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In the standard form, the transfer function is shown in constant gain, poles/zeros at origin, simple poles/zeros, and quadratic poles/zeros; each contributing uniquely to the system's overall response. The term represents the magnitude of the simple zero:

Equation 1

The Bode magnitude plot remains flat at low frequencies (approaching 0 dB) and begins to ascend at 20 dB/decade after a specific frequency known as the corner or break frequency, ω1. This is the frequency where the magnitude plot's slope changes and the actual response begins to deviate from the straight-line approximation. This deviation is quantified as 3 dB at ω=ω1.

The phase angle ϕ, expressed as:

Equation 2

Phase angle starts at 0° and approaches 90° asymptotically as the frequency increases. For frequencies much lower than the corner frequency (ω≪ω1), the term jω/ω1 is very small, so the magnitude is negligible, and the phase is essentially zero. As the frequency approaches ω1, leading to a -3 dB point in magnitude and a phase angle of 45°. For frequencies much higher than ω1 (ω≫ω1), the magnitude's slope changes to 20 dB/decade, and the phase settles at 90°.

Figure 1

Quadratic pole/zero:

The magnitude and phase angle of a quadratic pole is:

Equation 3

Equation 4

The amplitude plot for a quadratic pole has two parts: a flat response below the natural frequency ωn, and a -40 dB/decade slope above ωn, with the actual plot's peak varying with the damping factor ζ2. The phase plot for a quadratic pole decreases linearly with a slope of -90° per decade, starting at one-tenth of the natural frequency and ending at ten times that, influenced by the damping factor ζ2.

Tags
Transfer FunctionBode PlotMagnitude PlotCorner FrequencyBreak FrequencyPhase AngleSimple PoleQuadratic PoleFrequency ResponseDB decadeDamping FactorAmplitude PlotNatural Frequency

From Chapter 9:

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