Download In this mode the PW is connected to the grid and the CW is short

Complex Vector Model of the Brushless Doubly Fed
Machine in Unified Reference Frame
S. ATALLAH
D. BENATTOUS
M.-S. NAIT-SAID
Laboratoire des Systèmes Propulsion-Induction
Electromagnétiques LSP-IE’2000 Batna
EL-Oued, Algeria
[email protected]
Institute of Science Technology,
University Center of El-Oued
El-Oued, Algeria
[email protected]
Laboratoire des Systèmes PropulsionInduction Electromagnétiques LSPIE’2000 Batna, Algeria
[email protected]
Abstract—The Brushless Doubly Fed Machine (BDFM) is a
machine which incorporates the robustness of the squirrel cage
induction machine and the speed and power factor control of a
synchronous machine. In this paper, a detailed mathematical
derivation of the BDFM unified d–q reference frame model is
introduced. The model is based on coupled magnetic circuit
theory and complex space-vector notation. Then the obtained
dynamic model of the machine will be finally tested in simulation
workbench using MATLAB/SIMULINK in order to verify the
proposed model. This way, a simple dq model can be established,
which could be an interesting tool for control synthesis tasks.
Keywords-Brushless Doubly Fed Machine (BDFM), Cross
Coupling, Control Winding (CW), Power Winding (PW), Variable
Speed Generation, Unified Reference Frame Model
I.
NOMENCLATURE
LIST OF SYMBOLS
𝑓𝑝 (𝑓𝑐 ):
𝑖⃗𝑝 (𝑖⃗𝑐 ):
PW (CW) frequency.
𝑅𝑟 :
Stator PW (CW) current vector.
Rotor current vector.
Stator PW (CW) self-inductance.
Rotor self-inductance.
Unified frame stator PW (CW) to rotor coupling
inductance.
Number of rotor nests.
Power (control) winding pole pairs Number.
Real part.
Stator PW (CW) resistance.
Rotor resistance.
𝑇𝑒𝑚𝑝 (𝑇𝑒𝑚𝑐 ):
PW (CW) electromagnetic torque.
𝑇𝑒𝑚 :
⃗⃗𝑝 (𝑉
⃗⃗𝑐 ):
𝑉
Total electromagnetic torque.
𝔍𝑚 {… }:
Imaginary part.
𝜃𝑜𝑏𝑠𝑝 (𝜃𝑜𝑏𝑠𝑐 ):
Angle between the PW (CW) and the generic reference
𝑖⃗𝑟 :
𝐿𝑝 (𝐿𝑐 ):
𝐿𝑟 :
𝑀𝑝 (𝑀𝑐 ):
𝑁𝑟 :
𝑝𝑝 (𝑝𝑐 ):
ℛ𝑒 {… }:
𝑅𝑝 (𝑅𝑐 ):
Stator PW (CW) fed voltage vector.
𝜔𝑝 ( 𝜔𝑐 ):
frame.
Rotor shaft displacement between the rotor and the PW
reference axis.
Synchronous angular frequency of the PW (CW).
𝜔𝑟𝑝 :
Angular slip speed of the PW.
δ:
axis.
Initial angle between the rotor and the PW references
𝜃𝑟 :
𝜑
⃗⃗𝑝 (𝜑
⃗⃗𝑐 ):
𝜑
⃗⃗𝑟 :
Ω:
𝛾:
Stator PW (CW) flux linkage vector.
Rotor flux linkage vector.
Rotor’s mechanical angular speed.
Angle between the PW and the CW references axis.
SUBSCRIPTS
p, c, r:
𝑆𝑝 (𝑆𝑐 ):
Power winding, control winding, rotor.
Stator power (control) winding phase.
SUPERSCRIPTS
∗:
𝑑𝑞:
𝑑𝑞𝑝 :
Complex conjugate.
The direct and quadrant component on the power
winding flux frame.
Generic reference frame of 𝑃𝑝 -pole pairs.
𝑑𝑞𝑐 :
𝑥𝑦𝑝 :
𝛼𝛽𝑐 :
𝛼𝛽𝑝 :
Generic reference frame of 𝑃𝑐 -pole pairs.
Rotor reference frame.
Control winding reference frame.
Power winding reference frame.
ACRONYMS
BDFM:
CW:
PW:
WRIM:
Brushless Doubly Fed Machine.
Control winding
Power winding.
Wound Rotor Induction Machine.
II.
INTRODUCTION
When wind power generator is connected to the power grid,
the output frequency should be identical with the frequency of
the power grid. Wind energy capturing and conversion
efficiency can be improved by taking advantage of variablespeed constant-frequency (VSCF) method which uses the
Wound Rotor Induction Machine (WRIM, old material) [1].
But the main problem is that the slip rings and wound rotor
arrangement which limit its application in harsh environment.
Among the possible solutions for these shortcomings is the
introduction of the so-called Brushless Doubly Fed Machine
(BDFM), which can be seen as an advanced version of the
(WRIM), because it is based on the same principle of the slip
energy recovery used for the output control [2].
Fig.1. BDFM system
The Brushless Doubly-Fed Machine (BDFM) has the
potential to be employed as a variable speed generator such as
in wind power applications or as a motor in adjustable speed
drives [1] [2] [4]. The BDFM consists of two electrically
independent balanced three phase windings which have no
mutual couplings and wound on the same core in the stator.
The rotor circuit in BDFM is considered as a nested-loop type
which couples to both fields of the stator and is actually the
most well known one for BDFM. It consists of nests equally
spaced around the circumference whose number is equal to the
sum of the stator windings pole pairs Fig 1 [3].
A BDFM model was derived assuming that the machine
was composed of two superposed subsystems [5] [6]. Each
subsystem contained the dynamics of one of the two stator
windings (PW or CW) and the corresponding rotor dynamics.
The set of equations of the PW or CW subsystem were written
in two different synchronous reference frames related to each
pole-pair distribution. This leads to a couple of equations
describing the dynamics of two independent rotor currents
which correspond to two different synchronous reference
frames. The electromagnetic torque depends on the current
and the flux of both subsystems, as well as the so-called
‘synchronous angle’ between the two reference frames. But
the existence of multiple reference frames related to the two
stator windings and the rotor makes it difficult to exploit the
well known standard induction machine control strategies.
The main aim of this paper is to develop a mathematical
dynamic model for the BDFM based on the complex space
vector notation, leading to a unified dq reference frame model.
III.
CONCEPT OF THE CROSS COUPLING EFFECT
For the BDFM the major interest is the operation in
synchronous mode, the essential feature of synchronous
operation is the electromagnetic coupling of one stator winding
system with the other, exclusively through the rotor. Since the
stator windings which can be assumed to be sinusoidally
distributed for different pole-pair numbers, there is
(intentionally) no direct coupling between both stator windings
[7]. However, each stator winding can be coupled directly with
the rotor. The induced rotor currents from both stator windings
should be have an appropriate sequence and frequency. It
results that the rotor current creates the appropriate fields
Fig.2. BDFM coupling mechanism schematic
which can induce voltages in the power windings (stator)
initially due to control windings currents, and vice versa. This
indirect induction mechanism is referred to as cross coupling
effect which is illustrated in Fig 2 [8]. This concept
development assumes a linear magnetic circuit and deals solely
with synchronous operation of the BDFM in which the two
windings act in a complementary manner.
Asynchronous operation, in which the system consists of
two conflicting induction machines, is undesirable and is
avoided through proper control of the control winding voltage
(or current). When excited, each of the stator winding systems
produces a traveling flux wave in the airgap. Each of these can
be expressed in the form,
𝑏𝑝 (𝜃, 𝑡) = 𝐵𝑝 𝑐𝑜𝑠(𝜔𝑝 𝑡 − 𝑝𝑝 𝜃)
𝑏𝑐 (𝜃, 𝑡) = 𝐵𝑐 𝑐𝑜𝑠(𝜔𝑐 𝑡 − 𝑝𝑐 𝜃)
(1)
(2)
In order to obtain the desired cross-coupling effect, the PW
and the CW currents induce at the rotor bars must evolve with
the same frequency [2]. This operating restriction leads to the
so-called synchronous rotor speed, which is equal to,
𝛺=
𝜔𝑝 +𝜔𝑐
(3)
𝑝𝑝 +𝑝𝑐
The stator produced flux densities can be written in terms of
a particular rotor observer angle, p, and time by subsisting (4)
into (1) and (2).
𝜃 = 𝛺𝑡 + 𝜃̀
(4)
Which yields,
𝑏𝑝 (𝜃, 𝑡) = 𝐵𝑝 𝑐𝑜𝑠[𝜔𝑟𝑝 𝑡 − 𝑝𝑝 𝜃̀]
𝑏𝑐 (𝜃, 𝑡) = 𝐵𝑐 𝑐𝑜𝑠[𝜔𝑟𝑐 𝑡 − 𝑝𝑐 𝜃̀ ]
(5)
(6)
In which,
𝜔𝑟𝑝 =
𝑝𝑐 ω𝑝 −𝑝𝑝 ω𝑐
𝑝𝑝 +𝑝𝑐
= −𝜔𝑟𝑐
(7)
Thus, the effect of the stator operating at the speed given
by (7) which is the mean of the two stator-produced field
speeds is to produce flux velocities which are equal but in
opposite directions when viewed from the rotor. These flux
waves can be viewed collectively according to (8) where
𝑖𝑟1
1
𝑏−𝑃𝑝 … … 𝑏 −(𝑛−1)𝑃𝑝
𝑖𝑟2
𝑒 −𝑗𝑝𝑝 (𝜃𝑟 +𝛿) ( 𝑎(1 𝑏−𝑃𝑝 … … 𝑏−(𝑛−1)𝑃𝑝 ))} ⋮
⋮
𝑎2 (1 𝑏−𝑃𝑝 … … 𝑏−(𝑛−1)𝑃𝑝 )
(𝑖𝑟𝑛 )
ωrp = ωr ,
𝑏𝑠 (𝜃, 𝑡) = 𝐵𝑝 𝑐𝑜𝑠[𝜔𝑟 𝑡 − 𝑝𝑝 𝜃̀] + 𝐵𝑐 𝑐𝑜𝑠[𝜔𝑟 𝑡 + 𝑝𝑐 𝜃̀]
IV.
(8)
DYNAMIC MODEL OF THE BDFM
2𝜋
The objective of this section is to develop the unified dq
reference frame model of the BDFM based on the space vector
notation.
A. Stator Model
The PW flux linkage can be written as the contribution of
three components as,
Sp
Sp
Sp
(9)
φSp = φSp + φSc + φR
The first term can be expressed as,
𝑆𝑝
𝜑𝑆𝑝1
𝑆𝑝
𝜑𝑆𝑝
𝑙𝑝
𝑆𝑝
= (𝜑𝑆𝑝2 ) = (𝑚𝑝
𝑆𝑝
𝑚𝑝
𝜑
𝑆𝑝3
𝑚𝑝
𝑙𝑝
𝑚𝑝
0
𝐿𝑝
0
(10)
0
𝑖𝑆𝑝1
𝑖𝑆𝑝1
0 ) (𝑖𝑆𝑝2 ) = 𝐿𝑝 (𝑖𝑆𝑝2 )
𝑖𝑆𝑝3
𝑖𝑆𝑝3
𝐿𝑝
(11)
……
……
− 𝑝𝑝 [(𝜃𝑟 + 𝛿) + 𝛼𝑟 ]] … …
− 𝑝𝑝 (𝜃𝑟 +
……
− 𝑝𝑝 (𝜃𝑟 + 𝛿)] cos [2π − 𝑝 [(𝜃 + 𝛿) + 𝛼 ]]
𝑝
𝑟
𝑟
……
3
𝑖
𝑟1
[(𝜃
(𝑛
]
cos 𝑝𝑝 𝑟 + 𝛿) + − 1)𝛼𝑟
𝑖𝑟2
cos [2π
− 𝑝𝑝 [(𝜃𝑟 + 𝛿) + (𝑛 − 1)𝛼𝑟 ]]
⋮
3
⋮
[(𝜃
(𝑛
]]
cos [4π
−
𝑝
+
𝛿)
+
−
1)𝛼
𝑝
𝑟
𝑟
3
) (𝑖𝑟𝑛 )
2π
𝑀0𝑝 (cos[ 3
4π
cos[ 3
𝑆𝑝
𝜑𝑅 =
cos 𝑝𝑝 [(𝜃𝑟 + 𝛿) + 𝛼𝑟 ]
𝑀0𝑝 𝑗𝑝 (𝜃 +𝛿)
{ 𝑒 𝑝 𝑟 [1 𝑏𝑃𝑝 𝑏2𝑃𝑝 … … 𝑏(𝑛−1)𝑃𝑝 ] +
2
𝑖𝑟1
𝑖𝑟2
−𝑃
−2𝑃𝑝
𝑒 −𝑗𝑝𝑝(𝜃𝑟 +𝛿)[1 𝑏 𝑝 𝑏
… … 𝑏−(𝑛−1)𝑃𝑝 ]} ⋮
⋮
(𝑖𝑟𝑛 )
𝑆𝑝
𝑀0𝑝
2
(𝑛−1)𝑃𝑝
1
𝑏 …… 𝑏
{ 𝑒𝑗𝑝𝑝 (𝜃𝑟 +𝛿) (𝑎2 (1 𝑏𝑃𝑝 … … 𝑏(𝑛−1)𝑃𝑝 )) +
𝑎(1 𝑏𝑃𝑝 … … 𝑏(𝑛−1)𝑃𝑝 )
(16) yields,
𝜑
⃗⃗𝑠𝑝 = 𝐿𝑝 𝑖⃗𝑠𝑝 + 𝐾
𝜑
⃗⃗𝑠𝑐 = 𝐿𝑐 𝑖⃗𝑠𝑐 + 𝐾
𝑛
2
𝑛
𝑥𝑦𝑝
𝑀0𝑝 𝑒 𝑗𝑝𝑝 (𝜃𝑟 +𝛿) 𝑖⃗𝑅
𝑥𝑦𝑐
𝑀 𝑒 𝑗𝑝𝑐 [(𝜃𝑟 +𝛿)−𝛾] 𝑖⃗𝑅
2 0𝑐
(17)
(18)
The power and control windings voltage equation may be
done as followed,
(19)
Where,
𝑖⃗𝑠𝑝 = 23(𝑖𝑆𝑝1 + 𝑎𝑖𝑆𝑝2 + 𝑎2 𝑖𝑆𝑝3 )
𝑥𝑦𝑝
(12)
𝑖⃗𝑅
=
2
𝐾𝑛
(𝑖𝑟1 + 𝑖𝑟2 𝑏𝑃𝑝 + ⋯ + 𝑖𝑟𝑛 𝑏(𝑛−1)𝑃𝑝 )
𝑖⃗𝑠𝑐 = 23(𝑖𝑆𝑐1 + 𝑎𝑖𝑆𝑐2 + 𝑎2 𝑖𝑆𝑐3 )
𝑥𝑦𝑐
𝑖⃗𝑅
=
2
𝑘𝑛
(𝑖𝑟1 + 𝑖𝑟2 𝑏𝑃𝑐 + ⋯ + 𝑖𝑟𝑛 𝑏(𝑛−1)𝑃𝑐 )
(13)
(20)
(21)
(22)
(23)
B. Rotor Model
The rotor flux can be divided into three components,
𝑅
𝑅
𝜑𝑅 = 𝜑𝑅𝑅 + 𝜑𝑆𝑝
+ 𝜑𝑆𝑐
In the same way for the second and third components, results,
𝜑𝑅 =
(15)
1
𝑏𝑃𝑐 … … 𝑏(𝑛−1)𝑃𝑐
𝑖𝑆𝑐1
𝑀0𝑐
2
[(𝜃
𝑗𝑝
+𝛿)−𝛾]
𝑐
𝑟
𝜑𝑆𝑐 = 𝐿𝑐 (𝑖𝑆𝑐2 ) +
{𝑒
(𝑎 (1 𝑏𝑃𝑐 … … 𝑏(𝑛−1)𝑃𝑐 )) +
2
𝑖𝑆𝑐3
𝑎(1 𝑏𝑃𝑐 … … 𝑏(𝑛−1)𝑃𝑐 )
𝑖𝑟1
1
𝑏−𝑃𝑐 … … 𝑏−(𝑛−1)𝑃𝑐
𝑖𝑟2
𝑒 −𝑗𝑝𝑐[(𝜃𝑟 +𝛿)−𝛾] ( 𝑎(1 𝑏−𝑃𝑐 … … 𝑏−(𝑛−1)𝑃𝑐 ))} ⋮
(16)
⋮
𝑎2 (1 𝑏−𝑃𝑐 … … 𝑏−(𝑛−1)𝑃𝑐 )
(𝑖𝑟𝑛 )
𝑑𝑡
𝜑𝑅𝑝1 =
𝑃𝑝
𝑖𝑆𝑝1
1
𝑏𝑃𝑝 … … 𝑏(𝑛−1)𝑃𝑝
𝑀0𝑝
𝑗𝑝𝑝(𝜃𝑟 +𝛿) 𝑎 2 (1 𝑏 𝑃𝑝 … … 𝑏 (𝑛−1)𝑃𝑝 )
𝑖
𝜑𝑆𝑝 = 𝐿𝑝 ( 𝑆𝑝2 ) +
{𝑒
(
)+
2
𝑖𝑆𝑝3
𝑎(1 𝑏𝑃𝑝 … … 𝑏(𝑛−1)𝑃𝑝 )
𝑖𝑟1
1
𝑏−𝑃𝑝 … … 𝑏−(𝑛−1)𝑃𝑝
𝑖𝑟2
−𝑃
−(𝑛−1)𝑃
𝑝)
𝑒 −𝑗𝑝𝑝 (𝜃𝑟 +𝛿) ( 𝑎(1 𝑏 𝑝 … … 𝑏
)} ⋮
⋮
𝑎2 (1 𝑏−𝑃𝑝 … … 𝑏−(𝑛−1)𝑃𝑝 )
(𝑖𝑟𝑛 )
⃗⃗⃗𝑠𝑝,𝑠𝑐
⃗⃗𝑠𝑝,𝑠𝑐 = 𝑅𝑠𝑝,𝑠𝑐 𝑖⃗𝑠𝑝,𝑠𝑐 + 𝑑𝜑
𝑉
2π
𝛿)] cos [ 3
So we can write, for example, for the first component of last
flux vector,
𝑠
So (9) becomes,
Applying the three-phase space-vector definition to (15)
The second term equal zero because the two stator winding
sets have different numbers of poles (pp ≠ pc ) [8]. and the
last term is the contribution of the rotor current the cage rotor
having n rotor nest can be assumed as a system of n phases
[9]. The total stator flux linkage due to the rotor currents can
be derived as,
cos 𝑝𝑝 (𝜃𝑟 + 𝛿)
Where, 𝑏 = 𝑒𝑗𝛼𝑟 , 𝛼𝑟 = 2𝜋
, 𝑎 = 𝑒𝑗 3
𝑛
In the same manner for the CW flux linkage, we can write,
𝑚𝑝
𝑖𝑆𝑝1
𝑚𝑝 ) (𝑖𝑆𝑝2 )
𝑖𝑆𝑝3
𝑙𝑝
So, (10) becomes,
𝐿𝑝
𝑆𝑝
𝜑𝑆𝑝 = ( 0
0
(14)
(24)
One due to the rotor currents and two due to the PW and CW
currents, more detailed explanations of each term will be
developed separately in the following subsections.
Rotor flux linkage in the rotor nest due to rotor nest currents
𝜑𝑅𝑅
𝑀11
𝑀21
=(
⋮
𝑀𝑛1
𝑀12 … …
𝑀1𝑛
𝑖𝑟1
𝑖
𝑀22 … …
𝑀2𝑛
) ( 𝑟2 )
⋯ ………
⋮
⋮
⋯ … … 𝑀𝑛𝑛
𝑖𝑟𝑛
(25)
Applying the space-vector theory definition to (25) by
multiplying the first term of by (1, b Pp , … , b (n−1)Pp ) yields the
following serial equations,
𝑅
𝜑𝑅1
𝑅
𝑏 𝜑𝑅2
𝑀11
𝑅
𝑏2𝑃𝑝 𝜑𝑅3
𝑀
= {(1, 𝑏𝑃𝑝 , 𝑏2𝑃𝑝 , … , 𝑏(𝑛−1)𝑃𝑝 ) ( 21
⋮
⋮
⋮
𝑀𝑛1
⋮
𝑅
(𝑏 (𝑛−1)𝑃𝑝 𝜑𝑅𝑛
)
𝑃𝑝
𝑀12 … …
𝑀22 … …
⋯ ………
⋯ … …
𝑖𝑟1
𝑖𝑟2
𝑀1𝑛
𝑖𝑟3
𝑀2𝑛
)} ⋮
⋮
⋮
𝑀𝑛𝑛
⋮
(𝑖𝑟𝑛 )
With (1 + b 2Pp + ⋯ + b 2(n−1)Pp = 0) , the rotor nests flux
linkage due to the power winding current becomes,
𝑅
𝜑
⃗⃗𝑆𝑝
=
3 𝑀0𝑝
2 𝐾
𝛼𝛽𝑝
𝑒 −𝑗𝑝𝑝 (𝜃𝑟 +𝛿) 𝑖⃗𝑠𝑝
Flux linkage in the rotor nest due to control winding currents
(26)
Using the identity formulation 𝑏(𝑛+𝑚)𝑃𝑝 = 𝑏𝑚𝑃𝑝 , we can get
𝑅
𝜑𝑅1
𝑀11
𝑅
𝑏𝑃𝑝 𝜑𝑅2
𝑀21
𝑅
𝑀31
𝑏2𝑃𝑝 𝜑𝑅3
= (1, 𝑏𝑃𝑝 , 𝑏2𝑃𝑝 , … , 𝑏 (𝑛−1)𝑃𝑝 )
∗
⋮
⋮
⋮
⋮
⋮
⋮
(𝑀𝑛1 )
𝑅
(𝑏 (𝑛−1)𝑃𝑝 𝜑𝑅𝑛
)
𝑖𝑟1
𝑖𝑟2
𝑖𝑟3
(𝑛−1)𝑃
𝑝)
(1, 𝑏𝑃𝑝 , 𝑏2𝑃𝑝 , … , 𝑏
⋮
⋮
⋮
{
(𝑖𝑟ㅳ )}
And therefore,
𝑥𝑦𝑝
𝜑
⃗⃗𝑅𝑅 = 𝐿𝑟 𝑖⃗𝑅
(28)
With,
𝑀11
𝑀21
𝑀31
𝐿𝑟 = (1, 𝑏𝑃𝑝 , 𝑏2𝑃𝑝 , … , 𝑏(𝑛−1)𝑃𝑝 )
⋮
⋮
⋮
(𝑀𝑛1 )
= 𝑀11 + 𝑏𝑃𝑝 𝑀12 + 𝑏2𝑃𝑝 𝑀13 + ⋯ + 𝑏(𝑛−1)𝑃𝑝 𝑀1𝑛
(29)
Flux linkage in the rotor nest due to power winding currents
1
1
1
𝑖𝑆𝑝1
𝑀0𝑝
𝑏𝑃𝑝
𝑏𝑃𝑝
𝑏𝑃𝑝
𝑗𝑝𝑝 (𝜃𝑟 +𝛿)
=
{𝑒
(
) (𝑎2 𝑖𝑆𝑝2 ) +
⋮
⋮
⋮
2
𝑎𝑖𝑆𝑝3
𝑏(𝑛−1)𝑃𝑝 𝑏(𝑛−1)𝑃𝑝 𝑏(𝑛−1)𝑃𝑝
1
𝑒 −𝑗𝑝𝑝 (𝜃𝑟 +𝛿) (
𝑏
1
𝑏−𝑃𝑝
⋮
−(𝑛−1)𝑃𝑝
𝑏−𝑃𝑝
⋮
𝑏
−(𝑛−1)𝑃𝑝
𝑏
(30)
Applying the space-vector theory definition to (30) and after
multiplying each component of rotor flux successively by
(1, b Pp , b 2Pp , … , b (n−1)Pp ), we can get,
𝑅1
𝜑𝑆𝑝
1
1
1
𝑅2
𝑖𝑆𝑝1
𝑏 𝜑𝑆𝑝
𝑀0𝑝
𝑏2𝑃𝑝
𝑏 2𝑃𝑝
𝑏2𝑃𝑝
=
{ 𝑒 𝑗𝑝𝑝 (𝜃𝑟 +𝛿) (
) (𝑎2 𝑖𝑆𝑝2 ) +
⋮
⋮
⋮
⋮
2
𝑎𝑖𝑆𝑝3
⋮
𝑏2(𝑛−1)𝑃𝑝 𝑏2(𝑛−1)𝑃𝑝 𝑏2(𝑛−1)𝑃𝑝
(𝑛−1)𝑃𝑝 𝑅𝑛
𝑏
𝜑
(
𝑆𝑝 )
𝑃𝑝
𝑒 −𝑗𝑝𝑝 (𝜃𝑟 +𝛿) (
1
1
⋮
1
1
1
⋮
1
1
𝑖𝑆𝑝1
1
) ( 𝑎𝑖𝑆𝑝2 )}
⋮
𝑎2 𝑖𝑆𝑝3
1
2
{𝑒
𝑗𝑝𝑐 [(𝜃𝑟 +𝛿)−𝛾]
−𝑗𝑝𝑐 [(𝜃𝑟 +𝛿)−𝛾]
1
1
1
𝑖𝑆𝑐1
𝑏𝑃𝑐
𝑏𝑃𝑐
𝑏𝑃𝑐
(
) (𝑎2 𝑖𝑆𝑐2 ) +
⋮
⋮
⋮
𝑎𝑖𝑆𝑐3
𝑏(𝑛−1)𝑃𝑐 𝑏(𝑛−1)𝑃𝑐 𝑏(𝑛−1)𝑃𝑐
1
1
1
𝑖𝑆𝑐1
𝑏−𝑃𝑐
𝑏−𝑃𝑐
𝑏−𝑃𝑐
(
) ( 𝑎𝑖𝑆𝑐2 )}
⋮
⋮
⋮
𝑎2 𝑖𝑆𝑐3
𝑏−(𝑛−1)𝑃𝑐 𝑏−(𝑛−1)𝑃𝑐 𝑏−(𝑛−1)𝑃𝑐
(33)
The calculation of this magnetic coupling effect is vital to
determine the machine operation, since its existence produces
a cross coupling being well indicated by Fig.2 between both
stator windings through the rotor. Once the cross coupling is
produced, the current of each stator winding will not solely
depend on its own supply voltage, but it will also vary
according to the voltage of the other stator winding. On the
other hand, if the cross coupling does not produce the
electrical machine would operate like two independent
asynchronous machines with the same axis.
𝑅1
𝜑𝑆𝑐
1
1
1
𝑅2
𝑖𝑆𝑐1
𝑏𝑃𝑝 𝜑𝑆𝑐
𝑀0𝑐
𝑏𝑃𝑝 +𝑃𝑐
𝑏𝑃𝑝 +𝑃𝑐
𝑏𝑃𝑝 +𝑃𝑐
=
{ 𝑒 𝑗𝑝𝑐 [(𝜃𝑟 +𝛿)−𝛾] (
) (𝑎2 𝑖𝑆𝑐2 ) +
⋮
⋮
⋮
⋮
2
𝑎𝑖𝑆𝑐3
⋮
𝑏 (𝑛−1)(𝑃𝑝 +𝑃𝑐 ) 𝑏 (𝑛−1)(𝑃𝑝 +𝑃𝑐 ) 𝑏 (𝑛−1)(𝑃𝑝 +𝑃𝑐 )
𝑅𝑛
(𝑏(𝑛−1)𝑃𝑝 𝜑𝑆𝑐
)
𝑒 −𝑗𝑝𝑐 [(𝜃𝑟 +𝛿)−𝛾] (
1
𝑏𝑃𝑝 −𝑃𝑐
⋮
1
𝑏𝑃𝑝 −𝑃𝑐
⋮
𝑏(𝑛−1)(𝑃𝑝 −𝑃𝑐 ) 𝑏(𝑛−1)(𝑃𝑝 −𝑃𝑐 ) 𝑏
1
𝑏𝑃𝑝 −𝑃𝑐
⋮
𝑖𝑆𝑐1
) ( 𝑎𝑖𝑆𝑐2 )}
𝑎2 𝑖𝑆𝑐3
(𝑛−1)(𝑃𝑝 −𝑃𝑐 )
(34)
As shown in (34), the rotor flux vector due to CW currents
depends on the selected values of pp and pc pole-pairs. By
analyzing different combinations, there are two possible cases
[8],
 Possibility 1
𝑅
𝜑
⃗⃗𝑆𝑐
= 0 ⟹ 𝑏 (𝑃𝑝 +𝑃𝑐 ) & 𝑏 (𝑃𝑝 −𝑃𝑐 ) ≠ 1 ⟹ Inexistence of a cross
1
𝑖𝑆𝑝1
𝑏−𝑃𝑝
)} ( 𝑎𝑖𝑆𝑝2 )
⋮
𝑎2 𝑖𝑆𝑝3
−(𝑛−1)𝑃𝑝
=
𝑀0𝑐
From (33), after multiplying each component of rotor flux
successively by (1, b Pp , b 2Pp , … , b (n−1)Pp ) , which defines the
cross coupling, we can obtain,
The proportionality constant Lr corresponds on the
equivalent rotor self inductance. Note that its value is
expressed only in terms of rotor nest’s dimension.
𝑅
𝜑𝑆𝑝
𝑅
𝜑𝑆𝑐
𝑒
(27)
(32)
(31)
coupling between the both stator windings through the rotor
current.
 Possibility 2
𝑅
𝜑
⃗⃗𝑆𝑐
≠ 0 ⟹ 𝑏 (𝑃𝑝 +𝑃𝑐 ) 𝑜𝑟 𝑏 (𝑃𝑝 −𝑃𝑐 ) = 1 ⟹ The existence of a cross
coupling between the two stator windings through the rotor
current. There are two possible configurations,
2𝜋
 configuration1: 𝑏(𝑃𝑝−𝑃𝑐) = 𝑒 𝑗 𝑛 (𝑃𝑝−𝑃𝑐) = 1 ⟹ 𝑛 =
 configuration2: 𝑏(𝑃𝑝+𝑃𝑐) = 𝑒
2𝜋
𝑗 𝑛 (𝑃𝑝+𝑃𝑐 )
Where, 𝑞 = 0, ±1, ±2 … … … … …
=1⟹ 𝑛=
(𝑃𝑝−𝑃𝑐 )
𝑞
(𝑃𝑝 +𝑃𝑐 )
𝑞
(35)
(36)
So, to ensure this cross coupling effect, we should chose the
second configuration (36) which maximizes the number of
rotor nest’s i.e (𝑞 = 1). So, 𝑛 = 𝑝𝑝 + 𝑝𝑐 implying that,
𝑥𝑦𝑝
𝑖⃗𝑅
2
=
𝑘𝑛
(𝑖𝑟1 + 𝑖𝑟2 𝑏𝑃𝑝 + ⋯ + 𝑖𝑟𝑛 𝑏(𝑛−1)𝑃𝑝 )
(37)
We know that 𝑝𝑝 = 𝑛 − 𝑝𝑐 ⟹
𝑥𝑦𝑃
𝑖⃗𝑅
2
=
𝑘𝑛
=
=
So,
(𝑖𝑟1 + 𝑖𝑟2 𝑏𝑛−𝑃𝑐 + ⋯ + 𝑖𝑟𝑛 𝑏(𝑛−1)(𝑛−𝑃𝑐) )
2
𝑘𝑛
(𝑖𝑟1 + 𝑖𝑟2 𝑏
−𝑃𝑐
+ ⋯ + 𝑖𝑟𝑛 𝑏
(𝑛−1)(−𝑃𝑐 )
)
∗𝑥𝑦
𝑖⃗𝑅 𝑐
𝑥𝑦
𝑖⃗𝑅 𝑃
=
∗𝑥𝑦
𝑖⃗𝑅 𝑐
(38)
So we conclude that, in last configuration, one of the current
vectors behaves as the conjugate of the other. According to
this relation, it becomes straightforward to change from a pp −
type reference frame to a pc − type one or vice versa. This
one constitutes the key step for the derivation of the unified 𝑑𝑞
reference frame model.
Replacing with 𝑛 = 𝑝𝑝 + 𝑝𝑐 in (34), yields
𝑅1
𝜑𝑆𝑐
1 1
𝑅2
𝑏𝑃𝑝 𝜑𝑆𝑐
𝑀
1 1
= 0𝑐 { 𝑒 𝑗𝑝𝑐[(𝜃𝑟 +𝛿)−𝛾] (
⋮
𝟐
⋮ ⋮
⋮
1 1
(𝑛−1)𝑃𝑝 𝑅𝑛
𝜑𝑆𝑐 )
(𝑏
1
𝑖𝑆𝑐1
1
) (𝑎2 𝑖𝑆𝑐2 )}
⋮
𝑎𝑖𝑆𝑐3
1
(39)
And so,
𝑅
𝜑
⃗⃗𝑆𝑐
=
𝟑 𝑀0𝑐
𝟐 𝑲
∗𝛼𝛽𝑐
𝑒 𝑗𝑝𝑐[(𝜃𝑟 +𝛿)−𝛾] 𝑖⃗𝑠𝑐
(40)
𝑝𝑝 , 𝑝𝑐 pole-pairs
(45)
𝑀0(𝑝,𝑐)
As it can be observed the initial set of (44) is referred to
three different reference frames and tow possible pole-pairs
distributions may be considered, the goal is to get a set of
equations with a unified reference frame with a given polepairs distribution (e.g. 𝑝𝑝 ) which form the main aim of the
follows part.
V.
UNIFIED REFERENCE FRAME MODEL OF THE
BDFM
We can easily to write the previous system in a unified
reference frame model if we followed the steps given in
appendix IX.1.
By means of these vector transformations the machine
model (44) is expressed in a common dq-generic reference
frame where dq symbol indices have been removed to simplify
resulting expressions which are given as follows,
𝑑𝑡
𝑑𝑡
𝑑𝑡
3 𝑀0𝑝
2 𝐾
𝛼𝛽𝑝
𝑒 −𝑗𝑝𝑝 (𝜃𝑟 +𝛿) 𝑖⃗𝑠𝑝
+
3 𝑀0𝑐
2 𝐾
∗𝛼𝛽𝑐
𝑒 𝑗𝑝𝑐[(𝜃𝑟 +𝛿)−𝛾] 𝑖⃗𝑠𝑐
(41)
𝑛
3 𝑀0(𝑝,𝑐)
2
2
𝐾
(42)
The normalizing gain is identified as,
3
(43)
𝑛
𝜑
⃗⃗𝑐 = 𝐿𝑐 𝑖⃗𝑐 + 𝑀𝑐 𝑖⃗𝑟
⃗⃗⃗𝑟
⃗⃗𝑟 = 𝑅𝑟 𝑖⃗𝑟 + 𝑑𝜑
𝑉
+ 𝑗[𝜔𝑜𝑏𝑠𝑝 − 𝑝𝑝 Ω]𝜑
⃗⃗𝑟
(46)
𝑑𝑡
In (17), (18) and (41) in order to obtain the same equivalent
mutual inductance from rotor to stator as from stator to rotor,
the following constraint must be fulfilled,
𝐾=√
2
a 𝑝𝑐 pole-pairs
⃗⃗⃗𝑐
⃗⃗𝑐 = 𝑅𝑐 𝑖⃗𝑐 + 𝑑𝜑
𝑉
+ 𝑗[𝜔𝑜𝑏𝑠𝑝 − (𝑝𝑝 + 𝑝𝑐 )Ω]𝜑
⃗⃗𝑐
⃗⃗⃗𝑅
⃗⃗𝑅 = 𝑅𝑟 𝑖⃗𝑅 + 𝑑𝜑
𝑉
𝐾 𝑀0(𝑝,𝑐) =
√3𝑛
a pp pole-pairs
𝜑
⃗⃗𝑝 = 𝐿𝑝 𝑖⃗𝑝 + 𝑀𝑝 𝑖⃗𝑟
So the rotor nest’s voltage equation is given by
𝜑
⃗⃗𝑅 = 𝐿𝑟 𝑖⃗𝑅 +
𝑀𝑝,𝑐 =
assuming that the
⃗⃗⃗𝑝
⃗⃗𝑝 = 𝑅𝑝 𝑖⃗𝑝 + 𝑑 𝜑
𝑉
+ 𝑗𝜔𝑜𝑏𝑠𝑝 𝜑
⃗⃗𝑝
With, 𝑖⃗𝑠𝑐∗𝛼𝛽𝑐 = (𝑖𝑆𝑐1 + 𝑎2 𝑖𝑆𝑐2 + 𝑎𝑖𝑆𝑐3 ) = (𝑖𝑆𝑐1 + 𝑎−1 𝑖𝑆𝑐2 + 𝑎−2 𝑖𝑆𝑐3 )
{
The system defined by (44) is given in
following nomenclature,
𝛼𝛽𝑝
 𝑖⃗𝑠𝑝 ≡ 𝑖⃗𝑠𝑝 : PW reference frame in
distribution.
 𝑖⃗𝑠𝑐 ≡ 𝑖⃗𝑠𝑐𝛼𝛽𝑐 : CW reference frame in
distribution.
𝑥𝑦
 𝑖⃗𝑅 ≡ 𝑖⃗𝑅 𝑝 : Rotor references related to
distribution.
With,
Taking into account the obtained value from (43), we can
write the equations system from (17), (18), (19) and (41) as
follows,
⃗⃗⃗𝑠𝑝
⃗⃗𝑠𝑝 = 𝑅𝑠𝑝 𝑖⃗𝑠𝑝 + 𝑑𝜑
𝑉
⃗⃗𝑟 = 𝐿𝑟 𝑖⃗𝑟 + 𝑀𝑝 𝑖⃗𝑝 + 𝑀𝑐 𝑖⃗𝑐
{𝜑
This model is similar to the vector model of the induction
machine in presence of two stator winding. The expressions
related to stator power winding are the same as that of the
induction machine. In rotor flux equation, the influence of the
two stator currents is well represented. In stator control
winding, the factor [𝜔𝑜𝑏𝑠𝑝 − (𝑝𝑝 + 𝑝𝑐 )𝛺] characterizes the
relative angular velocity between the reference frames dq and
𝛼𝛽𝑝 [10].
VI.
TORQUE CALCULATION
The power absorbed by the machine caused by three
excitations PW, CW and rotor is given by,
𝑑𝑡
⃗⃗𝑝 . 𝑖⃗∗𝑝 } + ℛ𝑒 {𝑉
⃗⃗𝑐 . 𝑖⃗∗𝑐 } + ℛ𝑒 {𝑉
⃗⃗𝑟 . 𝑖⃗∗𝑟 }
𝑃𝑎𝑏𝑠 = ℛ𝑒 {𝑉
𝜑
⃗⃗𝑠𝑝 = 𝐿𝑝 𝑖⃗𝑠𝑝 + 𝑀𝑝 𝑒𝑗𝑝𝑝(𝜃𝑟 +𝛿) 𝑖⃗𝑅
⃗⃗𝑠𝑐 = 𝑅𝑠𝑐 𝑖⃗𝑠𝑐 +
𝑉
⃗⃗⃗𝑠𝑐
𝑑𝜑
𝑑𝑡
𝜑
⃗⃗𝑠𝑐 = 𝐿𝑐 𝑖⃗𝑠𝑐 + 𝑀𝑐 𝑒 𝑗𝑝𝑐[(𝜃𝑟 +𝛿)−𝛾] 𝑖⃗ ∗𝑅
⃗⃗⃗𝑟
⃗⃗𝑟 = 𝑅𝑟 𝑖⃗𝑅 + 𝑑𝜑
𝑉
𝑑𝑡
⃗⃗𝑟 = 𝐿𝑟 𝑖⃗𝑅 + 𝑀𝑝 𝑒 −𝑗𝑝𝑝 (𝜃𝑟 +𝛿) 𝑖⃗𝑠𝑝 + 𝑀𝑐 𝑒 𝑗𝑝𝑐[(𝜃𝑟 +𝛿)−𝛾] 𝑖⃗ ∗𝑠𝑐
{ 𝜑
(44)
(47)
Multiplying the voltage equations of (46) by 𝑖⃗∗𝑝 , 𝑖⃗∗𝑐 , 𝑖⃗∗𝑟
respectively and taking the real part we can write,
⃗⃗⃗𝑝 ∗
⃗⃗𝑝 . 𝑖⃗∗𝑝 } = 𝑅𝑝 𝑖𝑝2 + ℛ𝑒 {𝑑𝜑
ℛ𝑒 {𝑉
. 𝑖⃗𝑝 } + ℛ𝑒 {𝑗𝜔𝑜𝑏𝑠𝑝 𝜑
⃗⃗𝑝 . 𝑖⃗∗𝑝 }
𝑑𝑡
(48)
⃗⃗⃗𝑐 ∗
⃗⃗𝑐 . 𝑖⃗∗𝑐 } = 𝑅𝑐 𝑖𝑐2 + ℛ𝑒 {𝑑𝜑
ℛ𝑒 {𝑉
. 𝑖⃗𝑐 } + ℛ𝑒 {𝑗[𝜔𝑜𝑏𝑠𝑝 − (𝑝𝑝 + 𝑝𝑐 )Ω]𝜑
⃗⃗𝑐 . 𝑖⃗∗𝑐 }
𝑑𝑡
⃗⃗⃗𝑟
𝑑𝜑
2
∗
⃗⃗𝑟 . 𝑖⃗∗𝑟 } = 𝑅
ℛ𝑒 {𝑉
ℛ𝑒 {𝑗[𝜔𝑜𝑏𝑠𝑝 − 𝑝𝑝 Ω]𝜑
⃗⃗𝑟 . 𝑖⃗∗𝑟 }
⏟
𝑟 𝑖𝑟 + ℛ
⏟𝑒 { 𝑑𝑡 . 𝑖⃗𝑟 } + ⏟
𝑃𝐽𝑝𝑐𝑟
𝑃𝑎𝑝𝑐𝑟
(49)
(50)
𝑃𝑒𝑚𝑝𝑐𝑟
With,
𝑃𝑒𝑚𝑝𝑐𝑟 = 𝜔𝑜𝑏𝑠𝑝 ℛ𝑒 {𝑗𝜑
⃗⃗𝑝 . 𝑖⃗∗𝑝 } + [𝜔𝑜𝑏𝑠𝑝 − (𝑝𝑝 + 𝑝𝑐 )Ω]ℛ𝑒 {𝑗𝜑
⃗⃗𝑐 . 𝑖⃗∗𝑐 } + [𝜔𝑜𝑏𝑠𝑝 −
𝑝𝑝 Ω]ℛ𝑒 {𝑗𝜑
⃗⃗𝑟 . 𝑖⃗∗𝑟 }
(51)
By definition, the torque may be obtained from the
relationship of the total electromagnetic power at the rotor
shaft speed Ω,
𝑇𝑒𝑚 =
𝑃𝑒𝑚𝑝𝑐𝑟
(52)
Ω
A simple identity shows that,
ℛ𝑒 {𝑗𝑋⃗𝐴 . 𝑋⃗𝐵∗ } = 𝔍𝑚 {𝑋⃗𝐵 . 𝑋⃗𝐴∗ } = −𝔍𝑚 {𝑋⃗𝐴 . 𝑋⃗𝐵∗ }
(53)
So, Pempcr becomes,
𝑃𝑒𝑚𝑝𝑟 = 𝜔𝑜𝑏𝑠𝑝 𝔍𝑚 {𝑖⃗𝑝 . 𝜑
⃗⃗𝑝∗ } + [𝜔𝑜𝑏𝑠𝑝 − (𝑝𝑝 + 𝑝𝑐 )Ω]𝔍𝑚 {𝑖⃗𝑐 . 𝜑
⃗⃗𝑐∗ } + 𝑃𝑒𝑚𝑟
(54)
Where,
𝑃𝑒𝑚𝑟 = [𝜔𝑜𝑏𝑠𝑝 − 𝑝𝑝 Ω]𝔍𝑚 {𝑖⃗𝑟 . 𝜑
⃗⃗𝑟∗ }
(55)
The Conjugate of ⃗φ
⃗⃗r is,
𝜑
⃗⃗𝑟∗ = 𝐿𝑟 𝑖⃗∗𝑟 + 𝑀𝑝 𝑖⃗∗𝑝 + 𝑀𝑐 𝑖⃗∗𝑐
(56)
Fig3. Open loop speed scalar control scheme
presents open loop speed scalar control scheme from CW.
Note that two cases will be considered: CW short circuited and
CW fed controlled while PW is always grid supplied, relevant
parameter employed for simulation tasks are collected in
appendix IX.2.?
A. Singly fed induction mode operation
In this mode the PW is connected to the grid and the CW
is short-circuited. The existence of a single power supply in
the machine facilitates enormously the synchronization of the
both windings stator currents.
Fig4.a shows the simulated BDFM start-up speed-time
response under no-load condition, the obtained curve
resembles very closely to that of an induction motor. It will be
observed that once the synchronous speed is reached (Ω =
Replacing (56) in (55) yields,
𝑃𝑒𝑚𝑟 = [𝜔𝑜𝑏𝑠𝑝 − 𝑝𝑝 Ω]𝔍𝑚 {𝑀𝑝 𝑖⃗𝑟 . 𝑖⃗∗𝑝 + 𝑀𝑐 𝑖⃗𝑟 . 𝑖⃗∗𝑐 )}
(57)
⃗⃗⃗⃗ and 𝜑
⃗⃗⃗⃗ , we get also,
From the equation of 𝜑
𝑠𝑝
𝑠𝑐
𝑖⃗𝑟 =
⃗⃗⃗𝑐
𝜑
𝑀𝑐
𝑖⃗𝑟 =
−
⃗⃗⃗𝑝
𝜑
𝑀𝑝
𝐿𝑐
𝑖⃗
𝑀𝑐 𝑐
−
(58)
𝐿𝑝
𝑖⃗
𝑀𝑝 𝑝
(59)
(58) and (59) in (57) conduct to,
𝑃𝑒𝑚𝑟 = [𝜔𝑜𝑏𝑠𝑝 − 𝑝𝑝 Ω]𝔍𝑚 {𝜑
⃗⃗𝑝 . 𝑖⃗∗𝑝 + 𝜑
⃗⃗𝑐 . 𝑖⃗∗𝑐 }
(60)
Replacing (60) in (54) and after arrangement we get,
𝑃𝑒𝑚𝑝𝑐𝑟 = 𝑝𝑝 Ω𝔍𝑚 {𝜑
⃗⃗𝑝∗ . 𝑖⃗𝑝 } + 𝑝𝑐 Ω𝔍𝑚 {𝜑
⃗⃗𝑐 . 𝑖⃗∗𝑐 }
(61)
From (52) the electromagnetic torque, which is given by the
contribution of PW and CW, can be expressed as,
𝑇𝑒𝑚 = 𝑇𝑒𝑚_𝑝 + 𝑇𝑒𝑚_𝑐
(62)
Whereas,
𝑇𝑒𝑚 = 𝑝𝑝 𝔍𝑚 {𝜑
⃗⃗𝑝∗ . 𝑖⃗𝑝 } + 𝑝𝑐 𝔍𝑚 {𝜑
⃗⃗𝑐 . 𝑖⃗∗𝑐 }
(63)
In addition, we can express the electromagnetic torque by
the PW, CW and rotor currents as follows
𝑇𝑒𝑚 = 𝑝𝑝 𝑀𝑝 𝔍𝑚 {𝑖⃗𝑝 . 𝑖⃗∗𝑟 } + 𝑝𝑐 𝑀𝑐 𝔍𝑚 {𝑖⃗𝑟 . 𝑖⃗∗𝑐 }
(64)
VII. SIMULATIONS RESULTS
To test the BDFM, the model has been implemented using
MATLAB/SIMULINK package as shown in Fig3. Which
Fig4. Simulation results of singly fed induction mode operation
77.58 rad/𝑠𝑒𝑐), the frequency of CW is quite small near zero
as shown in Fig4.b.
Initially, the machine was running synchronously at 750
rpm (78.5rad/sec) with unload torque.
Fig4.c and Fig4.d shows the Temporal values of the
currents for both stator windings.
At t=3.5 seconds, the CW excitation voltage is applied.
The BDFM Speed decreases from synchronous to
subsynchronous regimes and the electromagnetic torque is
remained after the transient unchanged.
The starting torque-speed characteristic is also of great
interest. Simulation results are shown in Fig4.e and Fig4.f as
would be expected, the BDFM follows the torque-speed
characteristic of induction motor. Note that the total
electromagnetic torque, Tem produced by the machine is
composed of two components, Temp and Temc . Temp , is
produced by the PW pole-pairs system whileTemc , is due to the
CW pole-pairs system. Interaction between both torques can
be clearly observed.
B. Doubly-fed synchronous operation mode
In this case the controllability of the system is tested when
an external voltage is applied on the CW side which follows a
conventional Volt/Hertz law (Vc ⁄fc = constant).
Fig.5.a shows the variation of time speed response.
Switching from the short-circuit case to the step of fc = −4Hz
at t=3.5s, and after it increases once more until the step fc =
−2Hz at t = 6s.
Oscillations of the first transition in Fig.5 at t = 3.5s are
very high relatively to the second one which occurs at t=6s.
Step change for the first step explains the moment of CW
connection after its short circuit regime.
Fig.6 shows time responses of speed and torque
corresponding to the synchronous the subsynchronous and the
fault tolerant behavior of the BDFM system.
At t=5.0 seconds, load torque is applied up from zero to 2
Nm. Similar to conventional synchronous machines, in this
case, the rotor speed remains after the transient to its initial
value. Thus, the machine presents a synchronous operation at
speed of 750 rpm (78.5rad/sec), in which the rotor speed
depends only on the supply frequencies.
At t=7 seconds, we can see from Fig 6 the dynamic
responses for a sudden loss of the CW excitation when a short
circuit is applied to the CW terminals accompagned with
unload machine. An advantage of the BDFM drive system is
that a loss of synchronism does not lead to a catastrophic
situation and the machine can remain connected to the grid. As
a result, the drive system still operates in the singly-fed
induction mode and can be re-synchronized again.
VIII.
CONCLUSION
This paper has provided the detailed analysis of the
BDFM principle operations, in which its dynamic model has
been developed both in separate and in unified references
frame. The second one has been based on the generic dq
reference frame that will be used in vector control strategies.
This model has been validated in MATLAB/SIMULINK
packages where the BDFM has been controlled in open loop
Volt/Hertz. The simulation results attest the BDFM literature
assertion. As expected, the speed of BDFM can be controlled
through adjusting the voltage applied to the CW. The model
discussed above is an important part of this work, which offers
the basis of differ control strategy for the BDFM.
IX.
APPENDIX
IX.1. Transformation between different Reference Frames
The resulting model
(44) referred in three initial reference frames and two
possible pole pair distribution (shown in Fig.b-1):
Fig.5 Rotor speed and electromagnetic torque
Fig6. Speed and electromagnetic torque time response under load torque
Fig.IX.1. Unified reference frames (mechanical angle)

Coupling Relation 𝑥⃗ αβ𝑝 = 𝑓(𝑥⃗ αβ𝑐 )
It is assumed that the rotor of the BDFM fulfils the
equation (3) and maximizes the number of nests i.e. 𝒑𝒑 + 𝒑𝒄 =
𝒏, which implies that,
𝑥⃗ 𝑥𝑦𝑃 = 𝑥⃗ 𝑥𝑦𝑐
∗
IX.1.1
From Fig.IX.1. it can be deduced that:
𝑥⃗ xy𝑝 = 𝑥𝑒𝑗𝑃𝑝 𝛼
𝑥⃗
xy𝑐
= 𝑥𝑒
IX.1.2
𝑗𝑃𝑐 𝛼
IX.1.3
𝑥⃗ αβ𝑝 = 𝑒𝑗𝑃𝑝 (θr +δ) 𝑥⃗ xy𝑝
IX.1.4
𝑥⃗ αβ𝑐 = 𝑒𝑗𝑃𝑐(θr +δ−γ) 𝑥⃗ xy𝑐
IX.1.5
Combining IX.1.1, IX.1.4, IX.1.5 we get,
∗
𝑥⃗ αβ𝑝 = 𝑥⃗ αβ𝑐 𝑒𝑗θa
IX.1.6
With: θa = (𝑝𝑝 + 𝑝𝑐 )(θr + δ) − 𝑝𝑐 γ
IX.1.7
 vector transformations from original reference frames to
generic 𝑑𝑞𝑝 reference frame
We can define a generic 𝑑𝑞𝑝 reference frame with a
Pp pole-pair distribution and located at any mechanical position
(θobsp /pp ) from αβp , the vector transformation is defined as,
𝑥⃗ αβ𝑝 = 𝑒𝑗θobsp 𝑥⃗ 𝑑𝑞𝑝
IX. 1.6 & IX. 1.8 ⇒ 𝑥⃗ αβ𝑐 = 𝑥⃗
IX.1.8
𝑑𝑞𝑝 ∗ 𝑗(θa −θobsp)
.
𝑒
IX.1.9
𝑒 −𝑗𝑃𝑝 (θr +δ) 𝑥⃗ αβ𝑝
IX.1.10
IX. 1.8 in IX. 1.10 ⇒ 𝑥⃗ xy𝑝 = 𝑒𝑗[θobsp −𝑃𝑝(θr +δ)] 𝑥⃗ 𝑑𝑞𝑝
IX.1.11
IX. 1.4 ⇒ 𝑥⃗
xy𝑝
−𝑗𝑃𝑐 (θr +δ−γ) αβ𝑐
IX.1.12
IX. 1.9 In IX. 1.12 ⇒ 𝑥 xy𝑐 = 𝑒 −𝑗[θobsp −𝑃𝑝 (θr +δ)] 𝑥⃗ ∗𝑑𝑞𝑝
IX.1.13
IX. 1.5 ⇒ 𝑥⃗
xy𝑐
=
=𝑒
𝑥⃗
In this way any machine variable can be defined in a
generic dqp reference frame.
IX.2. BDFM Electrical Parameter for Simulation
TABLE 1. BDFM Electrical parameter
Rated voltage
Pole pairs number
Resistance(Ω)
Self inductance(mH)
Mutual inductance(mH)
PW
CW
𝑉𝑝 = 220𝑉
𝑉𝑐 = 220𝑉
𝑝𝑝 = 1
𝑝𝑐 = 3
𝑅𝑝 = 1.732
𝐿𝑝 = 714.8
𝑅𝑐 = 1.079
𝐿𝑝 = 121.7
𝑀𝑝 = 242.1
𝑀𝑐 = 59.8
R
𝑅𝑟 = 0.473
𝐿𝑟 = 132.6
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