9.5
Respuesta de frecuencia
Una función de transferencia presentada en su forma estándar integra la ganancia constante de los elementos, los ceros y los polos en el origen, los c…
Considere la función de transferencia de un sistema expresada en forma estándar.
La función de transferencia tiene una ganancia constante, un cero y un polo en el origen, un cero simple, un polo simple, un polo cuadrático y un cero cuadrático.
Los diagramas de Bode para factores individuales se trazan por separado y se combinan gráficamente.
Para el término de ganancia constante positiva, la ganancia de magnitud es constante y el ángulo de fase es cero e independiente de la frecuencia.
Para el término de ganancia constante negativa, la ganancia de magnitud sigue siendo la misma y el ángulo de fase es ±180 °.
Para una función de transferencia con un cero en el origen, el gráfico de magnitud representa una línea recta con una pendiente positiva de 20 dB/década. El gráfico de fase muestra un ángulo de fase constante de 90°.
Para un polo en el origen, la pendiente del gráfico de magnitud es de -20 dB/década y el desplazamiento de fase es de -90°.
Si hay más de un polo o cero en el origen, la pendiente del gráfico de magnitud y el desplazamiento de fase se multiplican por el número de polos o ceros presentes en el origen.
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Q1: What does a transfer function in standard form include?
A transfer function in standard form integrates constant gain, zeros and poles at the origin, simple zeros and poles, and quadratic poles and zeros. These elements define how a system responds across different frequencies. The transfer function is derived by normalizing polynomial coefficients and expressed as H(ω) to characterize system behavior.
Q2: How does constant gain affect magnitude and phase in Bode plots?
For positive constant gain K, the magnitude is expressed as 20 log₁₀K in decibels with a phase angle of 0°, both constant across frequency. For negative K, magnitude remains unchanged but phase becomes ±180°. When K equals 1, magnitude becomes zero decibels with zero phase angle.
Q3: What is the slope of the magnitude plot for a zero at the origin?
A zero at the origin produces a magnitude plot with a positive slope of 20 dB/decade and a constant phase angle of 90°. The magnitude increases linearly with frequency on a logarithmic scale. This contrasts with poles at the origin, which produce negative slopes and phase shifts.
Q4: How do multiple poles or zeros at the origin affect Bode plot slopes?
When multiple poles or zeros exist at the origin, the magnitude plot slope and phase shift are multiplied by the number present. For example, two poles at the origin produce a slope of -40 dB/decade and phase shift of -180°. This scaling relationship applies generally as (jω)ᴺ, where N is the integer count.
Q5: What role do poles and zeros play in transfer function behavior?
Poles and zeros are critical frequencies where the magnitude and phase of the system's output experience significant changes. They determine how the system responds at different frequencies. Understanding their locations and multiplicities is essential for predicting system stability and frequency response characteristics.
Q6: How are individual Bode plot factors combined graphically?
Bode plots for individual factors—constant gain, zeros, poles, and quadratic terms—are plotted separately then combined graphically. Each factor contributes its magnitude and phase response independently. The total system response is obtained by superimposing these individual contributions across the frequency range.
Q7: Why is the decibel scale used for magnitude in Bode plots?
The decibel scale, expressed as 20 log₁₀K, compresses the wide range of magnitude values into a manageable linear representation. This logarithmic scaling makes it easier to visualize system behavior across multiple decades of frequency. It also simplifies the graphical combination of individual transfer function factors.