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 contributing uniquely to the system's overall response. The term represents the magnitude of the simple zero:

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:

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°.

Quadratic pole/zero:

The magnitude and phase angle of a quadratic pole is:

The amplitude plot for a quadratic pole has two parts: a flat response below the natural frequency ω_n, and a -40 dB/decade slope above ω_n, 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.