By the end of this chapter, you will be able to:
Mastering these skills will give you a strong foundation for solving real-world electronics problems that involve complex numbers.
Complex numbers are fundamental in electronics engineering for analysing alternating current (AC) circuits, signal processing, and control systems. Representing complex numbers graphically on the Argand diagram provides intuitive insight into their magnitude and phase relationships, which are critical in designing and troubleshooting electronic circuits in Kenya. This chapter focuses on mastering Argand diagrams to visualize complex quantities effectively, preparing students for advanced applications such as impedance calculations and phasor analysis.
Argand diagrams are graphical representations of complex numbers on a two-dimensional plane, where the horizontal axis represents the real part and the vertical axis the imaginary part. In Kenyan electronics engineering, they are extensively used to interpret AC signals, visualize impedance, and analyze phase differences in circuits. Mastery of Argand diagrams enables engineers to simplify complex calculations and improve circuit design accuracy.
Complex numbers can be represented as points or vectors on the Argand plane, where the x-axis corresponds to the real part and the y-axis to the imaginary part. The position of the point or vector defines the complex number uniquely, with its length indicating magnitude and the angle with the positive x-axis representing the phase.
$$z = x + jy$$
where \(x\) is the real part and \(y\) is the imaginary part.
Example 1: Plot the complex number \(z = 3 + j4\) on the Argand diagram.
Given: \(x = 3\), \(y = 4\)
$$z = 3 + j4$$
Answer: The point is located 3 units right and 4 units up from the origin.
Example 2: Plot \(z = -2 + j5\).
Given: \(x = -2\), \(y = 5\)
$$z = -2 + j5$$
Answer: The point is 2 units left and 5 units up from the origin.
Example 3: Plot \(z = -3 - j4\).
Given: \(x = -3\), \(y = -4\)
$$z = -3 - j4$$
Answer: The point is 3 units left and 4 units down from the origin.
The magnitude (or modulus) of a complex number is its distance from the origin on the Argand diagram, found using Pythagoras' theorem. The argument (or angle) is the angle between the positive real axis and the line representing the complex number, measured in radians or degrees.
$$|z| = \sqrt{x^{2} + y^{2}}$$
$$\theta = \tan^{-1}(\frac{y}{x})$$
Example 1: Find the magnitude and argument of \(z = 3 + j4\).
Given: \(x = 3\), \(y = 4\)
$$|z| = \sqrt{3^{2} + 4^{2}}$$
$$= \sqrt{9 + 16}$$
$$= \sqrt{25} = 5$$
$$\theta = \tan^{-1}(\frac{4}{3})$$
$$= \tan^{-1}(1.3333) = 53.13^\circ$$
Answer: Magnitude = 5, Argument = 53.13°
Example 2: Find the magnitude and argument of \(z = -1 + j\sqrt{3}\).
Given: \(x = -1\), \(y = \sqrt{3}\)
$$|z| = \sqrt{(-1)^{2} + (\sqrt{3})^{2}}$$
$$= \sqrt{1 + 3} = \sqrt{4} = 2$$
$$\theta = \tan^{-1}(\frac{\sqrt{3}}{-1})$$
$$= \tan^{-1}(-\sqrt{3}) = -60^\circ$$
Since \(x\) is negative and \(y\) positive, the complex number lies in the second quadrant:
$$\theta = 180^\circ - 60^\circ = 120^\circ$$
Answer: Magnitude = 2, Argument = 120°
Example 3: Find magnitude and argument of \(z = -4 - j4\).
Given: \(x = -4\), \(y = -4\)
$$|z| = \sqrt{(-4)^{2} + (-4)^{2}}$$
$$= \sqrt{16 + 16} = \sqrt{32} = 5.657$$
$$\theta = \tan^{-1}(\frac{-4}{-4}) = \tan^{-1}(1) = 45^\circ$$
Since both \(x\) and \(y\) are negative, the complex number lies in the third quadrant:
$$\theta = 180^\circ + 45^\circ = 225^\circ$$
Answer: Magnitude = 5.657, Argument = 225°
Complex numbers are often converted between rectangular form \(x + jy\) and polar form \(r \angle \theta\) for ease of calculation in electronics. Polar form expresses the complex number by its magnitude and angle, which is useful for multiplication and division of complex numbers.
Rectangular to polar:
$$r = \sqrt{x^{2} + y^{2}}$$
$$\theta = \tan^{-1}(\frac{y}{x})$$
Polar to rectangular:
$$x = r \cos \theta$$
$$y = r \sin \theta$$
Example 1: Convert \(z = 1 + j1\) to polar form.
Given: \(x = 1\), \(y = 1\)
$$r = \sqrt{1^{2} + 1^{2}} = \sqrt{2} = 1.414$$
$$\theta = \tan^{-1}(\frac{1}{1}) = 45^\circ$$
Answer: \(z = 1.414 \angle 45^\circ\)
Example 2: Convert \(z = 5 \angle 60^\circ\) to rectangular form.
Given: \(r = 5\), \(\theta = 60^\circ\)
$$x = 5 \cos 60^\circ = 5 \times 0.5 = 2.5$$
$$y = 5 \sin 60^\circ = 5 \times 0.8660 = 4.33$$
Answer: \(z = 2.5 + j4.33\)
Example 3: Convert \(z = -3 + j3\sqrt{3}\) to polar form.
Given: \(x = -3\), \(y = 3\sqrt{3} = 5.196\)
$$r = \sqrt{(-3)^{2} + (5.196)^{2}} = \sqrt{9 + 27} = \sqrt{36} = 6$$
$$\theta = \tan^{-1}(\frac{5.196}{-3}) = \tan^{-1}(-1.732) = -60^\circ$$
Since \(x < 0\) and \(y > 0\), the point lies in the second quadrant:
$$\theta = 180^\circ - 60^\circ = 120^\circ$$
Answer: \(z = 6 \angle 120^\circ\)
Addition and subtraction of complex numbers can be visualized graphically on the Argand diagram by vector addition or subtraction. This is important in electronics when combining signals or impedances.
Given two complex numbers:
$$z_1 = x_1 + jy_1$$
$$z_2 = x_2 + jy_2$$
Their sum:
$$z = (x_1 + x_2) + j(y_1 + y_2)$$
Their difference:
$$z = (x_1 - x_2) + j(y_1 - y_2)$$
Example 1: Add \(z_1 = 3 + j4\) and \(z_2 = 1 + j2\).
Given: \(z_1 = 3 + j4\), \(z_2 = 1 + j2\)
$$z = (3 + 1) + j(4 + 2)$$
$$= 4 + j6$$
Answer: \(z = 4 + j6\)
Example 2: Subtract \(z_2 = 2 + j3\) from \(z_1 = 5 + j7\).
Given: \(z_1 = 5 + j7\), \(z_2 = 2 + j3\)
$$z = (5 - 2) + j(7 - 3)$$
$$= 3 + j4$$
Answer: \(z = 3 + j4\)
Example 3: Add \(z_1 = -3 + j4\) and \(z_2 = 5 - j2\).
Given: \(z_1 = -3 + j4\), \(z_2 = 5 - j2\)
$$z = (-3 + 5) + j(4 - 2)$$
$$= 2 + j2$$
Answer: \(z = 2 + j2\)
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Create a free accountThis chapter introduced the graphical representation of complex numbers using Argand diagrams, which plot complex numbers as points in a plane with real and imaginary axes. It then covered fundamental operations on complex numbers including addition, subtraction, multiplication, and division, demonstrating how these operations are performed algebraically. The chapter also explained the concept of the modulus and argument of a complex number, essential for understanding their geometric interpretation. Finally, De Moivre's theorem was presented as a powerful tool for raising complex numbers to integer powers and extracting roots, linking trigonometric functions with complex number operations. Through these topics, students learned both the theoretical and practical methods for manipulating complex numbers, which are crucial in electronics engineering calculations. The chapter emphasized step-by-step calculations to build proficiency in applying these concepts to real-world engineering problems.
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