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1-1-2015
A New Equivalent Circuit of a Salient-Pole
Synchronous Machine and Its Phasor
Interpretation
Said Ahmed-Zaid
Boise State University
Danyal Mohammadi
Boise State University
Ahmed M.A. Oteafy
Alfaisal University
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This is an author-produced, peer-reviewed version of this proceeding. The final, definitive version of this document can be found online at 2015 North
American Power Symposium (NAPS), published by IEEE. Copyright restrictions may apply. doi: 10.1109/NAPS.2015.7335151
A New Equivalent Circuit of a Salient-Pole
Synchronous Machine and its Phasor Interpretation
Said Ahmed-Zaid and Danyal Mohammadi
Ahmed M. A. Oteafy
Department of Electrical and Computer Engineering
Boise State University, Boise, Idaho, USA
Email: sahmedzaid@boisestate.edu
College of Engineering
Alfaisal University, Riyadh, KSA
Email: aoteafy@alfaisal.edu
Abstract—This paper examines a shortcoming of the classical
phasor diagram of a salient-pole synchronous machine based on
the well-established two-reaction theory. Unlike in the phasor
diagram of a smooth-air-gap machine, it is not possible to
readily identify the internally-developed electromagnetic power
of a salient-pole synchronous machine from this phasor diagram.
By de¿ning new machine reactances, a single equivalent circuit of
a salient-pole synchronous machine is proposed together with a
phasor diagram where the internally-developed electromagnetic
power is made apparent. The revised two-reaction theory is
illustrated using the mathematical model of a two-phase salientpole synchronous machine whose equations are manipulated
using complex space vectors instead of traditional matrix transformations.
II. T HEORETICAL R EVIEW
Let us consider a two-phase salient-pole synchronous machine with two stator windings (a,b) and one ¿eld winding
(f). The mathematical model of this machine comprises three
voltage equations and a second-order mechanical equation of
motion. Using Kimbark’s notation [7], the voltage equations
are expressed in motor notation as the following set of three
¿rst-order differential equations,
yd
ye
I. I NTRODUCTION
At the 2000 North American Power Symposium, a paper
by R. H. Park on the dq0 transformation ranked second in
a survey of papers having had the most impact in power
engineering over the last century [1]. The dq0 transformation,
also known as the Park transformation, is still today the basis
for the investigation of synchronous machine dynamics in
power systems.
Historically, the development of a versatile mathematical
model of a synchronous machine began with the two-reaction
theory of Blondel, Doherty and Nickle, and others [2-3], and
was later generalized by Park in his papers de¿ning an ideal
synchronous machine and outlining the theory of the dq0
transformation [4-5].
Subsequent re¿nements of Park’s theory have been made in
the development of equivalent damper windings in the direct
and quadrature axes, the quanti¿cation of magnetic saturation,
and the determination of machine parameters during subtransient, transient and steady-state analyses [6-9].
In this paper, we examine two shortcomings of the tworeaction theory. The ¿rst is a lack of a single equivalent
circuit for a salient-pole synchronous machine in steady state.
The second is the inability of the classical phasor diagram to
determine the internally-developed electromagnetic power of
the machine using relevant complex vectors.
We propose to remedy these shortcomings by analyzing a
two-phase salient-pole synchronous machine for simplicity.
Damper windings are not considered in this analysis as the
aim is to produce a new steady-state circuit and its phasor
diagram.
yi
gd
gw
ge
= Uv le +
gw
gi
= Ui li +
gw
= Uv ld +
together with the Àux-current relationships
6
5
65
6
5
Odd Ode Odi
ld
d
7 e 8 = 7 Oed Oee Oei 8 7 le 8
i
Oi d Oi e Oi i
li
(1)
(2)
(3)
(4)
where
Odd
Oee
Ode
Odi
Oei
Oi i
=
=
=
=
=
=
Ov + Op cos 2
Ov Op cos 2
Oed = Op sin 2
Oi d = Pi cos Oi e = Pi sin Oi
(5)
(6)
(7)
(8)
(9)
(10)
and where (w) = $w + r is the electrical angle of rotation
of the rotor shaft in steady state. By de¿ning the following
complex space vectors,
$
yv = yd + mye
$
v = d + me
$
lv = ld + mle
the voltage equations can be concisely formulated as
$
$ gv
$
yv = Uv lv +
gw
gi
yi = Ui li +
gw
(11)
(12)
(13)
(14)
(15)
This is an author-produced, peer-reviewed version of this proceeding. The final, definitive version of this document can be found online at 2015 North
American Power Symposium (NAPS), published by IEEE. Copyright restrictions may apply. doi: 10.1109/NAPS.2015.7335151
together with the Àux-current vector relations
$
$
$
v = Ov lv + Op hm2 lv + Pi hm li
(16)
1
$
$ 1
i =
(17)
Pi hm lv + Pi hm lv + Oi li
2
2
Assuming a magnetically-linear coupling ¿eld, the magnetic
coenergy of this machine is given by
1
1
1
0
Zp
=
(18)
d ld + e le + i li
2
2
2
It is straightforward to show that this coenergy can also be
expressed as
½
¾
1
1
$
$
0
(19)
v lv + i li
Zp = <h
2
2
Expanding this last expression yields
¾
½
1 1
$
$
$2 1
0
m2 2
m Zp = <h
Ov | lv | + Op h ( lv ) + Pi h lv li
2
2
2
1
1
1
$
$
+ Pi hm lv li + Pi hm lv li + Oi l2i
4½
4
2
¾
1
$
$ 2
m2 m = <h
Op h ( lv ) + Pi h lv li
2
1
1 $
(20)
+ Ov | lv |2 + Oi l2i
2
2
By taking the partial derivative of the coenergy with respect
to the mechanical rotor shaft angle p = (2@s), where
s is the number of poles per phase, we obtain the following
expression for the developed electromagnetic torque
³ s ´ CZ 0
0
CZp
p
Wh =
=
C
2
C
p
³s´
n
$ o
$
=
<h mOp hm2 ( lv )2 + mPi hm lv li (21)
2
III. C LASSICAL T WO -R EACTION T HEORY
If we de¿ne ¿ctitious dq space vectors rotating with the
rotor reference frame,
$
y$
= yv hm
(22)
gt
$
$ m
gt = v h
(23)
$ m
$
(24)
lgt = lv h
the stator voltage equation (14) can be transformed into the
following vector equation
$
$
$ ggt
y$
=
U
+
(25)
l
+ m$ gt
gt
v gt
gw
After substituting all complex space vectors with their rectan$
gular variables, that is, y$
gt = yg + myt , gt = g + mt and
$
lgt = lg +mlt , the following two real equations are obtained,
gg
(26)
$t
gw
gt
yt = Uv lt +
(27)
+ $g
gw
Expressed in dq variables, the original Àux-current relations,
$
$
$
v = Ov lv + Op hm2 lv + Pi hm li
(28)
1
1
$
$
(29)
i =
Pi hm lv + Pi hm lv + Oi li
2
2
yg
= Uv lg +
become
$
$
$
gt = Ov lgt + Op lgt + Pi li
1
$
$ 1
i =
Pi lgt + Pi lgt + Oi li
2
2
Expanding these equations in rectangular form,
g
t
i
= Og lg + Pi li
= Ot lt
= Pi lg + Oi li
(30)
(31)
(32)
(33)
(34)
where the d-axis and q-axis inductances, Og and Ot , are
respectively de¿ned as
Og
Ot
= Ov + Op
= Ov Op
(35)
(36)
The developed electromagnetic torque can be expressed in dq
variables as
³s´
n
$
$ o
Wh =
<h mOp (hm lv )2 + mPi (hm lv )li
2´
³s
n
$ o
$
=
(37)
<h mOp (lgt )2 + mPi lgt li
2
$
By replacing lgt with its rectangular coordinates,
$
(38)
lgt = lg + mlt
the standard form of the developed electromagnetic torque of
a salient-pole synchronous machine is obtained as
³s´
ª
©
Wh =
<h mOp (lg mlt )2 + mPi li (lg mlt )
2
³s´
=
(39)
(Pi lt li + 2Op lg lt )
2´
³s
=
(40)
(g lt t lg )
2
IV. C LASSICAL P HASOR D IAGRAM
The steady-state machine voltage equations are obtained
by setting the time derivatives of the d-axis and q-axis Àux
linkages to zero in Eqs. (26)-(27). Using capital letters to
denote steady-state quantities, these equations become
Yg
Yt
= Uv Lg $t
= Uv Lt + $g
(41)
(42)
These two equations can each be represented by a circuit in
the d-axis or q-axis as shown in Figure 1. The two circuits are
coupled via the steady-state speed voltages induced in opposite
axes, $g and $t .
The internally-developed electromagnetic power is obtained
by adding the power contributions of the two speed voltages
in the two subcircuits yielding
Sh
= $(g Lt t Lg ) = $p Wh
(43)
where $p = (2@s)$ is the mechanical rotor shaft speed in
steady state. This power expression can also be expressed as
Sh
= $g Lt $t Lg
= $(Og Lg + Pi Li )Lt $Ot Lt Lg
= ([g [t )Lg Lt + Hi g Lt
(44)
This is an author-produced, peer-reviewed version of this proceeding. The final, definitive version of this document can be found online at 2015 North
American Power Symposium (NAPS), published by IEEE. Copyright restrictions may apply. doi: 10.1109/NAPS.2015.7335151
Rs
Id
Rs
Iq
+
V. R EVISED P HASOR D IAGRAM
By de¿ning the following reactances,
+
−
Vd
+
+
ωΛ q
Vq
−
−
ωΛ d
−
(a)
Fig. 1.
(b)
d-axis and q-axis Circuits of a Synchronous Machine
where Hi g = $Pi Li , [g = $Og and [t = $Ot . This
last expression makes apparent the power developed due to
each of the reluctance term and the synchronous power of the
second term found in smooth-air-gap machines.
−X qI q
Vdq
Idq
Rs
=
jX +
*
jX − Idq
+
+
Vdq
Ei
−
−
+
−
+
jX dI d
jE fd
−
jE fd
Fig. 3. Proposed Single Equivalent Circuit of a Salient-Pole Synchronous
Machine
jI q
Idq
Id
Fig. 2.
[g + [t
(47)
2
[g [t
[ =
(48)
2
the steady-state voltage equations (41)-(42) can be manipulated to yield the following phasor equation which is represented by the single equivalent circuit shown in Figure 3:
$
$
$
Ygt = (Uv + m[ + )Lgt + m[ Lgt
+ mHi g
(49)
[+
Classical Phasor Diagram of a Salient-Pole Synchronous Machine
The classical phasor diagram is obtained by combining the
steady-state equations in both axes as
$
$
Ygt = Uv Lgt + m[g Lg + m[t (mLt ) + mHi g (45)
Figure 2 is a pictorial representation of Eq. (45) with the
stator resistance neglected. The phasor diagram in this picture
resembles Kimbark’s phasor diagram [7, p. 70] and differs
slightly from the classical one using space vectors instead of
$
complex quantities such as (mHi g ) instead of Hi g .
The input real power into the machine is equal to
1
$$
Slq>2! = 2 × <h{Ygt Lgt
} = Yg Lg + Yt Lt (46)
2
This input power is equal to the internally-developed electromagnetic Sh since stator resistance and magnetic core losses
are being neglected in the equivalent circuit representation.
The synchronous power term (Hi g Lt ) is recognizable from
the complex product of (mHi g ) and (mLt ) . However, the reluctance power term is not readily identi¿able in this classical
phasor diagram. In the next section, we propose a new singlecircuit representation of a salient-pole synchronous machine
along with a modi¿ed phasor diagram where the two terms
of the internally-developed electromagnetic power are readily
identi¿able.
$
If we de¿ne an internal voltage Hl as the sum of the de$
pendent voltage source (m[ Lgt ) and the independent voltage
source (mHi g ),
$
$
Hl = mHi g + m[ Lgt
= mHi g + m[ (Lg mLt )
= mHi g + m[ Lg + [ Lt
(50)
then we can easily verify that the internally-developed electromagnetic power Sh is equal to the real power absorbed by
this internal voltage by expanding the following equation
$ $
Sh = <h{ H l Lgt
}
¡
¢
= <h{ mHi g + m[ Lg + [ Lt (Lg mLt )}
= 2[ Lg Lt + Hi g Lt
(51)
= ([g [t )Lg Lt + Hi g Lt
(52)
A phasor diagram for the single equivalent circuit can be
drawn as shown in Figure 4 where the stator resistance has
been omitted for simplicity. The internally-developed power
Sh can now be identi¿ed as the real part of the complex
$
product of this internal voltage voltage Hl and the conjugate
$
of the current vector Lgt = (Lg + mLt ). By decomposing
the current vector into its real and imaginary components,
$
Lgt = Lg + mLt , the internally-developed power Sh becomes
readily available as the sum of three terms:
• a power term (Hi g Lt ) resulting from the complex product
of (mHi g ) and (mLt ) ;
• a power term ([ Lg Lt ) resulting from the complex
product of (m[ Lg ) and (mLt ) ; and
• a power term ([ Lg Lt ) resulting from the complex
product of ([ Lt ) and (Lg ) .
This is an author-produced, peer-reviewed version of this proceeding. The final, definitive version of this document can be found online at 2015 North
American Power Symposium (NAPS), published by IEEE. Copyright restrictions may apply. doi: 10.1109/NAPS.2015.7335151
−X +I q
Vdq
q−axis
X− Iq
jX −I d
jX +I d
Idq
Ei
jE fd
d−axis
If
jI q
Idq
I*
dq
Fig. 5. Torque Production in a Salient-Pole Synchronous Machine Interpreted
as the Pairwise Interaction of Three Complex Current Vectors
Id
Fig. 4.
Proposed Phasor Diagram of a Salient-Pole Synchronous Machine
VI. T ORQUE P RODUCTION IN A S ALIENT-P OLE
S YNCHRONOUS M ACHINE
In steady state, Equations (30)-(31) describing the Àuxcurrent relationships can be expressed as
$
$
$
$
+ Pi Li
(53)
gt = Ov Lgt + Op Lgt
1
$
$ 1
$
$
i =
+ Oi Li
(54)
Pi Lgt + Pi Lgt
2
2
r
r
$
$
where Li = Li hm0 and i = i hm0 .
In this form, it is apparent that there are three complex
current vectors in the air gap of a salient-pole synchronous
$ $
$
machine, namely, Lgt , Lgt
and Li . These three current vectors
are depicted in Figure 5 and are relative to a dq reference frame
$
attached to the rotor. The ¿eld current vector Li is oriented
along the ¿eld axis which is assumed to be colinear with the
$
$
are complex
direct axis. The stator current vectors Lgt and Lgt
conjugate vectors with respect to the direct axis.
In steady state, a previous expression of the electromagnetic
torque, Equation (37), can be successively rewritten as
³s´
o
n
$
$$
2
Wh =
+ mOp (Lgt
)
(55)
<h mPi Li Lgt
´
³ 2s ´ ³
$
$ $
$
× Lgt
(56)
Pi Li × Lgt + Op Lgt
=
2
where the cross product expressions have been intuitively
de¿ned as
n $o
n
$$o
$ $ $
= =p Li Lgt
(57)
Li × Lgt = <h m Li Lgt
n $ o
n
o
$
$
$$
2
Lgt
× Lgt = <h m(Lgt
)
= =p Lgt Lgt (58)
Noting that
1
$ $ 1 $
$
Li × Lgt + Lgt
× Li
(59)
2
2
the last torque expression, Equation (56), can be expanded as
³s´ μ1
$
$ $ 1
$
× Li
Pi Li × Lgt + Pi Lgt
Wh =
2
2
2
´
$
$
+Op Lgt
× Lgt
(60)
$
$
Li × Lgt =
Torque production in a salient-pole synchronous machine
can now be interpreted as the pairwise magnetic interaction of
three complex current vectors in the air gap of the machine.
$
$
The interaction of the conjugate current vectors Lgt and Lgt
is
responsible for the reluctance component of the torque which
is absent in round-rotor synchronous machines.
VII. C ONCLUSION
In this paper, we have extended the two-reaction theory of
a two-phase salient-pole synchronous machine by proposing
a single equivalent circuit in dq coordinates. A new phasor
diagram for this equivalent circuit identi¿es an internal voltage
responsible for the internally-developed electromagnetic power
in a straightforward manner. The torque production in a
salient-pole synchronous machine has also been explained
using the interaction of three complex current vectors in the
air gap of the machine.
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