A Bode plot is a graph of the frequency response of a system. In general, a Bode plot consists of two graphs: the Bode magnitude plot and the Bode phase plot. The ordinate of the graph denotes the frequency (usualy on a log scale) while the abscisa denotes the magnitude or the phase of response function. CircuitsU can generate two types of problems in which the student has to:

1. Draw the Bode magnitue plot of a given transfer function $H(\textcolor{blue}{s})$
2. Find the transfer function $H(\textcolor{blue}{s})$ correspoding to a given Bode magnitude plot

To draw the Bode plot of a transfer function $H(\textcolor{blue}{s})$, it is instrumental to first bring $H(\textcolor{blue}{s})$ to its standard form. The standard form of a transfer function is obtained by writting it as the ratio of two polynomials of the complex frequency $\textcolor{blue}{s}$, in which each polynomial is written as the product of first or second order polynomials in $\textcolor{blue}{s}$ with real coefficients: $$\begin{equation}H(\textcolor{blue}{s})=K_0 \textcolor{blue}{s}^{\pm N} \times \frac{ \left(1 + \frac{\textcolor{blue}{s}}{\omega_{11}}\right)^{n_{11}} \left(1 + \frac{\textcolor{blue}{s}}{\omega_{12}}\right)^{n_{12}} ...\left(1 + \frac{2 \zeta_{21} \textcolor{blue}{s}}{\omega_{21}} +\frac{\textcolor{blue}{s}^2}{\omega_{21}^2}\right)^{n_{21}}...} {\left(1 + \frac{\textcolor{blue}{s}}{\omega_{31}}\right)^{n_{31}} \left(1 + \frac{\textcolor{blue}{s}}{\omega_{32}}\right)^{n_{32}} ...\left[1 + \frac{2 \zeta_{41} \textcolor{blue}{s}}{\omega_{41}} +\frac{\textcolor{blue}{s}^2}{\omega_{41}^2}\right]^{n_{41}}...} \end{equation}$$ or, in terms of $\textcolor{blue}{s}=\textcolor{blue}{j} \omega$ $$\begin{equation}H(\textcolor{blue}{j} \omega)=K_0 (\textcolor{blue}{j} \omega)^{\pm N} \times \frac{ \left(1 + \frac{\textcolor{blue}{j} \omega}{\omega_{11}}\right)^{n_{11}} \left(1 + \frac{\textcolor{blue}{j} \omega}{\omega_{12}}\right)^{n_{12}} ... \left[1 + 2 \zeta_{21} \frac{\textcolor{blue}{j} \omega}{\omega_{21}} +\frac{(\textcolor{blue}{j} \omega _{21})^2}{\omega_{21}^2}\right]^{n_{21}}...} {\left(1 + \frac{\textcolor{blue}{j} \omega}{\omega_{31}}\right)^{n_{31}} \left(1 + \frac{\textcolor{blue}{j} \omega}{\omega_{32}}\right)^{n_{32}} ... \left[1 + 2 \zeta_{41} \frac{\textcolor{blue}{j} \omega}{\omega_{41}} +\frac{(\textcolor{blue}{j} \omega _{41})^2}{\omega_{41}^2}\right]^{n_{41}}...} \end{equation}$$ In the above expressions:

1. $K_0$ is a frequency-independent factor.
2. $\omega_{ij}$ are called break frequencies. The break frequencies can be either zeros if they appear in the numerator or poles if they appear in the denominator of the main fraction.
3. $\zeta_{ij}$ are the damping factors associated with the respective breaking frequencies. Notice that $ 0 \le \zeta_{ij} \lt 1 $, otherwise, this term has real roots and degenerates into two zeros or poles. The damping factors can be associated only with complex zeros and poles.
4. $n_{ij}$ are the orders associated with the respective breaking frequencies ($n_{ij} = 1, 2, 3,...$).
5. The roots of the second order polynomials (or quadratic terms) are complex, with a non-zero imaginary part. If the roots of the quadratic terms were real, they should be split into the product of two first order polynomials with real roots.

The easiest way to make the Bode magnitude plot for a given transfer function is to plot the absolute value of $20 \log |H(\textcolor{blue}{j} \omega)|$ as a function of $\omega$ using tools such as Python, Matlab, Excel, etc. The plot made in this way provides the exact values of the transfer function and is represetented in the figures below using dashed lines. However, if such tools are not available (which is unlikely ourdays!) it is possible to sketch the plot using a number of simple rules that will be discussed below.

In general, when we make the Bode magnitude plot, it is customary to represent $20 \log |H(\textcolor{blue}{j} \omega)|$ instead of $|H(\textcolor{blue}{j} \omega)|$. The units of $20 \log |H(\textcolor{blue}{j} \omega)|$ are $\: \textcolor{gray}{dB}$ (decibels). On a log scale, and if the break frequencies are well separated from each other, the Bode magnitude plot appears to be made of straight lines, for which reason we often start by approximating the plot with straight lines that are sometimes connected with round (smooth) curves at the vertexes. Such a plot is called the straight-line Bode magnitude plot.

The straight-line Bode magnitude plot approximates the exact plot relatively well if the the break frequencies are separated from each other by a factor of at least 5-10 fold (that means that each break frequency is at least 5-10 times larger than the previous one). In this case, the shape of the Bode magnitude plot around the break frequency is mostly given by the properties of that break frequency and is not influenced much by the neighboring break points. Therefore, it is useful to analyze the Bode magnitude plots given by the different types of break frequencies separately. The table below describes the shape of the Bode magnitude plot for different types of break points and gives a few examples.