MATERIALS AND METHODS

The basic block diagram of the proposed scheme is shown in Fig. 1

To have a better feeling of the method of analysis, the primitive machine model of the synchronous motor is drawn and it is shown in Fig. 2

In the following analysis, saturation is ignored but provision is made for inclusion of saliency and one number of damper winding on each axis. Following Park’s transform, a constant stator current of value is at a field angle ‘β’ can be represented by direct and quadrature axis currents as:

(1)

(2)

Designating steady state value by the subscript ‘0’ and small perturbation by Δ, the perturbation equations of the machines are:

(3)

(4)

Fig. 1:	Drive configuration for open-loop current-fed synchronous motor control

Fig. 2:	Primitive machine model of a synchronous motor

The transformed version of Eq. 3 and 4 are:

(5)

(6)

The generalized expression for electromagnetic torque of a primitive machine model is an established one and it is expressed as:

(7)

Small signal version of torque equation in time domain is expressed as:

(8)

Equation 8 after being transformed takes the shape as given by:

(9)

To tackle Eq. 9 in an easier form, it is expressed as:

(10)

where,

(11a)

(11b)

(11c)

(11d)

(11e)

The small perturbation model of the transformed voltage balance equations of F-coil, KD-coil and KQ are expressed as:

(12)

(13)

(14)

As the damper winding on d-axis and q-axis are short-circuited within themselves, ΔU_kd=0 and Δu_kq = 0. So in transformed version Δu_kd(s) = 0 and Δu_kq(s) = 0 as shown in Eq. 13 and 14. Furthermore in general, the voltage fed to the field winding is fixed. That is why Δu_f (s) = 0 as shown in Eq. 12.

From Eq. 12 and 13 it yields:

(15)

where,

(16a)

(16b)

(16c)

(16d)

(16e)

(16f)

Similarly Eq. 12 and 13 yields:

(17)

where,

(18a)

(18b)

From Eq. 14 it is obtained:

(19)

where,

(20a)

(20b)

(20c)

(20d)

Substituting Eq. 15, 17, 19 in Eq. 10, it yields:

(21)

Equation 21 can be re expressed as:

(22)

where,

(23a)

(23b)

(23c)

(23d)

(23e)

(23f)

(23g)

(23h)

(23i)

(23j)

(23k)

(23l)

Substituting the expressions for Δi_d (s) and Δi_q (s) from Eq. 3 and 4 in Eq. 22 we have:

(24)

Equation 24 can be re-expressed as:

(25)

where,

(26a)

(26b)

(26c)

(26d)

The torque dynamic equation of a synchronous motor can be written as:

(27)

where, ω = motor speed in mechanical rad sec^-1. J = polar moment of inertia of motor and load (combined). The small change in speed ‘ω’ equal to Δω can be related to small change in field angle, Δβ as given by:

(28)

The negative sign in equation physically indicates a drop in speed (ω) due to increase in field angle (β).

Based on Eq. 28, the following expression can be written:

(29)

The small-perturbation model of Eq. 27 can be written as:

(30)

Combining Eq. 29 and 30, it yields:

(31)

The transformed version of Eq. 31 with initial condition relaxed, comes out to be:

(32)

Substituting the expression for ΔT_e (s) from Eq. 25 in Eq. 32 we have:

(33)

Equation 33 gives, after manipulation, a transfer function T (s) expressed as:

(34)

where,

(35a)

(35b)

(35c)

(35d)

(35e)

(35f)

The expression for:

in Eq. 34 gives a light in the direction of analysis of steady state stability criterion. In fact the denominator polynomial of right-hand side of Eq. 34, set to zero, becomes the characteristic equation. A Routh-Herwitz analysis of the characteristic equation will ultimately lead to the status of steady state stability. The detailed R-H analysis, considering a suitable example and the associated interpretation of the results are presented in the next section.

RESULTS

The denominator polynomial of the right hand side of Eq. 34 being set to zero, takes the form as given by:

(36)

The Routh-Herwitz table for Eq. 36 is developed taking suitable values of the polynomial co-efficient (K₁ to K₆). The machine data based on which the K₁ to K₆ are calculated are presented in Appendix. The calculated values of K₁ to K₆ are as follows:

The coefficients of the first column of the R-H table do not show any change of sign. Therefore the steady state stability is assured.

Generally it is observed that at no-load, a synchronous motor without damper winding does not satisfy steady state stability criteria. At the no load condition with β₀ = 0, the transfer function expressed in Eq. 34 for a C.S.I. fed synchronous motor with damper winding is modified for the motor without damper winding as given by the expression:

(37)

The denominator of the right hand side of the Eq. 37, being set to zero leads to a characteristic equation expressed as:

(38)

where,

(39)

Equation 39 clearly shows that the pair of roots of the characteristic equation are pure imaginary and complex conjugate. Such roots lead to a marginally stable or undamped system which does not clearly indicate the steady state stability criteria.

However, with damper winding considered and at no load (β₀ = 0), the denominator polynomial of the right hand side of the Eq. 34 being set to zero is made subjected to R-H. analysis. With the R-H table formed the 1st column coefficient of that table do not show any change in sign. This assures that at no load also, the motor satisfies the steady state stability criterion. The values of the coefficient K₁ to K₆ for no load condition are calculated based on the machine data which are also present in the Appendix. The calculated values of K₁ to K₆ for no load condition are as follows:

DISCUSSION

From the study (Sergelen, 2007) we can observe that the study deals with the mathematical model of a synchronous motor energized through a power electronic converter and ultimately develops simulation results of time response of phase currents/phase voltages and motor speed. (This results are not in line with the results submitted in the present study because the objectives are different. However, a very good similarity is observed involving the fact that the study uses u-i relations and generalized torque equations in the primitive machine model and the same type of equations have also been used as a part of problem formulation in the present study.

Another comparison of the present study has been carried with Najafi and Kar (2007). In this study the main similarity lies in the method of modeling using voltage balance equations in the d-q model. As such there is no similarity in the results presented in the study, “Effect of Short-Circuit Voltage Profile on the Transient Performance of Saturated Permanent Magnet Synchronous Motors” and the present study. However, a minute observation on the literature on that study clearly indicates that the transient response on the performance figure takes torque angle, phase currents etc can lead to specific inference on stability criteria. Though the present study deals with the steady state stability analysis of a three phase synchronous motor the said similarity appears to be important as it falls in the broad category of stability studies on synchronous motors.

The results presented in the study, are similar as by the study of Kondo and Matsuoka (2007) uses block diagram approach for stability analysis of a permanent magnet Synchronous Motors for Railway Vehicle Traction in the Sudden Line Voltage Change and the formulation presented in the present study can easily lead to block diagram modeling in ‘S’ domain. Secondly both the studies are ultimately interested in the stability analysis but have some dissimilarity existing because the motor used in their work is permanent magnet synchronous motor and the study presented here uses synchronous motor consisting of physical field winding system. As a third aspect it is observed that, the study uses the concept of real part of the Eigen values to be negative for the assessment of the stability and the study presented here uses the same concept for stability assessment but in a slightly different form, that is on relative stability criteria based on Routh-Herwitz array.

The well known research study by Slemon et al. (1974) clearly indicates that the small perturbation model can be applied to a synchronous motor drive with current source inverter (the motor being without damper winding) for determination of steady state stability criteria. This particular study is in contrast of the proposed study because the authors of the present study have considered a synchronous motor with damper winding.

Jung et al. (2004) had drawn the attention to the authors of the present study. This study involves the stability improvement of a V/f controlled PWM inverter fed Induction motor using a dynamic current compensator. This particular study has been considered in contrast of the proposed study because induction motor can be treated as a singly excited magnetic field system where as synchronous motor considered by the authors of the present study can be treated as a multiple excited magnetic field system. Hence the opposition in the research philosophy is clearly observed.

The study of Tanneru et al. (2007) is related to induction motor as a component of the power system and transient modeling for such motor leading to the assessment of transient stability has been considered as the objective of the above study. This literature review is in contrast of the study carried by Tanneru et al. (2007) does not involve any analysis related to steady state stability criteria but it is clear that the model associated with the dynamics of Induction motor can be easily modified to be used in determination of the steady state stability criteria of the system.

CONCLUSIONS

This study highlights the following salient concluding points:

APPENDIX

List of symbols:

The machine data:

When machine is at load:

When machine is at no load: