BodePlots

Bode plots are graphical representations of the frequency response of a linear time-invariant (LTI) system. A Bode plot consists of two plots: the magnitude plot, showing the gain |G(jω)| in decibels versus the logarithm of angular frequency ω, and the phase plot, showing the phase angle ∠G(jω) in degrees versus log ω. The transfer function G(s) of the system, commonly expressed as a rational function of s, is evaluated on the imaginary axis s = jω to obtain G(jω). The magnitude in decibels is 20 log10|G(jω)|. Frequency is typically plotted on a log scale to cover multiple decades.

Breakpoints occur at poles and zeros of G(s). Each pole adds −20 dB/decade slope to the magnitude, each zero adds +20 dB/decade. The phase contribution of a pole is −90 degrees (asymptotically) and of a zero +90 degrees; the total phase is the sum. In practice, asymptotic Bode plots use straight-line approximations with corner frequencies where the slope changes.

Bode plots are widely used to assess stability and robustness in control design, by examining gain and phase margins, and to guide controller tuning. They are also used to analyze filters and electronic circuits, and to compare experimental data with theoretical models.

Limitations include reduced information about time-domain behavior, sensitivity to nonlinearity and non-minimum-phase systems, and potential inaccuracies away from breakpoints. They are most informative for LTI systems with transfer functions available in the frequency domain.