Editing Bode plot (section)

==Rules for handmade Bode plot==

For many practical problems, the detailed Bode plots can be approximated with straight-line segments that are [[asymptote]]s of the precise response. The effect of each of the terms of a multiple element [[transfer function]] can be approximated by a set of straight lines on a Bode plot. This allows a graphical solution of the overall frequency response function. Before widespread availability of digital computers, graphical methods were extensively used to reduce the need for tedious calculation; a graphical solution could be used to identify feasible ranges of parameters for a new design.

The premise of a Bode plot is that one can consider the log of a function in the form
:<math>f(x) = A \prod (x - c_n)^{a_n}</math>

as a sum of the logs of its [[zeros and poles]]:
:<math>\log(f(x)) = \log(A) + \sum a_n \log(x - c_n).</math>

This idea is used explicitly in the method for drawing phase diagrams. The method for drawing amplitude plots implicitly uses this idea, but since the log of the amplitude of each pole or zero always starts at zero and only has one asymptote change (the straight lines), the method can be simplified.

===Straight-line amplitude plot===
Amplitude decibels is usually done using <math>\text{dB} = 20 \log_{10}(X)</math> to define decibels. Given a transfer function in the form

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

where <math>x_n</math> and <math>y_n</math> are constants, <math>s = \mathrm{j}\omega</math>, <math>a_n, b_n > 0</math>, and <math>H</math> is the transfer function:
* At every value of ''s'' where <math>\omega = x_n</math> (a zero), '''increase''' the slope of the line by <math>20 a_n\ \text{dB}</math> per [[Decade (log scale)|decade]].
* At every value of ''s'' where <math>\omega = y_n</math> (a pole), '''decrease''' the slope of the line by <math>20 b_n\ \text{dB}</math> per decade.
* The initial value of the graph depends on the boundaries. The initial point is found by putting the initial angular frequency <math>\omega</math> into the function and finding {{nowrap|<math>|H(\mathrm{j}\omega)|</math>.}}
* The initial slope of the function at the initial value depends on the number and order of zeros and poles that are at values below the initial value, and is found using the first two rules.

To handle irreducible 2nd-order polynomials, <math>ax^2 + bx + c</math> can, in many cases, be approximated as <math>(\sqrt{a}x + \sqrt{c})^2 </math>.

Note that zeros and poles happen when <math>\omega</math> is ''equal to'' a certain <math>x_n</math> or <math>y_n</math>. This is because the function in question is the magnitude of <math>H(\mathrm{j}\omega)</math>, and since it is a complex function, <math>|H(\mathrm{j}\omega)| = \sqrt{H \cdot H^*}</math>. Thus at any place where there is a zero or pole involving the term <math>(s + x_n)</math>, the magnitude of that term is <math>\sqrt{(x_n + \mathrm{j}\omega)(x_n - \mathrm{j}\omega)} = \sqrt{x_n^2 + \omega^2}</math>.

===Corrected amplitude plot===

To correct a straight-line amplitude plot:
* At every zero, put a point <math>3 a_n\ \text{dB}</math> '''above''' the line.
* At every pole, put a point <math>3 b_n\ \text{dB}</math> '''below''' the line.
* Draw a smooth curve through those points using the straight lines as asymptotes (lines which the curve approaches).

Note that this correction method does not incorporate how to handle complex values of <math>x_n</math> or <math>y_n</math>. In the case of an irreducible polynomial, the best way to correct the plot is to actually calculate the magnitude of the transfer function at the pole or zero corresponding to the irreducible polynomial, and put that dot over or under the line at that pole or zero.

=== 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.