Bode Plot

Bode Plot

This is the main graphical representation of a frequency response. It has magnitudes and phases. For a given frequency, shows that your frequency response is in a graphical way. The Bode plot contains a magnitude and phase. The x-axis is typically in a logarithmic scale, and the y-axis is in a decibel scale

You basically just, solve for the frequency response of a given system (transfer function where $s=j\omega$) and plug in different values of $\omega$ and see what the frequency response is

General Rules

General rules for approximate graphing

1. Given a transfer function $G\left(s\right)$
2. Identify poles and zeroes
3. Transform into Bode forms (normalized forms) $G\left(s\right)=K_0{s}^p\frac{\left(\frac{s}{z_1}+1\right)\dots \left(\frac{s}{z_m}+1\right)}{\left(\frac{s}{p_1}+1\right)\dots \left({\left(\frac{s}{{\omega }_n}\right)}^2+2\zeta \frac{s}{{\omega }_n}+1\right)}$
4. List the breakpoints in ascending order
   1. For magnitude plots: $p_1<z_1<\dots <{\omega }_n$
   2. For phase plots: $0.1p_1,10p_1,0.1z_1,10z_1,0.1{\omega }_n,10{\omega }_n$ where you list them basically in the same order by also by the front magnitude
5. Continue plotting for subsequent breakpoints
   1. At each breakpoint, change the slope accordingly (for magnitudes)
      1. Ex. for a 1st order breakpoint, change the slope by 20, for a 2nd order, change the slope by 40
      2. The change is positive for zeros and negative for poles
      3. The last line needs a slope of -20
   2. At each point (0.1x and 10x), change the slope accordingly (for phases)
      1. For each 0.1x point, If it is for the 1st order term, change the slope by 45°, for a 2nd order, change the slope by 90°
      2. The change is positive for zeros and negative for poles
      3. For each 10x point, do it in negative way
      4. The final line must lie flat with a slope of -90°
6. If there is an $s^p$ in the TF, it changes the first line segment of the Bode plot
   1. Here, we draw the first line of the magnitude plot with a slope of 20p and its value $20{\log }_{10}{K}_0$ at $\omega =1$
   2. Draw the first line of the phase plot with a flat line with $90p$°
   3. If any break point coincides with the other one, apply the rules multiple times according to each rule