Method of torque notch minimization for quasi square wave back EMF permanent magnet synchronous machines with voltage source drive

A method for smoothing the torque characteristics of a multi-phase PM synchronous machine by controlled driving with a conventional voltage source inverter operated in accordance with a predetermined switching sequence wherein the sum of the excitation voltages applied to the incoming phase winding and outgoing phase winding are maintained constant at a predetermined value for the duration of each commutation period. In addition, the individual excitation voltages applied to the remaining phase active windings are also held constant during the commutation period.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to permanent magnet synchronous machines and, 
more particularly, to a method for smoothing the torque output of quasi 
square wave back EMF permanent magnet (PM) synchronous machines. 
2. Description of the Background 
Surface mount permanent magnet (PM) AC machines have concentrated, full 
pitch, stator phase windings and uniformly magnetized, approximately full 
pitch rotor magnets. 
When the excitation voltage is removed from the outgoing stator phase 
winding of PM-AC machine, and the excitation voltage is applied to the 
next incoming phase winding, there exists a difference between the rate at 
which the current falls in the outgoing phase winding and the rate of 
increase in the incoming phase winding. This differential results in a 
momentary torque notch (or depression) in the torque of the PM machine. 
Certain applications for PM machines require smooth torque characteristics. 
Hence, various methods have evolved for controlling both voltage source 
and current source drives in order to reduce the above-described torque 
notch. 
Several efforts have resulted in potential solutions for the torque ripple 
or notch problem in three phase square wave PM machines. 
For instance, in Le-Huy et al., "Minimization of Torque Ripple in Brushless 
DC Motor Drives," IEEE Trans. on IA, Vol. IA-22, No. 4, pp. 748-755 
(July/August 1986), modulation of the DC link current was proposed to 
compensate for back emf waveforms with less than 120.degree. flat top 
value. They did not, however, consider the current rise delay in the 
incoming phase. 
Murai et al., "Torque Ripple Improvement for Brushless DC Miniature Motors" 
IEEE Trans. on IA, Vol. 25, No. 3, pp. 441-450 (May/June 1989) developed 
pulse width modulation (PWM) methods and device conduction overlap periods 
to compensate for the difference in time required to turn-off the outgoing 
phase and fully turn-on the incoming phase. Their methods, however, 
resulted only in reduction of the torque ripple level, not in complete 
elimination of the commutation notching. 
Berendsen et al., "Commutation Strategies for Brushless D.C. Motors: 
Influence on Instant Torque," Conf. Proc. of 1990 Applied Power 
Electronics Conf., pp. 394-400 (March 1990) proposed a neutral voltage 
feedback/compensation method to PWM regulate the machine phase currents 
during phase commutation and thus minimize to any degree desired (subject 
to PWM switching speed limitations) the commutation torque ripple. They 
also discussed methods useful when the back emf waveform has flat top 
periods of less than 120.degree. duration. In both cases, however, an 
extra sensor, to sense the neutral connection voltage, is required in 
addition to the normal phase current sensors. 
Related co-pending U.S. patent application Ser. No. 07/939,123 discloses a 
quasi square wave brushless DC machine having five or more phases. The 
five (or more) phase design of the referenced machine yields an increased 
efficiency and/or torque density. Hence, the design holds great commercial 
and industrial promise. However, the existing solutions to the torque 
notch problem are inapplicable, inadequate, or excessive. 
It would be greatly advantageous to provide a more practical method for 
smoothing the torque output of quasi square wave back EMF permanent magnet 
(PM) synchronous machine, including the five-phase PM machine of related 
co-pending U.S. patent application Ser. No. 07/939,123 as well as 
conventional three-phase PM machines, for all applications requiring 
smooth torque characteristics. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to provide a method for smoothing 
the torque characteristics of a multi-phase PM machine. 
It is a more specific object to provide a method for smoothing torque as 
described above which may be employed in driving a conventional 
three-phase PM machine as well as a five-phase PM machine such as set 
forth in related co-pending U.S. patent application Ser. No. 07/939,123. 
It is a specific object to smooth the torque characteristics as described 
above by controlled driving of the PM machine with a conventional voltage 
source drive operated in accordance with a predetermined switching 
sequence. 
According to the present invention, the above-described and other objects 
are accomplished by providing a new method for powering a quasi square 
wave back EMF permanent magnet synchronous machine with a switched voltage 
source drive in order to minimize torque notch. The quasi square wave back 
EMF permanent magnet synchronous machine is preferably of the type having 
a rotor (or secondary in the case of a linear machine) with a plurality of 
permanent magnets spaced at equal intervals and a stator (or primary) 
having a plurality of stator windings forming an odd number N stator 
phases, where N.gtoreq.3. The voltage source drive is of conventional 
construction and uses 2N switch devices arranged in N parallely-connected 
switch legs for connection to an input voltage source. Each switch leg has 
an upper switch device coupled to each motor phase winding for positively 
driving said phase winding and a lower switch device coupled to each motor 
phase winding for negatively driving said phase winding. 
The method comprises gating the switch devices of the voltage source drive 
according to a predetermined sequence including a plurality of successive 
commutation periods. Each commutation period comprises applying an 
excitation voltage to an incoming phase winding of said stator while 
removing an excitation voltage from an outgoing phase winding of said 
stator. Throughout the duration of each commutation period, the sum of the 
excitation voltages applied to the incoming phase winding and outgoing 
phase winding are maintained constant at a predetermined value. 
The method of the invention may also include an additional step wherein the 
individual excitation voltages applied to the remaining phase windings 
(other than the incoming phase winding and outgoing phase winding) are 
also held constant for the duration of the respective commutation periods.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
Permanent magnet (PM) synchronous machines may come in axial or radial 
embodiments both having a stator and a permanent magnet rotor, or they may 
come in linear embodiments having a primary and permanent magnet 
secondary. In either case, the rotor/secondary comprises an even number of 
permanent magnets equally spaced with alternating magnetic orientations 
around (or along) a rotor/secondary core. The angular pitch between 
adjacent permanent magnets of opposite orientation defines one pole pitch. 
The stator/primary includes a plurality of stator windings forming a 
number N stator phases. In accordance with the present invention, the 
number N can be any odd number .gtoreq.3, i.e., 3, 5, 7, 9 . . . etc. The 
method of the invention will herein be discussed in the context of a 
rotary machine. However, the method is equally and directly applicable to 
linear machines and the operation of such is considered to be within the 
scope of the invention. 
The magnetic flux linking each stator winding in a PM AC machine emanates 
from two sources: 1) stator winding currents; and 2) rotor magnets. The 
flux linkages due to stator winding currents do not contribute to torque 
production if the rotor has no preferential magnetic path through its 
body, i.e., the rotor is not a salient structure. Hence, the invention 
contemplates a non-salient rotor. The preferred rotor employs true surface 
mounted magnets, not inset, with no iron magnet containment mechanisms. 
The stator winding flux linkages due to the rotor magnets are a function 
of the rotor angle measured with respect to some fixed point on the 
stator. The magnetic flux density distribution is periodic repeating over 
each pole pair of rotor and stator. Thus, the equivalent rotor electrical 
angle .theta..sub.e is related to the rotor mechanical angle .theta. by 
.theta..sub.e =Np.theta./2, where N.sub.p is the total number of magnetic 
poles within the machine. For a given phase stator winding (for example, 
.alpha.-phase), we designate the rotor magnet source flux linkages as 
.PSI..sub..alpha.m (.theta..sub.e). The emf e.sub. .alpha. in the phase 
winding due to the rotation of the rotor magnets is then, by Faraday's Law 
##EQU1## 
where .PSI.'.sub..alpha.m .tbd..differential..theta..sub.e ; d.theta./dt 
is the rotor angular mechanical velocity, which will be designated 
.omega.; and d.theta..sub.e /dt is the rotor angular electrical velocity, 
which will be designated .omega..sub.e. Since this is a synchronous 
machine, .omega..sub.e is also the fundamental radian frequency of the 
drive currents in the stator windings. 
The instantaneous electrical power which is converted to mechanical power 
P.sub..alpha. in the .alpha.-phase stator winding is i.sub..alpha. 
e.sub..alpha., where i.sub..alpha. is the winding current with reference 
positive direction into the winding terminal. The torque T.sub..alpha. on 
the structure (rotor or stator) due to .alpha.-phase action is then simply 
##EQU2## 
The total instantaneous torque for the machine is the summation of the 
contributions from each phase 
##EQU3## 
Each phase of the machine and drive is represented by the circuit model 
shown in FIG. 1. The half-bridge voltage source drive is fed by a DC bus 
of constant bus voltage V.sub.o. High and low side switching devices QH 
and QL are provided, and these may be any suitable gated solid-state 
switching elements such as bi-polar junction transistors, insulated-gate 
bi-polar transistors, field effect transistors, or gate turn-off 
thyristors. The high and low side switching devices QH and QL are each 
shunted by anti-parallel wheeling diodes DH and DL, respectively. These 
serve as wheeling paths for inductive currents in the phase winding when 
either QH or QL is turned-off while still conducting current. The 
resistance R is the total resistance of the .alpha.-phase winding, L is 
the self inductance of the winding, e.sub..alpha. is the back emf induced 
in the winding due to rotation of the rotor magnets, and v.sub.n is the 
instantaneous value of the voltage of the neutral connection with respect 
to the low side of the DC bus. The flux linkages within the .alpha.-phase 
winding due to stator currents are given by 
##EQU4## 
where the summation is taken over the remaining phases within the machine; 
and M.sub..alpha..beta. is the mutual inductance between .alpha. and 
.beta. windings. As stated previously, the reluctances of the flux paths 
through the rotor structure are assumed to be independent of rotor 
position. Thus, M.sub..alpha.,.beta. is a constant, and not a function of 
.theta.. 
For conventional three phase machines with concentrated windings the 
inductances of the windings, both self and mutual, can be expressed in 
matrix form as follows: 
##EQU5## 
where L.sub.o is the leakage inductance of any one winding, due to flux 
produced by the winding which does not couple to any other winding; and M 
is the magnitude of the mutual inductance between any two windings, due to 
flux which crosses the air gaps and passes through at least a portion of 
the rotor structure. The self inductance of any one winding includes a 
term of three times the magnitude of the mutual values. This factor of 
three is due to the concentrated nature of the windings. 
The applied voltage v.sub..alpha. to the reference terminal of the 
.alpha.-phase winding, dependent on the state of QH and QL or, if both 
off, on the direction of i.sub..alpha. (if non-zero), is given by 
##EQU6## 
where p is the operator d/dt, and .omega..sub.e .PSI.'.sub..alpha.m 
=e.sub..alpha. the winding back emf. In general, equation (6) is 
non-linear since the rotor speed, and thus .omega..sub.e is a variable 
quantity. In most cases, however, we can use the quasi-static 
approximation and assume that electrical quantities change much faster 
than mechanical quantities, and thus, treat the rotor speed as a constant 
over the short time periods in which we solve for electrical waveform 
variations. In this case, the back emf e.sub..alpha. and 
.PSI.'.sub..alpha.m both have the same shape over any given time interval. 
In a conventional three phase machine, three high side switching devices 
QHA, QHB, and QHC and three low side switching devices QLA, QLB, and QLC, 
respectively, correspond to each of the three phases A, B, and C, and the 
switch devices are switched according to a sequence such as shown in FIG. 
2. The sequential switching states represent a step-like drive (i.e., no 
PWM control) with a bus voltage V.sub.o that exceeds the peak line-to-line 
back emf of the machine at a given operating speed. The earmark 
characteristics of conventional quasi square wave back EMF PM synchronous 
machines are apparent in the back EMF voltage shape waveforms of FIG. 3. 
Namely, they exhibit a back EMF voltage waveform for each phase defined by 
trapezoidal cycles having plateaus of constant back EMF voltage levels. 
Dependent on machine design, the constant flat top portion can exceed the 
idealized 120.degree. value for the three-phase drive of FIG. 2. In FIG. 
3, the angular extent of the flat top portion of the waveform is 
approximately 140.degree., or 20.degree. longer than needed for the 
idealized 120.degree. drive condition. 
For purposes of explanation, we will consider the instantaneous torque 
production in one exemplary commutation interval during which high side 
switch QHA is turned off in the a-phase, and the high side switch QHB is 
turned on to drive the b-phase. Phase-c remains connected to the low side 
bus terminal throughout this interval via QLC. Phase-b is designated the 
incoming phase, and phase-a is designated the outgoing phase. The duration 
of this commutation process is assumed to be short enough such that 
a-phase current i.sub.a falls to zero before .PSI.'.sub.am starts to fall 
from its flat top peak value of .PSI.'.sub.om. 
The neutral connection voltage v.sub.n can be eliminated from the three 
circuit equations derived from equation (6) by use of the three wire 
restriction i.sub.a +i.sub.b +i.sub.c =0. Use of this condition results in 
the following two independent equations, valid for any a-phase and b-phase 
drive and back emf variation: 
EQU v.sub.a +v.sub.b =3(R+L.sub.3 p)(i.sub.a +i.sub.b)+.omega..sub.e 
(.PSI.'.sub.am +.PSI.'.sub.bm 2.PSI.'.sub.om) (7) 
and 
EQU v.sub.a -v.sub.b =(R+L.sub.e p)(i.sub.a -i.sub.b)+.omega..sub.e 
(.PSI.'.sub.am -.PSI.'.sub.bm) (8) 
where, during this commutation interval, v.sub.c =0, and .PSI.'.sub.cm 
=-.PSI.'.sub.om. Inductance L.sub.3 =L.sub.o +4M, where L.sub.o and M are 
defined in equation (5) above. For the specific case of step-like drive 
v.sub.b =V.sub.o and v.sub.a =0; and for the specified case of commutation 
completion during the flat top periods of emf variation, .PSI.'.sub.am 
=.PSI.'.sub.bm =.PSI.'.sub.om. We further assume that rotor speed 
.omega..sub.e does not vary appreciably over the commutation interval, so 
the phase winding back emf E.sub.o .tbd..omega..sub.e .PSI.'.sub.om is a 
constant. Equations (7) and (8) can now be easily solved for the time 
variation of i.sub.a and i.sub.b. 
FIG. 4 is a graph of the phase currents i.sub.a and i.sub.b in the 
three-phase example. Typical solutions to equations (7) and (8) are shown, 
where the quantities i.sub.n and t.sub.n are determined by 
EQU i.sub.n /I.sub.o =(2E.sub.o +4RI.sub.o)/(4E.sub.o +5RI.sub.o) (9) 
and 
EQU 1-e.sup.-t.sbsp.n.sup./.tau..sbsp.3 =3RI.sub.o /(4E.sub.o +5RI.sub.o) (10) 
where I.sub.o is the phase winding current immediately prior to commutation 
and .tau..sub.3 =L.sub.3 /R. Time t.sub.n is the time required for a-phase 
to turn-off, i.e., the time required for i.sub.a to fall from I.sub.o to 
zero. In general, this time t.sub.n is shorter, usually much shorter than 
the time required for b-phase current to rise from its a-phase turn-off 
time value i.sub.n to the steady-state resistance and voltage determined 
value I.sub.o =(V.sub.o -2E.sub.o)/2R. The machine torque during the 
commutation interval, from (3) above, is simply 
EQU T=N.sub.p .PSI.'.sub.om (i.sub.a +i.sub.b). (11) 
Thus, the torque notch has the same shape as the i.sub.a +i.sub.b curve for 
0.ltoreq.t.ltoreq.t.sub.n and the i.sub.b curve for t&gt;t.sub.n in FIG. 4. 
As given by (9) the minimum value of i.sub.a +i.sub.b =i.sub.n determines 
the minimum value of the torque T.sub.n at the notch point. Thus, equation 
(9) also gives T.sub.n /T.sub.o where T.sub.o is the nominal torque 
N.sub.p .PSI.'.sub.om I.sub.o. The torque notch is seen to approach 80% at 
low speeds where the winding resistive drop RI.sub.o is greater than the 
back emf E.sub.o and fall to near 50% at high speeds where the back emf 
E.sub.o is greater than the resistive drop RI.sub.o. 
The key constraint for torque notch compensation according to the present 
invention is clearer in view of equation (11) above. The current sum 
i.sub.a +i.sub.b is preferably kept constant throughout the commutation 
period. However, the variation of i.sub.a +i.sub.b is governed by equation 
(7) above. Hence, we can instead constrain the applied voltage sum v.sub.a 
+v.sub.b to its pre-commutation interval value. This way, i.sub.a +i.sub.b 
should not change during commutation. Just prior to commutation a-phase 
drive QHA is on, thus, v.sub.a =V.sub.o =2(E.sub.o +RI.sub.o); and b-phase 
is floating (i.e., QHB and QLB are both off and i.sub.b =0) so v.sub.b 
=2E.sub.o +RI.sub.o. Substitution of these values into equation (7) 
clearly shows that if the sum of the incoming and outgoing terminal 
voltages v.sub.a +v.sub.b sums to 4E.sub.o +3RI.sub.o, then p(i.sub.a 
+i.sub.b)=0 throughout the commutation and the machine torque remains 
constant. 
The method of the present invention may be generalized as maintaining the 
sum of the excitation voltages applied to the incoming phase winding and 
outgoing phase winding constant at a predetermined value for the duration 
of each commutation period. 
In practice, this means that the magnitude of the terminal voltage v.sub.b 
of the incoming phase b be boosted by an amount .DELTA.v above its nominal 
on-value while the voltage v.sub.a of the outgoing phase is reduced below 
its nominal on-value. 
The extent of the commutation interval is determined by equation (8). Let 
v.sub.b be the independent or controlling quantity. Further, let v.sub.b 
.tbd.V.sub.o +.DELTA.v. Thus, by the restriction on the sum v.sub.a 
+v.sub.b, v.sub.a becomes equal to V.sub.o -RI.sub.o -.DELTA.v. The 
solution to equation (8) now becomes 
EQU i.sub.a -i.sub.b =I.sub.o -(I.sub.ox +I.sub.o)(1-e.sup.-t/.tau..sbsp.3), 
(12) 
where I.sub.ox =2(RI.sub.o +.DELTA.v)/R. This solution is sketched in FIG. 
5 for different values of excess v.sub.b drive .DELTA.v. The commutation 
interval is over when i.sub.a -i.sub.b falls to the value -I.sub.o. At 
this time, QHA is safely switched off without resultant wheeling current 
through DLA since i.sub.a is now zero. The time required for complete 
commutation, t.sub.c, is seen from FIG. 5 to be a monotonically falling 
function of the incoming phase overdrive voltage .DELTA.v. A plot of 
t.sub.c /.tau..sub.3 is given in FIG. 6 as a function of .DELTA.v/RIo. 
One must choose a sufficient value of .DELTA.v to insure that the angle 
.omega..sub.e t.sub.c is less than the remaining angular extent of the 
flat top portion of .PSI.'.sub.am, measured from the time of commutation 
start. 
If the net or average DC bus voltage V.sub.o is attained by PWM of the 
drive switching devices then the needed values of v.sub.a and v.sub.b 
during commutation can be attained by modified pulse width modulation 
(PWM) of QHA and QHB respectively, if the actual DC bus voltage V.sub.om 
is greater than V.sub.o +.DELTA.v. If the DC bus voltage V.sub.o is not 
determined by PWM of the drive switching devices, then the .DELTA.v boost 
to the incoming phase must be supplied by additional circuitry. However, 
the most common method of drive control (i.e., regulation of I.sub.o) is 
PWM from a fixed voltage bus. Thus, modified PWM control of QHA and QHB 
during commutation is easily implemented with no additional hardware. An 
upper speed limit on this compensation method will be reached when the PWM 
determined value of V.sub.o is too close to V.sub.om to achieve the needed 
value .DELTA.v. 
Torque notch compensation can also be achieved in a five-phase machine in 
accordance with the method of the present invention. In an exemplary five 
phase machine as set forth in co-pending U.S. patent application Ser. No. 
07/939,123 with concentrated windings, the inductances of the windings 
(both self and mutual) can be expressed in matrix form as follows: 
##EQU7## 
where, as in the above-described three phase case, L.sub.o is the leakage 
inductance of any one winding and M is the magnitude of the mutual 
inductance between nearest neighbor phase windings. The factors of three 
and five in matrix (13) are due to the concentrated nature of the 
windings. For standard five phase drive, the high side switching devices 
QHA, QHB, QHC, QHD, and QHE and the low side switching devices QLA, QLB, 
QLC, QLD, and QLE, respectively, corresponding to each of the five phases 
A, B, C, D, and E are switched according to a sequence such as shown in 
FIG. 7. Again, the machine exhibits a back EMF voltage waveform for each 
phase defined by trapezoidal cycles having plateaus of constant back EMF 
voltage levels. 
Five-phase machines can be designed such that the angular extent (in 
electrical degrees) of the constant back EMF voltage level can be 
established within a range of between 120.degree. to nearly 180.degree.. 
The particular back EMF range depends upon the size and placement of the 
permanent magnets in the rotor and the construction of the stator 
windings. 
Idealized five phase rate of change of stator winding flux linkages due to 
the rotor magnets and accompanying step like drive conduction periods for 
the 10-device drive circuit are given in FIGS. 7 and 8. In a five phase 
machine with voltage source drive, each phase winding is typically excited 
over a range of only 144.degree. as shown. For the exemplary 160.degree. 
trapezoidal plateau shown in FIG. 8, there is approximately a 16.degree. 
overlapping period of constant back EMF during which the phase winding 
currents can be commutated under torque notch minimization control. As in 
the above-described three-phase method, the control scheme of the present 
invention is implemented during this period of overlap. For purposes of 
explanation, we will discuss the exemplary commutation period of time 
required to drive a-phase current to zero and to drive c-phase current to 
the steady state value I.sub.o. Throughout the commutation period, phase-b 
remains connected to the high side bus through QHB, and phases d and e 
remain connected to the low side through QLD and QLE, respectively. 
The five circuit equations derived from equation (6) can be reduced to four 
independent equations by eliminating the neutral connection voltage by use 
of the five wire restriction i.sub.a +i.sub.b +i.sub.c +i.sub.d +i.sub.e 
=0. We also use .PSI.'.sub.am =.PSI.'.sub.bm =.PSI.'.sub.cm 
=-.PSI.'.sub.dm =-.PSI.'.sub.em =.PSI.'.sub.om .tbd.E.sub.o 
/.omega..sub.e, and v.sub.d =v.sub.e =0. The resultant four equations are 
given by 
EQU [3(v.sub.a +v.sub.c)-2v.sub.b -8E.sub.o [/5=(R+L.sub.5m p)(i.sub.a 
+i.sub.c)+M.sub.5 pi.sub.b (14) 
EQU [4v.sub.b -(v.sub.a +v.sub.c)-4E.sub.o ]/5=M.sub.5 p(i.sub.a 
+i.sub.c)+(R+L.sub.5n p)i.sub.b (15) 
EQU v.sub.a -v.sub.c =(R+L.sub.5n p)(i.sub.a -i.sub.c)-M.sub.5 p(i.sub.d 
-i.sub.e) (16) 
and 
EQU 0=-M.sub.5 p(i.sub.a -i.sub.c)+(R+L.sub.5m p)(i.sub.d -i.sub.e) (17) 
where L.sub.5m =L.sub.o +4M, L.sub.5n =L.sub.o +8M, M.sub.5 =4M, with 
L.sub.o and M defined in (13). 
Note the couplings in these equations. Equations (14) and (15) stand as an 
independent set and (16) and (17) stand as an independent set. The set of 
(14) and (15) correspond to (7) in the three phase case and the set (16) 
and (17) correspond to (8) in the three phase case. The instantaneous 
torque in a five phase machine at the commutation interval of interest, 
with the five wire restriction i.sub.a +i.sub.b +i.sub.c +i.sub.d +i.sub.e 
=0, is (from equation (3)) 
EQU T=N.sub.p .PSI.'.sub.om (i.sub.a +i.sub.b +i.sub.c). (18) 
Thus, to maintain constant torque the high side connected terminal currents 
i.sub.a, i.sub.b, and i.sub.c must all sum to a constant. Following the 
three phase case, we see that this can be accomplished if constant 
solutions to (14) and (15) are possible. Again from the three phase case, 
we see that this is possible if the prior to commutation set of drive 
voltages v.sub.a +v.sub.c and v.sub.b are maintained throughout 
commutation. That is, prior to commutation v.sub.a =V.sub.o =2(E.sub.o 
+RI.sub.o), v.sub.b =V.sub.o, v.sub.c =2E.sub.0 +RI.sub.o. Thus, we must 
only maintain 
EQU v.sub.a +v.sub.c =4E.sub.o +3RI.sub.o (19) 
and 
EQU v.sub.b =V.sub.o =2(E.sub.o +RI.sub.o). (20) 
Note that condition (19) on the sum of the drive voltages for the incoming 
and the outgoing phases is exactly the same as that required in the three 
phase case. Substituting conditions (19) and (20) into (14) and (15) gives 
the desired p(i.sub.a +i.sub.c)=pi.sub.b =0 throughout commutation. 
The length of the commutation interval is determined by (16) and (17). As 
in the three phase case, we let the drive voltage for the incoming phase 
be the controlling or independent variable. We set v.sub.c =V.sub.o 
+.DELTA.v. From (19) we then have v.sub.a =V.sub.o -RI.sub.o -.DELTA.v (as 
in the three phase case). Subbing these values into (16) and (17) we 
obtain two coupled first order differential equations for the difference 
current sets i.sub.a -i.sub.c and i.sub.d -i.sub.e. These equations can be 
solved in closed form. The solution for i.sub.a -i.sub.c during 
commutation is given by 
##EQU8## 
The expressions for the constants K.sub.1, K.sub.2, .delta..sub.1, and 
.delta..sub.2 are quite complicated. But for the limiting case of 
5M&gt;&gt;L.sub.o, that is the leakage inductance component of each stator phase 
winding can be neglected, we have 
##EQU9## 
Solution (21) is plotted in FIG. 9 for the constants of (22)-(24) as a 
function of time normalized to .tau..sub.5 =4M/R and for different values 
of incoming phase overdrive voltage. Note the similarity to the 
corresponding three phase solution plotted in FIG. 5. The commutation 
interval ceases when i.sub.a reaches zero and i.sub.c simultaneously 
reaches I.sub.o. The compensation control is then switched off, that is 
QHA is turned-off and v.sub.c is lowered to V.sub.o, the nominal 
steady-state bus voltage needed to maintain I.sub.o in each of the active 
phase windings. 
A 10 hp 3600 rpm 4-pole, five phase machine is presently under development, 
and measured values for this prototype are: R=0.43.OMEGA. and 5M=4.4 mH. 
Rated phase current is 14A and rated DC bus voltage is 330 VDC. Thus, 
.tau..sub.5 =4M/R=8.2 msec. At one quarter speed operation, 30 Hz drive, 
assuming a 160.degree. flat top square wave back emf, We have a maximum 
time for compensation of [(160-144)/180]/(2.times.30)=1.5 msec, so that 
t.sub.c /.tau..sub.5 =1.5/8.2 =0.18. From FIG. 9, we see that this level 
of compensation requires a .DELTA.v/RI.sub.o near six or seven. Assuming 
the higher value and operation at rated torque (i.e., rated I.sub.o) would 
then require 7.times.6=42 volts. At one quarter speed and rated torque the 
nominal, steady-state bus voltage V.sub.o would be 92 VDC; thus, the 
ability to set v.sub.c =V.sub.o + .DELTA.v=134 V and v.sub.a =V.sub.o 
-RI.sub.o -.DELTA.v=44 V is assured. 
Having now fully set forth the preferred embodiments and certain 
modifications of the concept underlying the present invention, various 
other embodiments as well as certain variations and modifications of the 
embodiment herein shown and described will obviously occur to those 
skilled in the art upon becoming familiar with said underlying concept. It 
is to be understood, therefore, that within the scope of the appended 
claims, the invention may be practiced otherwise than as specifically set 
forth herein.