9.6
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 cuadrá…
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.
View the full transcript and gain access to JoVE Core videos
Q1: What happens to the magnitude and phase of a simple zero at the corner frequency?
At the corner frequency, the asymptotic magnitude deviates from the zero-slope line, and the phase approaches 45 degrees. Below this frequency, the magnitude plot remains flat with zero slope and phase near zero. Above the corner frequency, the magnitude increases at 20 dB/decade with phase approaching 90 degrees.
Q2: How does a simple pole Bode plot differ from a simple zero plot?
A simple pole is the reciprocal of a simple zero, meaning pole-based Bode plots mirror the simple zero plot reflected about the horizontal axis. Where zeros show positive magnitude slopes and phase increases, poles display negative magnitude slopes and phase decreases by the same magnitude.
Q3: What is the slope and phase response of a quadratic pole at higher frequencies?
At higher frequencies, a quadratic pole magnitude plot forms a straight line with a slope of -40 dB/decade and a phase of -180 degrees. This response is twice as steep as a simple pole because quadratic poles have two poles contributing to the overall system response.
Q4: How does the damping factor affect the quadratic pole response near the natural frequency?
The damping factor influences the peak amplitude and phase transition of a quadratic pole near the natural frequency. The actual plot's peak varies with the damping factor, and the phase plot slope is affected by this factor as it transitions from zero to -180 degrees across the frequency range.
Q5: What is the -3 dB point and why is it significant in Bode plots?
The -3 dB point occurs at the corner frequency where the magnitude plot's slope changes and the actual response begins to deviate from the straight-line approximation. This deviation quantifies where the system's gain transitions from flat response to the ascending or descending slope characteristic of poles or zeros.
Q6: How do multiple quadratic poles affect the overall Bode plot slope and phase shift?
For more than one quadratic pole, the slope of the magnitude line and phase shift are multiplied by the number of poles. This means two quadratic poles produce a -80 dB/decade slope and -360 degrees phase shift at high frequencies, compared to -40 dB/decade and -180 degrees for a single quadratic pole.
Q7: Why does the phase angle of a simple zero approach 90 degrees at high frequencies?
At high frequencies, the jω term in the transfer function dominates, making the simple zero behave like a pure differentiator. This causes the phase to approach 90 degrees asymptotically, representing the maximum phase lead that a simple zero can contribute to the system's frequency response.