By the end of this chapter, you will be able to:
Mastering these skills will help you solve real-world engineering problems involving complex numbers with confidence and precision.
Complex numbers are fundamental for industrial control and installation professionals in Kenya, especially in fields like signal processing, electrical systems, and control theory. These numbers allow engineers to analyze alternating current (AC) circuits, oscillating signals, and system stability efficiently. Understanding complex numbers is essential for solving practical engineering problems involving periodic motion, electrical impedance, and control system analysis.
Complex numbers combine real and imaginary components, enabling engineers to model and solve problems that cannot be addressed using real numbers alone. In industrial control and installation, complex numbers are used in circuit analysis, describing signals, and solving equations related to control systems.
A complex number is expressed as \( z = a + jb \), where \( a \) is the real part, \( b \) is the imaginary part, and \( j \) is the imaginary unit (\( j^2 = -1 \)). This notation is particularly useful in Kenyan industry for AC circuit calculations and signal analysis.
$$ z = a + jb $$
Example 1:
A technician measures a voltage signal expressed as \( 5 + j3 \) volts. Identify the real and imaginary parts.
Given: \( z = 5 + j3 \) volts
Example 2:
In a control system, the impedance is given by \( z = 8 - j6 \) ohms. Find the real and imaginary components.
Given: \( z = 8 - j6 \) ohms
Example 3:
A digital signal has a complex representation \( z = -2 + j7 \). State the real and imaginary parts.
Given: \( z = -2 + j7 \)
Example 4:
A frequency response is modeled as \( z = 0 + j4 \). What is the real part?
Given: \( z = 0 + j4 \)
Example 5:
In a university laboratory, a student records a measurement \( z = 6 \) (no imaginary part). Express as a complex number.
Given: \( z = 6 \)
Complex numbers can be visualized on the Argand diagram, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This helps engineers understand magnitude and phase relationships in signals and circuits.
$$ \text{On Argand diagram: } z = a + jb \text{ is plotted at } (a, b) $$
Example 1:
Plot \( z = 3 + j4 \) on the Argand diagram.
Given: \( a = 3 \), \( b = 4 \)
The point is at (3, 4).
Answer: Plotted at (3, 4) on the Argand diagram.
Example 2:
Plot \( z = -5 + j2 \) for a control signal.
Given: \( a = -5 \), \( b = 2 \)
The point is at (-5, 2).
Answer: Plotted at (-5, 2).
Example 3:
Plot \( z = 0 - j7 \) for an AC current.
Given: \( a = 0 \), \( b = -7 \)
The point is at (0, -7).
Answer: Plotted at (0, -7).
Example 4:
Plot \( z = 4 + j0 \) for a resistance value.
Given: \( a = 4 \), \( b = 0 \)
The point is at (4, 0).
Answer: Plotted at (4, 0).
Example 5:
Plot \( z = -3 - j6 \) for a university experiment.
Given: \( a = -3 \), \( b = -6 \)
The point is at (-3, -6).
Answer: Plotted at (-3, -6).
The modulus (magnitude) and argument (angle) describe the size and direction of a complex number. For \( z = a + jb \):
$$ |z| = \sqrt{a^2 + b^2} $$
$$ \theta = \tan^{-1}( \frac{b}{a} ) $$
Example 1:
Find the modulus and argument of \( z = 5 + j12 \).
Given: \( a = 5 \), \( b = 12 \)
$$ |z| = \sqrt{5^2 + 12^2} $$$$ |z| = \sqrt{25 + 144} $$$$ |z| = \sqrt{169} $$$$ |z| = 13 $$
$$ \theta = \tan^{-1}( \frac{12}{5} ) $$$$ \theta = \tan^{-1}(2.4) $$$$ \theta \approx 67.38^\circ $$
Answer: Modulus \( 13 \), Argument \( 67.38^\circ \)
Example 2:
Find modulus and argument for \( z = 7 - j24 \).
Given: \( a = 7 \), \( b = -24 \)
$$ |z| = \sqrt{7^2 + (-24)^2} $$$$ |z| = \sqrt{49 + 576} $$$$ |z| = \sqrt{625} $$$$ |z| = 25 $$
$$ \theta = \tan^{-1}( \frac{-24}{7} ) $$$$ \theta = \tan^{-1}(-3.4286) $$$$ \theta \approx -73.74^\circ $$
Answer: Modulus \( 25 \), Argument \( -73.74^\circ \)
Example 3:
Determine modulus and argument for \( z = -10 + j10 \).
Given: \( a = -10 \), \( b = 10 \)
$$ |z| = \sqrt{(-10)^2 + 10^2} $$$$ |z| = \sqrt{100 + 100} $$$$ |z| = \sqrt{200} $$$$ |z| \approx 14.14 $$
$$ \theta = \tan^{-1}( \frac{10}{-10} ) $$$$ \theta = \tan^{-1}(-1) $$$$ \theta \approx -45^\circ $$
Answer: Modulus \( 14.14 \), Argument \( -45^\circ \)
Example 4:
Calculate modulus and argument for \( z = 3 + j0 \).
Given: \( a = 3 \), \( b = 0 \)
$$ |z| = \sqrt{3^2 + 0^2} $$$$ |z| = \sqrt{9} $$$$ |z| = 3 $$
$$ \theta = \tan^{-1}( \frac{0}{3} ) $$$$ \theta = \tan^{-1}(0) $$$$ \theta = 0^\circ $$
Answer: Modulus \( 3 \), Argument \( 0^\circ \)
Example 5:
For \( z = 0 - j6 \), find modulus and argument.
Given: \( a = 0 \), \( b = -6 \)
$$ |z| = \sqrt{0^2 + (-6)^2} $$$$ |z| = \sqrt{36} $$$$ |z| = 6 $$
$$ \theta = \tan^{-1}( \frac{-6}{0} ) $$ \theta = -90^\circ$$
Answer: Modulus \( 6 \), Argument \( -90^\circ \)
The conjugate of a complex number \( z = a + jb \) is \( z^* = a, jb \). Conjugates are used in engineering to simplify division and represent physical quantities like power.
$$ z^* = a, jb $$
Example 1:
Find the conjugate of \( z = 4 + j9 \).
Given: \( z = 4 + j9 \)
$$ z^* = 4 - j9 $$
Answer: \( z^* = 4 - j9 \)
Example 2:
Determine the conjugate for \( z = -7 - j5 \).
Given: \( z = -7 - j5 \)
$$ z^* = -7 + j5 $$
Answer: \( z^* = -7 + j5 \)
Example 3:
A SACCO digital system records \( z = 12 + j0 \). Find the conjugate.
Given: \( z = 12 + j0 \)
$$ z^* = 12 - j0 = 12 $$
Answer: \( z^* = 12 \)
Example 4:
Find the conjugate of \( z = 0 + j8 \).
Given: \( z = 0 + j8 \)
$$ z^* = 0 - j8 = -j8 $$
Answer: \( z^* = -j8 \)
Example 5:
A university lab reports \( z = -3 + j4 \). State the conjugate.
Given: \( z = -3 + j4 \)
$$ z^* = -3 - j4 $$
Answer: \( z^* = -3 - j4 \)
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Create a free accountThis chapter introduced the concept of complex numbers, explaining their fundamental meaning and how they extend the real number system. Students learned how to state complex numbers in both standard and polar forms, as well as how to determine the modulus, which measures their magnitude. The chapter covered the graphical representation of complex numbers using the Argand diagram, allowing for a visual understanding of their position in the complex plane. Arithmetic operations such as addition, subtraction, multiplication, and division of complex numbers were thoroughly explored with step-by-step calculations. De Moivre's theorem was presented as a powerful tool for raising complex numbers to powers and extracting roots. Finally, the practical applications of complex numbers in engineering contexts were highlighted, showing their relevance in solving real-world problems involving electrical circuits, signal analysis, and structural computations.
Given: \( 7 - 4i \)
Step 1: Find modulus \( r \)$$ r = \sqrt{7^2 + (-4)^2} $$$$ r = \sqrt{49 + 16} $$$$ r = \sqrt{65} $$$$ r \approx 8.062 $$
Step 2: Find argument \( \theta \)$$ \theta = \arctan(\frac{-4}{7}) $$$$ \theta \approx \arctan(-0.571) $$$$ \theta \approx -29.74^\circ $$$$ \theta \approx -0.519 \text{ radians} $$
Step 3: Express in polar form$$ 7 - 4i = 8.062(\cos(-0.519) + i\sin(-0.519)) $$
Final Answer:
\(\boxed{8.062(\cos(-0.519) + i\sin(-0.519))}\)
Given: \( -5 + 12i \)
Step 1: Modulus$$ r = \sqrt{(-5)^2 + 12^2} $$$$ r = \sqrt{25 + 144} $$$$ r = \sqrt{169} $$$$ r = 13 $$
Step 2: Argument$$ \theta = \arctan(\frac{12}{-5}) $$$$ \theta = \arctan(-2.4) $$ Since the real part is negative and imaginary part is positive, angle is in 2nd quadrant:$$ \theta = 180^\circ + \arctan(\frac{12}{-5}) = 180^\circ - 67.38^\circ = 112.62^\circ $$$$ \theta \approx 1.967 \text{ radians} $$
Final Answer:
Modulus: \(\boxed{13}\)
Argument: \(\boxed{1.967 \text{ radians}}\)
A control panel at a Nairobi-based beverage factory uses an AC relay with an impedance of \(8 + 6i\) ohms. Calculate the modulus of the impedance and explain its significance in the control system. (4 marks)
Express the complex number \(7 - 24i\) in the form \(r(\cos\theta + i\sin\theta)\), giving \(r\) and \(\theta\) to two decimal places. (4 marks)
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