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# The Bode plot of second order systems

We first consider the transfer function of a 2nd order system (e.g., RCL circuit with voltage across the capacitor C as the output):

where is an arbitrary gain factor. If we let , we get the frequency transfer function

Specially, when , we have

with magnitude

The Bode plot of this transfer function can be found as

and

To obtain the asymptotic approximation of this function, consider the following three cases:
• When i.e., , we have

and

• When , we have

and

• When , we have

and

We see that the smaller , the larger , and the sharper and narrower the peak.

Next consider a second type of 2nd order system (e.g., RCL circuit with voltage across the resistor R as the output):

As the denominator of the frequency response function is already considered above, all we need to do now is to find the log-magnitude of the numerator:

The is a straight line with a slope of 20 dB/decade. In particular, when , it is , which cancels the denominator , in other words, at , and . The phase of the numerator is

Finally consider a second type of 2nd order system (e.g., RCL circuit with voltage across the inductor L as the output):

the log-magnitude of the numerator:

The is a straight line with a slope of 40 dB/decade. In particular, when , it is . The phase of the numerator is

Next: Peak frequency Up: active_filter Previous: Second order systems
Ruye Wang 2008-11-24