The switched reluctance motor (SRM) has come a long way since its inception about forty years ago. Its rugged and dynamic nature has made it one of the main research topics amongst engineering enthusiasts making it grow by leaps and bounds. Some look at it as an exciting alternative to the well established induction motor. As with all devices, it has its drawbacks and one of them is the mysterious behavior of its inductance profile. The inductance profile has been found to be non-linear by nature and contemporary research has deemed this characteristic of the motor as unfavorable. The purpose of this paper is to make a study on the various factors that contribute to the SRM's inductance profile and propose a way to translate this negative sentiment about the motor into something more positive. Furthermore, with relevant simulation studies, we propose to find a vantage point for the SRM and shed some light on the subject matter so that the engineering society can appreciate this inherent property of the motor.
1.2 SRM Characteristics
SRMs are unique in construction. They have rotors and stators just like any other induction motor. They deviate from other motors in the way the rotors and stators are constituted. There are no windings or magnets on the rotor but only in the stators. A schematic diagram is shown in Figure 1.
Figure1: A schematic diagram of a 8/6 SRM with 4 phases [ HYPERLINK \l "Moh07" 1 ]
Torque is produced in the SRM due to the varying reluctance along the magnetic circuit of the system. Energizing a stator phase compels a stator pole pair to move and align toward a rotor pole pair. In magnetic terms, the rotor-stator pair will align themselves in a configuration that gives the minimum reluctance and this in turn gives rise to torque in the system. Consecutive energization of successive phases produces movement in the rotor due to the development of torque as a byproduct 2]. The equation for electromagnetic torque T is represented as:
Where W represents co-energy and is represented in terms of stator flux and phase angle :
Moreover, flux linkage is composed of inductance L and current i to produce the following formula:
Utilizing the above equation, it is possible to extract inductive values for a system if the flux linkage and current characteristics are given. Moreover, applying equation (1.3) in (1.2), the expression for co-energy can be found:
Using the above equations, a torque formula can be derived by simplification as stated in [ HYPERLINK \l "RKr01" 3 ]:
EMBED Equation.DSMT4 MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 5)
and total torque T is given by:
EMBED Equation.DSMT4 MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 6)
Where the differential inductance is given by:
EMBED Equation.DSMT4 MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 7)
We can arrive to Equation 1.5 only when we consider that inductance is linearly varying with rotor position for a given value for current. But in real life, this is not the case since the inductance behaves in a non-linear manner.
An ideal inductance profile of an SRM is shown in Figure 2. Synchronization is essential when appropriate excitation currents are passed on to the stators relative to the rotor's angular position.
Figure SEQ Figure \* ARABIC 2: Ideal inductance profile CITATION Fer06 \l 1033 [HYPERLINK \l "Fer06"4]
The reluctance of the flux path between two opposite stator poles varies as a pair of rotor poles rotates in and out of alignment. The inductance of a phase winding is a maximum when the rotor is aligned and minimum when unaligned. Similar to an electrical circuit where current chooses a path with least resistance, a magnetic analogy can be derived while describing the events occurring in the SRM. Since reluctance is inversely proportional to inductance analogous to resistance in an electrical circuit, the rotor teeth will always tend to align with an energized phase in order to minimize the reluctance path CITATION Rak10 \l 1033 [HYPERLINK \l "Rak10"5].
It is clear from literature CITATION Vik08 \l 1033 [HYPERLINK \l "Vik08"2] to CITATION Rak10 \l 1033 [HYPERLINK \l "Rak10"5] that the phase inductance L of a SRM can be determined if we know the values for current I and rotor position θ, but the relationship is highly non-linear by nature.
To give a clearer picture, the SRM's working principles can be summarized by Table 1 which was adopted from CITATION Rak10 \l 1033 [HYPERLINK \l "Rak10"5].
Aligned Position
Unaligned Position
θ = 0, 180°
θ = ± 90°
Maximum Inductance
Minimum Inductance
Magnetic circuit likely to saturate
Magnetic circuit unlikely to saturate
Zero torque and stable equilibrium
Zero torque and unstable equilibrium
Table SEQ Table \* ARABIC 1: Properties of the SRM
If the behavior of the current were to be superimposed with that of the inductance, the curves would look like the ones in Figure 3. But we need to bear in mind that these curves represent a SRM that behaves ideally.
Figure SEQ Figure \* ARABIC 3: Ideal Inductance and Current Relationship in a SRM CITATION The10 \l 1033 [HYPERLINK \l "The10"6]
In stark contrast to the ideal behavior, an attempt to depict the real behavior of the SRM has been made in Figure 4. In real life implementation, even if a current profile shown in Figure 3 is achieved, an immediate change in inductance is not accomplished due to magnetic and inductive effects that try to resist such change. By the time a negative gradient in the inductance profile is attained, it will be time to power up the stator again CITATION Fer06 \l 1033 [HYPERLINK \l "Fer06"4].
Figure SEQ Figure \* ARABIC 4: Real behavior of phase inductance and phase current in a 6/4 SRM CITATION Fer06 \l 1033 [HYPERLINK \l "Fer06"4]
1.2.1 Factors contributing to non-linear inductance
There has been a multitude of literature discussing the causes behind the non-linear behavior of the SRM. In the beginning the researchers found it intriguing, but as time passed by it became more difficult to explain this property. Initially, when researchers found ripples in the torque profile, they condemned the inductance profile of the motor. But later it was found that the presence harmonics in the torque caused the ripples. This was attributed to the sharp switching of phase to phase voltages CITATION FSO01 \l 1033 [HYPERLINK \l "FSO01"7].
Other noteworthy causes are:
In the aligned position, the stator-rotor air-gap may contribute to variations in the slope of the inductance profile of the SRM, especially the maximum inductance CITATION MEh97 \l 1033 [HYPERLINK \l "MEh97"8].
Mutual inductance between phases can give rise to EMFs into non-active windings due to commutations in adjacent phases. This is undesirable as it tries to inhibit the motor's performance CITATION RMD81 \l 1033 [HYPERLINK \l "RMD81"9].
1.2.2 Equivalent Circuit and voltage equation for the SRM
According to CITATION RKr01 \l 1033 [HYPERLINK \l "RKr01"3] an adoption of a single phase equivalent circuit for the SRM can be made and the circuit is shown in Figure 5.
Figure SEQ Figure \* ARABIC 5: Equivalent circuit for single phase of SRM CITATION RKr01 \l 1033 [HYPERLINK \l "RKr01"3]
As stated in CITATION Skv02 \l 1033 [HYPERLINK \l "Skv02"10], the voltage balance equation for current flowing through stators in one phase of a SRM is written as:
EMBED Equation.DSMT4 MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 8)
Where Vph is the DC phase voltage, i is the instantaneous phase current, R is the resistance of the winding and EMBED Equation.DSMT4 represents the flux on the coil which is dependent on current and phase angle. From equation (1.3) we can arrive to the following expression by performing a derivative on the flux linkage CITATION FSO01 \l 1033 [HYPERLINK \l "FSO01"7]:
EMBED Equation.DSMT4 MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 9)
Hence the resulting voltage equation for a single phase stator of SRM becomes:
EMBED Equation.DSMT4 MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 10)
In the above equation, the three terms on the right hand side represent resistive voltage drop, inductive voltage drop and induced e.m.f. respectively CITATION RKr01 \l 1033 [HYPERLINK \l "RKr01"3] and it depicts the voltage equation for a single phase in a SRM.
1.2.3 Power and energy equations for a SRM
To understand the energy conversion process in a SRM, we need to establish a power balance relationship. Multiplying equation (1.10) by i on both sides, an expression for instantaneous input power Pi can be derived CITATION Skv02 \l 1033 [HYPERLINK \l "Skv02"10]:
EMBED Equation.DSMT4 MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 11)
Where ω = rotor's angular speed and combining with equation (1.5) we get the following abbreviated version of the equation:
EMBED Equation.DSMT4 MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 12)
In equation (1.11) the first term represents power loss in the stator winding, the second term denotes the rate of change of magnetic stored energy and the third term is the mechanical output power. In a SRM, the rate of change of magnetic stored energy is always more than the electromechanical energy conversion term. The most effective use of energy is made when the current is kept constant during a positive dl/dÏ´ slope. In the process, the magnetic stored energy may not be lost and can be retrieved by the source if appropriate power converters are used. Figure 6(a) depicts how energy is distributed in a linear SRM when it is not under magnetic saturation and Figure 6(b) shows when it works under magnetic saturation CITATION Skv02 \l 1033 [HYPERLINK \l "Skv02"10].
Figure SEQ Figure \* ARABIC 6: Energy partitioning in (a) linear SRM and (b) non linear SRM CITATION Skv02 \l 1033 [HYPERLINK \l "Skv02"10]
This is one of the reasons why energy ratio is used instead to evaluate a SRM's performance. It is defined by:
EMBED Equation.DSMT4 MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 13)
Where W is the energy converted to mechanical work and R is the energy returned to the DC supply. Energy ratio is synonymous to power factor for AC machines CITATION Skv02 \l 1033 [HYPERLINK \l "Skv02"10].
According to CITATION RKr01 \l 1033 [HYPERLINK \l "RKr01"3], the last term of equation (1.11) is termed as air gap power pa, and has another expression that is comprised of angular speed of the rotor ω and the torque T and is depicted by the following equation:
EMBED Equation.DSMT4 MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 14)
Equating the two gives us the equation for torque which was shown earlier in equation (1.5):
EMBED Equation.DSMT4 MACROBUTTON MTPlaceRef \* MERGEFORMAT SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTSec \c \* Arabic \* MERGEFORMAT 1. SEQ MTEqn \c \* Arabic \* MERGEFORMAT 15)
CHAPTER 2: RESULTS AND DISCUSSION
2.1 Simulation Characteristics
The model in Simulink was used to produce a variety of datasets that were stored inside variables in the Matlab workspace. This was achieved by varying the supplying voltage to the controller/ converter which in turn would provide varying phase currents to the stators of the switched reluctance motor. Voltages of 240V, 230V, 220V, 210V and 200V were used to determine rotor angle, inductance, and rate of change of inductance with respect to angle, mechanical output power, flux linkage, stored field energy and currents in both the phases. The values were tapped from relevant points in the circuit and stored inside variables. Table 2 shows mnemonics used to help generating the graphs in the following section. The suffix attached to the variable name corresponds to a particular voltage value for ease of coherence between the data.
V
Suffix
Ï´
L
EMBED Equation.DSMT4
I1
I2
Torque
Mechanical Power
Flux Linkage
Stored field energy
240
1
teta1
L1
dL1
I1
I12
T1
mpower1
linkage1
W1
230
2
teta2
L2
dL2
I2
I22
T2
mpower2
linkage2
W2
220
3
teta3
L3
dL3
I3
I32
T3
mpower3
linkage3
W3
210
4
teta4
L4
dL4
I4
I42
T4
mpower4
linkage4
W4
200
5
teta5
L5
dL5
I5
I52
T5
mpower5
linkage5
W5
Table SEQ Table \* ARABIC 2: Variables used in the simulation
2.2 Graphs generated from the Simulink model
2.2.1 Current
The Simulink model produced acceptable outputs that adhered to a 2 phase 4/2 switched reluctance motor. The current waveforms - two complete cycles- for phase 1for the five different voltage supplies have been depicted in Figure 7. The values for currents have been plotted against their corresponding rotor angles (Ï´). Linking the legend to the graphs, it can be seen that peak values for current is achieved with a higher voltage source, i.e. I1 > I2 > I3 > I4 > I5.
Figure SEQ Figure \* ARABIC 7: Phase 1 currents for the five voltages
In Figure 8, current waveforms for two phases have been plotted against rotor angle for one particular value of voltage, i.e. 240V. It can be seen that sharp switch off of current is not achieved due to the presence of commutational delays between the two phases. This is also a typical characteristic of the SRM. Moreover, the presence of ripples in the waveforms is an inherent characteristic contributed by the type of converter operating mode used- in this case it is the current control mode where current hysteresis is subjected to change by the regulator.
Phase 2
Phase 1
Figure SEQ Figure \* ARABIC 8: Currents in the two phases vs. Theta
2.2.2 Inductance
Figure 9 shows the inductance curves plotted against corresponding rotor angles. The behavior is typical of SRM inductance profiles. The phase inductance at any rotor position and phase current can be calculated from Eq. 1.3, using equation L(i,θ)= ψ(i,θ)/i. Fig.2.4 shows plot of L vs.rotor position. It can be seen that with increasing voltage supplies, which in turn translates to the input of higher currents, the magnetic saturation leads to a drop in effective inductance.
Figure SEQ Figure \* ARABIC 9: Inductance profiles for the five voltages ( L vs. Theta)
In Figure 10, the phase 1 current and inductance curves have been superimposed to emphasize on the relationship between the two. Since the current values exceed those of inductance, both the variables have been normalized and plotted, making their juxtaposition fit the same scale.
Figure SEQ Figure \* ARABIC 10: Phase 1 current superimposed with Phase 1 inductance
2.2.3 Flux Linkage
Figure 11 shows the measured flux linkage for phase winding against theta which also can be interpreted as a function of phase current at 240V. Fig. 12 shows the same flux linkage as a function of rotor position. These two fiures demonstrate the highly nonlinear relationship of flux linkage with phase current and rotor position. This intrinsic behavior is attributed to the magnetic saturation that the SRM undergoes in order to maximize utilization of the magnetic circuit, and hence, the flux-linkage is a nonlinear function of stator current and rotor position, as described by Skvarenina et al. CITATION Skv02 \l 1033 [HYPERLINK \l "Skv02"10]
Figure SEQ Figure \* ARABIC 11: Flux linkage VS Theta
Figure SEQ Figure \* ARABIC 12: Flux Linkage VS Current
2.2.4 Rate of change of inductance (dl/dÏ´)
At this point, we arrive at the main focus of this research paper. Exploring for new information on the inductance profile demanded that a unique characteristic of it be investigated. Figure 13 shows a complete cycle for dl/dÏ´ versus theta and Figure 14 superimposes the dl/dÏ´ profiles of all the voltage supplies to check for consistency or variations in it. In Figure 14, it can be seen that no matter what the input voltage/current rating, the overall envelope of dl/dÏ´ is more or less the same. Later, dl/dÏ´ will be juxtaposed with other variables to see their inter-relationships.
Figure SEQ Figure \* ARABIC 13: Rate of change of inductance w.r.t. theta VS Theta
Figure SEQ Figure \* ARABIC 14: Rate of change of inductance for 5 different voltage values
2.2.5 Current, inductance and rate of change of inductance (dL/dθ)
In this section it is important to highlight the relationship between the three entities. From theory of electrical circuits the following relationships between current and inducatnce can be established:
Looking at the equation for flux linkage in Equation 1.3, we can establish:
EMBED Equation.DSMT4 L α EMBED Equation.DSMT4 EMBED Equation.DSMT4
This inverse relationship exists because of the inherent nature of conductors. When current passes through a conductor, it develops the capacity to store energy in the form of a magnetic field. In another interpretation, inductance is stated as an opposing property of the conductance which is aimed to counteract the change in applied current over time. Moreover, it is also caused by the establishment of a magnetic field around a conductor CITATION All10 \l 1033 [HYPERLINK \l "All10"11].
Keeping the above description in mind, interpretation of the following graphs becomes easier. In Figure 15 the relationship between phase current and dl/dÏ´ is shown. The sharp change in dl/dÏ´ is characteristic to sharp changes in inductance. The moment inductance curve starts to move up, so does the dl/dÏ´ curve. In this region the sign of dl/dÏ´ is positive. Conversely, when the inductance curve starts to curve downwards, dl/dÏ´ attains a negative sign and its magnitude begins to decline. Values for both of them have been normalized for the sake of juxtaposition.
Figure SEQ Figure \* ARABIC 15: Inductance and dL/dθ VS Theta
A crucial behavior can be observed out from Figure 16 where dl/dÏ´ is put alongside phase current. As discussed in the previous page, feeding current into the rotor gives rise to an inductive property in it. Changes in current, as seen in the ripples at the peak, produce rapid changes in the rate of change of inductance. This shows that the inductance is trying to oppose the conductance of the rotor. The peak of dl/dÏ´ is achieved the moment when there is sharp fall in the current profile. A rise upwards in the dl/dÏ´ curve suggests that the curve of inductance has begun to rise as can be seen from Figure 17. dl/dÏ´ seizes to rise the moment the inductance profile as leveled off at the peak and when it begins to curve downwards, dl/dÏ´ falls sharply due to the change in sign.
Figure SEQ Figure \* ARABIC 16: Current and dL/dθ
Figure SEQ Figure \* ARABIC 17: Current, inductance and dL/dθ VS theta
2.2.6 Electrical torque
Figure 15 shows the characteristic output torque curve for a two phase SRM. Each lobe in the graph is contributed by one phase and therefore, since we have two phases in the motor and the torque curve shows the evidence of their presence.
Figure SEQ Figure \* ARABIC 18: Electrical torque VS Theta
The graph for phase torque is shown in Figure 16 and it follows the relationship given by equation 1.6 which is:
EMBED Equation.DSMT4
The equation states that torque is directly proportional to the square of current. Therefore, if we juxtapose the curve for torque along with the respective phase current, this relationship can be visually realized as shown in Figure 17.
Finally, in Figure 18 the electrical torque was superimposed with the two phase currents. It can be seen how the two lobes of the torque waveform coincide with their corresponding phase currents.
Figure SEQ Figure \* ARABIC 19: Phase torque VS Theta
Figure SEQ Figure \* ARABIC 20: Phase torque superimposed with phase current
Phase 1
Phase2
Figure SEQ Figure \* ARABIC 21: Electrical torque superimposed with phase currents
Phase 1 torque
Phase 1 inductance2.2.7 Phase torque and inductance
Figure SEQ Figure \* ARABIC 22: Torque and inductance
Figure 17 depicts the relationship between phase torque and inductance. The torque produced by a phase in which current flows, tends to move the rotor in such a direction so as to increase the phase inductance. In stator and rotor position terms, it is the aligned position. This means that the motoring torque can be produced only in the direction of rising inductance. The instantaneous torque is always changing, depending of the rotor position and the instantaneous phase current. Moreover, it is important to realize that the torque is independent of the direction of current flow. The motoring or braking torque production only relies on the rotor position. This accentuates the impact of switching angles of the power electronic switches for SRM control. This torque-speed characteristic of the SRM gave rise to a controlling technique where currents in the stator circuits are switched on and off in depending on the rotor position. This relatively simple type of control, the switched reluctance motor inherently begins to behave like a d.c. machine CITATION TJE89 \l 1033 [HYPERLINK \l "TJE89"12].
2.2.8 Torque, rate of change inductance and inductance (dL/dθ)
Figure 18 shows the relationship between phase torque, inductance and dl/dÏ´. The figure does not do just do justice on the implications of dl/dÏ´ on the phase torque. The graph was plotted after all the three values- phase torque, inductance and dl/dÏ´- had been normalized for the sake of comparison with each other and finding a relationship from them.
Figure SEQ Figure \* ARABIC 23: Phase torque, inductance and dl/dÏ´
From equation 1.6, we can derive that the magnitude of the instantaneous torque developed in the SRM is proportional to both I2 and dL/dθ. If the inductance is increasing with respect to the angle, and current flows in the phase winding, then the torque will be positive and the machine will operate in motoring mode. Hence, keeping an eye on the equation, it can be seen that when the motor phase is excited during a rising inductance region, part of the energy from the supply is converted to mechanical energy to produce the torque, and another part is stored in the magnetic field. If the supply is turned off during this region, then any stored magnetic energy is partly converted to mechanical energy and partly returned to the supply. However, a negative, or braking torque will be developed by the motor if the inductance is decreasing with respect to the rotor angle and, current flows in the phase winding. In this case energy flows back to the supply from both the stored magnetic energy and the mechanical load, which acts as a generator.
It can also be seen from Eq.1.6 that the sign (or direction) of the torque is independent of the direction of the current and is only dependent on the sign of dL/dθ. However, it was seen that the torque changes from positive to negative according to the sign of dL/dθ.
