Brushless Doubly Fed Reluctance Machines – a Tutorial R.E. Betz∗ and M.G. Jovanovi´c† ∗
Department of Electrical and Computer Engineering The University of Newcastle, 2308, Australia. email:
[email protected] †
Liverpool John Moores University, School of Electrical Engineering, Byrom St., Liverpool L3 3AF, UK email:
[email protected] Abstract This paper presents a brief tutorial on the basic principles of operation of the Brushless Doubly Fed Reluctance Machine (BDFRM). This machine is a relative of the Brushless Doubly Fed Induction Machine (BDFIM), but unlike this machine it has not been extensively studied in the literature. It is also closely related to the double fed wound rotor induction machine. The potential advantage of the BDFRM as compared to the BDFIM is that it does not have copper rotor losses and therefore should be more efficient. Furthermore, with suitable construction techniques the rotor should be simpler and more robust than that of the BDFIM. This paper will explain the basic principles of operation of the machine, develop a dynamic model, and finally examine some of its properties.
I. Introduction
T
HE brushless doubly fed induction machine (BDFIM) is a descendent of the classic cascaded induction machine. Cascaded induction machines were invented to allow limited speed control via rotor resistance control, but with a brushless structure. However, cascaded systems achieved this at the cost of using two machines. A significant development in the late 1960s and early 70s was the invention of the brushless doubly fed induction machine (BDFIM), also known as the self cascaded induction machine [1, 2]. This machine uses a special type of rotor and a double wound stator. Effectively the stator has two windings of different pole numbers. It was logically as though the two induction machines in the classic cascade design were collapsed into the same frame. The two stator windings of the original machines now share the same magnetic circuit, but they do not directly couple because the windings have different pole numbers.1 With a rotor designed as a number of nested coils, with the number of nests equal to the sum of the pole pairs of the stator windings it can be shown that there is rotor position dependent mutual coupling between the two stator windings. Therefore the machine can produce torque, since there is a change of co-energy with
1 It is also possible using parallel path windings for the two windings to share the same copper path.
rotor movement. In the 1970s work on the BDFIM did not progress much beyond the basic machine design and the construction of some demonstration machines. However, in the 1980s the advent of reliable power electronics and low cost computing resulted in a revival of interest in the machine. Since this time it has been used in some applications on a trial basis where modest speed ranges are required, although it is still not widely used [3]. The main motivation for the use of this and related machines is that they allow limited variable speed range operation with a partially rated inverter and the reliability of a brushless structure. Therefore, for large systems the overall machine-drive system cost may lower as compared to a conventional machine-inverter combination. At the same time as the BDFIM was being developed some work was also being carried out on the brushless doubly fed reluctance machine (BDFRM) [4]. However, this was very preliminary in nature and the machine was essentially ignored until the early 1990s [5–7]. The motivation for considering the BDFRM as compared to the BDFIM is that it offers the potential of higher efficiency. It is well known that the BDFIM has significant copper losses in the rotor which can be avoided with a reluctance rotor. In summary the motivations for studying the BDFRM are a mixture of academic, practical and eco-
Reluctance rotor Primary, p pole pairs
To grid
{
Secondary, q pole pairs
{
To inverter
Fig. 1. Conceptual diagram of the Brushless Doubly Fed Reluctance Machine.
nomic reasons: • The machine is intellectually extremely interesting because of its unusual principle of operation.
The BDFRM is potentially more efficient, robust and lower cost than a comparably rated BDFIM. •
One can gain limited speed operation using a partially rated inverter. Therefore one can lower the total drive system cost compared to conventional solutions in limited speed applications. •
• The machine structure allows user control of its mains power factor.
It has a brushless structure which increases its reliability compared to the double fed wound rotor induction machine.
In order to gain an understanding of the basic operational principles of this machine we shall make the linear magnetic circuit, sinusoidally distributed winding, and sinusoidal current assumptions normally used in machine analysis. In addition we shall assume that the inverse air-gap of the machine can be modelled as: g −1 (θ, θrm ) = m + n cos pr (θ − θrm ) where m ≥ n > 0 and: pr the number of rotor poles (not pole pairs) θ a mechanical angle around the machine θrm the mechanical rotor angle If the primary and secondary three phase windings are both assumed to have a sinusoidal distribution and are being fed with three phase sinusoidal currents with frequencies ωp and ωs respectively then it can be easily shown that the resultant three phase mmfs per air-gap produced are [8]: Fp (θ) = Fpm cos(ωp t − pθ) Fs (θ) = Fsm cos(ωs t − α − qθ)
•
The economic benefits of using this machine would most likely be in large drive systems. Typical applications are heating, ventilation, and air conditioning (HVAC), large wind turbines, and large pumps. The remainder of this paper will describe the basic operation principles of the BDFRM, develop the dynamic equations of the machine, and finally discuss some of its important properties. II. Fundamental Operation Principle Fig. 1 is a schematic diagram of the structure of a BDFRM machine. One can see that the machine has two windings denoted as the primary winding and secondary winding. The primary winding is connected directly to the mains supply. The secondary winding (also known as the control winding) is also usually connected to the grid, but indirectly via an inverter.2 Notice that the windings have different pole numbers, and therefore for ideal sinusoidally distributed windings there is no coupling between them. The rotor of the machine can be any type of reluctance rotor – from simple cut-out rotors through to an axially laminated rotor as would be used in a synchronous reluctance machine.
(1)
(2) (3)
where: p the pole pairs of the primary winding q the pole pairs of the secondary winding 3 Fpm = np IM 2 3 Fsm = ns Im 2 np np primary turns/phase/pole = p n (4) ns secondary turns/phase/pole = s q and np/s are the respective peak sinusoidal conductor densities in conductors/radian, and IM and Im are the peaks of the sinusoidal input currents, and α is an arbitrary phase shift in the secondary currents. Fig. 2 shows the inverse air-gap function and the sinusoidal mmf waveforms for a one pole pair primary and three pole pair secondary windings. The basic operation of the machine can be understood from the interaction of the mmf waveforms in (2) and (3) with the inverse air-gap function (1) to produce a flux density in the air-gap. The relevant expression for this interaction is: Bp,s (θ, θrm ) = µ0 g −1 (θ, θrm )Fp,s (θ)
(5)
2 It
is also possible to short the secondary windings and operate the machine as an induction machine.
If we set m = n = G/2 and (1) and (2) are substi-
g-1
m + n cos pr ( q - q rm )
Note 1: The spatial frequency is the term multiplying the θ terms in (6). Therefore this term is effectively the pole number of a component of the flux density field.
m+n
Conjecture 1: It is possible to choose a value for pr so that the side band components of the flux density produced by the stator primary windings have the same temporal and spatial frequency as the fundamental field of the secondary winding. m-n
0 0
p pr
q - q rm
2p pr
cos((ωp − pr ωrm )t − (p − pr )θ) = cos(ωs t − qθ) (7)
MMF/air gap 3 Fp ( q) = n p I M cos q 2
3 np IM 2
Considering one of the primary sidebands and the secondary fundamental flux density waveform the above conjecture means the following:
(setting α = 0). Fs ( q) =
3 ns Im 2
3 ns Im cos 3q 2
p
0
If the relevant components in (7) are equated then it is easy to show that the components are equal under the following conditions:
2p
q
Fig. 2. Inverse air-gap function and winding mmf waveforms for one and three pole pair windings with t = 0 and α = 0.
tuted into (5) then we can write: Bp (θ, θrm ) =
µ0 GFpm cos(ωp t − pθ) 2
1 + [cos((ωp − pr ωrm )t − (p − pr )θ) 2 + cos((ωp + pr ωrm )t − (p + pr )θ)]
(6)
where θrm = ωrm t and ωrm the rotor angular velocity in mechanical radians/sec. A number of remarks can be made about (6): Remark 1: Equation (6) has a similar form to a modulated waveform after mixing in a radio transmitter. However, in this case we not only have a frequency shift in relation to the temporal terms, but we also have a frequency shift in the spatial components. Remark 2: The resultant flux density waveform has a component at the same temporal and spatial frequencies as the winding mmf, as well as two sidebands whose temporal and spatial frequencies are dependent on the number of rotor poles (pr ).
ωp ± ωs p±q pr = |p ± q|
ωrm =
(8) (9)
Remark 3: The condition in (8) means that the rotor has to be rotating at this ωrm for the mixing action of the rotor to change the primary temporal frequency to be the same as that of the secondary. Remark 4: The condition in (9) causes the p pole pair field of the primary to be transformed to the q pole pair field of the secondary. Remark 5: The above has only been carried out for one sideband of the primary winding. Expressions which give exactly the same angular velocity and rotor pole relationships are found for the other primary and secondary sidebands. Remark 6: The only way that this machine can develop torque is if the flux linkage between the primary and secondary windings is a function of the rotor position. One may suspect that this is the case from (8). The assertion in Remark 6 can be formally proved by calculating the self and mutual inductances of the BDFRM [8]. These expressions indeed show that the mutual coupling inductances between the primary and the secondary contain dL/dθrm terms. Therefore there is a change of co-energy in the machine with rotor movement and consequently torque can be produced. Note 2: Only the mutual primary to secondary winding inductances are responsible for torque production. The fundamental component of the flux
q
qs qp
wp ip
ws
w
is
dp
ap
qp
ds w r - w
as qs
q
qr - q d
Fig. 3. Various reference frames used for the dynamic equations of BDFRM.
density for each winding (see (6) for the primary example) does not contribute to torque production and is effectively a leakage component. III. Dynamic Equations The dynamic equations for this machine are very unusual and interesting. The derivation is quite complex and for space reasons will not be developed in this paper. A very detailed development of the basic equations for the machine can be download at the address shown in reference [8]. In order to understand the equations one must understand the reference frames they are defined with respect to. Fig. 3 shows the relationship between the various reference frames. The stationary frame dynamic equations in space vector notation for the BDFRM can be shown to be [8]: v ps v ss
dλps = Rp ips + dt θr dλss = Rs iss + dt
const
θr const
+ jωr Lps i∗ss ejθr (10) + jωr Lps i∗ps ejθr (11)
where the fluxes are:
A “trick” can now be applied to these stationary frame equations – we shall refer the primary equation to a frame rotating at ω rad/sec, and the secondary equation to a frame rotating at ωr − ω rad/sec (see Fig. 3), and it can be shown that we get [8]: dλpr + jωλpr dt dλ + sr + j(ωr − ω)λsr dt + Lps i∗sr
v pr = Rp ipr +
(14)
v sr = Rs isr
(15)
λpr = Lp ipr
λsr = Ls isr +
Lps i∗pr
(16) (17)
Remark 8: The i∗sr in (16) appears to be in the reference frame of the secondary expression, but it can be shown that this current vector is actually in the primary reference frame. Similarly i∗pr in (17) is in the secondary reference frame. Given the general reference frame equations of (14)(17) we can choose ω = 0 so that the primary equation is in a stationary frame and the secondary equation is in a rotating frame at ωr . Using these frames it is possible after considerable manipulation and a few “tricks” to derive the steady state equations for the BDFRM, which can be written as: v ps = Rp ips + jωp (Lp − Lps )ips + jωp Lps (ips + i∗sr )
λps = Lp ips + Lps i∗ss ejθr
(12)
λss = Ls iss + Lps i∗ps ejθr
(13)
Lp the primary winding three phase inductance. Ls the secondary winding three phase inductance. Lps the primary to secondary three phase mutual inductance. and θr = pr θrm and ωr = dθr /dt.
Remark 7: The interpretation of the is∗s ejθr term in the primary equation is difficult.3 Even though it appears to be a conversion from one reference frame to another, it is not. It can be shown [8] that it is equal to iss ej(θp +θp0 ) , which is a vector in a stationary frame rotating at ωp rad/sec. Therefore Lps i∗ss ejθr represents the flux coupling to the primary from the secondary as a result of the flux density sidebands mentioned earlier, and it is rotating with the primary angular frequency. Similarly Lps i∗ps ejθr in the secondary equation is the flux linking from the primary to the secondary, and it is rotating with the secondary angular velocity.
(18) v sr Rs ∗ = i + jωp (Ls − Lps )i∗sr + jωp Lps (i∗sr + i∗sr ) s s sr (19) where s = −ωs /ωp . The equivalent circuit that can be drawn from (18) and (19) is shown in Fig. 4. Remark 9: If one compares Fig. 4 with the equivalent circuit of a double fed wound rotor induction machine one would find that they are identical in form. 3 The ∗
operator denotes the complex conjugate.
Rp
jw p ( L p - L ps )
jw p ( Ls - L ps ) i *sr
i ps v ps
Rs s
jw p L ps
which means that we have defined away the λpq flux component and therefore (23) can be simplified to: v *sr
Remark 10: One of the main differences between the BDFRM and induction machine steady state models are the values of some of the inductances. Due to the fact that a significant fundamental component of the winding fluxes does not couple to the other winding in the BDFRM the (Lp − Lps ) terms are much larger than the normal leakage flux of the double wound induction machine. Remark 11: The reference frame used for vector control is ω = ωp . Under this condition the primary equation is in the ωp frame, and the secondary equation is in the ωs frame. IV. Torque Expressions Another fundamental expression for any machine is the torque expression. Using the principle of virtual work, after considerable manipulation one can obtain the following expression for the torque of the machine in terms of primary values:
(21)
One can further manipulate (20) to get an expression in terms of the primary fluxes and the secondary currents. This expression is particularly useful since the secondary currents are the variables one has control over using an inverter, and the primary fluxes are fixed by the voltage and frequency of the mains supply. The expression is:
Remark 13: Equation (25) forms the basis for vector control of the BDFRM. One can see that the torque can be controlled by isq , whilst λpd is a constant related to the mains supply conditions. Therefore one has independent control of the torque. Note 3: The isq and isd currents in both (23) and (25) are “coupled” currents from the secondary winding to the primary winding. It should be stressed that these are secondary currents measured relative to the primary reference frame. Therefore isq is the q-axis component of the secondary winding current where the q-axis is a primary frame axis. V. Power Expressions Using the general expression for complex power: 3 − → S = P3φ + jQ3φ = (v pr i∗pr + v sr i∗sr ) 2
(26)
(22)
one can get the following expressions for the real and reactive powers into the BDFRM [8]:4 Pp Qp Ps Qs
= = = =
3 2 2 Rp ip + ωp Lps is ip sin(αs + αp ) 3 2 2ω p L2p ip + Lps is ip cos(αs + αp ) 3 2 Rs is + ωs Lps is ip sin(αs + αp ) 3 2 2 ωs Ls is + Lps is ip cos(αs + αp )
(27) It is possible to show that the torque expression (20) can be manipulated to give: Te =
3 pr Lps ip is sin(αs + αp ) 2
(23)
3 ωp Rp i2p + Te 2 pr 3 ωs Ps = Rs i2s + Te 2 pr
Pp =
(24)
(28)
which is very similar to the real primary and secondary power expressions in (27). By comparing we can see that:
Equation (23) can be further refined by using the frame of Remark 11 and aligning the primary reference frame so that it lies along the primary flux vector. We can write the primary flux vector as: λp = λpd + jλpq
(25)
(20)
Remark 12: Equation (21) is the same torque expression that one obtains for the induction machine.
Lps 3 Te = j pr λ∗pr i∗sr − λpr isr 4 Lp 3 Lps = pr [λpd isq + λpq isd ] 2 Lp
3 Lps pr λp isq 2 Lp
where λp the magnitude of the primary flux vector.
Fig. 4. Steady state model of the BDFRM
3 Te = j pr λpr i∗pr − λ∗pr ipr 4 3 = pr [λpd ipq − λpq ipd ] 2
Te =
s
4i p/s
denotes the current magnitudes.
(29) (30)
Remark 14: The total rotational power is simply the sum of the real powers minus the losses, therefore: Prot = P3φ − Plosses ωp ωs = Te + Te pr pr 1 = (ωp + ωs )Te pr = Te ωrm
VI. Conclusions
(31)
which conforms with the definition of output power. Remark 15: One can see from (29) and (30) that the contributions to the rotational output power from each winding are: ωp Te pr ωs = Te pr
Pprot =
(32)
Psrot
(33)
Remark 16: From (32) and (33) one can deduce that if ωs = −ωp and ωp > 0 then the rotational power from the secondary is negative. This means that power is flowing out of the secondary winding – i.e. the winding is regenerating, and all the power flowing into the primary is being returned to the mains via the secondary. The rotor is stationary under this condition (see (8) assuming the ‘+’ case). Remark 17: If ωs = 0 and ωp > 0 then we are feeding DC into the secondary winding. In this situation we can see from (32) that the primary is contributing to rotational output power but the secondary (from (33)) isn’t contributing anything. The machine is running as a synchronous machine in this case. Note 4: Remark 17 means that an inverter would not have to supply any rotational power. One can see from (33) that the secondary power is directly related to how much the secondary frequency varies from zero. This ability to control the machine speed over a limited range with a small amount of secondary power is one of the main motivations for the use of the machine. Remark 18: If ωs = ωp then the machine is running in a super synchronous mode of operation. Using the same frame alignment conditions as (25) the expression for the primary reactive power can be written as: Qp =
3 ωp λp (λp − Lps isd ) 2 Lp
Remark 19: Equation (34) indicates that one can control the primary reactive power via the secondary d -axis current. Comparison with (25) leads to the conclusion that the primary reactive power can be controlled independently of the torque.
(34)
This paper has provided a very brief introduction to the operation of this very interesting and unusual machine. Additional theoretical studies on the control properties of the machine have been carried out and, amongst other things, indicate that for the same torque output a BDFRM has to be larger than an induction machine [9, 10]. Much work has to be carried out on the analysis, design and experimental testing of this machine before the trade-offs in inverter size versus machine size and the subsequent cost issues can be quantified. Only then will it be clear whether this unusual machine has a viable application niche. References [1]
A. Broadway and L. Burbridge, “Self-cascaded machine: A low speed motor or a high frequency brushless alternator,” Proc. IEE, vol. 117, pp. 1277–1290, July 1970. [2] A. Broadway, B. Cook, and P. Neal, “Brushless cascade alternator,” Proc. IEE, vol. 121, pp. 1529–1535, December 1974. [3] B. Gorti, D. Zhou, R. Sp´ee, G. Alexander, and A. Wallace, “Development of a brushless doubly-fed machine for a limited speed pump drive in a waste water treatment plant,” Proceedings of the Annual meeting of the IEEE-IAS, vol. 1, pp. 523–529, Oct 1994. [4] A. Broadway, “Cageless induction machines,” Proc. IEE., vol. 118, pp. 1593–1600, November 1971. [5] F. Liang, L. Xu, and T. Lipo, “D-q analysis of a variable speed doubly AC excited reluctance motor,” Electric Machines and Power Systems, vol. 19, pp. 125– 138, March 1991. [6] L. Xu, F. Liang, and T. Lipo, “Transient model of a doubly excited reluctance motor,” IEEE Trans. on Energy Conversion, vol. 6, pp. 126–133, March 1991. [7] L. Xu and Y. Tang, “A novel wind-power generating system using field orientation controlled doubly-excited brushless reluctance machine,” Proceeding of the IEEE IAS Annual Meeting, pp. 408–418, 1992. [8] R. Betz and M. Jovanovi´ c, “Introduction to brushless doubly fed reluctance machines - the basic equations,” tech. rep., Dept. Elec. Energy Conversion, Aalborg University, Denmark., April 1998. Available at http://www.ee.newcastle.edu.au/users/staff/reb/Betz.html. [9] R. Betz and M. Jovanovi´ c, “Control aspects of brushless doubly fed reluctance machines,” Proceedings of the European Power Electroncis Conference (EPE’99), Sept. 1999. [10] R. Betz and M. Jovanovi´c, “The brushless doubly fed reluctance machine and the synchronous reluctance machine – a comparison,” Proceedings of the IEEE-IAS Annual Meeting, Oct. 1999.
