j in Polar Form)
In polar form, let
z = z ∠ θ.
Then since
j = 0 + 1 j = 1 ∠ 90 ° (34)
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008 we get
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008 Definition 15 (Division of Complex Numbers in Polar Form)
Let
z 1 = z 1 ∠θ 1 , z 2 = z 2 ∠ θ 2.
The quotient 2
1 z
z z = is the complex number satisfying
zz 2 = z 1. Hence,
2
1 (^2 21) z
z zz = z z = z ⇒ z = (38)
and
arg ( zz 2 ) = arg ( ) z + arg ( z 2 ) = arg ( z 1 ) ⇒ arg ( z ) = arg ( z 1 ) − arg ( z 2 ) (39)
Thus,
2
1 2 2
1 1 2
1 z
z z
z z
z
or
[ ( (^) 1 2 ) ( 1 2 2
1 2
(^1) cos jsin z
z z
z
= θ −θ + θ − θ )] (41)
Example 11 (Division by j in Rectangular Form)
Let
z = x + jy.
Then
y jx 1
y jx j j
x jy j j
x jy j
z = −
Prof. Bogdan Adamczyk Grand Valley State University EGR 214 Winter 2008 Example 12 (Division by j in Polar Form)
In polar form, let
z = z ∠ θ
then
= z 90 1 90
z j
z
Therefore, division by j is equivalent to clockwise rotation by 90 ° or π /2 radians , as
shown in Fig. 6.
Re z
Im z
(x,y)
x
y
y
-x
(y,-x)
Figure 6 Division by j
