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Bode Diagram

The frequency response function is complex-valued, it is convenient to write it in polar form:

$$H(j\omega) = \vert H(j\omega) \vert \angle H(j\omega) .$$

To sketch the frequency response, we sketch

• the magnitude $\vert H(j\omega) \vert$, and
• the phase $\angle H(j\omega)$

as a function of frequency $\omega$. This is called a Bode diagram.  

In our RC example, the frequency response function is

$$H(j\omega) = \frac{2}{2+j\omega} .$$

The magnitude is

$$\begin{array}{rl} \vert H(j\omega) \vert & = \vert \frac{2}{2... ...ega} \vert \\ \\ & = \frac{2}{\sqrt{\omega^2+2}} . \end{array}$$

The phase is

$$\begin{array}{rl} \angle H(j\omega) & = \angle \frac{2}{2+j\o... ...mega +2) \\ \\ & = - \tan^{-1}(\frac{\omega}{2}) . \end{array}$$

These are sketched in Figures 48 and 49.

  

Figure 48: Magnitude Bode diagram.

  

Figure 49: Phase Bode diagram.

The Bode diagrams show magnitude and phase as functions of frequency, usually plotted using a logarithmic scale for the frequency axis (x-axis). Phase angle is plotted in a linear scale (y-axis), see Figure 49. However, magnitude can vary greatly, and a special logarithmic scale called the decibel scale is used.   Given a real number R, the value of R in decibels is defined by

$$R \ {\rm (dB)} = 20 \log_{10} R .$$

In Figure 48, the magnitude (y-axis) is shown ranging from -80 dB to 0 dB (how many orders of magnitude is this?).

From our calculations we have

$$\vert H(j\omega) \vert \ {\rm (dB)} = 20 \log_{10}( 2 ) - 20 \log_{10} ( \sqrt{\omega^2 + 4} ) \ {\rm dB}.$$ (62)

It is useful to note that

$$20 \log_{10}( 2 ) \approx 6 \ {\rm dB}, \ \ {\rm and} \ 10 \log_{10}( 2 ) \approx 3 \ {\rm dB}.$$

The frequency at which the magnitude is 3 dB less than the maximum is called the 3 dB point.   We can find it as follows. Set

$$\max - 3 \ {\rm dB} = \vert H(j\omega_{{\rm 3 dB}}) \vert \ {\rm (dB)}$$

and solve for $\omega_{3dB}$, where in this case $\max = 0$ dB.

$$0 - 3 = 6 - 20 \log_{10} ( \sqrt{\omega_{3dB}^2 + 4} ) \ {\rm dB}$$

We get $\omega_{3dB} = 2$ rad/sec, or f_3dB = 0.318 Hz.

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