Modeling and Simulation of Current Source Inverter Fed Synchronous Motor in Complex Frequency Domain Taking the Transition Zone From Induction Motor to Synchronous Motor Mode into Account

Modeling of synchronous motor plays a dominant role in designing complicated drive system for different applications, especially large blower fans etc for steel industries. As synchronous motor has no inherent starting torque generally it is started as an induction motor with the help of a damper winding and it pulls into synchronism under certain conditions. The present paper exactly concentrates on this particular zone of transition from induction motor to synchronous motor mode for a current source inverter fed synchronous motor drive system. Due to complexity of synchronous motor in terms of number of windings and finite amount of air gap saliency, direct modeling of such transition zone in time domain becomes cumbersome at the first instance of modeling. That is why the modeling in complex frequency domain (s-domain) has been taken up using small perturbation model. Such a model clearly shows role of induction motor as noise function or disturbance function with respect to the open loop block diagram of synchronous motor. Such finding can be quantized in terms of important results and that is done in the present paper such that the results can help the designer for the successful design of a synchronous motor drive system.


INTRODUCTION
Many constant speed applications such as fans, fuel pump and compressors comprising a considerable amount of total electrical appliances (Isfahani and Vaez-Zadeh, 2011) basically need a 3 phase synchronous motor.Even though permanent magnet synchronous motors are widely used in such applications, current source inverter fed normal synchronous motors can also be applied in many constant speed applications (Knight and McClay, 2000;Weifu et al., 2012).The steady state stability study of a current source inverter fed synchronous motor was basically initiated in 1974 (Gordon et al., 1974) and after this as an extension, (Chattopadhyay et al., 2011) presented a detailed analysis of a current source inverter fed synchronous motor drive system taking damper windings into account in 2011 (Chattopadhyay et al., 2011).So far the research accuracy of the paper by (Chattopadhyay et al., 2011) is covered; it is not clear that exactly what is happening in the transition zone when the machine is jumping from induction motor action to synchronous motor action.Again the concentration on such detailed aspect is a matter of long discussion and in this context many researchers have tried to put sufficient light on the matter.The research paper (Xianhao et al., 2006) explains the analysis of magnetic fields and temperature fields for a salient pole synchronous motor in the process of steady state.They have used the d-q model of the synchronous motor but the role of field winding in transition from induction motor to synchronous motor is not reflected in the mathematical model.
A similar observation is valid on the other work Wang and Ren (2003), and it represents a good state variable model and its mathematic simulations in time domain.The research paper (Sergelen, 2007;Najafi and Kar, 2007) carries important works on the mathematical modeling of a salient pole synchronous motor supplied by a frequency converter and also the effect of short circuit voltage profile on the transient performance of permanent magnet synchronous motors .An important work on non-linear control of an inverter motor drive system with input filter (Marx et al., 2008) draws attention.In this paper the author has given a detailed signal analysis of the DC-link voltage stability.
Another interesting paper by Das and Casey (1999) and Al-Ohaly et al. (1997) clearly portrays the critical aspects of starting a large synchronous motor.Even though this particular work does not involve much mathematical analysis but the range of the slip presented in this paper with reference to pull in torque of a synchronous motor really may help a designer to select a particular synchronous motor for any specific application.
Based on the above said literature review, to the best of the authors understanding it reveals that researchers have not put sufficient light on the fact that exactly what happens to the mathematical model of a synchronous motor in time domain or complex frequency domain during the period when field winding is disconnected from the external resistance (generally 6-7 times of main field winding resistance to avoid effects due to George's phenomenon) and immediately thrown to the DC source.In the authors opinion investigation in such direction becomes very much crucial because before connecting the field winding to the DC source characterization of the machine can be done as 3 phase induction motor.Therefore, it will be very logical to get the semi induction semi synchronous machine character to be reflected in the resultant mathematical model.This particular work has been presented by the authors in the present paper.The strategy of formulation at this stage may not be necessary to feel the importance of the problem.But in the next section the mathematical methods will be explained in detail.

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 mac0068ine 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: Designating steady state value by the subscript '0' and small perturbation by∆, the perturbation equations of the machines are: The transformed version of Eq. ( 3) and ( 4) are: ( ) ( ) The generalized expression for electromagnetic torque of a primitive machine model is an established one and it is expressed as: Small signal version of torque equation in time domain is expressed as: Equation ( 8) after being transformed takes the shape as given by: To tackle Eq. ( 9) in an easier form, it is expressed as: where, ( ) The small perturbation model of the transformed voltage balance equations of F-coil, KD-coil and KQ are expressed as:  As the damper winding on d-axis and q-axis are shortcircuited within themselves, ∆U kd = 0 and ∆U kq = 0.So in transformed version ∆U kd (s) = 0 and ∆U kq (s) = 0 as shown in Eq. ( 13) and ( 14).Furthermore in general, the voltage fed to the field winding is fixed.It is a well known fact that a synchronous motor cannot start for itself and the easiest way to start a synchronous motor is to start it as an induction machine with the help of damper windings.But the problem is that we have to investigate what will be status of field winding of the synchronous motor when the damper winding is in action.As already the winding was physically embedded (existing), and during the running of the machine one cannot take it out.In other words when damper winding is in action field winding effect has to be inactivated.Such inactivation may be done by the following methods: (a) Field winding completely Open Circuit (b) Field Winding Short Circuit in itself The status of field winding in (a) can be looked upon as a transformer whose primary winding constitutes of 3 phase armature winding supplied from 415 V (L-L) ac and whose secondary winding is the field winding being open circuited.As generally in a normal synchronous machine of normal design N f /N a >> 1, where N f is number of field windings and N a is number of armature windings.The induced voltage in the open field terminal will be large and it may lead to hazardous conduction so far as operator safety is concerned.Hence this case is rejected.
Status of the field winding in (b): The induced emf in field winding due to transformer action will produce a single phase alternating current and in turn will produce a pulsating field in field winding.It is well known that a pulsating field m.m.f can be resolved as a combination of forward rotating and backward rotating m.m.f magnetic fields (strengths of each resolved component is half of the original pulsating m.m.f).The effect of backward rotating magnetic field will produce a torque opposite to the (asynchronous/induction) motor torque and it will dominate at some value of slip.Hence a situation may arise and motor may stall due to the negative effect of backward component.This phenomenon is known as George's phenomenon.
Hence such case cannot be completely accepted.However there is some remedial method.The field winding may be closed through an external resistance which is about 6-7 times of original field resistance; such that the magnitude of short circuit current diminishes and as a result effect of resolved backward component will be less or reduced.

What happens to change on field voltage (∆U f ):
In the current research problem ∆U f cannot be equal to zero because originally it was an induction motor with the field winding short circuited in it or closed through an external resistance of large value and at a later stage it was pulled into synchronism when dc supply is fed to the winding.
Quantitatively, ∆U f should depend on a particular property of induction motor and that property must be 'SLIP'.Here the technique of mathematical modeling appears as a novel approach and this approach forms the foundation of the proposed analysis.The proposed modeling considers that field winding is closed within itself.In other words the presence of external large resistance has not been considered in modeling to make the mathematical treatment comparatively easy.However it does not affect the accuracy of the system as the external resistance can be lumped or clubbed with the field winding.From Eq. ( 12) and ( 13) it yields: where, Similarly Eq. ( 12) and ( 13) yields: where, ( ) ( ) From Eq. ( 14) it is obtained: where, Substituting Eq. ( 15), ( 17), ( 19) in Eq. ( 10), it yields: Equation ( 21) can be re expressed as: Substituting the expressions for ∆I d (s) and ∆I q (s) from Eq. ( 3) and (4) in Eq. ( 23), we have: s s e ms m s ms m i s ns n s n s n i s T s s l s l s l s l Equation ( 24) can be re-expressed as: where, x s x s x s x l s l s l s l The torque dynamic equation of a synchronous motor can be written as: where, ω = Motor speed in mechanical rad./sec.J = Polar moment of inertia of motor and load (combined) The small change in speed 'ω' equal to ∆ω can be related to small change in field angle, ∆β as given by: 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: The small-perturbation model of Eq. ( 27) can be written as: Combining Eq. ( 29) and ( 30), it yields: The transformed version of Eq. ( 31), with initial condition relaxed, comes out to be: Substituting the expression for ∆Te(s) from Eq. ( 25) in (32), we have: The block diagram representation of the system obtained from the above equation is shown in Fig. 3.
The disturbance function can be taken separately and a detailed analysis in complex frequency domain is carried out as follows: where, and: ( ) ( ) Hence: where, (j ) F (j ) F (j ) (j ) where, where, The overall mathematical treatment basically interprets the role of induction motor action just before switching to synchronous motor action in a convenient mathematical from in the complex frequency domain.The motivation to formulate this problem in s domain (complex frequency domain) is not intentional rather it is a natural tendency because this paper may be treated as an extension of Chattopadhyay et al. (2011) in that paper the whole intention of the author were to investigate the steady state stability aspects using small perturbation model and that is why the formulation was can be treated as transfer function of the pure synchronous motor behavior of the same physical synchronous m/c similar to D(jω) formulation, T(s) can be converted to T(jω) after substitution of s = jω in the said function.The magnitude and phase angle plot of T(jω) against 'ω' are shown in Fig. 5 and 6.Even though the plots in Fig. 3 to 6 are based on the equation they need some physical explanation and the corresponding interpretation are presented in the next section.

RESULTS AND DISCUSSION
Magnitude of the |D(jω)| (Fig. 4) plotted against ω shows inverse functions and also a family of plot is obtained by varying slip.It is already known that: It may be recalled that during the period of induction machine action {ˠ # {˟{ + H˟ $ { ∆ {˟{ + ˘%{˟{ = ∆ˠ {˟{, where each function has been expressed in the equations.Hence F 3 (s) can be treated as an equivalent to additional transformed load torque.This additional function has appearance due to the role of the induction motor when synchronous motor has started as an induction machine with the help of damper windings we know that roughly induction motor torque is an increasing function of slip in the steady state zone.The transformations from time domain has been obtained using Laplace operator and ultimately the formulation is obtained in the complex frequency domain (in terms of kernel s).Hence the physical transformation from synchronous machine to induction machine during the starting period should not be confused with the mathematical transformations from time domain to Laplace domain and hence the role of slip remains same in Laplace domain as in time domain.The function {ˠ # {˟{ + H˟ $ { in Laplace domain appears due to the induced dynamics of the physical synchronous machine when it is exactly running as synchronous machine.Therefore the parameter slip , it has been observed that {ˠ # {˟{ + H˟2 at s = jω, has a numerator polynomial containing highest power ω 3 and denominator ω 5 hence such functions should show dominance near zero frequency.Now let us separately concentrate on the function F 3 (s) which has a real part containing highest power ω 2 and negative imaginary parts containing highest power ω 3 .Hence D(jω) which is product of F 3 (jω) and Figure 5 represents the plot of ∟ D(jω) against ω based on Eq. ( 65).It may be reminded that in Eq. ( 65) the angle of F 3 (s) at s = jω can be estimated very roughly as ˮIJ # Ӡ ӡ such rough expression will help the designer to predict the nature of the variation.The above said variation has a decaying nature with respect to ω. however the other part of the D(jω) function which is T(jω) = # { { at s = jω has an oscillatory behavior of its phase angle with respect to ω.This oscillatory behavior of ∟ T(jω) vs ω can be easily interpreted because this function can be treated roughly as tan -1 (ω) and that is why it is bounded between some specific values of angle (3.14).This behavior is plotted in Fig. 7.The plot of ∟ D (jω) vs ω involves the behavior of ∟ T (jω) vs ω but the dominance of ∟ F 3 (jω) is more, that is why the overall plot of ∟ D(jω) vs ω shows a dragging nature and it is plotted in Fig.

CONCLUSION
• Role of Induction motor in Laplace domain appears as an equivalent "system" consisting of multifrequencies.• Filtering of the equivalent "NOISE" may not be needed due to natural attenuation.• To avoid or minimize George's phenomenon during the transition zone, inclusion of external resistance in the field circuit basically leads to a complicated mathematical formulation, but clubbing this resistance along with the original field winding resistance develops the problem formulation in a more straight forward manner.• With reference to Fig. 3, the time-frequency contour of the function (in s-domain), D(s) can be developed using STOCKWELL-Transform(S-Transform) and such contour will be able to provide more design information to the designer.

List of symbols:
i d = Current in the D coil in p.u. i q = Current in the Q coil in p.u. i f = Current in the F coil in p.u. i kd = Current in the d axis damper coil (KD) in p.u. i kq = Current in the q axis damper coil (KQ) in p.u. L d = Self-inductance of D coil in p.u. L q = Self-inductance of Q coil in p.u. L md = Mutual inductance along d axis in p.u. L mq = Mutual inductance along q axis in p.u. L ff = Self-inductance of F (field) coil in p.u R f = Resistance of the F (field) coil in p.u. L kd = Self-inductance of the KD coil in p.u. R kd = Resistance of the KD coil in p.u L kq = Self-inductance of the KQ coil in p.u. R kq = Resistance of the KQ coil in p.u β = Angle between the field (rotor) m.m.f.axis and armature (stator) m.m.f.axis The machine data are given as (Xianhao et al., 2006): J = 8 p.u. L d = 1.17 p.u. L md = 1.03 p.u. L q = 0.75 p.u. L mq = 0.61 p.u. L kkd = 1.122 p.u. L kkq = 0.725 p.u. L ff = 1.297 p.u. R kd = 0.03 p.u. R kq = 0.039 p.u. R f = 0.0015 p.u.

9 )Fig. 1 :
Fig. 1: Drive configuration for open-loop current-fed synchronous motor control of turns in field winding Na = Number of turns in armature winding: k = (N f /Na)*(415/√3) N f = Number of turns in field winding Na = Number of turns in armature winding

Fig. 3 :
Fig. 3: Block diagram representation of the system with disturbance function D(s)

Fig. 6 :
Fig. 6: Magnitude of the transfer function plotted against ω finalized in's' domain.As an extension to such motivation it reveals that investigations can be made on the same machine just before switching to synchronous motor action (running as an induction motor).The overall model in complex frequency domain in the form of small perturbation model taking the role of induction motor into account does not lead to stability assessment model, however the nature if the function D(s) can be investigated in the marginal condition after substituting s = jω.After such mathematical substitution the function D(jω) involves magnitude and phase angle plot against 'ω' and such behavior are similar to analog filter characteristics.Hence the role of induction motor action for a physical synchronous motor may be looked upon as an analog filter behavior in the pure frequency domain in the concerned mathematical model.Those magnitude and phase angle plots of D(jω) are shown in Fig. 4 and 5.As the role of D(s) can be looked upon as a disturbance function in relation to the model given by equation D(S) =

Fig. 7 :
Fig. 7: Angle of the transfer function plotted against ω from the view point of engineering conception should not affect this function and furthermore it is well known that slip is the indication of an asynchronous machine.Explanation of the magnitude plot of D(jω) vs ω at a particular value of slip: Again we shall recall the function # { { . That is why in the plot of |D(jω)| vs ω shows inverse function at a particular value of sip with high dominance around zero frequency.
7. With reference to equation ∆β(jω) = T(jω ) ∆T L -D(jω) at this stage only the interpretation of the plot of T(jω) remains as pending because the other 3 plots (Fig. 4 to 7) have been interpreted.The pending interpretation for Fig. 6 is as follows: polynomial in ω, having highest power 5. Hence at the high frequency region, |T(jω)| decreases very fast.This plot is shown in Fig. 6.