Bode plot

The Bode plot for a linear system with frequency response function G(s) (s being the complex frequency in the Laplace domain) consists of a magnitude and a phase plot.

The Bode magnitude plot is the graph of the function |G(s=jω)| of frequency ω (with j being the imaginary unit). Usually the ω-axis of the magnitude plot is logarithmic and the magnitude is given in decibels, i.e. 20log10|G|.

The Bode phase plot is the graph of the phase, commonly expressed in degrees, of the response function arg(G(s=jω)) as a function of ω. The phase is plotted on the same logarithmic ω-axis as the magnitude plot, but the value is plotted on a linear vertical axis.

You can refer to the frequency diagram settings to generate a Bode plot.

Figure 47: Example of a Bode plot

Closed-loop stability

In most cases, control engineers utilize feedback when designing control systems. This is often accomplished using a PID controller system. Where there is regular feedback, control theory can be used to determine how the system responds to such feedback. In practically all such systems stability is important and control theory can help ensure stability is achieved.

We consider a system whose response function is G(s); when placed in a closed loop control with negative feedback H(s), the closed loop response function then becomes

Stability can be determined by examining the open loop response function GH(s).

Then assessment of the stability of a closed-loop negative feedback system is done by applying the gain margin and phase margin stability criterion to the Bode plot of the open-loop system (i.e. the same system without its feedback loop). Stability is determined by looking at:

• Gain margin

    The Bode phase plot locates the frequency where the phase of *GH(s=jω)* reaches -180 degrees. Using this frequency, the Bode magnitude plot finds the magnitude of *GH(s=jω)*. If *|GH(s=jω)|180*≥ 1, the control is unstable. If *|GH(s=jω)|180* < 1, instability does not occur, and the separation in dB of the magnitude of *|GH(s=jω)|180* from *|GH(s=jω)|*=1 is called the gain margin.
  

• Phase margin

    The Bode magnitude plot locates the frequency where the magnitude of *|GH(s=jω)|180* reaches unity. Using this frequency, the Bode phase plot finds the phase of *GH(s=jω)*. If the phase of *GH(s=jω)0dB* > -180, the instability condition cannot be met at any frequency, and the distance of the phase in degrees above -180 degrees is called the phase margin.
  

In the example below, for the frequency where the phase reaches -180 degrees, the Bode diagram shows that the magnitude of the red curve is greater than 0 dB and that stability is therefore not respected; while the magnitude of the blue curve is lower than 0 dB and that stability is therefore respected with a gain margin of around 5 dB.

Figure 48: Bode plot - Stability