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

Considere la función de transferencia en su forma estándar, con polos y ceros
Para una función de transferencia con un cero simple, la ganancia de magnitud a valores de frecuencia pequeños es una línea recta con pendiente cero, y la fase se aproxima a cero.
En la frecuencia de esquina, la magnitud asintótica se desvía de la línea de pendiente cero y la fase se acerca a los 45 grados.
A frecuencias más altas, el gráfico de magnitud forma una línea de +20 dB/década y la fase es de 90 grados.
Un polo simple es el recíproco de un cero simple. Esto significa que los diagramas de Bode basados en polos reflejan el diagrama de cero simple, reflejado alrededor del eje horizontal.
Considere una función de transferencia de polos cuadráticos.
A frecuencias más bajas, la ganancia y el ángulo de fase se acercan a cero.
En la frecuencia de esquina, la desviación de la magnitud asintótica depende del factor de amortiguación y el ángulo de fase es de casi -90 grados.
A frecuencias más altas, el gráfico de magnitudes forma una línea recta con una pendiente de -40 dB/década y una fase de -180 grados.
Para más de un polo cuadrático, la pendiente de la línea y el desplazamiento de fase se multiplican por el número de polos.

respuesta de frecuencia

En la forma estándar, la función de transferencia se muestra en ganancia constante, polos/ceros en el origen, polos/ceros simples y polos/ceros cuadráticos; cada uno de los cuales contribuye de manera única a la respuesta general del sistema. El término representa la magnitud del cero simple:
<img alt="Equation 1" src="/files/ftp_upload/15142/15142_Equation_1.svg" style="height: 36px;">
El gráfico de magnitud de Bode permanece plano en frecuencias bajas (acercándose a 0 dB) y comienza a ascender a 20 dB/década después de una frecuencia específica conocida como frecuencia de esquina o de ruptura, ω_1. Esta es la frecuencia con la que cambia la pendiente del gráfico de magnitud y la respuesta real comienza a desviarse de la aproximación en línea recta. Esta desviación se cuantifica en 3 dB en ω=ω_1.
El ángulo de fase ϕ, expresado como:
<img alt="Equation 2" src="/files/ftp_upload/15142/15142_Equation_2.svg" style="height: 60px;">
El ángulo de fase comienza en 0° y se acerca asintóticamente a 90° a medida que aumenta la frecuencia. Para frecuencias mucho más bajas que la frecuencia de esquina (ω≪ω_1), el término jω/ω_1 es muy pequeño, por lo que la magnitud es insignificante y la fase es esencialmente cero. A medida que la frecuencia se acerca a ω_1, se produce un punto de magnitud de -3 dB y un ángulo de fase de 45°. Para frecuencias mucho más altas que ω_1 (ω≫ω_1), la pendiente de la magnitud cambia a 20 dB/década y la fase se estabiliza en 90°.
<img alt="Figure 1" src="/files/ftp_upload/15142/15142_Figure_1.svg" style="height: 160px;">
Polo/cero cuadrático:
La magnitud y el ángulo de fase de un polo cuadrático es:
<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;">
El gráfico de amplitud para un polo cuadrático tiene dos partes: una respuesta plana por debajo de la frecuencia natural ωn y una pendiente de -40 dB/década por encima de ω_n, variando el pico real del gráfico con el factor de amortiguación ζ_2. La gráfica de fase para un polo cuadrático disminuye linealmente con una pendiente de -90° por década, comenzando en una décima parte de la frecuencia natural y terminando en diez veces esa frecuencia, influenciado por el factor de amortiguamiento ζ_2.

Respuesta de frecuencia

Considere la función de transferencia en su forma estándar, con polos y ceros
Para una función de transferencia con un cero simple, la ganancia de magnitud ...

Education

Educación

JoVE Core

Electrical Engineering

Ingeniería Eléctrica

Unit 1

Unidad 1

Frequency Response

Respuesta en frecuencia

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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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.

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Bode Plot Slope and Phase Changes

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