COMPLEX NUMBERS
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In this and the following chapters, we will present a very important topic: AC, or Alternating Current. The name alternating current is not very precise and normally covers circuits with sinusoidal voltages and currents; however, alternating current can also mean any arbitrary current waveform. The importance of AC voltage is that this kind of voltage is used for the main electric power source in homes and industry throughout the world. It is also the basis for many electronics, telecommunication, and industrial applications.
To handle sinusoidal waveforms and the circuits associated with them, we will use a simple and elegant method called the method of phasors. Phasors are based on the properties of complex numbers, which are ideal for representing sinusoidal quantities. In this chapter, we will summarize the principal facts about complex numbers and their operations. We will also show how TINA’s Interpreter makes it easy to do calculations with complex numbers.
Complex
numbers consist of two parts, a real part
(x), which is a real number, and a
so called imaginary part
(y), which is a real number multiplied by
, the imaginary unit. The complex
number z, therefore, can be described
as:
z
= x + jy
where
.
Examples
of complex numbers:
z1=
1+ j
z2=
4-2 j
z3=
3- 5j
Complex
numbers were originally introduced in the seventeenth century to represent the
roots of polynomials which could not be represented with real numbers alone. For
example, the roots of the equation x2 +2x +2 = 0 can only be
described as
and
, or using
the notation
, z1=
1+ j and z2= 1- j.
Using the new notation to investigate the properties of expressions,
mathematicians were able to prove theorems and solve problems which until then
had been difficult if not impossible to solve. This led to the elaboration of
complex algebra and complex functions, which now are widely used in mathematics
and engineering.
Geometric
representation of complex numbers
Rectangular form
Because a complex number can always be separated into its real and complex parts, we can represent a complex number as a point on a two-dimensional plane. The real part of a complex number is the projection of the point onto the real axis, and the imaginary part of the number is the projection onto the imaginary axis. When a complex number is represented as the sum of real and imaginary parts, we say it is in rectangular or algebraic form.
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The following figure shows the complex number z = 2 + 4j
Polar and exponential
form
As you can see from the figure above, the point A could also be represented by the length of the arrow, r (also called the absolute value, magnitude, or amplitude), and its angle (or phase), j relative in a counterclockwise direction to the positive horizontal axis. This is the polar form of a complex number. It is denoted as r Š j.
The next step is very important. A complex number in polar form can also be written in exponential form:
This
simple expression is distinctive in that it has an imaginary number in the
exponent instead of the usual real number. This complex exponential behaves very
differently from the exponential function with a real argument. While ex
grows rapidly in magnitude for increasing x>0 and
decreases for x<0, the function
has the
same magnitude (z =1) for any j.
Furthermore, its complex values lie on the unit circle.
Euler’s formula provides a unifying link among the rectangular, polar, and exponential forms of complex numbers:
z = x + j y = r e jj = r ( cos j + j sin j )
where
and j = tan-1 (y/x).
For our example above, z = 2 + 4j:
j
= tan-1 (4/2)
= 63.4°
therefore
.
Or
vice-versa:
You
will need to be adept at using both forms, depending on the application. For
example, addition or subtraction are obviously easier to do when the numbers are
in rectangular form, while multiplication and division are easier to do when the
numbers are in exponential form.
Operations with complex numbers
The
operations that can be done with complex numbers are similar to those for real
numbers. The rules and some new definitions are summarized below.
Operations
with j
The
operations with j simply follow from
the definition of the imaginary unit,
.
To
be able to work fast and accurately, you should memorize these rules:
j2
=
-1
j3
=
-j
j4
=
1
1/j = -j
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Complex
conjugate
The complex conjugate of a complex number is easily derived and is quite important. To obtain the complex conjugate of a complex number in rectangular form, simply change the sign of the imaginary part. To do so for a number in exponential form, change the sign of the angle of the complex number while keeping its absolute value the same.
The
complex conjugate of a complex number z
is often denoted by z*.
Given
the complex number z=a+ jb, its complex conjugate is z*=a
– jb.
If
z is given in exponential form,
, its complex conjugate is
Using the definitions above, it is easy to see that a complex number multiplied by its complex conjugate gives the square of the absolute value of the complex number:
z z*= r2 = a2 + b2
Also, by adding or subtracting any complex number and its conjugate, we get the following relations:
z +
z*
=
2a
therefore
Re
(z) = a = (z
+
z*)
/ 2
Similarly:
z -
z*
=
j2b
therefore
Im
(z) = b = (z
-
z*)
/ 2j
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Numerical
examples:
z*=
5 Š
- 53.13°
Addition
and subtraction
Addition and subtraction of complex numbers is straightforward—we only need to add the real and imaginary parts separately. For example, if
z1
=
3 – 4j
and
z2 = 2 + 3j
Obviously,
we should use the rectangular form for these operations. If the numbers are
given in exponential or polar form, we should transform them first to
rectangular form using Euler’s formula, as given earlier.
Multiplication
There
are two methods for multiplication of complex numbers--
Multiplication
of complex numbers given in rectangular form
To carry out the operation, simply multiply the real and imaginary parts of one number in turn by the real and imaginary parts of the other number and use the identity j2 = -1.
z1z2 = (a1 + jb1) (a2 + jb2)= a1 a2 + jb1a2+ jb2a1 – b1b2 = a1 a2– b1b2 + j(b1a2+ jb2a1)
When the complex numbers are given numerically, it is not necessary to use the formula above. For example, let
z1
=
3 – 4j
and z2 = 2 + 3j
With direct multiplication of the components:
z1z2 = (3 – 4j)( 2 + 3j)=6 - 8j +9j + 12 =18 + j
or using the formula: z1z2 = a1 a2– b1b2 + j(b1a2+ b2a1)
z1z2 = 6 +12 + j ( -8 + 9) = 18 + j
We think you’re more likely to make an error if you use the formula than if you multiply the components directly.
| {Solution
by TINA's Interpreter} z1:=3-4*j z2:=2+3*j z1*z2=[18+1*j] |
Multiplication
of complex numbers given in polar or exponential form
To carry out this operation, multiply the absolute values and add the angles of the two complex numbers. Let:
or in polar form
z1 z2 = r1 r2 Š j1+ j2
Note:
We have already used this rule when we calculated z
z*
above.
Since the angle of the conjugate has the opposite sign of the original angle, a
complex number multiplied by its own conjugate is always a real number; namely,
the square of its absolute value: z
z*
= r2
For
example, let:
z1
=
5 Š
30°
and z2
=
4 Š
-60°
then
z1
z2 =
20 Š
-30°
or
in exponential form
Multiplication
is obviously simpler when the numbers are in polar or exponential form.
However,
if the complex numbers are given in rectangular form, you should consider
performing the multiplication directly as shown above, since there are
additional steps if you convert the numbers to polar form before multiplying
them. Another factor to consider is whether you want the answers to be in
rectangular form or in polar/exponential form. For example, if the two numbers
are in rectangular form but you would like their product in polar form, it makes
sense to convert them immediately and then multiply them.
Division
There
are two methods for division of complex numbers--
Division
of complex numbers given in rectangular form
To carry out the operation, multiply the numerator and the denominator by the conjugate of the denominator. The denominator becomes a real number and the division is reduced to the multiplication of two complex numbers and a division by a real number, the square of the absolute value of the denominator.
z1
=
3 – 4j
and
z2 = 2 + 3j
Let’s check this result with TINA’s Interpreter:
| {Solution
by TINA's Interpreter} z1:=3-4*j z2:=2+3*j z1/z2=[-461.5385m-1.3077*j] |
Division
of complex numbers given in polar or exponential form
To carry out the operation, divide the absolute values (magnitudes) and subtract the angle of the denominator from the angle of the numerator. Let:
then using the rule of division of exponential functions
or in polar form
z1 / z2 = r1 / r2 Š j1 - j2
For
example, let:
z1
=
5 Š
30°
and z2
=
2 Š
-60°
then
z1
/
z2 = 2.5 Š
90°
or
in exponential and rectangular forms
Let’s check this result with TINA’s Interpreter:
| {Solution
by TINA's Interpreter} z1:=5*exp(j*degtorad(30)) z2:=2*exp(j*degtorad(-60)) z1/z2=[0+2.5*j] |
Division
is obviously simpler when the numbers are in polar or exponential form.
However,
if the complex numbers are given in rectangular form, you should consider
performing the division directly using the complex conjugate method as shown
above, since there are additional steps if you convert the numbers to polar form
before dividing them. Another factor to consider is whether you want the answers
to be in rectangular form or in polar/exponential form. For example, if the two
numbers are in rectangular form, but you would like their quotient in polar
form, it makes sense to convert them immediately and then divide them.
Now let us illustrate the use of complex numbers by more numerical problems. As usual, we will check our solutions using TINA’s Interpreter. The interpreter works with radians, but it has standard functions for the conversion of radians to degrees or vice-versa.
Example
1 Find
the polar representation:
z
= 12 – j 48
or
49.48 Š
-75.96°
| {Solution
by TINA's Interpreter} z:=12-j*48; abs(z)=[49.4773] arc(z)=[-1.3258] radtodeg(arc(z))=[-75.9638] |
Example
2 Find
the rectangular representation:
z
= 25 e j 125°
| {Solution
by TINA's Interpreter} z:=25*exp(j*(degtorad(125))); z=[-14.3394+20.4788*j] Re(z)=[-14.3394] Im(z)=[20.4788] |
Example
3
Find the polar representation of the following complex numbers:
z1
= 12 + j 48
z2 = 12 - j
48 z3
= -12 + j 48 z4
= -12 – j 48
The
absolute values of all four numbers are the same because the absolute value is
independent of the signs. Only the angles are different.
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{Solution
by TINA's Interpreter} z4:=-12-j*48: |
TINA’s
arc() function determines the angle of any complex number, automatically placing
it correctly in one of the four quadrants.
Be
careful, however, using the tan-1 function to find the angle, since
it is restricted to returning angles only in the first and fourth quadrants
(–90°
< j
< 90°).
Since
z1 is located in the first quadrant of the coordinate system, the
calculation is:
a1
= tan-1(48/12)= tan-1(4)= 75.96°
Since
z4 is located in the third quadrant of the coordinate system, tan-1
does not returns the angle correctly. The angle calculation is:
a4= 180°+75.96°
= 255.96°
or -360°+255.96°
= - 104.04°,
which is the same as calculated by TINA.
z2
is located in the fourth quadrant of the coordinate system The angle
calculation is:
a2= tan-1(-48/12)= tan-1(-4)= - 75.96°
z3
, however, is in the 2nd quadrant of the coordinate system, so tan-1 does
not return the angle correctly. The angle calculation is:
a3= 180°-75.96°
= 104.04°.
Example
4 We
have two complex numbers: z1=
4 – j 6 and z2 = 5 ej
45°.
Find
z3 = z1
+ z2;
z4 = z1
- z2; z5
= z1 * z2; z6
= z1 / z2
First
we solve the problem using TINA’s Interpreter
|
{Solution
by TINA's Interpreter} |
Notice how TINA effortlessly handles the two complex numbers given in different forms.
The solution is more complicated without the interpreter. So that we can compare the different methods of multiplication and division, we will first determine the polar form of z1 and the rectangular form of z2 .
Next,
we find the four solutions using the easiest forms first: rectangular for
addition and subtraction, and exponential for multiplication and division:
z3
= z1 + z2
= 4 – j 6 + 3.535 + j 3.535 = 7.535 – j
2.465
z4
= z1 – z2
= 4 – j 6 - 3.535 - j 3.535 = 0.465 – j
9.535
z5
= z1*z2
= 7.21*5*ej(-56.31°+45°)
= 36.05 e –j11.31°=36.03*(cos(-11.31°)+j*sin(-11.31°))
z5
= 35.33 – j 7.07
z6=
z1/z2=
(7.21/5)*e j (-56.31°-45°)
= 1.442 e – j 101.31°
= 1.442 (cos(-101.31°)+j*sin(-101.31°))
z6
= -0.2828 – j 1.414
which
agree with the results obtained with the TINA Interpreter.
The
multiplication carried out in rectangular form:
z5=z1*z2
= (4-j6)*3.535*(1+j)
= 7.07*(2-j3)*(1+j)=7.07*(2-j3+j2+3)=7.07*(5-j)=35.35-j7.07
Finally
the division carried out in rectangular form:
which
agree with the previous results.
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