Name: bode plot slope and phase changes
Uploaded: 2025-05-22T09:06:14.000Z
Duration: 1 min 23 s
Channel: JoVE

伝達関数を、極と零点を含む標準形式で考えてみましょう
単純なゼロをもつ伝達関数の場合、小さな周波数値での振幅ゲインは傾きがゼロの直線となり、位相はゼロに近づきます。
コーナー周波数では、漸近の大きさはゼロスロープ線から逸脱し、位相は45度に近づきます。
より高い周波数では、振幅プロットは+20dB/ディケードの線を形成し、位相は90度です。
単純極は、単純なゼロの逆数です。つまり、極ベースのボード線図は、水平軸を中心に反射する単純なゼロ プロットを反映します。
2 次極伝達関数について考えます。
周波数が低いと、ゲイン角と位相角はゼロに近づきます。
コーナー周波数では、漸近的な大きさの偏差は減衰係数に依存し、位相角はほぼ-90度です。
より高い周波数では、振幅プロットは-40 dB/decadeの傾きと-180度の位相を持つ直線を形成します。
2 次極が複数ある場合、ラインの傾きと位相シフトに極の数が乗算されます。

周波数応答

標準形式では、伝達関数は、一定のゲイン、原点の極/ゼロ、単純な極/ゼロ、および 2 次極/ゼロで示され、それぞれがシステムの全体的な応答に独自に貢献します。項は単純なゼロの振幅を表します:
<img alt="Equation 1" src="/files/ftp_upload/15142/15142_Equation_1.svg" style="height: 36px;">
ボード振幅プロットは、低周波数 (0 dB に近づく) では平坦なままで、コーナーまたはブレーク周波数 ω_1 と呼ばれる特定の周波数の後に 20 dB/decade で上昇し始めます。これは、振幅プロットの傾斜が変化し、実際の応答が直線近似から外れ始める周波数です。この偏差は、ω=ω_1 で 3 dB として定量化されます。
位相角 ϕ は次のように表されます:
<img alt="Equation 2" src="/files/ftp_upload/15142/15142_Equation_2.svg" style="height: 60px;">

位相角は 0° から始まり、周波数が増加するにつれて漸近的に 90° に近づきます。コーナー周波数よりはるかに低い周波数 (ω≪ω_1) の場合、項 jω/ω_1 は非常に小さいため、振幅は無視でき、位相は基本的にゼロです。周波数が ω1 に近づくと、振幅は -3 dB になり、位相角は 45° になります。ω1 よりはるかに高い周波数 (ω≫ω_1) の場合、振幅の傾斜は 20 dB/decade に変わり、位相は 90° に落ち着きます。
<img alt="Figure 1" src="/files/ftp_upload/15142/15142_Figure_1.svg" style="height: 160px;">
2 次極/零点:
2 次極の振幅と位相角は次のとおりです:
<img alt="Equation 3" src="/files/ftp_upload/15142/15142_Equation_3.svg" style="height: 42px;">
<img alt="Equation 4" src="/files/ftp_upload/15142/15142_Equation_4.svg" style="height: 60px;">
2 次極の振幅プロットには 2 つの部分があります。固有周波数 ω_n より下では平坦な応答、ω_n より上では -40 dB/decade の傾斜で、実際のプロットのピークは減衰係数 ζ_2 によって変化します。2 次極の位相プロットは減衰係数 ζ_2 の影響を受け、固有周波数の 10 分の 1 から始まり、その 10 倍で終わり、10 倍ごとに -90° の傾斜で直線的に減少します。

伝達関数を、極と零点を含む標準形式で考えてみましょう
単純なゼロを持つ伝達関数の場合、小さな...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Frequency Response

周波数特性

KEY TERMS AND CONCEPTS

主要用語および概念

Transfer function and Bode Plots-II

bode plot slope and phase changes

Bode plots show how a transfer function changes with frequency. The standard form includes constant gain, poles and zeros at the origin, simple poles and zeros, and quadratic poles and zeros. Each part shapes the system response in a different way.

A simple zero sets the magnitude trend first. The Bode magnitude plot stays flat at low frequencies, near 0 dB, and then rises at 20 dB per decade after the corner or break frequency, ω1. That corner frequency is where the straight-line approximation starts to differ from the actual curve, and the error is 3 dB at ω = ω1.

The phase angle for a simple zero also changes with frequency. It starts at 0° and rises toward 90° as frequency increases. When ω is much smaller than ω1, the term jω/ω1 is very small, so the magnitude is negligible and the phase is close to zero. At ω = ω1, the magnitude is at the -3 dB point and the phase is 45°. When ω is much larger than ω1, the magnitude slope is 20 dB per decade and the phase settles near 90°.

Quadratic poles have a different shape on Bode plots. Their magnitude is flat below the natural frequency, ωn, and drops at -40 dB per decade above ωn. The actual peak depends on the damping factor, ζ2. Their phase plot decreases linearly at -90° per decade from one-tenth of ωn to ten times ωn, and that shape is also affected by ζ2.

Bode Plot Slope and Phase Changes

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Bode Plot Slope and Phase Changes | 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

Bode Plot Slope and Phase Changes

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