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

데카르트 좌표로 설명된 복소수는 벡터로 묘사될 수 있습니다. 이것들은 극 형태로 표현될 수 있으며, 그 크기와 각도를 보여줍니다.
복소수가 함수에 입력되면 결과는 또 다른 복소수가 됩니다. 이것은 주요 기능, 즉 벡터 표현이 시작될 수 있는 함수의 영점을 강조합니다.
복합 요인의 분자 곱을 분모의 곱으로 나눈 값으로 정의된 함수를 생각해 보십시오.
각 복소 요인은 본질적으로 벡터이기 때문에 모든 점에서 함수의 크기는 0 길이의 곱을 극점 길이의 곱으로 나눈 값으로 계산됩니다.
함수의 각도는 극각의 합에서 0 각도의 합을 뺀 값으로 계산됩니다. 이러한 각도는 함수의 0 또는 극에서 주어진 점까지 그려진 벡터의 실수 축의 양의 확장에서 측정됩니다.
특정 지점에서 복잡한 함수를 풀기 위해 함수의 0과 극점에서 시작된 벡터가 계산되어 선택한 지점에서 끝납니다. 벡터 방정식을 대입하면 크기와 각도가 제공됩니다.

복소수의 벡터 표현

데카르트 좌표로 표현된 복소수는 벡터로도 시각화할 수 있습니다. 이러한 벡터는 극좌표 형태로 표현하여 크기와 각도를 강조할 수 있습니다. 복소수가 함수에 입력되면 출력은 또 다른 복소수로, 벡터 표현이 시작될 수 있는 함수의 영점을 강조합니다.

분자의 복소수 인자의 곱을 분모의 복소수 인자의 곱으로 나눈 것으로 정의된 함수를 생각해 보자. 벡터로 볼 때 각 복소수 인자는 함수의 크기를 0에서 벡터의 크기의 곱으로 계산하고, 극에서 벡터의 크기의 곱으로 나눈 값으로 정의할 수 있다.

함수의 모든 지점에서의 각도는 0점에서 벡터 각도의 합에서 극점에서 벡터 각도의 합을 뺀 값으로 결정됩니다. 이 각도는 양의 실수 축에서 함수의 0점 또는 극점에서 관심 지점까지 그려진 벡터까지 측정됩니다.

특정 지점에서 복소 함수를 풀기 위해 함수의 각 0과 극에서 선택한 지점까지 벡터를 그립니다. 그런 다음 이러한 벡터의 크기와 각도를 사용하여 해당 지점에서 함수의 값을 결정합니다. 함수의 크기는 0에서 벡터의 길이의 곱을 극에서 벡터의 길이의 곱으로 나눈 값입니다. 함수의 각도는 0에서 각도의 합에서 극에서 각도의 합을 빼서 구합니다.

이러한 벡터 방정식을 함수에 대입하면 지정된 지점에서 함수의 크기와 각도를 모두 얻을 수 있습니다. 이 접근 방식은 복소 함수에 대한 포괄적인 분석을 가능하게 하며, 극좌표 형태의 벡터 표현을 사용하여 함수의 동작과 속성에 대한 통찰력을 제공합니다.

데카르트 좌표로 설명된 복소수는 벡터로 묘사될 수 있습니다. 이것들은 극지 형태로 표현될 수 있으며, 그 규모와 ...

Education

교육

JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Root-Locus Method

근궤적법

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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.

Complex Numbers as Vectors in Polar Form

control system problem

Root Locus and Closed-Loop Stability

Root locus and closed-loop stability are key ideas in control systems. A basic thermostat is an open-loop system, so its transfer function poles affect the response but do not set stability. When feedback is added, the system becomes closed-loop, like an advanced thermostat that changes heating based on room temperature. In a closed-loop system, stability depends on the new poles of the closed-loop transfer function. If those poles move into the unstable region, the system can start to...

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...

root-locus method

Cruise Control Stability with Root Locus

Root locus helps show how cruise control stays stable as conditions change. In a car, cruise control keeps a set speed by adjusting the gas pedal automatically. It keeps checking the car’s speed and makes small pedal changes to hold that target speed. This system can be drawn as a block diagram, and its transfer function gives the math model. The model is useful when the car goes uphill, downhill, or faces strong wind resistance. To find the system poles for different pedal forces, the...

properties of the root locus

Root Locus Angle and Gain Rules

Root locus angle and gain rules help analyze higher-order control systems. The method works without factoring the denominator of the transfer function. It is especially useful for unity feedback systems, where the transfer function can be studied by tracking how poles and zeros shape the response. A pole is a point where the characteristic polynomial in the denominator of the transfer function equals zero. To check whether a point lies on the root locus, use the angle criterion. Add the angles...

construction of root locus

Root Locus Paths and Asymptotes

Root locus paths and asymptotes show how closed-loop poles move as gain changes. This method helps students track system stability by following the pole locations on a graph. It also shows where the poles start, where they may end, and how they travel in between. The number of root locus branches equals the number of closed-loop poles. The plot is symmetric about the real axis. For positive gain, the root locus lies on the real axis to the left of an odd number of finite open-loop poles or...

plotting and calibrating the root locus

Root Locus Breakaway and Stability Points

Root locus analysis tracks how system poles move as gain changes. It shows when poles leave the real axis, move into the complex plane, and affect stability. For high school students studying control systems, the key ideas are breakaway points, break-in points, and jω-axis crossings. Breakaway points are locations where the root locus leaves the real axis. Break-in points are locations where it returns to the real axis. At either type of point, the branches meet the real axis at an angle of...

root loci for positive-feedback systems

Root Locus Rules for Oscillator Feedback

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...

key terms and concepts

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

Complex Numbers as Vectors in Polar Form | JoVE K12

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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