Name: complex numbers as vectors in polar form
Uploaded: 2024-11-21T14:54:13.000Z
Duration: 1 min 16 s
Channel: JoVE

Números complexos, descritos em coordenadas cartesianas, podem ser representados como vetores. Estes podem ser expressos na forma polar, mostrando sua magnitude e ângulo.
Quando números complexos são inseridos em uma função, o resultado é outro número complexo. Isso destaca uma característica importante: o ponto zero da função, a partir do qual a representação vetorial pode se originar.
Considere uma função definida como o produto do numerador dos fatores complexos dividido pelo produto do denominador.
Como cada fator complexo é essencialmente um vetor, a magnitude da função em qualquer ponto é calculada como o produto de comprimentos zero dividido pelo produto dos comprimentos dos pólos.
O ângulo da função é calculado como a soma dos ângulos zero subtraídos da soma dos ângulos dos pólos. Esses ângulos são medidos a partir da extensão positiva do eixo real dos vetores desenhados do zero ou pólo da função até o ponto dado.
Para resolver uma função complexa em um ponto específico, são calculados vetores originados dos zeros e pólos da função, terminando no ponto escolhido. A substituição de equações vetoriais fornece a magnitude e o ângulo.

representação vetorial de números complexos

Números complexos, representados em coordenadas cartesianas, também podem ser visualizados como vetores. Esses vetores podem ser expressos em forma polar, enfatizando sua magnitude e ângulo. Quando um número complexo é inserido em uma função, a saída é outro número complexo, destacando o ponto zero da função a partir do qual a representação vetorial pode se originar.


 Considere uma função definida como o produto dos fatores complexos no numerador dividido pelo produto dos fatores complexos no denominador. Cada fator complexo, visto como um vetor, permite que a magnitude da função em qualquer ponto seja computada como o produto das magnitudes dos vetores dos zeros, dividido pelo produto das magnitudes dos vetores dos pólos.


 O ângulo da função em qualquer ponto é determinado pela soma dos ângulos dos vetores dos zeros menos a soma dos ângulos dos vetores dos pólos. Esses ângulos são medidos do eixo real positivo para os vetores desenhados dos zeros ou pólos da função até o ponto de interesse.


 Para resolver uma função complexa em um ponto específico, vetores são desenhados de cada zero e pólo da função até o ponto escolhido. As magnitudes e ângulos desses vetores são então usados ​​para determinar o valor da função naquele ponto. A magnitude da função é o produto dos comprimentos dos vetores dos zeros dividido pelo produto dos comprimentos dos vetores dos pólos. O ângulo da função é encontrado subtraindo a soma dos ângulos dos pólos da soma dos ângulos dos zeros.


 Ao substituir essas equações vetoriais na função, pode-se obter a magnitude e o ângulo da função no ponto especificado. Essa abordagem permite uma análise abrangente de funções complexas, fornecendo informações sobre seu comportamento e propriedades por meio do uso da representação vetorial na forma polar.

Representação vetorial de números complexos

Números complexos, descritos em coordenadas cartesianas, podem ser representados como vetores. Estes podem ser expressos na forma polar, mostrando sua magnitude e ...

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Electrical Engineering

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Unit 1

Unidade 1

Root-Locus Method

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Vector Representation of Complex Numbers

complex numbers as vectors in polar form

Complex numbers can be shown as vectors, not just as points in Cartesian coordinates. In polar form, a vector is described by its magnitude, or length, and its angle. This makes complex numbers easier to compare by size and direction.

The same idea also helps describe complex functions. When a complex number goes into a function, the output is another complex number. The function can be viewed as having a zero point, and the vector picture starts from that point.

For a function written as a product of complex factors in the numerator divided by a product of complex factors in the denominator, each factor can be treated like a vector. The magnitude of the function at any point is found by multiplying the lengths of the vectors from the zeros and dividing by the lengths of the vectors from the poles. A zero is a point where the function can start, and a pole is a point where the denominator creates a special point in the function.

The angle of the function is found in a similar way. It is the sum of the angles from the zeros minus the sum of the angles from the poles. These angles are measured from the positive real axis to the vectors drawn from each zero or pole to the point of interest.

To solve the function at a specific point, vectors are drawn from every zero and pole to that point. Their magnitudes and angles are then used to calculate the value of the function. By replacing the function with these vector equations, both the magnitude and angle at the chosen point can be found.

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vector representation of complex numbers

Complex numbers can be shown as vectors, not just as points in Cartesian coordinates. In polar form, a vector is described by its magnitude, or length, and its angle. This makes complex numbers easier to compare by size and direction. The same idea also helps describe complex functions. When a complex number goes into a function, the output is another complex number. The function can be viewed as having a zero point, and the vector picture starts from that point. For a function written as a...

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Root locus rules help describe how oscillator feedback systems behave as gain changes. The Hartley oscillator is one example of a positive feedback system. It keeps oscillating by feeding the output back to the input in phase, which reinforces the signal. Positive feedback systems can also be treated as negative feedback systems with the feedback signal inverted. In root locus analysis, the locus is the set of points on the s-plane where the transfer function angle equals 360 degrees. The...

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Learn how complex numbers can be treated as vectors in polar form, and how zeros, poles, magnitude, and angle help evaluate complex functions.

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JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Complex Numbers as Vectors in Polar Form

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