UNIT1: COMPLEX NUMBERS
Key unit competence
Perform operations on complex numbers in different forms and use them to solverelated problems in Physics, Engeneering, etc.
Introductory activity
Consider the extension of sets of numbers previously learnt from natural numbers
to real numbers. It is actually very common for equations to be unsolved in one
set of numbers but solved in another.
Let us find the solution of the following quadratic equations in the set of realnumbers:
1. 1 Algebraic form of Complex numbers and their geometric
representation1.1.1 Definition of complex number
Solution
Each of these numbers can be put in the form a + ib where a and b are real numbersas detailed in the following table:
Complex numbers are commonly used in electrical engineering, as well as in physics
as it is developed in the last topic of this unit. To avoid the confusion between i
representing the current and i for the imaginary unit, physicists prefer to use j torepresent the imaginary unit.
As an example , the Figure 1.1 below shows a simple current divider made up of acapacitor and a resistor. Using the formula, the current in the resistor is given by
Figure 1. 1 A generator and the R-C current divider
Figure 1. 2 Cycles of imaginary unit
From the figure 1.2, the following relations may be used:
1.1.2 Geometric representation of a complex number
The complex plane consists of two number lines that intersect in a right angle at
the point (0,0) . The horizontal number line (known as x − axis in Cartesian plane) isthe real axis while the vertical number line (the y − axis in Cartesian plane) is the
imaginary axis.
Every complex number z = a + bi can be represented by a point Z (a,b) in thecomplex plane.
The complex plane is also known as the Argand diagram. The new notation Z (a,b)
of the complex number z = a + bi is the geometric form of z and the point Z (a,b)is called the affix of z = a + bi . In the Cartesian plane, (a,b)is the coordinate of the
Figure 1. 3 The complex plane containing the complex number z = a + bi
Complex impedances in series
In electrical engineering, the treatment of resistors, capacitors, and inductors can
be unified by introducing imaginary, frequency-dependent resistances for the latter
two (capacitor and inductor) and combining all three in a single complex number
called the impedance. If you work much with engineers, or if you plan to become
one, you’ll get familiar with the RC (Resistor-Capacitor) plane, just as you will with
the RL (Resistor-Inductor) plane.
Each component (resistor, an inductor or a capacitor) has an impedance that can be
represented as a vector in the RX plane. The vectors for resistors are constantregardless of the frequency.
together when coils and capacitors are in series. Thus , X = XL + XC .
In the RX plane, their vectors add, but because these vectors point in exactly
opposite directions inductive reactance upwards and capacitive reactance
downwards, the resultant sum vector will also inevitably point either straight up ordown (Fig. 1.4).
Figure 1. 4 Pure inductance and pure capacitance represented by reactance vectors that point straight
up and down.Example 1.2
Solution
1.1.3 Operation on complex numbers
1.1.3.1 Addition and subtraction in the set of complex numbers
Adding impedance vectors
If you plan to become an engineer, you will need to practice adding and subtracting
complex numbers. But it is not difficult once you get used to it by doing a few sample
problems. In an alternating current series circuit containing a coil and capacitor,
there is resistance, as well as reactance.
Whenever the resistance in a series circuit is significant, the impedance vectors
no longer point straight up and straight down. Instead, they run off towards the
“northeast” (for the inductive part of the circuit) and “southeast” (for the capacitivepart). This is illustrated in Figure 1.5.
Figure 1. 5 Resistance with reactance and impedance vectors pointing “northeast “or “southeast.”
When vectors don’t lie along a single line, you need to use vector addition to be
sure that you get the correct resultant. In Figure 1.6, the geometry of vector additionis shown by constructing a parallelogram, using the two vectors Z1 = R1 + jX1 and
Z2=R2+jX2 as two of the sides. Then, the diagonal is the resultant
Figure 1. 6 Vector addition of impedances Z1 = R1 + jX1 and Z2 = R2 + jX2
Formula for complex impedances in series RLC circuits
Figure 1. 7 A series RLC circuit
Example 1.4
A resistor, coil, and capacitor are connected in series with , R = 45Ω XL= 22Ω andXC = − 30Ω. What is the net impedance, Z ?
SolutionConsider the resistor to be part of the coil (inductor), obtaining two complex vectors,
45 + 22 j and0 − 30 j. Adding these gives the resistance component of
(45 + 0)Ω = 45Ω, and the reactive component of (22 j − 30 j)Ω = −8 jΩ. Therefore
the net impedance is Z = (45 −8 j)Ω .
Application activities 1.3
Represent graphically the following complex numbers, and then deduce the
numerical answers from the diagrams.
1.1.3.2 Conjugate of a complex number
Activity 1.5
In the complex plane,
1. Plot the affix of complex number z = 2 + 5i
2. Find the image P'of the point P affix of z by the reflection across the real
axis. What is the coordinate of P' ?
3. Write the complex number z ' associated to P' and discuss the relationship
between z and z ' of P' ?
4. Write algebraically the complex number z ' associated to P' and discuss therelationship between z and z '
Every complex number z a bi = + has a corresponding complex number z − called
Figure 1. 8 Reflection of affix about the real axis
1.1.3.3 Multiplication and powers of complex number
1.1.4 Modulus of a complex number
Figure 1. 9 Modulus of z = a + bi
Activity 1.11
Given the quadratic equation z2 − (1+ i) = 0 , you can write it as z2 =1+ i . Calculate
the square root of 1+ i to get the value of z and discuss how to solve equations of
the form Az2 +C = 0 where A and C are complex numbers and A is differentfrom zero. Express in words the formula used.
Solving simple quadratic equations in the set of complex numbers recalls theprocedure of how to solve the quadratic equations in the set of real numbers
From affix of a complex number z = a + bi , there is a connection between its
modulus and angle between the corresponding vector and positive x − axis as
illustrated in figure 1.10. This angle is called the argument of z and denoted byarg ( z) .
1.3 Exponential form of complex numbers
1.3.1 Definition of exponential form of a complex number
In electrical engineering, the treatment of resistors, capacitors and inductors can be
unified by introducing imaginary, frequency-dependent resistances for capacitor,
inductors and combining all three (resistors, capacitors, and inductors) in a single
complex number called the impedance. This approach is called phasor calculus. As
we have seen, the imaginary unit is denoted by j to avoid confusion with i which
is generally in use to denote electric current. Since the voltage in an AC circuit isoscillating, it can be represented as
Example 1.15
a) Express the complex numbers in exponential form
Given that the same current must flow in each element, the resistor and capacitor
are in series such that the common current can often be taken to have the referencephase.
1.3.3 Application of complex numbers in Physics
Activity 1.17
Conduct a research from different books of the library or browse internet to
discover the application of complex numbers in other subjects such as physics,applied mathematics, engineering, etc
Complex numbers are applied in other subjects to express certain variables or
facilitate the calculation in complicated expressions. They are mostly used in electrical
engineering, electronics engineering, signal analysis, quantum mechanics, relativity,
applied mathematics, fluid dynamics, electromagnetism, civil and mechanical
engineering. Let look at an example from civil and mechanical engineering.
An alternating current is a current created by rotating a coil of wire through amagnetic field.
Figure1.11: generation of alternating current
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