Bodeplotten

Bodeplotten, also known as Bode plotting, is the practice of creating Bode plots, graphical representations of a system's frequency response. In control theory and signal processing, Bode plots illustrate how a linear time-invariant system responds to sinusoidal inputs over a range of frequencies. A standard presentation includes two plots: the magnitude plot, in decibels versus frequency on a logarithmic scale, and the phase plot, in degrees versus frequency on a logarithmic axis.

Construction: For a transfer function G(s), evaluate G(jω). The magnitude |G(jω)| is plotted as 20 log10|G(jω)| in dB, and the phase ∠G(jω) is plotted in degrees. The frequency axis is typically logarithmic, and breakpoints occur at the system's poles and zeros where the slope or phase changes.

Asymptotic approximation: For simple poles and zeros, the magnitude slope changes by ±20 dB per decade at each break, and the phase changes by ±90 degrees. A pole contributes -20 dB per decade and -90°, a zero +20 dB per decade and +90°.

Applications: Bode plots are used to assess stability margins (gain and phase margins), design compensators (lead, lag, and PID controllers), and predict closed-loop performance from frequency response. They complement other tools such as Nyquist and Nichols charts.

Limitations: They assume linear time-invariant, rational transfer functions; less informative for nonminimum-phase, time-varying, or highly nonlinear systems, and when model accuracy is poor.

Example: G(s) = K/(τs+1) yields a flat magnitude at low frequencies, a -20 dB per decade slope after ω = 1/τ, and a phase shift from 0° toward -90° as frequency increases.

Software: MATLAB, Python SciPy, and other numerical tools can generate exact or asymptotic Bode plots.