Complex Numbers in FE Electrical and Computer
If you are a recent college graduate excited to attempt your FE electrical and computer exam to begin a promising career, Complex Numbers in FE Electrical and Computer will be your primary tool. For instance, You’re handed a complex circuit to analyze and optimize, and you quickly realize that this task is no easy feat. As you pore over the complex equations and intricate diagrams, you can’t help but wonder: is there a better way to approach this problem?
This is where complex numbers applications make your job easier. Complex Numbers in FE Electrical and Computer are the secret weapon of engineers. From a straightforward definition of complex numbers theory, these numbers take the form a + bi, where a and b are real numbers, and i is the imaginary unit (i.e., the square root of -1). The idea may appear abstract at first glance. However, advanced complex numbers are the key to unlocking the power of circuit analysis, signal processing, control theory, and many other modern applications.
With complex numbers applications, circuit analysis becomes a breeze. Instead of working with cumbersome trigonometric functions and vectors, you can learn and master Complex Numbers in FE Electrical and Computer exams to use complex impedance and admittance to simplify your calculations and gain deeper insights into the circuits’ behaviour and patterns.
What are complex numbers used for? To your surprise, Complex numbers are also used in Fourier analysis and signal processing, allowing engineers to quickly analyze and manipulate complex waveforms. And in the field of control theory, complex analysis is indispensable for designing stable, robust control systems that can handle the challenges of the modern world.
If you look forward to creating a firm grip on Complex Numbers in FE Electrical and Computer exam preparation, you landed at the right place. Join me as we dive deeper into this fascinating topic and explore how complex numbers shape this tech-driven world.
Basic Concepts of Complex Numbers
The Complex Numbers in FE Electrical and Computer involves a lot of calculation. Before moving forward to the more technical concepts and complex numbers applications it is mandatory to discuss how these numbers are manipulated in simple mathematical calculations and the core mathematical concepts behind them. Let’s dicuss this in detail.
1. The Imaginary Unit
Understanding imaginary numbers is vital to understand the complex number system.
Let’s consider this analogy that I often tell my students to help them understand the concept.
A group of detectives investigating a crime scene. As they comb through the evidence, they come across a room with a peculiar trait: anyone who enters the room wearing a white shirt comes out wearing a black shirt or if someone enters the room wearing a black short comes out wearing a black shirt too. As they try to piece together what happened, they realize that they can’t determine the original color of the person’s shirt just by looking at them coming out of the room. After all, the black shirt could have been the original shirt, or it could have been the result of the room’s mysterious transformation.
In mathematics, a similar phenomenon occurs with the square root of a positive number. Just like the detectives couldn’t determine the original color of the shirt, we can’t determine whether the square root of a positive number is positive or negative. That’s why we use both the + and – signs when taking the square root of a positive number, to account for both possibilities.
But what about the square root of a negative number? As no person can come out of the room wearing a white shirt, how it could be possible for someone to claim he/she came out of that room wearing a white shirt? Just like the detectives needed to think outside the box to solve the mystery of the room, mathematicians had to do the same when dealing with imaginary numbers.
Square root is the reverse of square. It means sending a person wearing black shirt back to the room to determine which shirt he/she is wearing. But you can’t send back a person wearing white shirt as he/she have not been there and its an imagination in that particular case. Similarly, The square root of a negative number can’t be a real number, so we use imaginary numbers to represent the square root of a negative number.
So the next time you see the + and – sign in front of the square root of a positive number or the letter “i” in front of the square root of a negative number, remember the detective story. Just as the colour of a shirt could be transformed by a mysterious room, the square has a tendency to change the sign of a number to positive irrespective of its original sign before the squaring took place. So taking the square root of a negative number which is not a square of any number in the first place, is dealt as an imaginary number.
2. Real and Imaginary Parts of Complex Numbers
In complex numbers theory, these numbers are the resultant of real and imaginary numbers, the real part refers to the coefficient of the real unit, and the imaginary part refers to the coefficient of the imaginary unit i. The imaginary unit is defined as the square root of -1.
A complex number is expressed as Z = a + bi, where a and b are real constants. The real part is a, and the imaginary part is bi. The real part represents the complex plane’s horizontal axis (x-axis), while the imaginary part represents the vertical axis (y-axis).
For example, let’s take the complex number Z = 3 + 2i. Here, the real part is 3, and the imaginary part is 2.
3. Modulus and Argument of Complex Numbers
We can also find the magnitude (Modulus) and Argument/Phase (Angle) of Z using its real and imaginary parts. The magnitude of a complex number is given by the square root of the sum of the squares of its real and imaginary parts. For Z = 3 + 2i, the magnitude is found simply as vector magnitude:
The phase of Z is the angle it makes with the positive x-axis (Real axis) on the complex plane, measured counterclockwise. It is given by vector calculation like Vectors (Tangent of the ratio of perpendicular to base) just replacing perpendicular with complex part and base with real part:
, and its phase is approximate 𝜃=0.588 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 𝑜𝑟 33.69°
counterclockwise from the positive x-axis.
4. The Polar Form of Complex Numbers
The polar form of a complex number is the exponential form of a complex number.
Express the complex number in polar form as:
Where |z| is the magnitude or modulus of the complex number, 𝜃 is the phase or angle of the complex number, e is Euler’s number or exponential function (approximately equal to 2.71828), and 𝑖 is the imaginary unit.
Operations with Complex Numbers
Complex numbers can be added, subtracted, multiplied, and divided like real numbers. However, because complex numbers have real and imaginary parts, we must be careful when performing these operations.
Addition and Subtraction
To add or subtract complex numbers, we simply add or subtract their real and imaginary parts separately. For example, let’s say we want to add Z = 2+3i to another complex number W = 4-2i:
𝑍 + 𝑊 = (2 + 3𝑖) + (4 − 2)𝑖=6 + 𝑖
Similarly, to subtract Z from W, we can do:
𝑊 − 𝑍 = (4 − 2𝑖) − (2 + 3𝑖) = 2 − 5𝑖
The complex conjugate of a complex number is simply the same as the original number, but with the sign of its imaginary part flipped. We denote the complex conjugate of Z as Z*, so for example
𝑍∗ = 2 − 3𝑖
The complex conjugate is useful when performing certain operations with complex numbers, such as division, as shown in the coming sections.
Rationalize the Denominator of Complex Numbers
Sometimes, we want to simplify a fraction that has a complex number in its denominator. For example, let’s say we want to simplify the fraction 1 / (2+3𝑖), the solution will be found by multiplying and dividing it by its complex conjugate.
Multiplication and Division
To multiply or divide complex numbers, we use the distributive property and the fact that 𝑖2 =−1. For example, to multiply Z and W:
𝑍 x 𝑊 = (2 + 3𝑖) x (4 − 2𝑖) = 8 + 12𝑖 − 4𝑖 − 6𝑖2 = 14 + 8𝑖
To divide Z by W, we first need to find the complex conjugate of W, which is simply the same as W but with the sign of its imaginary part flipped:
𝑊∗ = 4 + 2𝑖
To divide Z by W, we first need to find the complex conjugate of W, which is simply the same as W but with the sign of its imaginary part flipped.
Now, we can use the fact that Z x Z∗ = |Z|2 for any complex number Z to simplify the division or (𝑎 + 𝑏) (𝑎 − 𝑏) = 𝑎2 − 𝑏2
Powers and Roots of Complex Numbers
They also have powers and roots, just like real numbers. To find the power of a complex number, we use the fact that (𝑎 + 𝑏𝑖)𝑛 = 𝑎𝑛 + 𝑛𝑎(𝑛−1) 𝑏𝑖 − 𝑛(𝑛−1) / 2! 𝑎(𝑛−2) 𝑏2…, where n is a non-negative integer.
For example, to find (2 + 3𝑖)3
(2 + 3𝑖)3 = 23 + 3×22×3𝑖 − 3×2×32 − 27𝑖 = − 46 − 9𝑖
Or you can also expand it by a cubic formula (𝑎 + 𝑏)3 = 𝑎3 − 3𝑎2𝑏 + 3𝑎𝑏2 + 𝑏3 with every
𝑖2 = −1 ( breaking 𝑖𝑂𝐷𝐷 = 𝑖1𝑖𝐸𝑉𝐸𝑁 = 𝑖1 x (𝑖2)𝑛 )
Finding the roots of a complex number is a bit more complicated. To find the nth roots of a complex number 𝑧 , we first find its modulus |𝑧| and argument 𝜃 using the formulas:
The nth root is found by:
Where K is an integer between 0 and n-1 (for phase shift of an angle in respective planes)
Then, the two square roots of Z are:
Applications of Complex Numbers in Electrical and Computer Engineering
Now the real question arrives, what are complex numbers used for and what is the roadmap and area to study the application and usage of Complex Numbers in FE Electrical and Computer preparation. Let’s discuss each aspect one by one.
Circuit Analysis using Complex Numbers
Complex numbers are used extensively in circuit analysis for calculating voltage, current, and power. The voltage and current in a circuit are typically represented as complex numbers. The use of complex numbers enables the representation of phasors, the sinusoidal voltage, and current signals in a circuit. The complex representation of the phasors simplifies the analysis of circuits, especially those containing reactive components such as inductors and capacitors.
Complex numbers are used to analyze and design circuits, especially AC signals. For example, in a series RL circuit with a voltage source, the current can be described by a complex number expression. The real part represents the resistive component, and the imaginary part represents the reactive component. The impedance of the circuit can also be expressed as a complex number, where the magnitude represents the magnitude of the impedance and the angle represents the phase shift between the voltage and current. We can easily calculate and analyze the circuit’s behavior using complex numbers to represent a circuit’s voltage, current, and impedance.
Consider we have a series RL circuit with a voltage source of amplitude V and frequency ω, and the resistance and inductance are R and L, respectively. The current through the circuit can be described by the following complex number expression
where j is the imaginary unit (j2 = -1), the real part of this expression represents the resistive component of the current, while the imaginary part represents the reactive component.
The impedance of the circuit can be expressed as 𝑍 = 𝑅 + 𝑗𝜔𝐿,
where Z is a complex number that represents the impedance of the circuit, and the magnitude of Z represents the magnitude of the impedance. The angle of Z represents the phase shift between the voltage and current.
Impedance and Admittance of Circuits
Impedance and admittance are essential concepts in circuit analysis, and complex numbers play a vital role in their calculation. Impedance is the measure of opposition to the current flow in an AC circuit and is represented as a complex number. The impedance of a circuit is the ratio of the phasor voltage to the phasor current.
Similarly, admittance measures how easily current flows through a circuit and is represented as a complex number. Admittance is the inverse of impedance.
Impedance and admittance are essential parameters in circuit analysis and design. Impedance is the opposition of a circuit element to the flow of an AC current, while admittance is the ease with which a circuit element allows the flow of an AC current. Both impedance and admittance can be expressed as complex numbers, where the magnitude represents the magnitude of the impedance or admittance and the angle represents the phase shift. By using complex numbers to represent impedance and admittance, we can easily calculate the behavior of circuits that involve AC signals.
Suppose we have a circuit with impedance Z. The admittance Y of the circuit is defined as the reciprocal of Z, that is:
Z and Y can be expressed as complex numbers, where the magnitude represents the magnitude of the impedance or admittance, and the angle represents the phase shift.
For example, if 𝑍 = 3 − 4𝑖 , then the magnitude of Z is
and the angle of Z is 𝜃 = (4 / 3) = −53.13°. The admittance Y is then Y = 1 / (3−4𝑖)
After rationalizing the denominator by multiplying and dividing by complex conjugate = 3+4𝑖 / 25 = 0.12 + 0.16𝑖
Fourier Analysis and Signal Processing
Complex numbers are used in Fourier analysis and signal processing to represent complex signals composed of a combination of sinusoidal signals. The Fourier analysis involves breaking down a complex signal into its component sinusoidal signals, which can be analyzed separately. Complex numbers enable the representation of these sinusoidal signals in a simplified manner, making the analysis of complex signals more manageable.
Fourier analysis is a mathematical technique to analyze and decompose complex signals into simpler components. It is widely used to analyze and design filters, modulators, demodulators, and other signal-processing systems. Complex numbers are used in Fourier analysis to represent the complex amplitudes and phases of each signal frequency component. By using complex numbers to represent the frequency components of a signal, we can easily analyze and design signal processing systems that involve complex signals.
Suppose we have a signal x(t) that can be decomposed into a sum of complex exponential functions:
where 𝑐𝑛 is the complex amplitude of the nth frequency component, 𝜔𝑛 is the frequency of the nth component, and 𝑗 is the imaginary unit. The complex amplitude 𝑐𝑛 and frequency 𝜔𝑛 can be calculated using the Fourier transform of 𝑥(𝑡)
where T is the period of the signal.
For example, if x(t) is a rectangular pulse with amplitude 1 and duration T, then the Fourier transform of x(t) is
Complex Analysis in Control Theory
Complex analysis is used in control theory to study the behavior of systems with complex transfer functions. Control systems typically have transfer functions that are represented as complex numbers. Using complex numbers in control theory enables studying control systems’ stability, frequency response, and transient response.
For example, consider a control system with the transfer function
The roots of the denominator polynomial (S2 + 2𝑠 + 5) are complex numbers. By analyzing the system’s behavior in the complex plane, we can determine its stability and frequency response.
We can use the quadratic formula to find the roots:
where a = 1, b = 2, and c = 5. Substituting these values, we get:
Therefore, the roots of the denominator polynomial are -1 + 2i and -1 – 2i, which are complex numbers.
To determine the system’s stability, we need to check whether the real parts of the roots are negative. In this case, the real part of both roots is -1, less than zero, so the system is stable.
To determine the system’s frequency response, we can plot the magnitude and phase of the transfer function H(s) on the complex plane. We can write H(s) in polar form as 𝐻(𝑠) = |𝐻(𝑠)| x 𝑒𝑗𝜃 where |H(s)| is the magnitude of H(s), and θ is the phase angle which can be found by the process we discussed in above sections.
If your studying for your FE exam, the Complex Numbers in FE Electrical and Computer exam is the most critical topic that you can’t skip in any case. The concept is fairly simple as we have almost discussed all the ins and outs of complex numbers applications.
Once you are done with the complex numbers theory and understand their mathematical operations, the next step is to explore what are complex numbers used for. As we have also discussed, Complex numbers are crucial in electrical and computer engineering. They are used extensively in circuit analysis, impedance and admittance calculations, Fourier analysis and signal processing, and control theory. The use of complex numbers enables the representation of signals and systems in a simplified manner, making their analysis and design more manageable.