9.6
Transfer function and Bode Plots-II
In the standard form, the transfer function is shown in constant gain, poles/zeros at origin, simple poles/zeros, and quadratic poles/zeros; each cont…
Consider the transfer function in its standard form, with poles and zeros
For a transfer function with a simple zero, the magnitude gain at small frequency values is a straight line with zero slope, and the phase approaches zero.
At the corner frequency, the asymptotic magnitude deviates from the zero-slope line, and the phase approaches 45 degrees.
At higher frequencies, the magnitude plot forms a +20 dB/decade line, and the phase is 90 degrees.
A simple pole is the reciprocal of a simple zero. This means that pole based Bode plots mirror the simple zero plot, reflected about the horizontal axis.
Consider a quadratic pole transfer function.
At lower frequencies, the gain and phase angle approaches zero.
At the corner frequency, the asymptotic magnitude deviation depends on the damping factor, and the phase angle is nearly -90 degrees.
At higher frequencies, the magnitude plot forms a straight line with a slope of -40 dB/decade and a phase of -180 degrees.
For more than one quadratic pole, the slope of the line and phase shift are multiplied by the number of poles.
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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.