The second order equation is a common approximation used for feedback
system analysis because it describes a two-pole circuit, which is the
most common approximation used. All real circuits are more complex
than two poles, but except for a small fraction, they can be
represented by a two-pole equivalent. The second order equation is
extensively described in electronic and control literature [ 6 ] .
After algebraic manipulation Equation 5–16 is presented in the form
of Equation 5–17.
Equation 5–17 is compared to the second order control Equation
5–18, and the damping ratio, ΞΆ, and natural frequency, w N are
obtained through like term comparisons.
Comparing these equations yields formulas for the phase margin and
per cent overshoot as a function of damping ratio.
When the two poles are well separated, Equation 5–21 is valid.
The salient equations are plotted in Figure 5–18, which enables a
designer to determine the phase margin and overshoot when the gain
and pole locations are known.
Figure 5–18. Phase Margin and Overshoot vs Damping Ratio
Enter Figure 5–18 at the calculated damping ratio, say 0.4, and
read the overshoot at 25% and the phase margin at 42°. If a designer
had a circuit specification of 5% maximum overshoot, then the damping
ratio must be 0.78 with a phase margin of 62°.Enter Figure 5–18 at
the calculated damping ratio, say 0.4, and read the overshoot at 25%
and the phase margin at 42°. If a designer had a circuit
specification of 5% maximum overshoot, then the damping ratio must be
0.78 with a phase margin of 62°.
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