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

デカルト座標で記述される複素数は、ベクトルとして表すことができます。これらは極性の形で表現でき、その大きさと角度を示しています。
複素数が関数に入力されると、結果は別の複素数になります。これは、ベクトル表現の起点となる関数のゼロ点という重要な特徴を強調しています。
複素因数の分子積を分母の積で割ったものとして定義される関数について考えてみます。
各複素因数は基本的にベクトルであるため、任意の点での関数の大きさは、ゼロの長さを極の長さの積で割った積として計算されます。
この関数の角度は、極角度の合計からゼロ角度の合計を引いた値として計算されます。これらの角度は、関数のゼロまたは極から特定の点まで描画されたベクトルの実軸の正の延長から測定されます。
特定の点で複素関数を解くために、関数の零点と極に由来するベクトルが計算され、選択した点で終了します。ベクトル方程式を代入すると、大きさと角度が得られます。

複素数のベクトル表現

直交座標で表された複素数は、ベクトルとして視覚化することもできます。これらのベクトルは、大きさと角度を強調した極形式で表現できます。複素数を関数に入力すると、出力は別の複素数となり、ベクトル表現の元となる関数のゼロ点が強調されます。 
分子の複素因数の積を分母の複素因数の積で割ったものとして定義された関数を考えてみましょう。各複素因数をベクトルとして見ると、関数の任意の点における大きさは、ゼロからのベクトルの大きさの積を極からのベクトルの大きさの積で割ったものとして計算できます。 
関数の任意の点の角度は、ゼロからのベクトルの角度の合計から、極からのベクトルの角度の合計を引いて決定されます。これらの角度は、関数のゼロまたは極から対象の点に引かれたベクトルまでの正の実軸から測定されます。 
特定の点で複雑な関数を解くには、関数の各ゼロ点と極から選択した点までベクトルを描きます。次に、これらのベクトルの大きさと角度を使用して、その点における関数の値を決定します。関数の大きさは、ゼロ点からのベクトルの長さの積を極点からのベクトルの長さの積で割ったものです。関数の角度は、ゼロ点からの角度の合計から極からの角度の合計を減算して求められます。 
これらのベクトル方程式を関数に代入すると、指定された点における関数の大きさと角度の両方を取得できます。このアプローチにより、極形式のベクトル表現を使用することで複雑な関数の包括的な分析が可能になり、関数の動作と特性に関する洞察が得られます。

デカルト座標で記述される複素数は、ベクトルとして表すことができます。これらは極性の形で表現でき、その大きさと...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Root-Locus Method

根軌跡法

KEY TERMS AND CONCEPTS

主要用語と概念

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 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

Complex Numbers as Vectors in Polar Form

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