🔸Application in electronics engineering:
Information that expresses a single dimension, such as linear distance, is called a scalar quantity in mathematics. Scalar numbers are the kind of numbers students use most often🙃. In relation to science, the voltage produced by a battery, the resistance of a piece of wire (ohms), and current through a wire (amps) are scalar quantities.😌
When electrical engineers analyzed alternating current circuits, they found that quantities of voltage, current and resistance (called impedance in AC) were not the familiar one-dimensional scalar quantities that are used when measuring DC circuits. These quantities which now alternate in direction and amplitude possess other dimensions (frequency and phase shift) that must be taken into account.
In order to analyze AC circuits, it became necessary to represent multi-dimensional quantities. In order to accomplish this task, scalar numbers were abandoned andcomplex numberswere used to express the two dimensions of frequency and phase shift at one time.
In mathematics, i is used to represent imaginary numbers🙂. In the study of electricity and electronics, j is used to represent imaginary numbers so that there is no confusion with i, which in electronics represents current. It is also customary for scientists to write the complex number in the form a+jb.
In electrical engineering, the Fourier transform😍 is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus. This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, andvideosignals.
Complex numbers are used a great deal in electronics. The main reason for this is they make the whole topic of analyzing and understanding alternating signals much easier. This seems odd at first, as the concept of using a mix of real and ‘imaginary’ numbers to explain things in the real world seem crazy!. To help you get a clear picture of how they’re used and what they mean we can look at a mechanical example…
We can now reverse the above argument when considering a.c. (sine wave) oscillations in electronic circuits. Here we can regard the oscillating voltages and currents as ‘side views’ of something which is actually ‘rotating’ at a steady rate. We can only see the ‘real’ part of this, of course, so we have to ‘imagine’ the changes in the other direction. This leads us to the idea that what the oscillation voltage or current that we see is just the ‘real’ portion’ of a ‘complex’ quantity that also has an ‘imaginary’ part. At any instant what we see is determined by aphase anglewhich varies smoothly with time.
We can now consider oscillating currents and voltages as being complex values that have a real part we can measure and an imaginary part which we can’t. At first it seems pointless to create something we can’t see or measure, but it turns out to be useful in a number of ways.
1️⃣ It helps us understand the behaviour of circuits which contain reactance (produced by capacitors or inductors) when we apply a.c. signals.😊
2️⃣ It gives us a new way to think about oscillations. This is useful when we want to apply concepts like the conservation of energy to understanding the behaviour of systems which range from simple a mechanical pendulums to a quartz-crystal oscillator.
@EngMath_aut
Information that expresses a single dimension, such as linear distance, is called a scalar quantity in mathematics. Scalar numbers are the kind of numbers students use most often🙃. In relation to science, the voltage produced by a battery, the resistance of a piece of wire (ohms), and current through a wire (amps) are scalar quantities.😌
When electrical engineers analyzed alternating current circuits, they found that quantities of voltage, current and resistance (called impedance in AC) were not the familiar one-dimensional scalar quantities that are used when measuring DC circuits. These quantities which now alternate in direction and amplitude possess other dimensions (frequency and phase shift) that must be taken into account.
In order to analyze AC circuits, it became necessary to represent multi-dimensional quantities. In order to accomplish this task, scalar numbers were abandoned andcomplex numberswere used to express the two dimensions of frequency and phase shift at one time.
In mathematics, i is used to represent imaginary numbers🙂. In the study of electricity and electronics, j is used to represent imaginary numbers so that there is no confusion with i, which in electronics represents current. It is also customary for scientists to write the complex number in the form a+jb.
In electrical engineering, the Fourier transform😍 is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus. This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, andvideosignals.
Complex numbers are used a great deal in electronics. The main reason for this is they make the whole topic of analyzing and understanding alternating signals much easier. This seems odd at first, as the concept of using a mix of real and ‘imaginary’ numbers to explain things in the real world seem crazy!. To help you get a clear picture of how they’re used and what they mean we can look at a mechanical example…
We can now reverse the above argument when considering a.c. (sine wave) oscillations in electronic circuits. Here we can regard the oscillating voltages and currents as ‘side views’ of something which is actually ‘rotating’ at a steady rate. We can only see the ‘real’ part of this, of course, so we have to ‘imagine’ the changes in the other direction. This leads us to the idea that what the oscillation voltage or current that we see is just the ‘real’ portion’ of a ‘complex’ quantity that also has an ‘imaginary’ part. At any instant what we see is determined by aphase anglewhich varies smoothly with time.
We can now consider oscillating currents and voltages as being complex values that have a real part we can measure and an imaginary part which we can’t. At first it seems pointless to create something we can’t see or measure, but it turns out to be useful in a number of ways.
1️⃣ It helps us understand the behaviour of circuits which contain reactance (produced by capacitors or inductors) when we apply a.c. signals.😊
2️⃣ It gives us a new way to think about oscillations. This is useful when we want to apply concepts like the conservation of energy to understanding the behaviour of systems which range from simple a mechanical pendulums to a quartz-crystal oscillator.
@EngMath_aut
🔸Applications in Fluid Dynamics:
complex functions are used to describe potential flow in two dimensions. Fractals.
Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set
Fluid Dynamics and its sub disciplines aerodynamics, hydrodynamics, and hydraulics have a wide range of applications. For example, they are used in calculating forces and moments onaircraft, the mass flow of petroleum through pipelines, and prediction of weather patterns.
The concept of a fluid is surprisingly general. For example, some of the basic mathematical concepts in traffic engineering are derived from considering traffic as a continuous fluids. @EngMath_aut
complex functions are used to describe potential flow in two dimensions. Fractals.
Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set
Fluid Dynamics and its sub disciplines aerodynamics, hydrodynamics, and hydraulics have a wide range of applications. For example, they are used in calculating forces and moments onaircraft, the mass flow of petroleum through pipelines, and prediction of weather patterns.
The concept of a fluid is surprisingly general. For example, some of the basic mathematical concepts in traffic engineering are derived from considering traffic as a continuous fluids. @EngMath_aut
🔸In Civil and Mechanical Engineering:
The concept of complex geometry and Argand plane is very much useful in constructing buildings and cars. This concept is used in 2-D designing of buildings and cars. It is also very useful in cutting of tools. Another possibility to use complex numbers in simple mechanics might be to use them to represent rotations.
@EngMath_aut
The concept of complex geometry and Argand plane is very much useful in constructing buildings and cars. This concept is used in 2-D designing of buildings and cars. It is also very useful in cutting of tools. Another possibility to use complex numbers in simple mechanics might be to use them to represent rotations.
@EngMath_aut
Forwarded from ریاضیات مهندسی (NaCm)