Complex numbers are essential in electronics engineering, particularly in analyzing alternating current (AC) circuits, signal processing, and control systems. Their representation on the Argand diagram provides a powerful visual tool for understanding the magnitude and phase relationships of electrical quantities. This chapter focuses on mastering Argand diagrams to represent complex numbers graphically, enabling effective problem-solving in practical electronics engineering tasks encountered in Kenyan industries.
Argand diagrams graphically represent complex numbers as points or vectors in a two-dimensional plane, where the horizontal axis is the real part and the vertical axis is the imaginary part. This visual approach is invaluable in electronics engineering for analyzing impedance, voltage, and current phasors. Kenyan electronics technicians use Argand diagrams extensively when working on AC power systems, filter design, and signal analysis.
Complex numbers have the form \( z = x + jy \), where \( x \) is the real part and \( y \) is the imaginary part. On the Argand diagram, \( x \) is plotted along the horizontal (real) axis, and \( y \) along the vertical (imaginary) axis. The point \( (x, y) \) represents the complex number \( z \).
The magnitude \( r \) and angle \( \theta \) of the complex number, known as polar form, correspond to the vector length and angle from the positive real axis, respectively.
The polar form is given by:
$$ r = \sqrt{x^{2} + y^{2}} $$
$$ \theta = \tan^{-1}(\frac{y}{x}) $$
Example 1: A signal voltage is represented by the complex number \( z = 3 + j4 \) volts. Find its magnitude and angle.
Given: \( x = 3 \, V \), \( y = 4 \, V \)
$$ r = \sqrt{3^{2} + 4^{2}} $$
$$ r = \sqrt{9 + 16} $$
$$ r = \sqrt{25} = 5 \, V $$
$$ \theta = \tan^{-1}(\frac{4}{3}) $$
$$ \theta = \tan^{-1}(1.333) = 53.13^\circ $$
Answer: Magnitude \( r = 5 \, V \), Angle \( \theta = 53.13^\circ \)
Example 2: For the complex current \( I = -2 + j2 \, A \), determine the magnitude and phase angle.
Given: \( x = -2 \, A \), \( y = 2 \, A \)
$$ r = \sqrt{(-2)^{2} + 2^{2}} $$
$$ r = \sqrt{4 + 4} = \sqrt{8} = 2.828 \, A $$
$$ \theta = \tan^{-1}(\frac{2}{-2}) $$
$$ \theta = \tan^{-1}(-1) = -45^\circ $$
Since \( x < 0 \) and \( y > 0 \), angle is in the second quadrant:
$$ \theta = 180^\circ - 45^\circ = 135^\circ $$
Answer: Magnitude \( r = 2.828 \, A \), Angle \( \theta = 135^\circ \)
Example 3: Represent the impedance \( Z = 5 - j12 \, \Omega \) on the Argand diagram by finding its magnitude and phase angle.
Given: \( x = 5 \, \Omega \), \( y = -12 \, \Omega \)
$$ r = \sqrt{5^{2} + (-12)^{2}} $$
$$ r = \sqrt{25 + 144} = \sqrt{169} = 13 \, \Omega $$
$$ \theta = \tan^{-1}(\frac{-12}{5}) $$
$$ \theta = \tan^{-1}(-2.4) = -67.38^\circ $$
Answer: Magnitude \( r = 13 \, \Omega \), Angle \( \theta = -67.38^\circ \)
Example 4: An AC voltage is \( V = -7 - j24 \, V \). Calculate its magnitude and phase angle.
Given: \( x = -7 \, V \), \( y = -24 \, V \)
$$ r = \sqrt{(-7)^{2} + (-24)^{2}} $$
$$ r = \sqrt{49 + 576} = \sqrt{625} = 25 \, V $$
$$ \theta = \tan^{-1}(\frac{-24}{-7}) $$
$$ \theta = \tan^{-1}(3.4286) = 73.74^\circ $$
Since both \( x < 0 \), \( y < 0 \), angle is in the third quadrant:
$$ \theta = 180^\circ + 73.74^\circ = 253.74^\circ $$
Answer: Magnitude \( r = 25 \, V \), Angle \( \theta = 253.74^\circ \)
Example 5: For the complex number \( z = 0 + j10 \), find the magnitude and angle.
Given: \( x = 0 \), \( y = 10 \)
$$ r = \sqrt{0^{2} + 10^{2}} = 10 $$
Angle is:
$$ \theta = \tan^{-1}(\frac{10}{0}) $$
Since division by zero is undefined and \( y > 0 \),
$$ \theta = 90^\circ $$
Answer: Magnitude \( r = 10 \), Angle \( \theta = 90^\circ \)
Plotting complex numbers on the Argand diagram involves marking the point corresponding to its real and imaginary parts or drawing the vector from the origin to that point. This visual representation aids in understanding phase relations in AC circuits and impedance matching.
Example 1: Plot the complex number \( z = 4 + j3 \) on an Argand diagram.
Given: \( x = 4 \), \( y = 3 \)
Answer: Point plotted at (4,3) with vector representing \( z = 4 + j3 \).
Example 2: Plot \( I = -3 + j5 \) representing current phasor in amperes.
Given: \( x = -3 \), \( y = 5 \)
Answer: Vector in the second quadrant representing \( I = -3 + j5 \, A \).
Example 3: Plot impedance \( Z = 6 - j8 \, \Omega \).
Given: \( x = 6 \), \( y = -8 \)
Answer: Vector in the fourth quadrant representing \( Z = 6 - j8 \, \Omega \).
Example 4: Plot \( z = -5 - j5 \).
Given: \( x = -5 \), \( y = -5 \)
Answer: Vector in the third quadrant representing \( z = -5 - j5 \).
Example 5: Plot \( z = 0 + j7 \).
Given: \( x = 0 \), \( y = 7 \)
Answer: Vector on positive imaginary axis representing \( z = j7 \).
In electronics engineering, converting complex numbers between rectangular form \( (x + jy) \) and polar form \( (r \angle \theta) \) is essential for circuit analysis and signal representation. The Argand diagram facilitates this conversion by interpreting the complex number as a vector with magnitude and angle.
Conversion formulas:
From rectangular to polar:
$$ r = \sqrt{x^{2} + y^{2}} $$
$$ \theta = \tan^{-1}(\frac{y}{x}) $$
From polar to rectangular:
$$ x = r \cos \theta $$
$$ y = r \sin \theta $$
Example 1: Convert \( z = 3 + j4 \) to polar form.
Given: \( x = 3 \), \( y = 4 \)
$$ r = \sqrt{3^{2} + 4^{2}} = 5 $$
$$ \theta = \tan^{-1}(\frac{4}{3}) = 53.13^\circ $$
Answer: \( z = 5 \angle 53.13^\circ \)
Example 2: Convert \( z = 10 \angle 30^\circ \) to rectangular form.
Given: \( r = 10 \), \( \theta = 30^\circ \)
$$ x = 10 \cos 30^\circ = 10 \times 0.866 = 8.66 $$
$$ y = 10 \sin 30^\circ = 10 \times 0.5 = 5 $$
Answer: \( z = 8.66 + j5 \)
Example 3: Convert \( z = -5 + j12 \) to polar form.
Given: \( x = -5 \), \( y = 12 \)
$$ r = \sqrt{(-5)^{2} + 12^{2}} = \sqrt{25 + 144} = \sqrt{169} = 13 $$
$$ \theta = \tan^{-1}(\frac{12}{-5}) = \tan^{-1}(-2.4) = -67.38^\circ $$
Since \( x < 0 \), \( y > 0 \), angle is in second quadrant:
$$ \theta = 180^\circ - 67.38^\circ = 112.62^\circ $$
Answer: \( z = 13 \angle 112.62^\circ \)
Example 4: Convert \( z = 20 \angle 210^\circ \) to rectangular form.
Given: \( r = 20 \), \( \theta = 210^\circ \)
$$ x = 20 \cos 210^\circ = 20 \times (-0.866) = -17.32 $$
$$ y = 20 \sin 210^\circ = 20 \times (-0.5) = -10 $$
Answer: \( z = -17.32 - j10 \)
Example 5: Convert \( z = 0 + j15 \) to polar form.
Given: \( x = 0 \), \( y = 15 \)
$$ r = \sqrt{0^{2} + 15^{2}} = 15 $$
Angle:
Since \( x = 0 \) and \( y > 0 \),
$$ \theta = 90^\circ $$
Answer: \( z = 15 \angle 90^\circ \)
In electronics, adding and subtracting complex numbers corresponds to combining phasors representing voltages or currents. Graphically, this is vector addition or subtraction on the Argand diagram, which helps visualize resultant magnitudes and phase shifts.
Addition and subtraction in rectangular form:
$$ z_1 = x_1 + j y_1, \quad z_2 = x_2 + j y_2 $$
$$ z_1 + z_2 = (x_1 + x_2) + j(y_1 + y_2) $$
$$ z_1 - z_2 = (x_1 - x_2) + j(y_1 - y_2) $$
Example 1: Add \( z_1 = 4 + j3 \) and \( z_2 = 1 + j2 \).
Given: \( z_1 = 4 + j3 \), \( z_2 = 1 + j2 \)
$$ z = (4 + 1) + j(3 + 2) = 5 + j5 $$
Calculate magnitude and angle:
$$ r = \sqrt{5^{2} + 5^{2}} = \sqrt{25 + 25} = \sqrt{50} = 7.07 $$
$$ \theta = \tan^{-1}(\frac{5}{5}) = 45^\circ $$
Answer: \( z = 5 + j5 \), magnitude \( 7.07 \), angle \( 45^\circ \)
Example 2: Subtract \( z_2 = 3 + j4 \) from \( z_1 = 7 + j1 \).
Given: \( z_1 = 7 + j1 \), \( z_2 = 3 + j4 \)
$$ z = (7 - 3) + j(1 - 4) = 4 - j3 $$
Calculate magnitude and angle:
$$ r = \sqrt{4^{2} + (-3)^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5 $$
$$ \theta = \tan^{-1}(\frac{-3}{4}) = -36.87^\circ $$
Answer: \( z = 4 - j3 \), magnitude \( 5 \), angle \( -36.87^\circ \)
Example 3: Add \( z_1 = -2 + j5 \) and \( z_2 = 4 - j3 \).
Given: \( z_1 = -2 + j5 \), \( z_2 = 4 - j3 \)
$$ z = (-2 + 4) + j(5 - 3) = 2 + j2 $$
Calculate magnitude and angle:
$$ r = \sqrt{2^{2} + 2^{2}} = \sqrt{4 + 4} = \sqrt{8} = 2.828 $$
$$ \theta = \tan^{-1}(\frac{2}{2}) = 45^\circ $$
Answer: \( z = 2 + j2 \), magnitude \( 2.828 \), angle \( 45^\circ \)
Example 4: Subtract \( z_2 = -3 + j7 \) from \( z_1 = 1 - j4 \).
Given: \( z_1 = 1 - j4 \), \( z_2 = -3 + j7 \)
$$ z = (1 - (-3)) + j(-4 - 7) = 4 - j11 $$
Calculate magnitude and angle:
$$ r = \sqrt{4^{2} + (-11)^{2}} = \sqrt{16 + 121} = \sqrt{137} = 11.705 $$
$$ \theta = \tan^{-1}(\frac{-11}{4}) = -70.02^\circ $$
Answer: \( z = 4 - j11 \), magnitude \( 11.705 \), angle \( -70.02^\circ \)
Example 5: Add \( z_1 = 0 + j6 \) and \( z_2 = 8 + j0 \).
Given: \( z_1 = j6 \), \( z_2 = 8 \)
$$ z = (0 + 8) + j(6 + 0) = 8 + j6 $$
Calculate magnitude and angle:
$$ r = \sqrt{8^{2} + 6^{2}} = \sqrt{64 + 36} = \sqrt{100} = 10 $$
$$ \theta = \tan^{-1}(\frac{6}{8}) = 36.87^\circ $$
Answer: \( z = 8 + j6 \), magnitude \( 10 \), angle \( 36.87^\circ \)
Find the magnitude and angle of the complex impedance \( Z = 8 + j15 \, \Omega \). (5 marks)
Convert the complex current \( I = 12 \angle 120^\circ \, A \) to rectangular form. (5 marks)
Plot the complex voltage \( V = -6 + j8 \, V \) on an Argand diagram and determine its magnitude and phase angle. (6 marks)
Add the phasors \( V_1 = 5 + j12 \, V \) and \( V_2 = -3 + j4 \, V \), then find the magnitude and angle of the resultant voltage. (6 marks)
Subtract the current phasor \( I_2 = 7 - j24 \, A \) from \( I_1 = 10 + j10 \, A \) and determine the magnitude and phase angle of the difference. (6 marks)
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Create a free accountThis chapter introduced the representation of complex numbers on Argand diagrams, where each complex number is shown as a point or vector in a plane defined by real and imaginary axes. It explained how to plot complex numbers and interpret their geometric properties such as magnitude and direction. The chapter then covered the fundamental operations on complex numbers including addition, subtraction, multiplication, and division, demonstrating how these operations combine algebraic manipulation with 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 form and exponential expressions of complex numbers. Throughout, the chapter emphasized the practical applications of these concepts in solving engineering problems involving oscillations, waves, and alternating currents. The combination of graphical and algebraic methods provided a comprehensive understanding of complex number manipulation.