By the end of this chapter, you will be able to:
Mastering these skills will help you handle complex calculations with ease, making you a more effective and precise engineer in your trade.
Complex numbers are fundamental in automotive mechatronics, especially when dealing with alternating current circuits, signal processing, and control systems. Understanding complex numbers allows engineers to analyze and design systems involving oscillations, vibrations, and electromagnetic waves in vehicle electronics. This chapter introduces complex numbers, their representation, and their application in solving engineering problems relevant to automotive systems in Kenya.
Complex numbers extend the real numbers by including an imaginary unit \( j \) where \( j^2 = -1 \). They are expressed in the form \( z = a + jb \), where \( a \) and \( b \) are real numbers, \( a \) is the real part, and \( b \) is the imaginary part. This concept is critical in automotive mechatronics for analyzing AC circuits, motor control, and sensor signal processing.
A complex number is written as
$$ z = a + jb $$
where:
Example 1: Identify the real and imaginary parts of the complex number \( z = 5 + j3 \).
Given: \( z = 5 + j3 \)
Real part: \( \text{Re}(z) = 5 \)
Imaginary part: \( \text{Im}(z) = 3 \)
Answer: Real part = 5, Imaginary part = 3
Example 2: For the complex number \( z = -2 - j7 \), find the real and imaginary parts.
Given: \( z = -2 - j7 \)
Real part: \( \text{Re}(z) = -2 \)
Imaginary part: \( \text{Im}(z) = -7 \)
Answer: Real part = -2, Imaginary part = -7
Example 3: Determine the real and imaginary parts of \( z = 0 + j4 \).
Given: \( z = 0 + j4 \)
Real part: \( \text{Re}(z) = 0 \)
Imaginary part: \( \text{Im}(z) = 4 \)
Answer: Real part = 0, Imaginary part = 4
Example 4: Find the real and imaginary parts of \( z = 8 \) (a purely real number).
Given: \( z = 8 \)
Real part: \( \text{Re}(z) = 8 \)
Imaginary part: \( \text{Im}(z) = 0 \)
Answer: Real part = 8, Imaginary part = 0
Example 5: For \( z = -j5 \), express the complex number in standard form and find its parts.
Given: \( z = -j5 = 0 - j5 \)
Real part: \( \text{Re}(z) = 0 \)
Imaginary part: \( \text{Im}(z) = -5 \)
Answer: Real part = 0, Imaginary part = -5
Addition and subtraction of complex numbers involve combining their real and imaginary parts separately. For two complex numbers \( z_1 = a + jb \) and \( z_2 = c + jd \),
$$ z_1 + z_2 = (a + c) + j(b + d) $$
$$ z_1 - z_2 = (a, c) + j(b, d) $$
Example 1: Add \( z_1 = 3 + j4 \) and \( z_2 = 1 - j2 \).
Given: \( z_1 = 3 + j4 \), \( z_2 = 1 - j2 \)
$$ z_1 + z_2 = (3 + 1) + j(4 - 2) $$
$$ = 4 + j2 $$
Answer: 4 + j2
Example 2: Subtract \( z_2 = 5 + j3 \) from \( z_1 = 7 + j8 \).
Given: \( z_1 = 7 + j8 \), \( z_2 = 5 + j3 \)
$$ z_1 - z_2 = (7 - 5) + j(8 - 3) $$
$$ = 2 + j5 $$
Answer: 2 + j5
Example 3: Calculate \( (4 - j6) + (-2 + j3) \).
Given: \( z_1 = 4 - j6 \), \( z_2 = -2 + j3 \)
$$ z_1 + z_2 = (4 - 2) + j(-6 + 3) $$
$$ = 2 - j3 $$
Answer: 2 - j3
Example 4: Find \( (6 + j7) - (3 + j10) \).
Given: \( z_1 = 6 + j7 \), \( z_2 = 3 + j10 \)
$$ z_1 - z_2 = (6 - 3) + j(7 - 10) $$
$$ = 3 - j3 $$
Answer: 3 - j3
Example 5: Add \( z_1 = -4 + j5 \) and \( z_2 = 4 - j5 \).
Given: \( z_1 = -4 + j5 \), \( z_2 = 4 - j5 \)
$$ z_1 + z_2 = (-4 + 4) + j(5 - 5) $$
$$ = 0 + j0 $$
Answer: 0
To multiply two complex numbers \( z_1 = a + jb \) and \( z_2 = c + jd \), apply the distributive property and use \( j^2 = -1 \):
$$ z_1 \times z_2 = (a + jb)(c + jd) = ac + ajd + jbc + j^2 bd = (ac, bd) + j(ad + bc) $$
Example 1: Multiply \( z_1 = 2 + j3 \) and \( z_2 = 1 + j4 \).
Given: \( z_1 = 2 + j3 \), \( z_2 = 1 + j4 \)
$$ z_1 \times z_2 = (2 \times 1 - 3 \times 4) + j(2 \times 4 + 3 \times 1) $$
$$ = (2 - 12) + j(8 + 3) $$
$$ = -10 + j11 $$
Answer: -10 + j11
Example 2: Multiply \( z_1 = 3 - j2 \) and \( z_2 = 4 + j5 \).
Given: \( z_1 = 3 - j2 \), \( z_2 = 4 + j5 \)
$$ z_1 \times z_2 = (3 \times 4 - (-2) \times 5) + j(3 \times 5 + (-2) \times 4) $$
$$ = (12 + 10) + j(15 - 8) $$
$$ = 22 + j7 $$
Answer: 22 + j7
Example 3: Multiply \( z_1 = -1 + j6 \) and \( z_2 = 2 - j3 \).
Given: \( z_1 = -1 + j6 \), \( z_2 = 2 - j3 \)
$$ z_1 \times z_2 = (-1 \times 2 - 6 \times (-3)) + j(-1 \times (-3) + 6 \times 2) $$
$$ = (-2 + 18) + j(3 + 12) $$
$$ = 16 + j15 $$
Answer: 16 + j15
Example 4: Multiply \( z_1 = 0 + j7 \) and \( z_2 = -4 + j0 \).
Given: \( z_1 = j7 \), \( z_2 = -4 \)
$$ z_1 \times z_2 = (0 \times -4 - 7 \times 0) + j(0 \times 0 + 7 \times -4) $$
$$ = 0 + j(-28) $$
$$ = -j28 $$
Answer: -j28
Example 5: Multiply \( z_1 = 5 + j0 \) and \( z_2 = 0 + j3 \).
Given: \( z_1 = 5 \), \( z_2 = j3 \)
$$ z_1 \times z_2 = (5 \times 0 - 0 \times 3) + j(5 \times 3 + 0 \times 0) $$
$$ = 0 + j15 $$
Answer: j15
To divide \( z_1 = a + jb \) by \( z_2 = c + jd \), multiply numerator and denominator by the conjugate of the denominator \( c, jd \):
$$ \frac{z_1}{z_2} = \frac{a + jb}{c + jd} \times \frac{c, jd}{c, jd} = \frac{(a + jb)(c, jd)}{c^2 + d^2} = \frac{(ac + bd) + j(bc, ad)}{c^2 + d^2} $$
Example 1: Divide \( z_1 = 3 + j4 \) by \( z_2 = 1 - j2 \).
Given: \( z_1 = 3 + j4 \), \( z_2 = 1 - j2 \)
$$ c^2 + d^2 = 1^2 + (-2)^2 = 1 + 4 = 5 $$
$$ \text{Numerator} = (3 \times 1 + 4 \times (-2)) + j(4 \times 1 - 3 \times (-2)) = (3 - 8) + j(4 + 6) = -5 + j10 $$
$$ \frac{z_1}{z_2} = \frac{-5 + j10}{5} = -1 + j2 $$
Answer: -1 + j2
Example 2: Divide \( z_1 = 5 + j6 \) by \( z_2 = 2 + j3 \).
Given: \( z_1 = 5 + j6 \), \( z_2 = 2 + j3 \)
$$ c^2 + d^2 = 2^2 + 3^2 = 4 + 9 = 13 $$
$$ \text{Numerator} = (5 \times 2 + 6 \times 3) + j(6 \times 2 - 5 \times 3) = (10 + 18) + j(12 - 15) = 28 - j3 $$
$$ \frac{z_1}{z_2} = \frac{28 - j3}{13} = \frac{28}{13} - j\frac{3}{13} $$
$$ = 2.154 - j0.231 $$
Answer: 2.154 - j0.231
Example 3: Divide \( z_1 = -1 + j7 \) by \( z_2 = 3 - j4 \).
Given: \( z_1 = -1 + j7 \), \( z_2 = 3 - j4 \)
$$ c^2 + d^2 = 3^2 + (-4)^2 = 9 + 16 = 25 $$
$$ \text{Numerator} = (-1 \times 3 + 7 \times (-4)) + j(7 \times 3 - (-1) \times (-4)) = (-3 - 28) + j(21 - 4) = -31 + j17 $$
$$ \frac{z_1}{z_2} = \frac{-31 + j17}{25} = -1.24 + j0.68 $$
Answer: -1.24 + j0.68
Example 4: Divide \( z_1 = 4 - j5 \) by \( z_2 = 1 + j1 \).
Given: \( z_1 = 4 - j5 \), \( z_2 = 1 + j1 \)
$$ c^2 + d^2 = 1^2 + 1^2 = 2 $$
$$ \text{Numerator} = (4 \times 1 + (-5) \times 1) + j((-5) \times 1 - 4 \times 1) = (4 - 5) + j(-5 - 4) = -1 - j9 $$
$$ \frac{z_1}{z_2} = \frac{-1 - j9}{2} = -0.5 - j4.5 $$
Answer: -0.5 - j4.5
Example 5: Divide \( z_1 = 6 + j0 \) by \( z_2 = 0 + j3 \).
Given: \( z_1 = 6 \), \( z_2 = j3 \)
$$ c^2 + d^2 = 0^2 + 3^2 = 9 $$
$$ \text{Numerator} = (6 \times 0 + 0 \times 3) + j(0 \times 0 - 6 \times 3) = 0 - j18 $$
$$ \frac{z_1}{z_2} = \frac{-j18}{9} = -j2 $$
Answer: -j2
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Create a free accountThis chapter introduced the concept of complex numbers, defining them as numbers comprising a real part and an imaginary part. It explored expressing complex numbers using their conjugates, arguments, and moduli, providing a foundation for understanding their geometric and algebraic properties. The representation of complex numbers on the Argand diagram was examined, illustrating how they can be plotted as points or vectors in a plane. Arithmetic operations involving complex numbers, including addition, subtraction, multiplication, and division, were applied and explained in detail. The chapter also covered the use of De Moivre's theorem to simplify the computation of powers and roots of complex numbers. Finally, the practical applications of complex numbers in engineering contexts were discussed, highlighting their importance in solving real-world problems. This comprehensive approach ensures a thorough grasp of both the theory and practical utility of complex numbers in engineering mathematics.
Find the modulus and argument of the complex number \(z = 3 + 4i\). (2 marks)
Given the complex number \(z = -1 + \sqrt{3}i\), find its conjugate and calculate the product \(z \times \overline{z}\). (3 marks)
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