chapter 5-3.ppt

Chapter 5 Frequency Response Methods Nyquist Criterion -1 (+) (-) 0 Bode Analysis The Nyquist criterion provides us with suitable information concerning the absolute stability and, furthermore, can be utilized to define and ascertain the relative stability of a system. Relative Stability and the Nyquist Criterion The Nyquist stability criterion is defined in terms of (-1,0) point on the polar plot or the 0-dB, -1800 point on the Bode diagram or log-magnitude-phase diagram. Clearly the proximity of the -locus to the this stability point is a measure of the relative stability of a system. Relative Stability and the Nyquist Criterion Example The gain margin is defined as The phase margin is Gain margin and phase margin Gain margin Phase margin Example Consider a second-order system, where the loop transfer function is Let Rearranging above equation, we obtain The relationship between the damping ratio and the phase margin The relationship between the damping ratio and the phase margin The phase margin for this system is The slope of the linear approximation is equal to 0.01, and therefore an approximate linear relationship between the damping ratio and the phase margin is The relationship between the damping ratio and the phase margin where the phase margin is measured in degrees. Above approximation is suitable for the second-order system and may be used for higher-order systems if one can assume that the transient response of the system is primarily due to a pair of dominant under-damped roots. Consider system with a loop transfer function Example Time-Domain Performance Criteria Specified in The Frequency Domain The transient performance of a feedback system can be estimated from the closed-loop frequency response. The open- and closed-loop frequency responses for a unity feedback system are related as follows: Let Squaring the above equation and rearranging, we obtain The Constant-M Loci Let Squaring the above equation and r