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The cascade induction machine: a reliable and controllable motor or generator
Electric Power Systems Research 68 (2004) 193–207
The cascade induction machine: a reliable and controllable motor or generator D. Picovici∗ , D. Levy, A.E. Mahdi, T. Coffey Department of Electronic & Computer Engineering, University of Limerick, Limerick, Ireland Received 14 October 2002; received in revised form 13 May 2003; accepted 9 June 2003
Abstract This paper discusses and analyses a set of two induction machines, with 2p and 2q pole-pairs, respectively, connected in cascade. It highlights the design and development principles of a single unit version of a system consisting of two wire-wound rotor induction machines with their rotors connected in cascade. Presented performance analysis shows the described cascade machine, which is brushless and with no slip rings or commutators, as a reliable, efficient and practical machine that could replace the induction machine, which has a wire-wound rotor. An example of the conversion of a standard squirrel cage six pole induction machine into a (6 + 2) pole induction machine, without rewinding the stator, is also presented together with simulation and measurements for asynchronous and synchronous modes of operation. © 2003 Elsevier B.V. All rights reserved. Keywords: Wire-wound rotor; Commutators; Squirrel cage induction machine
1. Introduction The cascade machine consists of two wire-wound rotor induction machines with their rotors connected in cascade. If the two machines have a different number of pole-pairs, then they can be constructed as a single unit version with a single stator and a squirrel cage rotor. Otherwise, if the two machines have the same number of pole-pairs, the single unit version is not possible [1–9]. To date, reported work on the development of the cascade induction machine [1–9] has shown that the two machines making the cascade can be of the same size or different sizes, and can run in the same direction or in opposite directions. It has also been shown that the difference in the number of poles between the two machines is not the only parameter that affects the characteristic of the cascade unit. Several parameters are involved in the construction of a cascade machine in order to be useful and adaptable for a specific application. This may explain why such a machine is yet to be made commercially available as ‘off-the-shelf’ product. This paper analyses two sets of induction machines
∗
Corresponding author. Tel.: +353-61-202925; fax: +353-61-338176. E-mail address: [email protected] (D. Picovici).
0378-7796/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2003.06.008
connected in cascade and highlights the parameters affecting the cascade operational efficiency. Knowing these parameters and understanding well the operating and design principles will result in the development of an efficient practical machine. An example on the conversion of an ordinary squirrel cage induction machine into a cascade machine is also presented.
2. The equivalent circuit of a multipole machine The transformation α = pθ transforms a ‘p’ pole-pair machine into a single pole-pair machine. Here, α and θ represent the electrical and mechanical angles, respectively, of a ‘p’ pole-pair machine. According to this transformation, the ‘p’ pole-pair are ‘folded’ together and the input current of a ‘p’ pole-pair machine is equal to p times the input current of a single pole-pair machine for the same air gap flux. The input impedance of the ‘p’ pole machine is therefore equal to the voltage per pole divided by the total current, or 1/p times the impedance of a single pole-pair machine. The slip of the ‘p’ pole-pair machine is therefore equal to (ω − pΩ)/ω, where Ω is the machine’s mechanical speed, ω is its line frequency and ‘pΩ’ is its ‘folded’ speed or the electrical speed.
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Fig. 1. Cascade machine equivalent circuit with reference to their stators.
3. The equivalent circuit of a cascade electrical machine For a symmetrical grid, symmetrical polyphase machine, the equivalent electrical circuit per phase for a set of two wire-wound induction machines connected in cascade through their rotor’s terminals with reference to their stators, is shown in Fig. 1. One machine is assumed to have ‘p’ pole-pair and the other ‘q’ pole-pair. In this equivalent circuit, the two machines are shown to be coupled through a transformer with voltage ratio y and current ratio x. In practice, x = y = 1 and therefore the two machines are coupled directly without a transformer. Referring to Fig. 1, the circuits components of a ‘p’ pole-pair machine can be expressed, in terms of those of the resulting single pole-pair machine, as follows: [Rf + jω1 (Lf − M)] A1 = =Rf1 + jω1 (Lf1 − M1 ) p A2 =
jω1 M//Ri =jω1 M1 //Ri1 p
A3 =
[Ra /s1 + jω1 (La − M)] Ra1 = + jω1 (La1 − M1 ) p s1
Similarly for the ‘q’ pole-pair: B1 =
[Rf + jω2 (Lf − M)] =Rf2 + jω2 (Lf2 − M2 ) q
B2 =
[jω2 M//Ri ] =jω2 M2 //Ri2 q
B3 =
[Ra /s2 + jω2 (La − M)] Ra2 = + jω2 (La2 − M2 ) q s2
where ω1 is the frequency of the stator of the p pole-pair machine, which is assumed to be a constant grid frequency and ω2 is the frequency of the q pole-pair machine which therefore is a dependent variable and function of the speed. In general, the machine has two ports which can be used as inputs or outputs similar to the wire wound machine. For ¯ 2 = 0. For synchronous operaasynchronous operation, E ¯ tion, E2 can be a voltage with a fixed frequency such as a dc voltage for constant speed operation or a variable frequency voltage for a variable speed application. ¯ 1 is ω1 and the frequency If the frequency of the voltage E ¯ of the voltage E2 is ω2 , then power can flow between the two
machines as shown in Fig. 1, if and only if, the frequencies of the two rotor voltages or currents are equal, thus, |ω1 − pΩ| = |ω2 + qΩ|
(3.1)
Eq. (3.1) assumes the two machines are connected mechanically back to back, they are rotating in opposite directions. For synchronous operation, ω1 and ω2 are independent variables and the machine speed Ω becomes a dependent variable which depends on the phase between ¯ 1 and E ¯ 2 with reference to rotating frames of references E ω1 and ω2 , respectively. On the other hand, in asynchronous operation, ω2 is a dependent variable. In that case, E2 = 0.
4. The isosynchronous machine For the case p = q and ω2 = ω1 , the speed Ω is an independent variable in the Eq. (3.1). For this special particular case, the machine is called an isosynchronous machine. In practical application, two machines are coupled back to back ¯1 = E ¯ 2 . Torque and produce zero net torque for the case E is produced only when a phase difference between |E1 | and |E2 | exists. Normally, the torque is controlled by a phase shifter or by adjusting mechanically one stator relative to the other as shown in Fig. 3. The torque/speed characteristic is shown in Fig. 5. Maximum torque is produced from the set if the angle of the electrical phase difference between the two machine stators is equal to 180◦ . The two machines can also be coupled front to back if double axis is available. It is worth emphasising, that this type of operation of the set is unique since both inputs accept voltage at the same frequency. However, the set works in the asynchronous mode contrary to the cascade set withp = q. The latter can operate only in the synchronous mode if two input voltages are present. This is the reason why the p = q set is called an isosynchronous machine. Unfortunately, the set cannot be constructed in a single frame.
5. The cascade machine analysis Analysis of the cascade machine equivalent electrical circuit shown in Fig. 1 is given in Appendix B for asynchronous and synchronous modes of operation. The two machines
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Fig. 2. Approximate equivalent circuit of the cascade machine at light load.
equivalent circuit shown are referred to their stators. The rotor’s frequency for this type of application is calculated from Eq. (3.1).
Ra2 B3 ∼ = s2
ω1 − pΩ = ω2 + qΩ
Then
(5.1)
In order to understand the behaviour of the assembly, an analytical analysis needs to be carried out for an approximate equivalent circuit shown in Fig. 2. Let A1 = 0
(5.2)
B 2 = jω2 M2
(5.3)
5.1. Asynchronous Operation For asynchronous operation E2 = 0. In this mode, the current I3 can be shown to be: I3 =
E1 A3 + B 3 (s2 /s1 ) + (s2 /s1 )(B 1 B 2 /B 1 + B 2 )
where ω1 − (p + q)Ω s1 = s2 ω1
(5.4)
(5.5)
where the ratio s1 /s2 is called the ‘cascade slip’ which is equivalent to the slip of a p + q pole-pair machine, where s1 and s2 are the slip of each separate machine defined as ω1
− pΩ ω1
(5.6)
ω2
+ qΩ ω2
(5.7)
s1 =
s2 =
In asynchronous operation ω1 is a fixed independent variable but ω2 is a dependent variable of the machine speed Ω. By eliminating the speed Ω from Eqs. (5.1), (5.6) and (5.7), ω2 as a function of the cascade slip can be obtained: s1 (5.8) ω2 = ω1 s2 As an approximation, let B1 be equal to the control resistance connected in series with the stator resistance of the p pole-pair machine. Let
B 1 =R Ra1 A3 ∼ = s1
(5.9) (5.10)
I¯3 = =
(5.11)
E1 ((Ra1 + Ra2 )/s1 ) + (s2 /s1 )(jω2 M2 R/(R + jω2 M2 )) E1 ((Ra1 + Ra2 )/s1 ) + (jω1 M2 R/(s1 /s2 ))/(R/(s1 /s2 ) + jω1 M2 ) (5.12)
At the cascade slip (s1/s2) = 0, I3 is not purely reactive due to the losses in the 2p pole machine and in the rotors. Eq. (5.12) shows clearly the disadvantage of this machine where the rotors resistances together with 2p pole stators resistance limits its output power and reduces its efficiency if it is not designed properly. At slip s1 = 0 of the 2p pole machine, the current I3 = 0. The torque is therefore equal to zero together with the field of the 2q pole machine. Therefore, around s1 = 0, the set behaves as a 2p pole machine and its characteristic is uncontrollable with the resistance R. For better efficiency and higher output power, the stator of the 2p pole machine should have as low an impedance as possible and the 2q pole machine should have as high an impedance ω1 M2 as possible. Therefore, the feeding of the set should be carried out from the higher pole-pair side, i.e. p > q. The torque of the 2p pole machine T1 = Re(jM1 )
E E I3 = Re(I3 ) jω1 M1 ω1
(5.13)
T1 therefore depends on the real part of I3 which can be calculated to be: Re
I3 E1
=
(Ra1 + Ra2 /s1 )(R/(s1 /s2 ))2 + ω12 M22 ((Ra1 + Ra2 )/s1 + R/(s1 /s2 )) (Ra1 + Ra2 )/s1 )2 (R/(s1 /s2 ))2 + ω12 M22 ((Ra1 + Ra2 )/s1 + R/(s1 /s2 ))2 (5.14)
For s1 < 0, s1 /s2 is already negative. Therefore, Re(I3 ) is negative and the mechanical power is negative (generative).
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At s1 /s2 < 0, Re(I3 /E1 ) is negative within the region given by: −Q −
Q2
R/(s1 /s2 ) −1< < −Q + Q2 − 1 ω1 M 2
(5.15)
where
Q=
ω1 M2 2((Ra1 + Ra2 )s1 )
(5.16)
where s1 is positive. Q is a goodness factor for the simplified linear model. Furthermore, from Eq. (5.15), Q should be greater than one in order that the torque T1 and therefore the cascade mechanical power, becomes negative at (s1 /s2 ) < 0 within that region. The electrical power input to the machine, with no iron losses, according to the simplified model: = Re(E1 · I1∗ ) = E1 Re(I3 )
(5.17)
Therefore, the region given in Eq. (5.15) is for power generation at negative cascade slip. The torque of the 2q pole machine:
jM2 RI3 I3∗ R + jω2 M2 ω1 M22 R ∗ = −I3 I3 s1 /s2 (R/(s1 /s2 ))2 + ω12 M 2
T2 = −Re
(5.18)
T2 is equal to zero at s1 = 0 (where I3 = 0) and at the cascade slip s1 /s2 = 0. At negative cascade slip, s1 /s2 < 0, T2 reverses its sign and becomes positive. The mechanical power T2 (−pΩ) is therefore negative (generative). As previously stated, at s1 = 0, I3 = 0, the 2q pole machine is not excited. Therefore, around s1 = 0, the control resistance R has no effect on regulation. A good cascade machine should not reach the slip s1 ≤ 0 As a motor, it works at (s1 /s2 ) ≥ 0 and as a generator at (s1 /s2 ) < 0 where s1 > 0 or very close to one at (s1 /s2 ) = 0.
5.2. Synchronous operation ¯ 2 = (ω2 /ω1 )E2 ejϕ = At synchronous operation E (s1 /s2 )E2 ejϕ , where ϕ is the phase difference between the voltage of the 2p pole stator and the 2q pole stator with reference to the two rotating frames ω1 and ω2 , respectively. ¯ 2 is normally proportional to the speed in order to prevent E saturation of the iron core. The angle ϕ depends on the load and should be within the region 2π ≥ ϕ ≥ 0 otherwise the machine will be asynchronous at a given frequency ω2 supplied to the 2q pole machine. In this mode, the machine becomes doubly excited from the two stators of the 2p and 2q pole. The analysis is similar to the asynchronous mode using the superposition theory. For the reasons explained in the asynchronous mode of operation, most of the output power is supplied by the 2p pole machine where p > q. If ω2 = 0, the 2q pole machine is excited by a dc voltage. The cascade will run at synchronous speed when the cascade slip s1 /s2 = 0 only. The dc voltage with reference to a ¯ 2 = E2 ejϕ . Again rotating frame of reference ω1 will be E the phase ϕ will depend on the load. E2 is normally supplied from a well controlled current source in order to prevent saturation. 6. Conversion of a 2p = 6 pole machine into a (p + q) = (3+1) pole-pair machine A Brook Hausend =1/821837, three pole-pair machine with data given in Appendix C is to be converted to a (3 + 1) pole-pair cascade without rewinding the stator. The rotor shorted circuit bars are to be replaced by a graded winding. 6.1. Stator The stator winding of that machine consists of six coils per phase connected in series in order to produce three pole-pair. Therefore, the number of slots in the stator is 36. Line voltage was 415 V. After conversion, the coils are now connected
Fig. 3. Stator windings connection of the converted three pole-pair machine into (p = 3 + q = 1). The figure is useful for conversion of a similar stator machine.
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in parallel for each pole-pair instead of in series. This is in order to allow a second input for the 2q pole machine as shown in Fig. 3. The impedance of each phase is therefore reduced by 1/q. The voltage E1 is now reduced to 415/3 = 138 V and the current =I1 is now three times the original phase current. In p = 3, pole-pair equivalent circuit, all components are therefore divided by nine. In the q = 1, pole-pair equivalent circuit, all components are divided by three. In order for the air gap flux to remain the same as the original three pole-pair in the q = 1 pole-pair, the voltage E2 should be 415/9 = 46 V. The ampere turns in the q = 1 side is obviously three times than that on the p = 3 side. Furthermore, in the cascade application, ignoring the rotor’s resistances in Fig. 2, the iron laminations should support the double flux
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of the machine. Thus, instead of M = M1 p = M2 q, we have in the cascade M/2 air gap mutual inductance. If the machine is double excited from both sides, the input voltages should now be reduced to E1 = (415/3)(1/2) = 70 V and to E2 = 23 V, respectively at 50 Hz frequency in order to prevent saturation. Appendix C shows the machine data before and after conversion. (Also see Sections 6.2 and 6.3). 6.2. Rotor The air gap flux due to the 2p pole machine is approximately equal to E1 /ω1 p. From Eq. (5.12), ignoring Ra1 + Ra2 , the air gap flux due 2q pole machine is ∼ = E1 /(ω1 p) = 0.11 (web turns per pole-pair). The total
Fig. 4. The rotor with four concentric coils grouping of graded windings shorted circuit (photography from two angles) The figure is useful for conversion of an ordinary squirrel cage rotor into a cascade machine rotor.
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instantaneous air gap flux with reference to a static frame of reference: E1 j(ω1 +qΩ)t E1 j(ω1 −pΩ)t (6.2.1) e + Flux = e ω1 p ω1 p where E2 E1 = p q
(6.2.2)
At Ωt= θ, t = 0 the absolute value of the flux is in the form: √ E1 −jpθ 2E1 +jqθ |= +e (1+Cos(p + q)θ)1/2 (6.2.3) e ω1 p ω1 p At any time t, other than t = 0, (6.2.3) is simply shifted by ω1 t. Therefore, the shape of the rotor shorted circuit conductor should be according to Eq. (6.2.3) instead of E1 /ω1 p = constant, independent of θ in the original standard induction machine. For the present conversion, Eq. (6.2.3) is approximated by grouping concentric coils of graded winding (Fig. 4). 6.3. Simulation of the converted machine The converted machine data was simulated using Fortran NAG and the results were plotted for ((p = 3) + (q = 1)) and for ((p = 1) + (q = 3)) connected in asynchronous mode. As expected, the ((p = 1) + (q = 3)) has very little power due to its very low goodness factor.
At frequency of 50 Hz, the converted cascade set is still not very useful as a motor, or as a generator since for the ((p = 3) + (q = 1)) machine, Ra1 + Ra2 = 2 ohms, M2 = 8.45 × 10−2 Hy, s1 = 1 ω = 2π50 the goodness factor is only Q = 6.3 at most. For the ((p = 1)+(q = 3)) machine, the goodness factor is only 2.1. Obviously, 50 Hz operation of the above cascade is not practical. When the frequency is increased to 100 Hz, a more practical machine will result with almost identical power output to the original machine before conversion. Figs. 5–8 show the characteristics of the machine before conversion and Figs. 9–12 show the characteristics of the cascade ((p = 3) + (q = 1)) after conversion, operating from 100 Hz and E1 = 140 V in asynchronous mode for different control resistors 0 < R < 10. The characteristics are not different from the standard wire wound induction machine. The synchronous speed (slip s1 /s2 = 0) of the cascade is 1500 rpm (157 rad/s). The slip s1 = 0 is at speed 2000 rpm (209 rad/s) which is far from the synchronous speed of the cascade and therefore, s1 can be approximated as s1 ∼ = 1 in the practical operating speed of the cascade. Therefore, ω1 is adequately high. The losses (Ra1 + Ra2 )/s1 in Eq. (5.12) can be neglected and the equivalent electrical circuit of the cascade becomes close to the ideal wire wound induction machine. In synchronous mode of operation, Figs. 13–16 shows the characteristics of the machine converted into a cascade where ω1 = 2π100 and ω2 is varied according to the machine speed Eq. (5.1). The characteristics shown are for phase (3π/2) ≥ ϕ ≥ 0.
Fig. 5. Torque/speed characteristic, per phase, of the original p = 3 pole-pair machine, the characteristic is useful for optimal stable coupling of the machine to the load as a motor or optimal coupling of the machine to a turbine as a generator.
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Fig. 6. Electrical power/speed characteristic per phase of the original p = 3 pole-pair machine. The characteristic is useful for predicting the power consumption as a motor or, the power fed to the grid as a generator.
Fig. 7. Mechanical power/speed characteristic per phase of the original p = 3 pole-pair machine. The characteristic is useful for predicting the optimal output mechanical power as a motor or, the optimal input mechanical power from a turbine as a generator.
7. Simulation results
7.1. Asynchronous mode (100 Hz grid)
Results of the simulation of the converted six poles, 50 Hz, and 5.5 kW squirrel cage induction machine into a (3 + 1) pole-pair, 100 Hz cascade machine is given in this section for asynchronous and synchronous modes.
Fig. 9 shows the torque/speed characteristic. The control resistance is shown as a parameter 10 ≥ R ≥ 0 ohms. Optimal resistance for maximum starting torque is seen to be compatible with wire wound rotor machines. Above slip
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Fig. 8. Efficiency/speed characteristic of the original p = 3 pole-pair machine. The characteristic is useful for predicting the optimal operating efficiency of the machine as a motor or, as a generator.
s1 /s2 = 0 (speed = 1500 rpm), the machine is generating. At much higher speed, (where s1 = 0 at speed 2000 rpm or 209 rad/s) the machine will again be generating, however its characteristic is non-controllable by R since the 2q machine produces no torque.
Fig. 10 shows the electrical power input to the cascade. This figure is similar to Fig. 9. The mechanical power is shown in Fig. 11 and the machine efficiency in Fig. 12. At the practical operating region 1 ≥ s1 /s2 ≥ 0 (up to around 157 rad/s as a motor), the efficiency is around 55%
Fig. 9. Torque/speed characteristic per phase of the converted machine in asynchronous mode. Variable resistance, at the q = 1 pole-pair side, as a parameter. The characteristic is useful for optimal stable coupling of the machine to a load, as a motor or, to a turbine as a generator.
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Fig. 10. Electrical power/speed characteristic per phase of the converted machine in asynchronous mode. Variable resistance, at the q = 1 pole-pair side, as a parameter. The characteristic is useful for predicting the power consumption as a motor or, the power fed to the grid as a generator.
Fig. 11. Mechanical power/speed characteristic per phase of the converted machine in asynchronous mode. Variable resistance, at the q = 1 pole-pair side, as a parameter. The characteristic is useful for predicting the optimal output mechanical power as a motor or, the optimal input mechanical power from a turbine as a generator.
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Fig. 12. Efficiency/speed characteristic of the converted machine in asynchronous mode. Variable resistance, at the q = 1 pole-pair side, as a parameter. The characteristic is useful for predicting the optimal operating efficiency of the machine as a motor or, as a generator.
Fig. 13. Torque/speed characteristic per phase of the converted machine in synchronous speed mode. The characteristic is useful for optimal stable coupling of the machine to the load as a motor or, to a turbine as a generator. The characteristic indicates the range of operation as a motor or as a generator, for a given speed. The parameter is the phase ϕ = 0, 90, 180 and 270◦ .
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Fig. 14. Electrical power/speed characteristic per phase of the converted machine in synchronous mode. The characteristic is useful for determining the range of speed as a motor or a generator. As a generator, the range of speed is very limited in contrast to the motor. The parameter is the phase ϕ = 0, 90, 180 and 270◦ .
compared to 75% of the original machine before conversion. The efficiency of the converted cascade can be improved if the supply frequency is increased above 100 Hz. However, as a generator, its efficiency is much greater than a motor and it can reach 90% at 100 Hz excitation as Fig. 12 indicates.
7.2. Synchronous mode (ω1 = 2π100, ω2 = variable) Fig. 13 shows the torque/speed characteristic with phase ϕ as a parameter. For a given speed, ϕ will adapt itself automatically according to the load on the machine. The
Fig. 15. Mechanical power/speed characteristic per phase of the converted machine in synchronous mode. The characteristic is useful for determining the maximum output load possible as a motor, or, input drive power from a turbine as a generator before leaving the synchronous mode. The phase as a parameter ϕ = 0, 90, 180 and 270◦ .
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Fig. 16. Efficiency/speed characteristic of the converted machine in synchronous mode. The characteristic is useful for determining the maximum efficiency possible before leaving synchronous mode as a motor or as a generator. The phase as a parameter, ϕ = 0, 90, 180 and 270◦ .
range of the maximum torque that the machine can provide at different speed can be found from Fig. 13. Fig. 14 shows the total electrical power from both inputs to the machine. It is similar to Fig. 13. The mechanical power is shown in Fig. 15 and the machine efficiency in Fig. 16. As expected, due to the two supply inputs, the efficiency in the synchronous mode is higher than in the asynchronous mode and it is close to the original machine efficiency before conversion.
8. Experimental results The converted cascade characteristics in asynchronous mode were measured. The q = 1 pole-pair side was terminated by a variable resistance. The machine was fed from a constant voltage at a clean 100 Hz frequency rotating converter. The machine was mechanically loaded with an induction motor used as a variable load in the motoring
Fig. 17. Measured torque/speed characteristic, asynchronous mode. Variable resistance as a parameter. Two extremes are shown, zero and infinite resistances.
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Fig. 18. Measured electrical power/speed characteristic, asynchronous mode, variable resistance as a parameter. Two extremes are shown, zero and infinite resistance.
Fig. 19. Measured mechanical power/speed characteristic, asynchronous mode, variable resistance as a parameter. Two extremes are shown, zero and infinite resistance.
region and as an over speed motor simulating a turbine in the generating mode region. The torque/speed characteristic is shown in Fig. 17 for two parameters. Load resistance = 0 and infinity (open circuit). By comparison with the ideal characteristic Fig. 9 obtained from simulation, it can be seen that there is not too much difference. The only difference in shape was due to the finite number of copper bias used in the rotor (three in each group instead of infinite). Similarly, with the electrical power in the motoring and generating regions in Fig. 18. The converted machine has, as expected, more losses than the ideal, simulated machine in Fig. 10. Fig. 19 shows the mechanical power obtained from Fig. 17 by multiplying the torque by the speed. The ideal characteristic obtained from simulation is shown in Fig. 11.
9. Conclusion The cascade machine is a controllable induction machine similar to the wire- wound rotor induction machine, however it is much more reliable and cheaper due to its cage rotor or brushless construction. A six pole or more induction machine can be converted into a cascade machine without rewinding the stator, but by modification of the cage rotor structure. However, by doing so, the efficiency of the machine is reduced significantly and the resulting cascade becomes unpractical unless its operating frequency is increased. In asynchronous operation, the input voltage should be connected to the stator with the higher number of poles, in order to maximise its efficiency, otherwise, the resulting cascade is not practical.
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Acknowledgements This material is based upon work supported by Enterprise Ireland under the Strategic Research Program 1997. The authors would like to pay tribute to their colleague and co-author of this paper, Dr. D.Levy, who sadly passed away last year, and acknowledge his significant contribution to this work. He is deeply missed.
E1 = A1 I1 + (I1 − I3 )A2
(A.1)
E2 = B 1 I2 + B 2 (xI3 + I2 )
(A.2)
Vr = A2 (I1 − I3 ) − A3 I3 s1
(A.3)
Vr = xB 3 I3 + B 2 (xI3 + I2 ) s2
(A.4)
j
where in practice Appendix A. List of symbols
x=y=1
(A.5)
solving for I3 Machine parameters A1 , A2 , A3 2p pole machine parameters (complex variables) B 1, B 2, B 3 2q pole machine parameters (complex variables) Rf , Rf1 , Rf2 2 pole, 2p pole and 2q pole stator resistances (ohms) 2 pole, 2p pole and 2q pole stator Lf , Lf1 , Lf2 self inductances (henries) M, M1 , M2 2 pole, 2p pole and 2q pole rotor/stator mutual inductances (henries) Ra , Ra1 , Ra2 2 pole, 2p pole and 2q pole rotor resistances (ohms) La , La1 , La2 2 pole, 2p pole and 2q pole rotor self inductances (henries) 2 pole, 2p pole and 2q pole iron losses Ri , Ri1 , Ri2 equivalent resistances (ohms) Other variables ¯ 1 , E1 E 2p pole machine input voltage (V) ¯ 2 , E2 E 2q pole machine input voltage (synchronous voltage) (V) ¯ 1 (A) I¯1 , I1 current of E ¯I2 , I2 current of E2 (A) 2p pole machine rotor current (A) I¯3 , I3 p number of pole-pairs q number of pole-pairs Re() real part s1 slip of the 2p pole machine s2 slip of the 2q pole machine α electrical angle (rad) ¯ 2 and E ¯ 1 relative to two ϕ Phase between E rotating frames at speed ω2 and ω1 (rad) ω Supply angular frequency (rad/s) ω1 2p pole stator angular frequency (rad/s) ω2 2q pole stator angular frequency (rad/s) Ω machine speed (rad/s) θ mechanical Angle (rad) Appendix B. Linear Analysis of the Cascade Machine The set of equations describing the cascade machine in Fig. 1 are as follows:
I3 =
A2 s1 I1 − (B 2 s2 /y)I2 s1 (A2 + A3 ) + s2 (x/y)(B 2 + B 3 )
(A.6)
From Eqs. (A.1)–(A.6) a set of two equations with two unknowns I1 , I2 will result A22 s1 A 2 B 2 s2 A1 + A2 − D , I1 E1 yD = xs2 B 22 I2 xB 2 A2 s1 E2 , B1 + B2 − yD D (A.7) where D is defined as
x D=s1 (A2 + A3 ) + s2 (B 2 + B 3 ) y I3 =
(A.8)
A2 s1 B 2 s2 I1 − I2 D yD
(A.9)
Solving for the currents xs2 B 22 A 2 B 2 s2 I1 1 B 1 + B 2 − yD , − yD = 2 xB 2 A2 s1 I2 A 2 s1 − , A 1 + A2 − D D E1 (A.10) × E2 where is the determinant of the matrix. The mechanical power of the 2p pole-pair machine = T1 pΩ
(A.11)
The mechanical power of the 2q pole-pair machine = T2 (−qΩ) = −T2 qΩ
(A.12)
The total mechanical power = (pT1 − qT2 )Ω
(A.13)
where T1 and T2 are the torques of 2p pole and 2q pole machines, respectively. T1 = Re[jM1 (I1 − I3 )I3∗ ] = Re[jM1 I1 I3∗ ]
(A.14)
T2 = −Re[jM2 · xI3 (xI3 + I2 )∗ ] = −Re[jM2 · xI3 I2∗ ] (A.15)
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The total torque of the cascade T = pT1 − qT2 = pM1 Re[jI1 I3∗ ] + xqM2 Re[jI3 I2∗ ]
(A.16)
The input electrical power per phase Re(E1 I1∗ ) = E1 Re(I1∗ ) Similarly, the input power at the E2 port can be calculated for synchronous operation. The efficiency is defined as η = mechanical power/electrical power when the cascade is operating as a motor and 1/η when the cascade is operating as a generator.
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Machine paramter
Locked rotor values (ohm/phase)
Full-speed values (ohm/phase)
R X1 XM RM X2 R2 S
3.18 4.97 79.7 – 5.511 3.111 1.00
3.18 5.10 79.7 688.7 6.837 2.982 0.06
References Appendix C. Book hausend 3 poled pairs standard induction machine # 1/821837 data sheet Customer : University of Limerick Order no. – Serial no. 821837 Frame size D132MB kW 5.5 rpm Hertz 50 Phase Volts 415 Poles FLC (A) 12.6 FLT (Nm)
940 3 6 55.9
[1] A.R.W. Broadway, L. Burbridge, Self-cascade machine: a low-speed motor or high frequency brushless alternator, Proc. IEEE 117 (7) (1970) 1277–1290. [2] B.H. Smith, Synchronous behaviour of doubly fed twin stator induction machine, IEEE Trans. Power Appar. Syst. PAS-86 (10) (1967) 1227– 1236. [3] B.H. Smith, Theory and performance of a twin stator induction machine PAS-85 (2) (1996) 123–131. [4] H. Owen Whiney, Induction Machine, US Patent 4,228,391, 14 October 1980. [5] R. Li, R. Spéc, A.K. Wallace, G.C. Alexandra, Synchronous drive performance of brushless doubly-fed motors, 1992 IEEE Industry Applications Soc. Annual Meet., Houston, 1992, pp. 631–637. [6] D. Levy, Analysis of double stator induction machine used for VSCF small scale hydro/wind electric power generator, Electr. Power Syst. Res. 11 (3) (1987) 205–223. [7] L.J. Hant, The cascade induction motor, IEEE J. 52 (1914) 406–434. [8] F. Creedy, Some developments in multi-speed cascade induction motors, IEEE J. 59 (1921) 511–537. [9] A. Kusko, C.B. Somuah, Speed control of a single–frame cascade induction motor with slip power pump back, IEEE transactions on industry applications IA-14 (2) (1978) 97–105.
The cascade induction machine: a reliable and controllable motor or generator D. Picovici∗ , D. Levy, A.E. Mahdi, T. Coffey Department of Electronic & Computer Engineering, University of Limerick, Limerick, Ireland Received 14 October 2002; received in revised form 13 May 2003; accepted 9 June 2003
Abstract This paper discusses and analyses a set of two induction machines, with 2p and 2q pole-pairs, respectively, connected in cascade. It highlights the design and development principles of a single unit version of a system consisting of two wire-wound rotor induction machines with their rotors connected in cascade. Presented performance analysis shows the described cascade machine, which is brushless and with no slip rings or commutators, as a reliable, efficient and practical machine that could replace the induction machine, which has a wire-wound rotor. An example of the conversion of a standard squirrel cage six pole induction machine into a (6 + 2) pole induction machine, without rewinding the stator, is also presented together with simulation and measurements for asynchronous and synchronous modes of operation. © 2003 Elsevier B.V. All rights reserved. Keywords: Wire-wound rotor; Commutators; Squirrel cage induction machine
1. Introduction The cascade machine consists of two wire-wound rotor induction machines with their rotors connected in cascade. If the two machines have a different number of pole-pairs, then they can be constructed as a single unit version with a single stator and a squirrel cage rotor. Otherwise, if the two machines have the same number of pole-pairs, the single unit version is not possible [1–9]. To date, reported work on the development of the cascade induction machine [1–9] has shown that the two machines making the cascade can be of the same size or different sizes, and can run in the same direction or in opposite directions. It has also been shown that the difference in the number of poles between the two machines is not the only parameter that affects the characteristic of the cascade unit. Several parameters are involved in the construction of a cascade machine in order to be useful and adaptable for a specific application. This may explain why such a machine is yet to be made commercially available as ‘off-the-shelf’ product. This paper analyses two sets of induction machines
∗
Corresponding author. Tel.: +353-61-202925; fax: +353-61-338176. E-mail address: [email protected] (D. Picovici).
0378-7796/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2003.06.008
connected in cascade and highlights the parameters affecting the cascade operational efficiency. Knowing these parameters and understanding well the operating and design principles will result in the development of an efficient practical machine. An example on the conversion of an ordinary squirrel cage induction machine into a cascade machine is also presented.
2. The equivalent circuit of a multipole machine The transformation α = pθ transforms a ‘p’ pole-pair machine into a single pole-pair machine. Here, α and θ represent the electrical and mechanical angles, respectively, of a ‘p’ pole-pair machine. According to this transformation, the ‘p’ pole-pair are ‘folded’ together and the input current of a ‘p’ pole-pair machine is equal to p times the input current of a single pole-pair machine for the same air gap flux. The input impedance of the ‘p’ pole machine is therefore equal to the voltage per pole divided by the total current, or 1/p times the impedance of a single pole-pair machine. The slip of the ‘p’ pole-pair machine is therefore equal to (ω − pΩ)/ω, where Ω is the machine’s mechanical speed, ω is its line frequency and ‘pΩ’ is its ‘folded’ speed or the electrical speed.
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Fig. 1. Cascade machine equivalent circuit with reference to their stators.
3. The equivalent circuit of a cascade electrical machine For a symmetrical grid, symmetrical polyphase machine, the equivalent electrical circuit per phase for a set of two wire-wound induction machines connected in cascade through their rotor’s terminals with reference to their stators, is shown in Fig. 1. One machine is assumed to have ‘p’ pole-pair and the other ‘q’ pole-pair. In this equivalent circuit, the two machines are shown to be coupled through a transformer with voltage ratio y and current ratio x. In practice, x = y = 1 and therefore the two machines are coupled directly without a transformer. Referring to Fig. 1, the circuits components of a ‘p’ pole-pair machine can be expressed, in terms of those of the resulting single pole-pair machine, as follows: [Rf + jω1 (Lf − M)] A1 = =Rf1 + jω1 (Lf1 − M1 ) p A2 =
jω1 M//Ri =jω1 M1 //Ri1 p
A3 =
[Ra /s1 + jω1 (La − M)] Ra1 = + jω1 (La1 − M1 ) p s1
Similarly for the ‘q’ pole-pair: B1 =
[Rf + jω2 (Lf − M)] =Rf2 + jω2 (Lf2 − M2 ) q
B2 =
[jω2 M//Ri ] =jω2 M2 //Ri2 q
B3 =
[Ra /s2 + jω2 (La − M)] Ra2 = + jω2 (La2 − M2 ) q s2
where ω1 is the frequency of the stator of the p pole-pair machine, which is assumed to be a constant grid frequency and ω2 is the frequency of the q pole-pair machine which therefore is a dependent variable and function of the speed. In general, the machine has two ports which can be used as inputs or outputs similar to the wire wound machine. For ¯ 2 = 0. For synchronous operaasynchronous operation, E ¯ tion, E2 can be a voltage with a fixed frequency such as a dc voltage for constant speed operation or a variable frequency voltage for a variable speed application. ¯ 1 is ω1 and the frequency If the frequency of the voltage E ¯ of the voltage E2 is ω2 , then power can flow between the two
machines as shown in Fig. 1, if and only if, the frequencies of the two rotor voltages or currents are equal, thus, |ω1 − pΩ| = |ω2 + qΩ|
(3.1)
Eq. (3.1) assumes the two machines are connected mechanically back to back, they are rotating in opposite directions. For synchronous operation, ω1 and ω2 are independent variables and the machine speed Ω becomes a dependent variable which depends on the phase between ¯ 1 and E ¯ 2 with reference to rotating frames of references E ω1 and ω2 , respectively. On the other hand, in asynchronous operation, ω2 is a dependent variable. In that case, E2 = 0.
4. The isosynchronous machine For the case p = q and ω2 = ω1 , the speed Ω is an independent variable in the Eq. (3.1). For this special particular case, the machine is called an isosynchronous machine. In practical application, two machines are coupled back to back ¯1 = E ¯ 2 . Torque and produce zero net torque for the case E is produced only when a phase difference between |E1 | and |E2 | exists. Normally, the torque is controlled by a phase shifter or by adjusting mechanically one stator relative to the other as shown in Fig. 3. The torque/speed characteristic is shown in Fig. 5. Maximum torque is produced from the set if the angle of the electrical phase difference between the two machine stators is equal to 180◦ . The two machines can also be coupled front to back if double axis is available. It is worth emphasising, that this type of operation of the set is unique since both inputs accept voltage at the same frequency. However, the set works in the asynchronous mode contrary to the cascade set withp = q. The latter can operate only in the synchronous mode if two input voltages are present. This is the reason why the p = q set is called an isosynchronous machine. Unfortunately, the set cannot be constructed in a single frame.
5. The cascade machine analysis Analysis of the cascade machine equivalent electrical circuit shown in Fig. 1 is given in Appendix B for asynchronous and synchronous modes of operation. The two machines
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195
Fig. 2. Approximate equivalent circuit of the cascade machine at light load.
equivalent circuit shown are referred to their stators. The rotor’s frequency for this type of application is calculated from Eq. (3.1).
Ra2 B3 ∼ = s2
ω1 − pΩ = ω2 + qΩ
Then
(5.1)
In order to understand the behaviour of the assembly, an analytical analysis needs to be carried out for an approximate equivalent circuit shown in Fig. 2. Let A1 = 0
(5.2)
B 2 = jω2 M2
(5.3)
5.1. Asynchronous Operation For asynchronous operation E2 = 0. In this mode, the current I3 can be shown to be: I3 =
E1 A3 + B 3 (s2 /s1 ) + (s2 /s1 )(B 1 B 2 /B 1 + B 2 )
where ω1 − (p + q)Ω s1 = s2 ω1
(5.4)
(5.5)
where the ratio s1 /s2 is called the ‘cascade slip’ which is equivalent to the slip of a p + q pole-pair machine, where s1 and s2 are the slip of each separate machine defined as ω1
− pΩ ω1
(5.6)
ω2
+ qΩ ω2
(5.7)
s1 =
s2 =
In asynchronous operation ω1 is a fixed independent variable but ω2 is a dependent variable of the machine speed Ω. By eliminating the speed Ω from Eqs. (5.1), (5.6) and (5.7), ω2 as a function of the cascade slip can be obtained: s1 (5.8) ω2 = ω1 s2 As an approximation, let B1 be equal to the control resistance connected in series with the stator resistance of the p pole-pair machine. Let
B 1 =R Ra1 A3 ∼ = s1
(5.9) (5.10)
I¯3 = =
(5.11)
E1 ((Ra1 + Ra2 )/s1 ) + (s2 /s1 )(jω2 M2 R/(R + jω2 M2 )) E1 ((Ra1 + Ra2 )/s1 ) + (jω1 M2 R/(s1 /s2 ))/(R/(s1 /s2 ) + jω1 M2 ) (5.12)
At the cascade slip (s1/s2) = 0, I3 is not purely reactive due to the losses in the 2p pole machine and in the rotors. Eq. (5.12) shows clearly the disadvantage of this machine where the rotors resistances together with 2p pole stators resistance limits its output power and reduces its efficiency if it is not designed properly. At slip s1 = 0 of the 2p pole machine, the current I3 = 0. The torque is therefore equal to zero together with the field of the 2q pole machine. Therefore, around s1 = 0, the set behaves as a 2p pole machine and its characteristic is uncontrollable with the resistance R. For better efficiency and higher output power, the stator of the 2p pole machine should have as low an impedance as possible and the 2q pole machine should have as high an impedance ω1 M2 as possible. Therefore, the feeding of the set should be carried out from the higher pole-pair side, i.e. p > q. The torque of the 2p pole machine T1 = Re(jM1 )
E E I3 = Re(I3 ) jω1 M1 ω1
(5.13)
T1 therefore depends on the real part of I3 which can be calculated to be: Re
I3 E1
=
(Ra1 + Ra2 /s1 )(R/(s1 /s2 ))2 + ω12 M22 ((Ra1 + Ra2 )/s1 + R/(s1 /s2 )) (Ra1 + Ra2 )/s1 )2 (R/(s1 /s2 ))2 + ω12 M22 ((Ra1 + Ra2 )/s1 + R/(s1 /s2 ))2 (5.14)
For s1 < 0, s1 /s2 is already negative. Therefore, Re(I3 ) is negative and the mechanical power is negative (generative).
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At s1 /s2 < 0, Re(I3 /E1 ) is negative within the region given by: −Q −
Q2
R/(s1 /s2 ) −1< < −Q + Q2 − 1 ω1 M 2
(5.15)
where
Q=
ω1 M2 2((Ra1 + Ra2 )s1 )
(5.16)
where s1 is positive. Q is a goodness factor for the simplified linear model. Furthermore, from Eq. (5.15), Q should be greater than one in order that the torque T1 and therefore the cascade mechanical power, becomes negative at (s1 /s2 ) < 0 within that region. The electrical power input to the machine, with no iron losses, according to the simplified model: = Re(E1 · I1∗ ) = E1 Re(I3 )
(5.17)
Therefore, the region given in Eq. (5.15) is for power generation at negative cascade slip. The torque of the 2q pole machine:
jM2 RI3 I3∗ R + jω2 M2 ω1 M22 R ∗ = −I3 I3 s1 /s2 (R/(s1 /s2 ))2 + ω12 M 2
T2 = −Re
(5.18)
T2 is equal to zero at s1 = 0 (where I3 = 0) and at the cascade slip s1 /s2 = 0. At negative cascade slip, s1 /s2 < 0, T2 reverses its sign and becomes positive. The mechanical power T2 (−pΩ) is therefore negative (generative). As previously stated, at s1 = 0, I3 = 0, the 2q pole machine is not excited. Therefore, around s1 = 0, the control resistance R has no effect on regulation. A good cascade machine should not reach the slip s1 ≤ 0 As a motor, it works at (s1 /s2 ) ≥ 0 and as a generator at (s1 /s2 ) < 0 where s1 > 0 or very close to one at (s1 /s2 ) = 0.
5.2. Synchronous operation ¯ 2 = (ω2 /ω1 )E2 ejϕ = At synchronous operation E (s1 /s2 )E2 ejϕ , where ϕ is the phase difference between the voltage of the 2p pole stator and the 2q pole stator with reference to the two rotating frames ω1 and ω2 , respectively. ¯ 2 is normally proportional to the speed in order to prevent E saturation of the iron core. The angle ϕ depends on the load and should be within the region 2π ≥ ϕ ≥ 0 otherwise the machine will be asynchronous at a given frequency ω2 supplied to the 2q pole machine. In this mode, the machine becomes doubly excited from the two stators of the 2p and 2q pole. The analysis is similar to the asynchronous mode using the superposition theory. For the reasons explained in the asynchronous mode of operation, most of the output power is supplied by the 2p pole machine where p > q. If ω2 = 0, the 2q pole machine is excited by a dc voltage. The cascade will run at synchronous speed when the cascade slip s1 /s2 = 0 only. The dc voltage with reference to a ¯ 2 = E2 ejϕ . Again rotating frame of reference ω1 will be E the phase ϕ will depend on the load. E2 is normally supplied from a well controlled current source in order to prevent saturation. 6. Conversion of a 2p = 6 pole machine into a (p + q) = (3+1) pole-pair machine A Brook Hausend =1/821837, three pole-pair machine with data given in Appendix C is to be converted to a (3 + 1) pole-pair cascade without rewinding the stator. The rotor shorted circuit bars are to be replaced by a graded winding. 6.1. Stator The stator winding of that machine consists of six coils per phase connected in series in order to produce three pole-pair. Therefore, the number of slots in the stator is 36. Line voltage was 415 V. After conversion, the coils are now connected
Fig. 3. Stator windings connection of the converted three pole-pair machine into (p = 3 + q = 1). The figure is useful for conversion of a similar stator machine.
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in parallel for each pole-pair instead of in series. This is in order to allow a second input for the 2q pole machine as shown in Fig. 3. The impedance of each phase is therefore reduced by 1/q. The voltage E1 is now reduced to 415/3 = 138 V and the current =I1 is now three times the original phase current. In p = 3, pole-pair equivalent circuit, all components are therefore divided by nine. In the q = 1, pole-pair equivalent circuit, all components are divided by three. In order for the air gap flux to remain the same as the original three pole-pair in the q = 1 pole-pair, the voltage E2 should be 415/9 = 46 V. The ampere turns in the q = 1 side is obviously three times than that on the p = 3 side. Furthermore, in the cascade application, ignoring the rotor’s resistances in Fig. 2, the iron laminations should support the double flux
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of the machine. Thus, instead of M = M1 p = M2 q, we have in the cascade M/2 air gap mutual inductance. If the machine is double excited from both sides, the input voltages should now be reduced to E1 = (415/3)(1/2) = 70 V and to E2 = 23 V, respectively at 50 Hz frequency in order to prevent saturation. Appendix C shows the machine data before and after conversion. (Also see Sections 6.2 and 6.3). 6.2. Rotor The air gap flux due to the 2p pole machine is approximately equal to E1 /ω1 p. From Eq. (5.12), ignoring Ra1 + Ra2 , the air gap flux due 2q pole machine is ∼ = E1 /(ω1 p) = 0.11 (web turns per pole-pair). The total
Fig. 4. The rotor with four concentric coils grouping of graded windings shorted circuit (photography from two angles) The figure is useful for conversion of an ordinary squirrel cage rotor into a cascade machine rotor.
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instantaneous air gap flux with reference to a static frame of reference: E1 j(ω1 +qΩ)t E1 j(ω1 −pΩ)t (6.2.1) e + Flux = e ω1 p ω1 p where E2 E1 = p q
(6.2.2)
At Ωt= θ, t = 0 the absolute value of the flux is in the form: √ E1 −jpθ 2E1 +jqθ |= +e (1+Cos(p + q)θ)1/2 (6.2.3) e ω1 p ω1 p At any time t, other than t = 0, (6.2.3) is simply shifted by ω1 t. Therefore, the shape of the rotor shorted circuit conductor should be according to Eq. (6.2.3) instead of E1 /ω1 p = constant, independent of θ in the original standard induction machine. For the present conversion, Eq. (6.2.3) is approximated by grouping concentric coils of graded winding (Fig. 4). 6.3. Simulation of the converted machine The converted machine data was simulated using Fortran NAG and the results were plotted for ((p = 3) + (q = 1)) and for ((p = 1) + (q = 3)) connected in asynchronous mode. As expected, the ((p = 1) + (q = 3)) has very little power due to its very low goodness factor.
At frequency of 50 Hz, the converted cascade set is still not very useful as a motor, or as a generator since for the ((p = 3) + (q = 1)) machine, Ra1 + Ra2 = 2 ohms, M2 = 8.45 × 10−2 Hy, s1 = 1 ω = 2π50 the goodness factor is only Q = 6.3 at most. For the ((p = 1)+(q = 3)) machine, the goodness factor is only 2.1. Obviously, 50 Hz operation of the above cascade is not practical. When the frequency is increased to 100 Hz, a more practical machine will result with almost identical power output to the original machine before conversion. Figs. 5–8 show the characteristics of the machine before conversion and Figs. 9–12 show the characteristics of the cascade ((p = 3) + (q = 1)) after conversion, operating from 100 Hz and E1 = 140 V in asynchronous mode for different control resistors 0 < R < 10. The characteristics are not different from the standard wire wound induction machine. The synchronous speed (slip s1 /s2 = 0) of the cascade is 1500 rpm (157 rad/s). The slip s1 = 0 is at speed 2000 rpm (209 rad/s) which is far from the synchronous speed of the cascade and therefore, s1 can be approximated as s1 ∼ = 1 in the practical operating speed of the cascade. Therefore, ω1 is adequately high. The losses (Ra1 + Ra2 )/s1 in Eq. (5.12) can be neglected and the equivalent electrical circuit of the cascade becomes close to the ideal wire wound induction machine. In synchronous mode of operation, Figs. 13–16 shows the characteristics of the machine converted into a cascade where ω1 = 2π100 and ω2 is varied according to the machine speed Eq. (5.1). The characteristics shown are for phase (3π/2) ≥ ϕ ≥ 0.
Fig. 5. Torque/speed characteristic, per phase, of the original p = 3 pole-pair machine, the characteristic is useful for optimal stable coupling of the machine to the load as a motor or optimal coupling of the machine to a turbine as a generator.
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199
Fig. 6. Electrical power/speed characteristic per phase of the original p = 3 pole-pair machine. The characteristic is useful for predicting the power consumption as a motor or, the power fed to the grid as a generator.
Fig. 7. Mechanical power/speed characteristic per phase of the original p = 3 pole-pair machine. The characteristic is useful for predicting the optimal output mechanical power as a motor or, the optimal input mechanical power from a turbine as a generator.
7. Simulation results
7.1. Asynchronous mode (100 Hz grid)
Results of the simulation of the converted six poles, 50 Hz, and 5.5 kW squirrel cage induction machine into a (3 + 1) pole-pair, 100 Hz cascade machine is given in this section for asynchronous and synchronous modes.
Fig. 9 shows the torque/speed characteristic. The control resistance is shown as a parameter 10 ≥ R ≥ 0 ohms. Optimal resistance for maximum starting torque is seen to be compatible with wire wound rotor machines. Above slip
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Fig. 8. Efficiency/speed characteristic of the original p = 3 pole-pair machine. The characteristic is useful for predicting the optimal operating efficiency of the machine as a motor or, as a generator.
s1 /s2 = 0 (speed = 1500 rpm), the machine is generating. At much higher speed, (where s1 = 0 at speed 2000 rpm or 209 rad/s) the machine will again be generating, however its characteristic is non-controllable by R since the 2q machine produces no torque.
Fig. 10 shows the electrical power input to the cascade. This figure is similar to Fig. 9. The mechanical power is shown in Fig. 11 and the machine efficiency in Fig. 12. At the practical operating region 1 ≥ s1 /s2 ≥ 0 (up to around 157 rad/s as a motor), the efficiency is around 55%
Fig. 9. Torque/speed characteristic per phase of the converted machine in asynchronous mode. Variable resistance, at the q = 1 pole-pair side, as a parameter. The characteristic is useful for optimal stable coupling of the machine to a load, as a motor or, to a turbine as a generator.
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Fig. 10. Electrical power/speed characteristic per phase of the converted machine in asynchronous mode. Variable resistance, at the q = 1 pole-pair side, as a parameter. The characteristic is useful for predicting the power consumption as a motor or, the power fed to the grid as a generator.
Fig. 11. Mechanical power/speed characteristic per phase of the converted machine in asynchronous mode. Variable resistance, at the q = 1 pole-pair side, as a parameter. The characteristic is useful for predicting the optimal output mechanical power as a motor or, the optimal input mechanical power from a turbine as a generator.
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Fig. 12. Efficiency/speed characteristic of the converted machine in asynchronous mode. Variable resistance, at the q = 1 pole-pair side, as a parameter. The characteristic is useful for predicting the optimal operating efficiency of the machine as a motor or, as a generator.
Fig. 13. Torque/speed characteristic per phase of the converted machine in synchronous speed mode. The characteristic is useful for optimal stable coupling of the machine to the load as a motor or, to a turbine as a generator. The characteristic indicates the range of operation as a motor or as a generator, for a given speed. The parameter is the phase ϕ = 0, 90, 180 and 270◦ .
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203
Fig. 14. Electrical power/speed characteristic per phase of the converted machine in synchronous mode. The characteristic is useful for determining the range of speed as a motor or a generator. As a generator, the range of speed is very limited in contrast to the motor. The parameter is the phase ϕ = 0, 90, 180 and 270◦ .
compared to 75% of the original machine before conversion. The efficiency of the converted cascade can be improved if the supply frequency is increased above 100 Hz. However, as a generator, its efficiency is much greater than a motor and it can reach 90% at 100 Hz excitation as Fig. 12 indicates.
7.2. Synchronous mode (ω1 = 2π100, ω2 = variable) Fig. 13 shows the torque/speed characteristic with phase ϕ as a parameter. For a given speed, ϕ will adapt itself automatically according to the load on the machine. The
Fig. 15. Mechanical power/speed characteristic per phase of the converted machine in synchronous mode. The characteristic is useful for determining the maximum output load possible as a motor, or, input drive power from a turbine as a generator before leaving the synchronous mode. The phase as a parameter ϕ = 0, 90, 180 and 270◦ .
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Fig. 16. Efficiency/speed characteristic of the converted machine in synchronous mode. The characteristic is useful for determining the maximum efficiency possible before leaving synchronous mode as a motor or as a generator. The phase as a parameter, ϕ = 0, 90, 180 and 270◦ .
range of the maximum torque that the machine can provide at different speed can be found from Fig. 13. Fig. 14 shows the total electrical power from both inputs to the machine. It is similar to Fig. 13. The mechanical power is shown in Fig. 15 and the machine efficiency in Fig. 16. As expected, due to the two supply inputs, the efficiency in the synchronous mode is higher than in the asynchronous mode and it is close to the original machine efficiency before conversion.
8. Experimental results The converted cascade characteristics in asynchronous mode were measured. The q = 1 pole-pair side was terminated by a variable resistance. The machine was fed from a constant voltage at a clean 100 Hz frequency rotating converter. The machine was mechanically loaded with an induction motor used as a variable load in the motoring
Fig. 17. Measured torque/speed characteristic, asynchronous mode. Variable resistance as a parameter. Two extremes are shown, zero and infinite resistances.
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Fig. 18. Measured electrical power/speed characteristic, asynchronous mode, variable resistance as a parameter. Two extremes are shown, zero and infinite resistance.
Fig. 19. Measured mechanical power/speed characteristic, asynchronous mode, variable resistance as a parameter. Two extremes are shown, zero and infinite resistance.
region and as an over speed motor simulating a turbine in the generating mode region. The torque/speed characteristic is shown in Fig. 17 for two parameters. Load resistance = 0 and infinity (open circuit). By comparison with the ideal characteristic Fig. 9 obtained from simulation, it can be seen that there is not too much difference. The only difference in shape was due to the finite number of copper bias used in the rotor (three in each group instead of infinite). Similarly, with the electrical power in the motoring and generating regions in Fig. 18. The converted machine has, as expected, more losses than the ideal, simulated machine in Fig. 10. Fig. 19 shows the mechanical power obtained from Fig. 17 by multiplying the torque by the speed. The ideal characteristic obtained from simulation is shown in Fig. 11.
9. Conclusion The cascade machine is a controllable induction machine similar to the wire- wound rotor induction machine, however it is much more reliable and cheaper due to its cage rotor or brushless construction. A six pole or more induction machine can be converted into a cascade machine without rewinding the stator, but by modification of the cage rotor structure. However, by doing so, the efficiency of the machine is reduced significantly and the resulting cascade becomes unpractical unless its operating frequency is increased. In asynchronous operation, the input voltage should be connected to the stator with the higher number of poles, in order to maximise its efficiency, otherwise, the resulting cascade is not practical.
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Acknowledgements This material is based upon work supported by Enterprise Ireland under the Strategic Research Program 1997. The authors would like to pay tribute to their colleague and co-author of this paper, Dr. D.Levy, who sadly passed away last year, and acknowledge his significant contribution to this work. He is deeply missed.
E1 = A1 I1 + (I1 − I3 )A2
(A.1)
E2 = B 1 I2 + B 2 (xI3 + I2 )
(A.2)
Vr = A2 (I1 − I3 ) − A3 I3 s1
(A.3)
Vr = xB 3 I3 + B 2 (xI3 + I2 ) s2
(A.4)
j
where in practice Appendix A. List of symbols
x=y=1
(A.5)
solving for I3 Machine parameters A1 , A2 , A3 2p pole machine parameters (complex variables) B 1, B 2, B 3 2q pole machine parameters (complex variables) Rf , Rf1 , Rf2 2 pole, 2p pole and 2q pole stator resistances (ohms) 2 pole, 2p pole and 2q pole stator Lf , Lf1 , Lf2 self inductances (henries) M, M1 , M2 2 pole, 2p pole and 2q pole rotor/stator mutual inductances (henries) Ra , Ra1 , Ra2 2 pole, 2p pole and 2q pole rotor resistances (ohms) La , La1 , La2 2 pole, 2p pole and 2q pole rotor self inductances (henries) 2 pole, 2p pole and 2q pole iron losses Ri , Ri1 , Ri2 equivalent resistances (ohms) Other variables ¯ 1 , E1 E 2p pole machine input voltage (V) ¯ 2 , E2 E 2q pole machine input voltage (synchronous voltage) (V) ¯ 1 (A) I¯1 , I1 current of E ¯I2 , I2 current of E2 (A) 2p pole machine rotor current (A) I¯3 , I3 p number of pole-pairs q number of pole-pairs Re() real part s1 slip of the 2p pole machine s2 slip of the 2q pole machine α electrical angle (rad) ¯ 2 and E ¯ 1 relative to two ϕ Phase between E rotating frames at speed ω2 and ω1 (rad) ω Supply angular frequency (rad/s) ω1 2p pole stator angular frequency (rad/s) ω2 2q pole stator angular frequency (rad/s) Ω machine speed (rad/s) θ mechanical Angle (rad) Appendix B. Linear Analysis of the Cascade Machine The set of equations describing the cascade machine in Fig. 1 are as follows:
I3 =
A2 s1 I1 − (B 2 s2 /y)I2 s1 (A2 + A3 ) + s2 (x/y)(B 2 + B 3 )
(A.6)
From Eqs. (A.1)–(A.6) a set of two equations with two unknowns I1 , I2 will result A22 s1 A 2 B 2 s2 A1 + A2 − D , I1 E1 yD = xs2 B 22 I2 xB 2 A2 s1 E2 , B1 + B2 − yD D (A.7) where D is defined as
x D=s1 (A2 + A3 ) + s2 (B 2 + B 3 ) y I3 =
(A.8)
A2 s1 B 2 s2 I1 − I2 D yD
(A.9)
Solving for the currents xs2 B 22 A 2 B 2 s2 I1 1 B 1 + B 2 − yD , − yD = 2 xB 2 A2 s1 I2 A 2 s1 − , A 1 + A2 − D D E1 (A.10) × E2 where is the determinant of the matrix. The mechanical power of the 2p pole-pair machine = T1 pΩ
(A.11)
The mechanical power of the 2q pole-pair machine = T2 (−qΩ) = −T2 qΩ
(A.12)
The total mechanical power = (pT1 − qT2 )Ω
(A.13)
where T1 and T2 are the torques of 2p pole and 2q pole machines, respectively. T1 = Re[jM1 (I1 − I3 )I3∗ ] = Re[jM1 I1 I3∗ ]
(A.14)
T2 = −Re[jM2 · xI3 (xI3 + I2 )∗ ] = −Re[jM2 · xI3 I2∗ ] (A.15)
D. Picovici et al. / Electric Power Systems Research 68 (2004) 193–207
The total torque of the cascade T = pT1 − qT2 = pM1 Re[jI1 I3∗ ] + xqM2 Re[jI3 I2∗ ]
(A.16)
The input electrical power per phase Re(E1 I1∗ ) = E1 Re(I1∗ ) Similarly, the input power at the E2 port can be calculated for synchronous operation. The efficiency is defined as η = mechanical power/electrical power when the cascade is operating as a motor and 1/η when the cascade is operating as a generator.
207
Machine paramter
Locked rotor values (ohm/phase)
Full-speed values (ohm/phase)
R X1 XM RM X2 R2 S
3.18 4.97 79.7 – 5.511 3.111 1.00
3.18 5.10 79.7 688.7 6.837 2.982 0.06
References Appendix C. Book hausend 3 poled pairs standard induction machine # 1/821837 data sheet Customer : University of Limerick Order no. – Serial no. 821837 Frame size D132MB kW 5.5 rpm Hertz 50 Phase Volts 415 Poles FLC (A) 12.6 FLT (Nm)
940 3 6 55.9
[1] A.R.W. Broadway, L. Burbridge, Self-cascade machine: a low-speed motor or high frequency brushless alternator, Proc. IEEE 117 (7) (1970) 1277–1290. [2] B.H. Smith, Synchronous behaviour of doubly fed twin stator induction machine, IEEE Trans. Power Appar. Syst. PAS-86 (10) (1967) 1227– 1236. [3] B.H. Smith, Theory and performance of a twin stator induction machine PAS-85 (2) (1996) 123–131. [4] H. Owen Whiney, Induction Machine, US Patent 4,228,391, 14 October 1980. [5] R. Li, R. Spéc, A.K. Wallace, G.C. Alexandra, Synchronous drive performance of brushless doubly-fed motors, 1992 IEEE Industry Applications Soc. Annual Meet., Houston, 1992, pp. 631–637. [6] D. Levy, Analysis of double stator induction machine used for VSCF small scale hydro/wind electric power generator, Electr. Power Syst. Res. 11 (3) (1987) 205–223. [7] L.J. Hant, The cascade induction motor, IEEE J. 52 (1914) 406–434. [8] F. Creedy, Some developments in multi-speed cascade induction motors, IEEE J. 59 (1921) 511–537. [9] A. Kusko, C.B. Somuah, Speed control of a single–frame cascade induction motor with slip power pump back, IEEE transactions on industry applications IA-14 (2) (1978) 97–105.