ANALYZE THE STABILITY OF DC SERVO MOTOR USING NYQUIST PLOT
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The document presents a case study analyzing the stability of a DC servo motor using Nyquist plots, highlighting key concepts such as transfer functions, stability analysis, and characteristics of DC motors. It explains the Nyquist stability criterion, methods for stability analysis, and the importance of gain and phase margins. The study aims to illustrate how variations in motor characteristics and control parameters affect stability and transient behavior in DC servo motor systems.
ANALYZE THE STABILITY OF DC SERVO MOTOR USING NYQUIST PLOT
- 1. Raghu Engineering College Department of Electrical & Electronics Engineering(Autonomous) Accredited by NBA & NAAC with ‘A Grade, Permanently Affiliated JNTU Kakinada Dakamarri (v), Bheemunipatnam Mandal, Visakhapatnam, Andhra Pradesh 531162 A CASE STUDY ON ANALYZE THE STABILITY OF DC SERVO MOTOR USING NYQUIST PLOT BY P.SANJAY KUMAR 18981A0241 BACHELOR OF TECHNOLOGY IN ELECTRICAL AND ELECTRONICS ENGINEERING
- 2. CONTENT :- • Transfer function of a dc servo motor • Stability analysis of dc servo motor • Characteristics of a dc servo motor • Nyquist Stability Criterion • Stability Analysis using Nyquist Plots • Rules for Drawing Nyquist Plots • Relative Stability from the Nyquist Plot • Representation of the plotting
- 3. Transfer Function of a Field Controlled DC Motor: • The speed of a dc motor can be varied by varying the field current. The speed can be increased beyond base speed by decreasing the field current. • In this type of control constant torque operation is not possible as the armature current would increase to dangerous values at low fluxes. • It is therefore necessary to maintain the armature current at a constant value at all flux levels. • The armature is also supplied by means of a phase controlled rectifier to maintain constant armature current.
- 4. TRANSFER FUNCTION OF A DC SERVO MOTOR:- TRANSFER FUNCTION OF A DC SERVO MOTOR
- 5. Stability analysis of dc servo motor: The stability boundary of the system in terms of the time delay is theoretically determined and an expression is obtained to compute the delay margin in terms of system parameters. The delay margin is defined as the maximum amount of time delay for which the DC motor speed control system is marginally stable.
- 6. Characteristics of a dc servo motor :- The torque developed by the motor 𝑇𝑑 = 𝐾 𝜑𝑖 1 In a Transfer Function of a Field Controlled DC Motor as discussed above, the armature current is constant and field current is variable. Therefore we have 𝑇𝑑 = 𝐾2 𝐼𝑓 2 The equation of the Transfer Function of a Field Controlled DC Motor is give 𝑟𝑓 𝑖 𝑓 + 𝐿 𝑓(ⅆ𝑖 𝑓ⅆ 𝑡) 3 The dynamic equation of the motor is 𝐽 𝑑𝜔 𝑑𝑡 + 𝑓. 𝜔 = 𝐾2 𝐼𝑓 4 The Laplace transforms of Eqs 2-3 with zero initial conditions are given by
- 7. (𝐿 𝑓s + 𝑟𝑓) 𝐼𝑓(s) = 𝐸𝑓 𝑠 (𝐽𝑠+𝑓) w(s) = 𝐾2 𝐼𝑓(s) Eliminating It(s) from Eq. 6.26 and simplifying we get w(s) 𝐸 𝑓 𝑠 = 𝐾2 (𝐿 𝑓s + 𝑟 𝑓)(𝐽𝑠+𝑓) = 𝐾 𝑚 (𝑇 𝑓s +1)(𝑇 𝑚 𝑠+1) where Km = K2/(rf.f) motor gain constant Tf = Lf/rf field time constant Tm = J/f mechanical time constant.
- 8. Nyquist Stability Criterion • The Nyquist stability criterion works on the principle of argument • . It states that if there are P poles and Z zeros are enclosed by the ‘s’ plane closed path, • then the corresponding G(s)H(s) plane must encircle the origin P−Z times. • So, we can write the number of encirclements N as, • N=P−Z
- 9. •If the enclosed ‘s’ plane closed path contains only poles, •then the direction of the encirclement in the G(s)H(s) • plane will be opposite to the direction of the enclosed closed path in the ‘s’ plane. •If the enclosed ‘s’ plane closed path contains only zeros, • then the direction of the encirclement in the G(s)H(s) • plane will be in the same direction as that of the enclosed closed path in the ‘s’ plane.
- 10. •The zeros of the characteristic equation are same as that of •the poles of the closed loop transfer function. We know that the open loop control system is stable if there is no open loop pole in the the right half of the ‘s’ plane. •The Poles of the characteristic equation are same as that of the poles of the open loop transfer function. P=0⇒N=−Z
- 11. Stability Analysis using Nyquist Plots • From the Nyquist plots, we can identify whether the control system is stable, marginally stable or unstable based on the values of these parameters. • Gain cross over frequency and phase cross over frequency • Gain margin and phase margin
- 12. Rules for Drawing Nyquist Plots •Locate the poles and zeros of open loop transfer function G(s)H(s) in ‘s’ plane. •Draw the polar plot by varying ωω from zero to infinity. If pole or zero present at s = 0, • then varying ωω from 0+ to infinity for drawing polar plot. •Draw the mirror image of above polar plot for values of ωω ranging from • −∞ to zero (0− if any pole or zero present at s=0). The number of infinite radius half circles will be equal to the number of poles or zeros at origin. The infinite radius half circle will start at the point where the mirror image of the polar plot ends. And this infinite radius half circle will end at the point where the polar plot starts.
- 13. Relative Stability from the Nyquist Plot • The relative stability is indicative of how various poles of the system affect the transient behaviour of a system, in other words, how oscillatory the transient behaviour of the system is. • A practical system has damped oscillations before reaching the new steady-state behaviour. This behaviour of the system can be identified by the term relative stability. • The damping ratio of the system decreases. On the other hand if the locus is away from (-1, 0) the system has more damping and less oscillatory behaviour and has better stability conditions. • The gain margin of a system is used to describe the nearness of the point of intersection of the locus and the x- axis to (-1, 0) point. • The phase margin of the system Φm is also used to describe the closeness of the Nyquist locus to the system for different values of gain constant. • The frequency at which the magnitude is unity is called the gain cross over frequency. The phase margin is the additional phase lag at the gain cross-over frequency to make the locus pass through the point (-1, 0), so that the system is on the verge of instability.
- 14. REPRESENTATION OF THE PLOTTING Stable system a>1,ᴓm is postive Stable system a<1,ᴓm is negative
- 15. REFERENCES :- • P. Antsaklis, J. Baillieul, “Special issue on networked control systems”, IEEE Transactions on Automatic Control, Vol. 49, pp. 1421–1597, 2004. • M.Y. Chow, “Special section on distributed network-based control systems and applications”, IEEE Transactions on Industrial Electronics, Vol. 51, pp. 1126–1279, 2004. • W. Zhang, M.S. Branicky, S.M. Phillips, “Stability of networked control systems”, IEEE Control System Magazine, Vol. 21, pp. 84–99, 2001. • D.S. Kim, Y.S. Lee, W.H. Kwon, H.S. Park, “Maximum allowable delay bounds of networked control systems”, Control Engineering Practice, Vol. 11, pp. 1301–1313, 2003.