Editing Bode plot (section)

=== Straight-line phase plot ===

Given a transfer function in the same form as above,

:<math>H(s) = A \prod \frac{(s - x_n)^{a_n}}{(s - y_n)^{b_n}},</math>

the idea is to draw separate plots for each pole and zero, then add them up. The actual phase curve is given by
:<math>\varphi(s) = -\arctan \frac{\operatorname{Im}[H(s)]}{\operatorname{Re}[H(s)]}.</math>

To draw the phase plot, for ''each'' pole and zero:
* If <math>A</math> is positive, start line (with zero slope) at 0°.
* If <math>A</math> is negative, start line (with zero slope) at −180°.
* If the sum of the number of unstable zeros and poles is odd, add 180° to that basis.
* At every <math>\omega = |x_n|</math> (for stable zeros <math>-\operatorname{Re}(z) < 0</math>), ''increase'' the slope by <math>45 a_n</math> degrees per decade, beginning one decade before <math>\omega = |x_n|</math> (e.g., <math>|x_n|/10</math>).
* At every <math>\omega = |y_n|</math> (for stable poles <math>-\operatorname{Re}(p) < 0</math>), ''decrease'' the slope by <math>45 b_n</math> degrees per decade, beginning one decade before <math>\omega = |y_n|</math> (e.g., <math>|y_n|/10</math>).
* "Unstable" (right half-plane) poles and zeros (<math>\operatorname{Re}(s) > 0</math>) have opposite behavior.
* Flatten the slope again when the phase has changed by <math>90 a_n</math> degrees (for a zero) or <math>90 b_n</math> degrees (for a pole).
* After plotting one line for each pole or zero, add the lines together to obtain the final phase plot; that is, the final phase plot is the superposition of each earlier phase plot.