Name: transfer function gain and origin effects
Uploaded: 2024-07-08T13:20:02.000Z
Duration: 1 min 19 s
Channel: JoVE

Considérons la fonction de transfert d’un système exprimée sous forme standard.
La fonction de transfert a un gain constant, un zéro et un pôle à l’origine, un zéro simple, un pôle simple, un pôle du second degré et un zéro du second degré.
Les diagrammes de Bode pour les facteurs individuels sont tracés séparément et combinés graphiquement.
Pour le terme de gain constant positif, le gain d’amplitude est constant et l’angle de phase est nul et indépendant de la fréquence.
Pour le terme de gain constant négatif, le gain d’amplitude reste le même et l’angle de phase est de ±180°.
Pour une fonction de transfert avec un zéro à l’origine, le graphique de magnitude représente une droite avec une pente positive de 20 dB/décennie. Le diagramme de phase représente un angle de phase constant de 90°.
Pour un pôle à l'origine, la pente du graphique de magnitude est de -20 dB/décennie et le déphasage est de -90°.
S’il y a plus d’un pôle ou d’un zéro à l’origine, la pente du diagramme de magnitude et le déphasage sont multipliés par le nombre de pôles ou de zéros présents à l’origine.

réponse en fréquence

Une fonction de transfert présentée sous sa forme standard intègre le gain constant des éléments, les zéros et les pôles à l'origine, les zéros et pôles simples, et les pôles et zéros quadratiques. La fonction de transfert peut s'écrire H(ω) :
<img alt="Equation 1" src="/files/ftp_upload/15141/15141_Equation_1.svg" style="height: 40px;">
La fonction de transfert, H(ω), souvent exprimée sous la forme standard, est dérivée en normalisant les coefficients polynomiaux de la fonction de transfert. Les pôles (jω) et les zéros (jω) sont des fréquences critiques où l'amplitude et la phase de la sortie du système subissent des changements significatifs.
Gain, K :
La fonction de transfert a un terme constant K, avec une amplitude de 20log10K et un angle de phase de 0° pour une valeur positive de K. L'amplitude et la phase sont constantes avec la fréquence. Si K est négatif, la grandeur reste inchangée, mais la phase est de ±180°. Lorsque le gain K = 1, l'amplitude deviendra nulle ainsi que l'angle de phase. En termes de tracé de Bode, la magnitude est exprimée en décibels (dB) sous la forme 20log_10K, et sa phase reste à 0∘ ou 180∘ selon le signe de K.
Pôle/Zéro à l'Origine :
Un pôle ou un zéro à l’origine a un impact déterminant sur l’intrigue. Un zéro (jω)^+1 à l'origine a une magnitude de 20 log1_0 ω et la phase est de 90°. La pente du tracé d'amplitude est de 20 dB/décennie et la phase est constante avec la fréquence.
Pour un pôle (jω)^-1 à l'origine, la magnitude est de -20 dB/décennie et la phase est de -90°. Généralement, pour (jω)^N, où N est un nombre entier, le tracé de magnitude aura une pente de -20NdB/décennie tandis que la phase est de 90N°.
La grandeur constante K, pôles/zéros aux origines ont l'angle de phase qui ne change pas avec la fréquence. Dans le cas de zéros/pôles, l'angle de phase change lorsque le nombre de pôles/zéros change à l'origine.

Réponse en fréquence

Considérons la fonction de transfert d’un système exprimée sous forme standard.
La fonction de transfert a un gain constant, un zéro et un pôle à l’origine, un ...

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Electrical Engineering

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Transfer function and Bode Plots-I

transfer function gain and origin effects

Transfer function gain and poles or zeros at the origin shape a system's Bode plot. The transfer function is often written in standard form after the polynomial coefficients are normalized. In that form, the constant gain, zeros, poles, and quadratic terms all appear clearly. The poles and zeros mark critical frequencies where the output magnitude and phase change noticeably.

The constant gain K creates a fixed magnitude and phase term. Its magnitude is 20 log 10 K in decibels, and its phase is 0° when K is positive. If K is negative, the magnitude stays the same, but the phase becomes ±180°. When K = 1, the magnitude is 0 dB and the phase angle is 0°.

A zero at the origin has a strong effect on the Bode plot. For a zero at the origin, the magnitude is 20 log 10 ω, and the phase is 90°. The magnitude rises with a slope of 20 dB/decade, and the phase stays constant as frequency changes. A pole at the origin acts in the opposite way. Its magnitude slope is -20 dB/decade, and its phase is -90°.

More generally, a factor of (jω)^N changes the plot by an integer amount. For a zero at the origin, the magnitude slope becomes 20N dB/decade and the phase is 90N°. For a pole at the origin, the magnitude slope becomes -20N dB/decade, and the phase is also 90N°. In both cases, the constant gain K and the poles or zeros at the origin keep a phase angle that does not change with frequency. The phase changes when the number of poles or zeros at the origin changes.

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Explore how constant gain, poles, and zeros at the origin affect transfer functions and Bode plots, including dB slope and phase changes.

Transfer Function Gain and Origin Effects | 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

Transfer Function Gain and Origin Effects

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