# Use of complex numbers in electrical engineering. 2.1: Complex Numbers 2022-10-25

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Complex numbers are a mathematical concept that have found numerous applications in electrical engineering. They consist of a real part and an imaginary part, and can be represented in the form a + bi, where a is the real part and bi is the imaginary part. The imaginary unit, denoted by "i", is defined as the square root of -1.

One of the main uses of complex numbers in electrical engineering is in the analysis and design of electrical circuits. Complex numbers are particularly useful in this context because they allow engineers to represent both the magnitude and phase of a quantity, such as an electrical current or voltage.

For example, consider an electrical circuit that contains an inductor, which is a passive electrical component that stores energy in the form of a magnetic field. The current through an inductor is not in phase with the voltage across it, meaning that the phase angle between the current and voltage is not zero. By representing the current and voltage as complex numbers, it is possible to easily calculate the phase angle between them, which is important for understanding the behavior of the circuit.

Another important use of complex numbers in electrical engineering is in the analysis of alternating current (AC) systems. AC systems are commonly used to distribute electrical power over long distances, and the voltage and current in these systems oscillate at a specific frequency. By representing the voltage and current as complex numbers, it is possible to analyze the phase relationship between them, as well as the power being transmitted in the system.

Complex numbers are also used in electrical engineering to represent the impedance of electrical components, such as resistors, capacitors, and inductors. Impedance is a measure of the opposition to the flow of electrical current in a circuit, and it is complex-valued because it includes both resistance (a real quantity) and reactance (an imaginary quantity). By representing impedance as a complex number, it is possible to easily calculate the total impedance of a circuit, which is important for understanding the behavior of the circuit.

In summary, complex numbers are an essential tool in electrical engineering, with numerous applications in the analysis and design of electrical circuits, the analysis of AC systems, and the representation of impedance. Their ability to represent both magnitude and phase makes them a powerful tool for understanding and predicting the behavior of electrical systems.

## How complex numbers are used in electrical engineering problems? Certain fractals are plotted in the complex plane, e. So complex numbers are absolutely crucial to understanding modern applications of electricity. . These linear transformations are the symmetries of the differential equations. Yeah, theyre not exactly the same and are used in different contexts, but they're so closely related that they should occupy the same part of your brain. Sometimes the word phasor is used in electrical applications where the angle of the vector represents phase shift between waveforms. Give an example where they arise like the child that throws a ball, the equation, and what happens when we ask the time of an impossible height.

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## Application of Complex Numbers (Electrical Engineering) By then, using i for current was entrenched and electrical engineers chose i for writing complex numbers. The quantity θ is the complex number's angle. In Electrical Engineering there are different ways to represent a complex number either graphically or mathematically. Complex numbers are broadly used in physics, normally as a calculation tool that makes things easier due to Euler's formula. Theroot locus,Nyquist plot, andNichols plottechniques all make use of the complex plane. Because no real number satisfies this equation, i is called an imaginary number.

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## Can Someone Explain how Complex Numbers are Used in Electrical Engineering? : math Frequency and phase shift are two of these dimensions that come into play. In the previous tutorials including this one we have seen that we can use phasors to represent sinusoidal waveforms and that their amplitude and phase angle can be written in the form of a complex number. Now you can perform arithmetic on electrical circuit by manipulating complex numbers. That's not much help. Just like the example of giving directions from one city to another, AC quantities in a single-frequency circuit have both amplitude analogy: distance and phase shift analogy: direction.

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## math The imaginary component represents the transfer of reactive power. It allows us to deal with alternating current without having to solve an integro differential equation every time. It is fuzzy, but somewhere along the line, mathematicians added the concept of non-integer or rational numbers, e. Application of complex number in Computer Science. Conjugate Complex Numbers The conjugate of a complex number is an important element used in Electrical Engineering to determine the apparent power of an AC circuit using rectangular form.

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## Why complex numbers are used/introduced in electricity This phenomenon is known as wave-particle duality. A positive a is an unstable system that will spin out of control and collapse. In the vector representation, the rectangular coordinates are typically referred to simply as X and Y. Although imaginary numbers are not commonly used in daily life, in engineering and physics they are in fact used to represent physical quantities such as impedance of RL, RC, or RLC circuit. Thanks a lot and have a nice day. Although that can be supplanted via an appropriate matrix 0, -1; 1, 0. In order to analyze AC circuits, it became necessary to represent multi-dimensional quantities.

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## Why are imaginary numbers used in electrical engineering To multiply together two vectors in polar form, we must first multiply together the two modulus or magnitudes and then add together their angles. The magnitude of the complex number represents the resulting magnitude of voltage in response to a unit current. This can be done very nicely with complex numbers. This representation is known as the Cartesian form of z. Think of a "change" as AC. One such way that uses the cosine and sine rule is called the Cartesian or Rectangular Form.

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## Complex Numbers & Phasors in Polar and Rectangular Form If I draw a line with a certain length magnitude and angle direction , I have a graphic representation of a complex number which is commonly known in physics as a vector: Figure below A vector has both magnitude and direction Like distances and directions on a map, there must be some common frame of reference for angle figures to have any meaning. If there are multiple frequencies, you can study the network at each frequency, and add the results super impose them. Many things in EE, as well as mechanics, and physics in general, go round in a circle, simple harmonic motion. Without complex numbers, how would you describe how voltage and current across a capacitor behave? It seems to me you've done your homework. Thanks a lot, Sara While I agree with most of the comments already made, I would like to add, or emphasize, that complex numbers were introduced by Steinmetz in his theory of AC circuits because they can convert complicated problems into simpler, algebraic ones. Good luck on your interview! However, it is especially important for the electrical engineer to master this topic. Hopefully this is a useful example.

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## Application of complex number in engineering Hence you get the Fourier transform. If your choice in your R 2 vector space is for magnitudes of vectors multiply as scalars and angles of those vectors to add, that will be much more helpful in describing how things like current and voltage interact and combine. For example, in electronics, the state of a circuit element is defined by the voltage V and the current I. But Inductors and Capacitors have imaginary impedance. What career uses imaginary numbers? Consequently, multiplying a complex number by i rotates the number's position by 90 degrees. However this is a very complicated topic and you need to read a couple of books to grasp it fully.

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