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## Bode Diagram

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

To sketch the frequency response, we sketch
• the magnitude , and
• the phase
as a function of frequency . This is called a Bode diagram.

In our RC example, the frequency response function is

The magnitude is

The phase is

These are sketched in Figures 48 and 49.

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

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

 (62)

It is useful to note that

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

and solve for , where in this case dB.

We get rad/sec, or f3dB = 0.318 Hz.

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