Bode plot

In electronics, a Bode plot (or Bode diagram) is a plot of the circuit gain in decibels (or phase, in degrees) on a linear-scale $y$-axis (recall decibels are themselves logarithmic values) plotted against frequency on a log-scale $x$-axis. This is the standard way of plotting the frequency response of a circuit.

Gain plot

A first-order low-pass gain plot is given by the below. We can refer to this chart when drawing them out ($\mathcal{N}$ is the numerator, $\mathcal{D}$ is the denominator). Some observations:

• The 20 dB per decade is mainly a result of the decibel scaling ($20\log f$). If our poles/zeros are duplicated (i.e., $(s+1000{)}^2$) then we get 40 dB per decade. This scales as we continue.
  • We equivalently scale by 6 dB per octave.
• The contributions are summed up to get an overall Bode plot.
  • What this means is that we should build multiple different plots then combine them.
  • Why is this? We can take the logarithm of $T(j\omega )$: by log properties multiplication becomes addition and division becomes subtraction.
• Putting everything in standard form (the rightmost format) is easiest to do. This lets us figure out the corner frequency pretty easily and gives us a uniquely easy point of reference for non-constant poles or zeroes (0 dB).
• It’s good practice to start from the left and combine moving right.
• There’s a pole at the corner frequency, which results in the corner of the plot. In practice, the gain is around 3 dB less than the usual flat gain to the left of the corner.
  • This straight line approximation allows us to roughly draw the Bode plot only with straight lines. The 3 dB point is at ${\omega }_0=\frac{1}{\tau }$. At $\omega =\frac{1}{2\tau }$, we’re roughly at 1 dB. Then, we can say to the left of $\omega =\frac{1}{10\tau }$ we’re at 0 dB.

Phase plot

The first-order phase function $\theta (\omega )=\angle K-\arctan (\frac{\omega }{\alpha })$ is given by: We can refer to this plot when drawing phase contributions: