Proc. Japan Acad.,
30
7.
Sakae
YAMAMURA,
(Communicated
Abstract:
some
extent.
saliency
Almost
theory.
be
As the rotor
have
been
generator,
M. J. A.
12, 1992)
of electric
industries
is generated
salient-pole
synchronous
generators
inductances
of the three
phase
of generator
provided
the spiral
March
energy
rotates,
analysis
not
In this paper
synchronous
all electric
which are either
and it makes
simulations
[Vol. 68(B),
Spiral Vector Theory of Salient-Pole
Synchronous Machine
By
nous generators,
68, Ser. B (1992)
vector
performance
difficult.
by the
conventional
method
will be applied
and adequate
analytical
winding
Even
theories,
solutions
by synchro-
or have
saliency
vary
adequate
such
to analyses
to
due to the
computer
as the
d,q
axis
of the salient-pole
and computer
simultations
will
derived.
Key
salient-pole
words:
Spiral
vector;
synchronous
computer
simulation
of synchronous
machine;
analysis
of
machine.
1. Introduction.
The spiral vector method was proposed by the author (1)-(6). It can
unify AC circuit theories of steady state and transient state. It is most suitable to analyses
of AC machine transients.
Analytical
results
obtained
have led to superior
control
performances
of AC motors. 1 4) The paper will now apply the method to analyses of the
salient-pole synchronous
machine, whose analyses have been exclusively made by Park's
equations based on the d,q axis theory, but they have neither provided adequate analytical
solutions nor computer simulations
of the synchronous
machine. In this paper it will be
shown that the spiral vector method provides both of them.
2. Steady state analysis of salient-pole synchronous machine.
Fig. 1 shows the model of
the salient-pole synchronous
machine for the analysis which follows. It is the permanentmagnet-excited
type or the field winding type. Self inductances
of the armature windings
are functions of spacial angle e, which is the angle between the center line of winding of
phase
a and the
direct
axis of the
where
w is synchronous
given
by
speed
La=L+L'
Lb=L+L'
L,=L+L'
(2)
These
(3)
Then
8 = wt + cod [elect,
(1)
tween
salient-pole.
two
inductances
phases
of the
Then
radians]
self inductances
of phases
a, b and c are
cos (2d)=L+L'
cos (2wt+2cpd)
cos (2d-47r/3)=L+L'
cos (2wt+2~od-4,r/3)
cos (26+4,r/3)=L+L'
cos (2wt+2'pd+4,r/3)
contain
among
rotor.
we have
three
double
phases
Mao=Mav+M'
Mbc=Mati+M'
Mca=Mati+M'
frequency
a,
b and
components.
c are
similarly
cos (28+)
cos (2d+I'-4ir/3)
cos (2e+ik+47r/3)
Mutual
given
inductances
as
below.
be-
No. 3]
Spiral
Fig.
1.
Vector
Analytical
Angle ip is phase difference
(2) and mutual inductances
flux linkage /2ga of phase
(4)
Theory
of Salient-Pole
model
between
of the
Synchronous
salient-pole
double frequency
synchronous
variations
of eq. (3) and is to be determined
a is
~oa = - Laid-
Mab2b-
Machine
31
motor.
of self inductances
shortly
in this paper.
of eq.
Main
Mcaic +2 cos B
The last term of this equation is flux linkage from the magnetic poles of the rotor, which
are the permanent magnet or the field winding with constant DC current. For symmetrical
or balanced steady state operation three-phase armature currents are given by
(5)
is=1I1aI
cos (wt+So~)
ib = '
IIa i cos (wt +cp, - 21r/3)
i~=1/ G IIaI cos (wt+cp,+21r/3)
Here Ia is circular vector of current
eq. (5), eq. (4) becomes
Aga=-(L-May) 4IIaI cos ((,t+cp)+(sl2)I'aI[L'
cos (cot+2Sod-oi)
+M' cos (cat+2co +Jr-cp,+2ir/3)+M' cos (wt +2cod+ -5o1+2ir/3)
+(f
/2)IIa)[L' cos (3wt+2wd+wl)+M' cos (3wt+2Wd+Sol+f -21x/3)
+M' cos (3wt+2~od+cpl+fr)]+2 cos (wt+cpd)
(6)
Triple
for
of phase a. Inserting
the
frequency
terms
triple-frequency
are given
terms
in the brancketed
to
L'=
(7)
disappear
ill-',
the
term
second
following
Inserting
(9)
M av=L
eqs.
the end
are
conditions
cos (2,r/3)= -L/2
(7) and (8), Xga of eq. (6) now
Aga=-(3L/2)f
IIa cos (cot+~o,)+2 cos (cat+5o,)
and in or der
sufficient.
ifr= -2,r/3
Salient-pole synchronous
machines are constructed
so that these
From symmetry
of the machine structure
we have
(8)
from
conditions
becomes
(3L' /2)4/f
Ila cos (cat+25P-So,)
are satisfied.
32
S. YAMAMURA
Now variables are expressed
(10)
[Vol. 68(B),
in circular vectors, and then eq. (9) becomes
2Qa=-(3L/2)
T Ia~et(at+oj)-(3L'/2) /
Ia~e~(at+c1+2~d-2oi)+Ae'(mt+td)
_-(3L/2)ia-(3M'/2)iae'(2i
-2rt)-Aejt+cd)
Here ia= V
l a = \/72 11 e3t+ ~1>is circular vector of current of phase a. Circuit
equation of phase a of the synchronous generator is now given by
(11)
va=-Rlia-11pia+PAoa
= -Rlia-(11+3L/2)pia-(3L'
/2)&(2~d-2ca)pia+ea
Here 11 is primary leakage inductance and ea is internal induced voltage given below.
(12)
ea=jc~Ae"°t+~d)=w2e'(mt+~),
= Od+ 2
Equation (11) contains phase a only, being segregated from other phases. In order for
phase a to represent primary three phases, subscript a is changed to 1 and eq. (11) becomes
(13)
v1=-R,i1--(l
+3L/2)pi1+(3L'/2)ef(2~-2~')pi1+e1
In this equation inductance becomes eq. (14), which is a complex inductance.
(14)
ll+3L/2-(3L'/2)e'(2rP-2pi)=Ls-(3L'/2)e~c2~-24'1)
Ld=t,+3L/2
Under steady states operator p becomes jw and variables are circular vectors, which
are now expressed by capital letters with • on top. Then eq. (13) becomes
(15)
V1=-R1I1-jw[LS-(3L'/2)
cos (2w-2W1)-j(3M'/2)
sin (2w-2cpd)]I,+E1
= - [R1 + w(3M'/2) sin (2~o-2c1)]Il- jw[L8+(3L'/2) cos (2~o-2cp1)]I+EI
Vector diagram corresponding to eq. (15) at t=0, is shown in Fig. 2. Resistance is
increased by w(3M'/2) x sin(2q -2q 1) due to saliency. And motor torque is also increased
due to saliency by
(16)
3(P/2)(3L'/2)
sin (2cp-2cp1)1I,12 [N-m]
where P is number of poles. This is saliency torque or reluctance
Fig.
2.
Circular
synchronous
vector
generator.
diagram
at
t = 0 of
the
torque.
salient-pole
No. 3]
Spiral
Vector
Theory
of Salient-Pole
Synchronous
Machine
33
3. Transient state analysis of salient-pole synchronous machine. Under transient state
there are plural number of characteristic roots, which are generally complex. Under
symmetrical (or balanced) operation state variables of three phases are symmetrical.
Under asymmetrical operation asymmetrical state variables are transformed by the
spiral-vector symmetrical-component method into symmetrical spiral vectors. 6) Symmetrical three phase currents under transient state are assumed to be expressed by the
following equations.
2a=y " I'a Ie-2t cos (w't+co1)
2b-~ u I'a Ie-2t cos (w't+Sol-2,c/3)
ic =~ ~' IIa le-1t cos (w't+sp,+2,r/3)
(17)
Inserting
(18)
eq.
(17), main
flux
linkage
Aga--(L-Mav)e-2tV
of phase
a of eq. (4) becomes
tal cos (w't+cp1)-(4
/2)I'al e-at[L' cos (2w-w't
+2Sod-cp,)-M' cos (2w-w'+2cod-cpl+ ~+2,r/3)-M'
cos (2w-w't
+2cpd-coi+tfr+2ic/3)]-(f
/2)IIal e-2t[L' cos (2w+ w't+2o4 +cpi)
-M' cos (2w+w't+2cod+w,+~-2n/3)-M'
cos (2w+w't+2~od+Pl
+~)]+2
cos (wt+~o)
In order for 2w+ w' frequency terms to disappear, the condition of eq. (7) is sufficient. And
the Aga of eq. (18) becomes
alga= - (3L/2)e-2tf
-(3L'/2)e-'t4IIaL
(19)
Expressing
(20)
this
in
spiral
vectors,
it
cos (wt+~o)
becomes
' ga=-(3L/2)~IIaIe-ate'c~+~~~_(3L/2)~(IaIe-ate'c2wt+2icm-t+P1)
+2e'tcmt+g&=-(3L/2)ia-(3M'/2)e'26ia
+2e'°
Here is = VH
conjugate.
Now circuit
(21)
equation
(Ia I cos (w't +(Soi)
cos (2w-w't+2cp-cp,)+i
Ia
J(oI+ ~i) is
equation
of phase
spiral
vector
of current
of phase
ia*
is its
a is
va = - Rlia -1iPia + p2ga
= - Rlia - (l +3L /2)Pia - (3L' /2)p(e
) i282*)
+p(2
e20)
This equation contains only variables of phase a, which is segregated
Changing subscript
a to 1, eq. (23) becomes
(22)
a and
from other
phases.
- (l~+3L /2)pil - (3L' /2)p(e'2ai*) +p(2 e'°)
v1= -R,i
is valid both for steady and transient
states.
When the generator is permanent-magnet-excited
flux A is constant.
Then eq. (22) becomes
This
e,-v,
(23)
where
e1 is internal
= -R1i+(1,+3L2)pi,
induced
voltage
or its field current
+(3M'/2)p(e
of eq. (12) with
i2gji)
1 replacing
a.
is kept constant,
34
S. YAMAMURA
[Vol. 68(B),
Let us write the general solution of eq. (23) as follows;
(24)
i, = i1t +
1I, ej (0t+Sid)
The first term is the transient state term and the second term is the steady state term,
which was obtained in the previous section. Inserting eq. (24) into eq. (23), we get
(25)
0=R~i1 t+ (l +3L/2)pi1 t+(3L'/2)p(e'2c~t+~d)i*)
The last term is a time function, which behaves as a forcing function. As a first
approximation, let us assume ilt = Ae- for the forcing function. Then eq. (25) becomes
(26)
R1i1t+(11+3L/2)pi1t=-(3L'/2)p[Ae(-2+2l
°t+j2pd]
The forcing function is a spiral vector and general solution of this equation is given, as
follows.
(27)
= Ae-~t _ R (3L' /2) (- 2+ J2cv)A
1+(-A+52w)(11+3L/2)
ec-1+j2w)t+2rpd
Here
(28)
~= 1 R1
,+3L/2
and A is an arbitrary constant, which is to be determined by an initial condition. We can
repeat the process by inserting the solution of eq. (27) into the forcing function of eq. (26).
And then we get the same general solution with A slightly modified, which contains an
arbitrary constant A. And so eq. (27) is a transient solution of good approximation. General
solution of eq. (24) now becomes
(29)
i1_Ae-~t_
R (3L' /2)(- 2 + J2w)A
1+(-2+j2w)(l1+3L/2)
e(- +j2a)t+j2~od+4/~~I,~ejcwt+o1)
The first and second terms are transient terms and contain A. If at t=0 the following
condition is satisfied
(30)
i,=f
iI,Ie'~'
A becomes zero and no transient occurs. Eq. (30) is the transientless condition of
synchronousmachine control.
4. Transientstateanalysisof salient-pole
synchronous
machinewithfieldwinding. When
the synchronousmachineis of the field windingtype and transient current flowsin the
winding,transient state analysisbecomesa little more complicated.Analysisin section3,
whichwas madeabout the machinewith constant fluxlinkageof the fieldwinding,is valid
up to eq. (20) for analysis in this section.
Whentransient current flowsin the fieldwinding,the last term of eq. (20)becomes
(31)
2=M1ife'a
Thus eq. (22) becomes
(32)
v1= -R1i1-(11+3L/2)pi1-(3L'/2)p(e'2eii)+Mfp(if e'°)
No. 3]
Spiral VectorTheoryof Salient-PoleSynchronousMachine
35
The circuit equation of the field windingis in a similar way derived, as follows,
(33)
of=RfI f+(1f+Mf)pif-(3Mf/2)p(i* e'°)
Eqs. (32) and (33) are circuit equations to be solved simultaneously.
5. Computersimulationof the synchronousgenerator. In the spiral vector method
performanceequations are written in terms of original state variables, without any
variable transformation. Thus the equations are very suitable to make computer
simulation of the synchronous machine, which can be incooperated with ease into
simulationof the power system. In the conventionaltheories, suchas the d, q axistheory
or Park's equations, it seems that adequate computer simulationis not available.
Computersimulationof the synchronousgenerator willbe given, as an example,of
equations (32) and (33) in section 4. They are modifiedas below
(34)
i, = -
(35)
2f-
Fig.
3.
vl -
(l
corresponding
6.
Block
salient-pole
is shown
has
block
Conclusion.
synchronous
that
provided
conventional
The
-
diagram
2L,
Rf
if+Mf
of
f i f eye
Ld
2f + 3
Mf
p
2 if+Mf
computer
simulation
(&oi*)
of
salient-pole
generator.
diagram
of these
spiral
vector
machine.
Both
analytical
power
computer
simulations
theories
L8 p
of
f+Mf)p
synchronous
The
?~- 3 !-_ (e'2ei*)+
L8p
based
steady
of the method
on
of the
the
equations
method
has
is given
been
and transient
is much
synchronous
d, q axis
in
Fig.
applied
to
states
better
than
machine,
of
the
analyzed.
It
d, q axis method.
It
have
the
3.
which
analyses
been
was
missing
in
method.
References
1)
2)
3)
4)
5)
6)
S. Yamamura: AC Motors for High Performance Applications. Marcel Dekker, Inc.,
New York, Basel (1986).
Modern theory of ac motors. Proc. BICM, p. 887, TEE of China, Beijing (1987).
ibid., ICEMD, Rumania (1986).
Theory of ac motor analysis and control. ICEM, vol. 1, Pisa, Italy, p. 1 (1988).
Spiral vector theory of ac circuit and ac machine. Proc. Japan Acad., 65B,142-145
(1989).
Spiral vector method and symmetrical component method. ibid., 67B, 1-6 (1991).