THIS PPT IS ABOUT THE ANALYZE THE STABILITY OF DC SERVO MOTOR USING NYQUIST PLOT AND IN THIS PPT WE CAN ALSO SEE THE DIFFERENT CHARACTERISTICS EQUATION FOR THE DC SERVO MOTOR AND THE EXAMPLE GRAPHS ARE ALSO SHOWN IN THIS PPT AND THIS PPT IS SO USEFUL FOR THE CONTROL SYSTEM STUDENTS AND ANALYSIS OF THE EQUATIONS ARE ALSO AVAILABLE IN THIS PPT

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.