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Inverter controlled characteristics of variable frequency induction motors
H. Raphael, A. Skjellnes
INVERTER CONTROLLED CHARACTERISTICS OF VARIABLE FREQUENCY INDUCTION MOTORS H. Raphael is leader of the Electrical Machinery and Power Electronics Group of the Electrical Power Engineering Laboratories, the Norwegian Institute of Technology, Trondheim, Norway. A. Skjellnes is senior research engineer with the Norwegian Research Institute for Electricity Supply, Trondheim, Norway. SUMMARY This paper deals with the theory of asynchronous motors running at variable frequency. Motor characteristics are found by the help of a computer programme where higher harmonics are neglected, but the influence of saturation is considered. It is shown that it is impossible to get optimum performance if the voltage is a function of the frequency only. A regulator which controls the voltage as a function of both frequency and load to keep constant flux gives torque and current characteristics equal to those of a dc-machine. This paper presents two alternatives for such a regulator. Braking methods are discussed and a regulator principle for a four-quadrant drive without current sensing is shown. ZUSAMMENFASSUNG Die Theori des Kafiglaufermotors, der von einer Quelle mit variabler Frequenz und Spannung gespeisst wird, soll erlautert werden. Die Kennlinien werden mit Hilfe eines Rechenprogramms gefunden. Die Oberharmonischen der Grundwelle werden in diesem Programm vernachlassigt, aber der Einfluss der magnetischen Sattigung wird berucksichtigt. Es wird gezeigt, dass optimaler Motorbetrieb nicht moglich ist,wenn die Spannung nur als eine Funktion der Frequenz angepasst wird. Die Kennlinien des Kafiglaufermotors mit konstanter Durchflutung werden mit denen eines Gleichstrommotors verglichen. Zwei prinzippielle Losungen der Durchflutungsregulierung werden beschrieben. Verschiedene Verfahren des Bremsens werden diskutiert, und ein Aufbau einer Umkehrregelung ohne Strommessung wird gezeigt.
THEORY OF MOTOR OPERATION The aim of this section is to find an optimal relationship between voltage and frequency for an inverter-fed squirrel cage motor drive. The inverter is considered as a general energy-converting component, and this paper does not deal with inverter principles or construction. Therefore the analysis supposes sinusoidal voltages. The harmonic components, which are thus neglected do not alter the principal results. They give rise to some additional losses
713
H. Raphael, A. Skjellnes
and except at very low speed quite negligible torque components. It will be shown how the effect of saturation is taken into account. Under the mentioned condition, the conventional equivalent circuit, figure 1, is applicable, where all circuit parameters are referred to the stator. In a physical machine they are dependent upon voltage, current, slip and frequency. The relations are complex, and in order to achieve the goal of the analysis by tolerable efforts the following simplifications are required: ~ ~
=.
1.
Iron losses are neglected, that is
2.
The stator and rotor parameters are kept constant and equal to their values at rated voltage, frequency and load.
The iron losses may be seperat~ly calculated when the flux density is determined. The roughest assumption is to keep the rotor parameters L2 and R2 constant. The analysis will thus give the most reliable results for the speed region near synchronism. Fortunately this region is of the greatest interest for the inverter fed motor drive. Magnetizing inductance LM is dependent upon the relationship between the induced voltage E and the stator frequency W\. The following phasor equations give the electrical balance per phase: -+
Stator voltage:
E+
U l
Rl'!l + jWILl'!l
( 1)
-+
Magnetizing current: Rotor current:
-+
I
2
!M
E
(2)
jwlLM
!l - ! M
( 3)
E
..
STATOR
•
ROTOR
Figure 1: Asynchronous motor, single phase equivalent 714
H. Raphael, A. Skjellnes
Induced voltage:
( 4)
Total power and torque is given by the following classical equations: Airgap power:
Po
Rotor losses:
P
2
Mechanical power: Mechanical torque:
P
R 2 2 m'S-.I 2
( 5)
m.R I 2 2 2
(6)
L
To
I-s 2 m·---·R 2 I 2 s R I Po 2 2 m·p·---s,w W l s
(7)
(8)
where m and p are number of phases and pairs of poles respectively. Ws is mechanical synchronous speed and the slip is defined by
s = On the basis of these equations an iterative computer programme is developed which determines the motor characteristics. Input values are stator voltage and frequency. In the following it is first shown how the motor parameters may be found. Then it is demonstrated how the programme is used to find the optimal voltage-frequency relationship, and the consequences of this relationship to the regulator system are outlined. Determination of Motor Parameters Measurements are performed at standstill, at rated voltage, frequency and load and at no load with variable voltage. The latter is used to find the saturation curve for the main flux path. Standstill Measurement. - With locked rotor a reduced stator voltage at rated frequency, which gives about rated stator current, is used. Input voltage, current and power as well as the torque are measured. At standstill and low voltage the magnetizing current is neglected. The following equations determine the stator parameters. Suffix S indicates standstill and N rated value. Rotor losses:
P
WIN
2S
T .--S P
(9)
715
H. Raphael, A. Skjellnes
Losses in stator: Stator resistance:
Pl,loss = m.Rl1 l R
2
(10)
Pl,lOSS l
m.1lQ ,.
Reactive power: Q =v(m.Uls1 1S ) 1S = m·w
lN
(11)
2 2
- P 1S
(L +L ) .1 2 l 16
2
2r
( 12)
Assuming the same leakage inductance in stator and rotor, equation (12) gives Stator inductance:
L
l
(13)
The values given by (11) and (13) are constant parameters in the computer programme. No-Load Measurement. - The motor is running without any load on a rated frequency variable voltage supply. The voltage is changed from about half to about 1.5 of the rated voltage, and corresponding values of voltage U10 and current 110 are fed into the computer. By help of equations (1), (2) and the values found from (11) and (13) the magnetizing inductance is known, and corresponding values of the ratio E/wl and LM are stored. The saturation of the motor is thus taken care of by this array. Rated Load Measurement. - The machine is then connected to a rated frequency, rated voltage supply, and loaded to rated current. The corresponding values of input power, speed and torque are measured. The rotor parameters can then be calculated. First stator current phasor is found relative to the voltage: (14)
(15)
Then equation (1) is used to find the rated induced voltage EN and (2) gives together with the stored array from the no-load measurement the magnetizing current. Finally equation (3) determines the rated rotor current. The rotor impedance at rated load is (referred to stator)
716
H. Raphael, A. Skjellnes
(16)
and the rotor parameters are (17)
(18)
These values (17) and (18) are also constant parameters in the computer programme. As already mentioned this is physically the worst approximation, but fortunately an inverter fed asynchronous machine normally operates near rated rotor conditions. It has been evaluated that this method of determining the motor parameters gives an uncertainty of about 10%. This accuracy is sufficient to give principally correct results. Machine Characteristics at Variable Frequency It is well known that the available torque of an inverter fed squirrel cage motor drops considerabily at low speed if the voltage is kept directly proportional to the frequency. If constant torque is wanted over the entire speed range, or perhaps even higher torque at low speed, which is the case in traction applications etc., other frequencyvoltage relationships have to be used. To obtain the desired performance two approaches are tried. One assumes that the voltage function is independent of the load, and the second approach is to keep the main flux constant for all load conditions. Voltage Independant of Load. - An easy function to realize in a regulator is the u, linear relationship:
U 1
=
U +k 0
!f
N
U 1N
(19)
out of which the proportional equation (20) Figure 2: Voltage-frequency functions
is already mentioned.
717
H. Raphael, A. Skjellnes
Figure 2 shows both this function (20) and the more general form (19). As indicated the direct proportional equation leads to low available torque at low speeds. Motor characteristics have been computed for a 4 kW, 230 V, 50 Hz motor assuming the voltage following equation 20. It is clearly seen in figure 3 how the torque characteristic drops at low speeds. This simple voltage-frequency relationship is therefore only applicable when the motor works under easy starting conditions and rated torque is not required at low speeds. It is evident that better performance might be obtained with the voltage following equation 19. The function starts with a constant voltage Uo and the slope k is chosen to give rated voltage at rated frequency. The computer programme is used to determine these constants in such a way that approximately the same pull-out torque is available at all speeds. At each frequency the calculation starts with a voltage according to equation (20), finds the pull-out torque and adjusts the input voltage until the pull out torque has the same value as at rated frequency and rated voltage. The results are shown in figure 4. It is seen that even though the additional constant voltage is
IU,=!- U,NI fN
2
I FIELD
"\ \
WEAKENING
i
•
/
\_L
--- .J ---
-1
-2
T 11 Elf
Figure 3: Motor characteristics with voltage proportional to frequency 718
H. Raphael, A. Skjellnes
L . .!L TN' l'N
ECN) U'N
T
3
.
i / f
+
f O. SSt) U'N N
U,= U'N WEAKENING.
\
I
""-'2
U,=(O.12
I FIELD
\
'-.
.... \
I /
/
\
\
/
/
-I- - -_ 0 -1
f= O.1fN
f=O.5fN
-2
T
-3
Figure 4:
I
/
I, Elf
Motor characteristics at constant pull-out torque
relatively small, O.IZ·U N , the current characteristic at low speed rises considerably. This is due to the greater induction, Elf, which consecutively leads to still heavier saturation. It is worth noting the change of the current characteristic shape at low speed which is also result of the heavy sa~uration. The no-load current is greater than the· current at rated torque. One can easily see that the main flux, Elf, drops much faster when the motor is loaded at low frequencies than at rated frequency. The voltage drop over the stator resistance is relatively greater. This comoensates the influence of the variable saturation and the' resultant stator current at very low frequencies will be relatively independent of the load. As the maximum available voltage often is equal to the rated motor voltage, the motor operates in the field weakening region for speeds above the rated value. With this voltage-frequency relationship the instantaneous overload capability is the same at all speeds, i.e. the natural maximum for an asynchronous motor. Due to the strong rise of current at low speed the motor should then only run at these speeds for short intervals even under easy load conditions. The overload capability is of 719
H. Raphael, A. Skjellnes
course limited by the temperature rise in the motor. As a compromise between the simple voltage-frequency relationship (20) and the found function that gives constant pull-out torque, other functions of the form of equation (19) might be cr.osen. These will result in weaker torque characteristics but better current conditions. However it is obvious that such a simple function as (19) cannot in general lead to optimal performance for the motor drive. Load-Dependent Voltage Function. - The ideal way of running a machine is when the main flux is kept constant at the value optimized in the construction, regardless of input frequency and load conditions. Such an ideal operation is possible if the main flux either is directly measured or is found by help of motor current and stator parameters as shown in the next section. The computer programme is finally used to find the machine characteristics under the condition of constant flux. As figure 5 indicates, the same characteristics are obtained at all frequencies. Remarkable is the much higher pullout torque. This is due to the fact that the influence of the stator impedance is now eliminated. The voltage-load
T . I,
r,;' r,;;
4
U, U'N
/
/
/ I
i
.... .( ....
J n nsN
Figure 5:
720
Motor characterics at constant flux
H. Raphael, A. Skjellnes
dependence, which is considerabily more perceptible at low frequencies than at high, is clearly seen. The diagram indicates that if the flux is kept constant in an asynchronous machine, it will at all speeds below the field weakening region have the same capability of overload as a dc-machine. The current and heat conditions will also be approximately the same. In the field weakening region however, the pullout torque drops inversely as the speed to the second power, while the dc-motor shows only a fall inversely as the speed. FEEDBACK SYSTEM WITH FLUX REGULATOR If a load-independent function of the type equation (19) is used, either an open frequency-adjusted system or a closed loop regulation is applicable. The control equipment will then be of quite ordinary and well-known type with which this paper does not deal. The more ideal operation mode of constant flux necessitates a closed loop system where the flux either can be measured directly or calculated by help of current and voltage measurements and known motor parameters. This section treats both these two possibilities. Feedback System with Flux Density Measurement The principle of the flux regulator system which forms the inner loop of the motor drive is shown on Figure 6. The semiconductor power converter SPC must have independent voltage and frequency control. The main problem concerning
OUTER LOOP
f SPC
SPC: SEMICONDUCTOR POWER CONVERTER
U,
ASM: ASYNCHRONOUS MOTOR MS: MAGNETIC SENSORS SC: SIGNAL CONVERTER FR: FLUX REGULATOR
FR +
FLUX REFERENCE
Figure 6:
Block diagram of flux regulator loop 721
H. Raphael, A. Skjellnes
the inner loop is the proper function of the magnetic sensors MS and the corresponding signal converter SC. If induction coils are used, and are laid in stator slots with one pole width separation, a voltage proportional to the derivative of the flux is measured. By sensing in all three phases, then integrating and rectifying, a dcsignal suitable for the regulator is obtained. The integration gives a delay inversely proportional to the frequency. Therefore the regulator has to be very slow to achieve stability at low speed. This system has been tested on an experimental set-up for one 4 kW motor and worked satisfactorily for frequencies down to about 10 Hz. If a magnet sensing device such as a Hall probe is used for the flux measurement, the time delay is omitted and the flux regulator will most probably work well on very low frequencies too. ' The use of some type of magnetic sensors necessitates an option to the normal standard motor. This option might either be furnished by the manufacturer of the motor or the motor must later on be opened. Anyhow, it means additional costs that will make the asynchronous motor drive a little less advantageous in the competition with the dcmotor drive. Analog Calculation of Induced Voltage The instantaneous voltage balance in each stator phase may be expressed by the simple equation ( 21)
When the instantaneous phase voltage and current are
CONV
3 PHASE B PHASE C
---------
JJ2 PHASES
v.
Figure 7: Analog calculation of induced voltage
722
H. Raphael, A. Skjellnes
measured, then converted to adequate analog voltages Vu and VI' an ac cignal VEA proportional to the induced voltage can be found as indicated in figure 7 (left hand side) . A converter that performes a 3-phase to 2~phase transformation is the heart of the calculator. The two signals Vd and Ve are 90 0 out of phase, thus, by help of Pythagoras' law, giving an output.dc-voltage VEO as is indicated on the right hand side of figure 7. At each operating frequency the calculated value VED is proportional to the induced voltage. By dividing this value by a voltage proportional to the frequency, a signal directly proportional to the flux is produced which is suitable as feedback to the flux regulator. The analog system needs multipliers and many operational amplifiers to replace the magnetic sensors. Even if these semiconductor devices are cheap, some engineering and constructional work is needed, and the stator parameters must be known. Both measuring the phase currents and performing the division with the voltage proportional to the frequency, provides problems at very low frequencies. A general recommendation in the choice between the two outlined alternatives is therefore not possible. BRAKING
LCI
~---------,
r----J I
I
I
IL
:
J(}- ~-----,
' I •
~
I
eR:
:
~
I
Line
SO/60Hz
ICI
i
Q
I
b
I
I
o
I
o
.J...
LCR: LINE COMMUTATED RECTIFIER 1CI INDEPENDENT COMMUTATING INVERTER
I I I
6
I
~
...J....
LCI: LINE COMMUTATED INVERTER ASM ASYNCHRONOUS MOTOR
Figure 8: Principle of converter fed motor drive with braking alternatives 723
H. Raphael, A. Skjellnes
The basic problem in all braking is the transfer of the kinetic energy of the moving parts into other forms of energy. A general block diagram of a converter fed asynchronous motor drive is shown in figure 8. The parts necessary for the normal motor operation are fully outlined. It must be remembered that the line commutated rectifier LCR only permits current flow in one direction, from the line to de stage of the converter. The independent commutating inverter ICI, however, has current paths in both directions due to the incorporated free-wheeling diodes. Braking may be performed by one of the following methods. Plugging. - This method is commonly used when braking mains fed motors. It is well known that braking with counter-rotating field gives a very bad current to torque relationship. That means the converter has to be heavily oversized if reasonable braking torque is needed. Both the kinetic energy and the energy fed from the line will be dissipated as heat in the motor during braking, thus gi ving a considerable temperature rise. Plugging is eVidently not a desirable braking method for converter fed motor drives. Dc-braking. - This is the extremal case of the counterrotating field method. The rotational energy must still be converted to heat in the rotor, but the energy fed from the inverter will be much lower than in the case of plugging. For squirrel cage motors the braking torque will be highly dependent upon rotor speed. Thus dc-braking is hardly applicable to this type of drive. Dynamic braking with resistors. - The machine is running as a generator with low negative slip. Magnetizing energy must be fed from the inverter. The kinetic energy is dissipated outside the motor in resistors either on the ac side, point a figure 8, or on the de side, point b. Additional logic functions are necessary to operate the switches which may of course be of solid state type. A semiconductor switch is easier to realize on the ac side, but in this case the maximum braking current and hence the available torque will be proportional to Ul/RB1. The inverter will therefore be heavily loaded during light braking at high speed, which may lead to commutation failure. Apart from this fact, braking on resistors provides a good technical and economic solution. Regenerative braking. - If the inverter works with a fixed dc voltage, and an adequate battery is available, regenerative braking is possible at no extra cost. See figure 8. During braking the machine feeds the energy into the battery, and at motor operation the line commutated rectifier works as a battery charger too. The motor drive will also be uninterruptible because the battery automatically replaces the rectifier if the line falls out. The braking energy can also be fed back to the mains supply,
724
H. Raphael, A. Skjellnes
but this necessitates an additional line commutated inverter LeI, as indicated with broken lines on figure 8. This solution makes very smooth and accurate control of the braking torque possible at all speeds, but leads to higher system costs. However, the price of a line commutated inverter working on the mains supply is moderate compared to an independant commutating inverter that needs special commutating circuits and fast thyristors. FOUR QUADRANT DRIVE In the previous section different braking methods have been discussed. The phase sequence in the inverter and thus the direction of rotation can easily be changed at signal level. The problems for the control equipment is at all operational modes to limit the stator current and to control the additional braking equipment in such a way that smooth and reliable operation is achieved. At very low speed stable operation is difficult. This is due to the nature of the asynchronous machine and the problems are amplified if small input signals have to be used as divisors or if complex signal treatment is necessary. In addition, harmonic contents of the inverter output will give torque pulsations. If the motor drive is to operate at very low speed the inverter should be of the pulse width Qodulation type.
L....-
nia -+-.:.=......
SR: SPEED REGULATOR FSU: FIRING & SEQUENCE UNIT SPC: SEMICONDUCTOR POWER CONVERTER
...
ASM: ASYNCHRONOUS MOTOR MS: MAGNETIC SENSOR SC: SIGNAL CONVERTER FR: FLUX REGULATOR
Figure 9: Principle of speed regulator
725
H. Raphael, A. Skjellnes
In a four quadrant drive the sensing of the speed direction is necessary, which is automatically taken care of by a dctachometer. Due to the nature of the asynchronous machine separate current sensing and ~egulation can be omitted if the rotor frequency is limited to the maximum permissible value and the machine operates at constant flux. This can be obtained by a positive feedbach system shown on figure 9 which operates as follows: The speed reference is as normally compared with the real speed, and an error signal is fed to the speed regulator SR. However,the output of this regulator is strictly limited to a value proportional to the allowable speed drop determined by the maximum rotor frequency. The output signal of the regulator is added to the ceal speed signal thus giving the synchronous speed signal n~. The firing and sequence unit FSU determines the rotat~onal direction by the sign of ns and controls the output frequency of the semiconductor power converter which is proportional to the input value Insl. When a speed drop is commanded by the regulator the output frequency of SPC falls, making ASM working as a generator. If regenerative braking is provided there are no problems, the energy is automatically fed into the battery or by the antiparallel inverter to the mains supply. If braking resistors are used, a logic circuit must operate the switches. This can be done by sensing on the internal dcvoltage of the converter, point b of figure 8. A voltage rise indicates generator mode, and consequently the switches must be operated. While both positive and negative slip are limited by the speed difference ~n the stator current is under control in generator mode also. The voltage control is done by the flux regulator as previously explained. CONCLUSIONS It is possible to realize an asynchronous motor variable speed drive with performance compatible to a qualified dc-motor drive. Options to the standard are necessary, or motor parameters must be well known and complex regulator functions must be provided. The advantages of the squirrelcage motor is the ruggedness, free from brushes and commutators, and the much lower price compared with the dc motor. On the other side the converter and regulator system is much more complex and expensive than the rectifier for the dc-motor. Calculations indicate that the asynchronous motor drives are able to compete from the range of about ,0 kW and up when the development costs are written off. In addition they often provide system advantages such as less maintenance, less space and no positioning problems.
726
H. Raphael, A. Skjellnes
References /1/
Schonnung, A & Stemmler, H.: Geregelter Drehstrom Umkehrantrieb mit gesteuertem Umrichter nach dem Unterschwingungsverfahren. Brown Boveri Mitteilungen. Bd.51 Nr.8/9 1964.
/2/
Lipo, Th. & Krause, P.C.: Stability Analysis of a Rect1fier Inverter Induction Motor Drive. IEEE Trans. vol. PAS 88 Nr.l 1969.
/3/
Lipo Th. & Nelson, R.: Stability Analysis of a Symmetrical Induction Machine. IEEE Trans. vol. PAS 88 Nr.ll 1969.
/4/
Raphael, H., Fylling, H.B., Skjellnes, A.E., Undeland, T.M.: Str~mretterstyrte Asynkronmotorer. EFl teknisk rapport 1742 1973 (Norwegian).
/5/
Murphy. J.: Thyristor Control of A.C.Motors. Pergamon Press 1973 P. 104/122.
/6/
Barton, T.H. & Stefanovic, V.R.: Static Torque Characteristics of Induction Motor with a Variable Frequency Supply. Conference Paper C 73 199-7. IEEE 1973 Winter Power Meeting, New York Jan.28. - Feb.2.1973
727
INVERTER CONTROLLED CHARACTERISTICS OF VARIABLE FREQUENCY INDUCTION MOTORS H. Raphael is leader of the Electrical Machinery and Power Electronics Group of the Electrical Power Engineering Laboratories, the Norwegian Institute of Technology, Trondheim, Norway. A. Skjellnes is senior research engineer with the Norwegian Research Institute for Electricity Supply, Trondheim, Norway. SUMMARY This paper deals with the theory of asynchronous motors running at variable frequency. Motor characteristics are found by the help of a computer programme where higher harmonics are neglected, but the influence of saturation is considered. It is shown that it is impossible to get optimum performance if the voltage is a function of the frequency only. A regulator which controls the voltage as a function of both frequency and load to keep constant flux gives torque and current characteristics equal to those of a dc-machine. This paper presents two alternatives for such a regulator. Braking methods are discussed and a regulator principle for a four-quadrant drive without current sensing is shown. ZUSAMMENFASSUNG Die Theori des Kafiglaufermotors, der von einer Quelle mit variabler Frequenz und Spannung gespeisst wird, soll erlautert werden. Die Kennlinien werden mit Hilfe eines Rechenprogramms gefunden. Die Oberharmonischen der Grundwelle werden in diesem Programm vernachlassigt, aber der Einfluss der magnetischen Sattigung wird berucksichtigt. Es wird gezeigt, dass optimaler Motorbetrieb nicht moglich ist,wenn die Spannung nur als eine Funktion der Frequenz angepasst wird. Die Kennlinien des Kafiglaufermotors mit konstanter Durchflutung werden mit denen eines Gleichstrommotors verglichen. Zwei prinzippielle Losungen der Durchflutungsregulierung werden beschrieben. Verschiedene Verfahren des Bremsens werden diskutiert, und ein Aufbau einer Umkehrregelung ohne Strommessung wird gezeigt.
THEORY OF MOTOR OPERATION The aim of this section is to find an optimal relationship between voltage and frequency for an inverter-fed squirrel cage motor drive. The inverter is considered as a general energy-converting component, and this paper does not deal with inverter principles or construction. Therefore the analysis supposes sinusoidal voltages. The harmonic components, which are thus neglected do not alter the principal results. They give rise to some additional losses
713
H. Raphael, A. Skjellnes
and except at very low speed quite negligible torque components. It will be shown how the effect of saturation is taken into account. Under the mentioned condition, the conventional equivalent circuit, figure 1, is applicable, where all circuit parameters are referred to the stator. In a physical machine they are dependent upon voltage, current, slip and frequency. The relations are complex, and in order to achieve the goal of the analysis by tolerable efforts the following simplifications are required: ~ ~
=.
1.
Iron losses are neglected, that is
2.
The stator and rotor parameters are kept constant and equal to their values at rated voltage, frequency and load.
The iron losses may be seperat~ly calculated when the flux density is determined. The roughest assumption is to keep the rotor parameters L2 and R2 constant. The analysis will thus give the most reliable results for the speed region near synchronism. Fortunately this region is of the greatest interest for the inverter fed motor drive. Magnetizing inductance LM is dependent upon the relationship between the induced voltage E and the stator frequency W\. The following phasor equations give the electrical balance per phase: -+
Stator voltage:
E+
U l
Rl'!l + jWILl'!l
( 1)
-+
Magnetizing current: Rotor current:
-+
I
2
!M
E
(2)
jwlLM
!l - ! M
( 3)
E
..
STATOR
•
ROTOR
Figure 1: Asynchronous motor, single phase equivalent 714
H. Raphael, A. Skjellnes
Induced voltage:
( 4)
Total power and torque is given by the following classical equations: Airgap power:
Po
Rotor losses:
P
2
Mechanical power: Mechanical torque:
P
R 2 2 m'S-.I 2
( 5)
m.R I 2 2 2
(6)
L
To
I-s 2 m·---·R 2 I 2 s R I Po 2 2 m·p·---s,w W l s
(7)
(8)
where m and p are number of phases and pairs of poles respectively. Ws is mechanical synchronous speed and the slip is defined by
s = On the basis of these equations an iterative computer programme is developed which determines the motor characteristics. Input values are stator voltage and frequency. In the following it is first shown how the motor parameters may be found. Then it is demonstrated how the programme is used to find the optimal voltage-frequency relationship, and the consequences of this relationship to the regulator system are outlined. Determination of Motor Parameters Measurements are performed at standstill, at rated voltage, frequency and load and at no load with variable voltage. The latter is used to find the saturation curve for the main flux path. Standstill Measurement. - With locked rotor a reduced stator voltage at rated frequency, which gives about rated stator current, is used. Input voltage, current and power as well as the torque are measured. At standstill and low voltage the magnetizing current is neglected. The following equations determine the stator parameters. Suffix S indicates standstill and N rated value. Rotor losses:
P
WIN
2S
T .--S P
(9)
715
H. Raphael, A. Skjellnes
Losses in stator: Stator resistance:
Pl,loss = m.Rl1 l R
2
(10)
Pl,lOSS l
m.1lQ ,.
Reactive power: Q =v(m.Uls1 1S ) 1S = m·w
lN
(11)
2 2
- P 1S
(L +L ) .1 2 l 16
2
2r
( 12)
Assuming the same leakage inductance in stator and rotor, equation (12) gives Stator inductance:
L
l
(13)
The values given by (11) and (13) are constant parameters in the computer programme. No-Load Measurement. - The motor is running without any load on a rated frequency variable voltage supply. The voltage is changed from about half to about 1.5 of the rated voltage, and corresponding values of voltage U10 and current 110 are fed into the computer. By help of equations (1), (2) and the values found from (11) and (13) the magnetizing inductance is known, and corresponding values of the ratio E/wl and LM are stored. The saturation of the motor is thus taken care of by this array. Rated Load Measurement. - The machine is then connected to a rated frequency, rated voltage supply, and loaded to rated current. The corresponding values of input power, speed and torque are measured. The rotor parameters can then be calculated. First stator current phasor is found relative to the voltage: (14)
(15)
Then equation (1) is used to find the rated induced voltage EN and (2) gives together with the stored array from the no-load measurement the magnetizing current. Finally equation (3) determines the rated rotor current. The rotor impedance at rated load is (referred to stator)
716
H. Raphael, A. Skjellnes
(16)
and the rotor parameters are (17)
(18)
These values (17) and (18) are also constant parameters in the computer programme. As already mentioned this is physically the worst approximation, but fortunately an inverter fed asynchronous machine normally operates near rated rotor conditions. It has been evaluated that this method of determining the motor parameters gives an uncertainty of about 10%. This accuracy is sufficient to give principally correct results. Machine Characteristics at Variable Frequency It is well known that the available torque of an inverter fed squirrel cage motor drops considerabily at low speed if the voltage is kept directly proportional to the frequency. If constant torque is wanted over the entire speed range, or perhaps even higher torque at low speed, which is the case in traction applications etc., other frequencyvoltage relationships have to be used. To obtain the desired performance two approaches are tried. One assumes that the voltage function is independent of the load, and the second approach is to keep the main flux constant for all load conditions. Voltage Independant of Load. - An easy function to realize in a regulator is the u, linear relationship:
U 1
=
U +k 0
!f
N
U 1N
(19)
out of which the proportional equation (20) Figure 2: Voltage-frequency functions
is already mentioned.
717
H. Raphael, A. Skjellnes
Figure 2 shows both this function (20) and the more general form (19). As indicated the direct proportional equation leads to low available torque at low speeds. Motor characteristics have been computed for a 4 kW, 230 V, 50 Hz motor assuming the voltage following equation 20. It is clearly seen in figure 3 how the torque characteristic drops at low speeds. This simple voltage-frequency relationship is therefore only applicable when the motor works under easy starting conditions and rated torque is not required at low speeds. It is evident that better performance might be obtained with the voltage following equation 19. The function starts with a constant voltage Uo and the slope k is chosen to give rated voltage at rated frequency. The computer programme is used to determine these constants in such a way that approximately the same pull-out torque is available at all speeds. At each frequency the calculation starts with a voltage according to equation (20), finds the pull-out torque and adjusts the input voltage until the pull out torque has the same value as at rated frequency and rated voltage. The results are shown in figure 4. It is seen that even though the additional constant voltage is
IU,=!- U,NI fN
2
I FIELD
"\ \
WEAKENING
i
•
/
\_L
--- .J ---
-1
-2
T 11 Elf
Figure 3: Motor characteristics with voltage proportional to frequency 718
H. Raphael, A. Skjellnes
L . .!L TN' l'N
ECN) U'N
T
3
.
i / f
+
f O. SSt) U'N N
U,= U'N WEAKENING.
\
I
""-'2
U,=(O.12
I FIELD
\
'-.
.... \
I /
/
\
\
/
/
-I- - -_ 0 -1
f= O.1fN
f=O.5fN
-2
T
-3
Figure 4:
I
/
I, Elf
Motor characteristics at constant pull-out torque
relatively small, O.IZ·U N , the current characteristic at low speed rises considerably. This is due to the greater induction, Elf, which consecutively leads to still heavier saturation. It is worth noting the change of the current characteristic shape at low speed which is also result of the heavy sa~uration. The no-load current is greater than the· current at rated torque. One can easily see that the main flux, Elf, drops much faster when the motor is loaded at low frequencies than at rated frequency. The voltage drop over the stator resistance is relatively greater. This comoensates the influence of the variable saturation and the' resultant stator current at very low frequencies will be relatively independent of the load. As the maximum available voltage often is equal to the rated motor voltage, the motor operates in the field weakening region for speeds above the rated value. With this voltage-frequency relationship the instantaneous overload capability is the same at all speeds, i.e. the natural maximum for an asynchronous motor. Due to the strong rise of current at low speed the motor should then only run at these speeds for short intervals even under easy load conditions. The overload capability is of 719
H. Raphael, A. Skjellnes
course limited by the temperature rise in the motor. As a compromise between the simple voltage-frequency relationship (20) and the found function that gives constant pull-out torque, other functions of the form of equation (19) might be cr.osen. These will result in weaker torque characteristics but better current conditions. However it is obvious that such a simple function as (19) cannot in general lead to optimal performance for the motor drive. Load-Dependent Voltage Function. - The ideal way of running a machine is when the main flux is kept constant at the value optimized in the construction, regardless of input frequency and load conditions. Such an ideal operation is possible if the main flux either is directly measured or is found by help of motor current and stator parameters as shown in the next section. The computer programme is finally used to find the machine characteristics under the condition of constant flux. As figure 5 indicates, the same characteristics are obtained at all frequencies. Remarkable is the much higher pullout torque. This is due to the fact that the influence of the stator impedance is now eliminated. The voltage-load
T . I,
r,;' r,;;
4
U, U'N
/
/
/ I
i
.... .( ....
J n nsN
Figure 5:
720
Motor characterics at constant flux
H. Raphael, A. Skjellnes
dependence, which is considerabily more perceptible at low frequencies than at high, is clearly seen. The diagram indicates that if the flux is kept constant in an asynchronous machine, it will at all speeds below the field weakening region have the same capability of overload as a dc-machine. The current and heat conditions will also be approximately the same. In the field weakening region however, the pullout torque drops inversely as the speed to the second power, while the dc-motor shows only a fall inversely as the speed. FEEDBACK SYSTEM WITH FLUX REGULATOR If a load-independent function of the type equation (19) is used, either an open frequency-adjusted system or a closed loop regulation is applicable. The control equipment will then be of quite ordinary and well-known type with which this paper does not deal. The more ideal operation mode of constant flux necessitates a closed loop system where the flux either can be measured directly or calculated by help of current and voltage measurements and known motor parameters. This section treats both these two possibilities. Feedback System with Flux Density Measurement The principle of the flux regulator system which forms the inner loop of the motor drive is shown on Figure 6. The semiconductor power converter SPC must have independent voltage and frequency control. The main problem concerning
OUTER LOOP
f SPC
SPC: SEMICONDUCTOR POWER CONVERTER
U,
ASM: ASYNCHRONOUS MOTOR MS: MAGNETIC SENSORS SC: SIGNAL CONVERTER FR: FLUX REGULATOR
FR +
FLUX REFERENCE
Figure 6:
Block diagram of flux regulator loop 721
H. Raphael, A. Skjellnes
the inner loop is the proper function of the magnetic sensors MS and the corresponding signal converter SC. If induction coils are used, and are laid in stator slots with one pole width separation, a voltage proportional to the derivative of the flux is measured. By sensing in all three phases, then integrating and rectifying, a dcsignal suitable for the regulator is obtained. The integration gives a delay inversely proportional to the frequency. Therefore the regulator has to be very slow to achieve stability at low speed. This system has been tested on an experimental set-up for one 4 kW motor and worked satisfactorily for frequencies down to about 10 Hz. If a magnet sensing device such as a Hall probe is used for the flux measurement, the time delay is omitted and the flux regulator will most probably work well on very low frequencies too. ' The use of some type of magnetic sensors necessitates an option to the normal standard motor. This option might either be furnished by the manufacturer of the motor or the motor must later on be opened. Anyhow, it means additional costs that will make the asynchronous motor drive a little less advantageous in the competition with the dcmotor drive. Analog Calculation of Induced Voltage The instantaneous voltage balance in each stator phase may be expressed by the simple equation ( 21)
When the instantaneous phase voltage and current are
CONV
3 PHASE B PHASE C
---------
JJ2 PHASES
v.
Figure 7: Analog calculation of induced voltage
722
H. Raphael, A. Skjellnes
measured, then converted to adequate analog voltages Vu and VI' an ac cignal VEA proportional to the induced voltage can be found as indicated in figure 7 (left hand side) . A converter that performes a 3-phase to 2~phase transformation is the heart of the calculator. The two signals Vd and Ve are 90 0 out of phase, thus, by help of Pythagoras' law, giving an output.dc-voltage VEO as is indicated on the right hand side of figure 7. At each operating frequency the calculated value VED is proportional to the induced voltage. By dividing this value by a voltage proportional to the frequency, a signal directly proportional to the flux is produced which is suitable as feedback to the flux regulator. The analog system needs multipliers and many operational amplifiers to replace the magnetic sensors. Even if these semiconductor devices are cheap, some engineering and constructional work is needed, and the stator parameters must be known. Both measuring the phase currents and performing the division with the voltage proportional to the frequency, provides problems at very low frequencies. A general recommendation in the choice between the two outlined alternatives is therefore not possible. BRAKING
LCI
~---------,
r----J I
I
I
IL
:
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' I •
~
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eR:
:
~
I
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SO/60Hz
ICI
i
Q
I
b
I
I
o
I
o
.J...
LCR: LINE COMMUTATED RECTIFIER 1CI INDEPENDENT COMMUTATING INVERTER
I I I
6
I
~
...J....
LCI: LINE COMMUTATED INVERTER ASM ASYNCHRONOUS MOTOR
Figure 8: Principle of converter fed motor drive with braking alternatives 723
H. Raphael, A. Skjellnes
The basic problem in all braking is the transfer of the kinetic energy of the moving parts into other forms of energy. A general block diagram of a converter fed asynchronous motor drive is shown in figure 8. The parts necessary for the normal motor operation are fully outlined. It must be remembered that the line commutated rectifier LCR only permits current flow in one direction, from the line to de stage of the converter. The independent commutating inverter ICI, however, has current paths in both directions due to the incorporated free-wheeling diodes. Braking may be performed by one of the following methods. Plugging. - This method is commonly used when braking mains fed motors. It is well known that braking with counter-rotating field gives a very bad current to torque relationship. That means the converter has to be heavily oversized if reasonable braking torque is needed. Both the kinetic energy and the energy fed from the line will be dissipated as heat in the motor during braking, thus gi ving a considerable temperature rise. Plugging is eVidently not a desirable braking method for converter fed motor drives. Dc-braking. - This is the extremal case of the counterrotating field method. The rotational energy must still be converted to heat in the rotor, but the energy fed from the inverter will be much lower than in the case of plugging. For squirrel cage motors the braking torque will be highly dependent upon rotor speed. Thus dc-braking is hardly applicable to this type of drive. Dynamic braking with resistors. - The machine is running as a generator with low negative slip. Magnetizing energy must be fed from the inverter. The kinetic energy is dissipated outside the motor in resistors either on the ac side, point a figure 8, or on the de side, point b. Additional logic functions are necessary to operate the switches which may of course be of solid state type. A semiconductor switch is easier to realize on the ac side, but in this case the maximum braking current and hence the available torque will be proportional to Ul/RB1. The inverter will therefore be heavily loaded during light braking at high speed, which may lead to commutation failure. Apart from this fact, braking on resistors provides a good technical and economic solution. Regenerative braking. - If the inverter works with a fixed dc voltage, and an adequate battery is available, regenerative braking is possible at no extra cost. See figure 8. During braking the machine feeds the energy into the battery, and at motor operation the line commutated rectifier works as a battery charger too. The motor drive will also be uninterruptible because the battery automatically replaces the rectifier if the line falls out. The braking energy can also be fed back to the mains supply,
724
H. Raphael, A. Skjellnes
but this necessitates an additional line commutated inverter LeI, as indicated with broken lines on figure 8. This solution makes very smooth and accurate control of the braking torque possible at all speeds, but leads to higher system costs. However, the price of a line commutated inverter working on the mains supply is moderate compared to an independant commutating inverter that needs special commutating circuits and fast thyristors. FOUR QUADRANT DRIVE In the previous section different braking methods have been discussed. The phase sequence in the inverter and thus the direction of rotation can easily be changed at signal level. The problems for the control equipment is at all operational modes to limit the stator current and to control the additional braking equipment in such a way that smooth and reliable operation is achieved. At very low speed stable operation is difficult. This is due to the nature of the asynchronous machine and the problems are amplified if small input signals have to be used as divisors or if complex signal treatment is necessary. In addition, harmonic contents of the inverter output will give torque pulsations. If the motor drive is to operate at very low speed the inverter should be of the pulse width Qodulation type.
L....-
nia -+-.:.=......
SR: SPEED REGULATOR FSU: FIRING & SEQUENCE UNIT SPC: SEMICONDUCTOR POWER CONVERTER
...
ASM: ASYNCHRONOUS MOTOR MS: MAGNETIC SENSOR SC: SIGNAL CONVERTER FR: FLUX REGULATOR
Figure 9: Principle of speed regulator
725
H. Raphael, A. Skjellnes
In a four quadrant drive the sensing of the speed direction is necessary, which is automatically taken care of by a dctachometer. Due to the nature of the asynchronous machine separate current sensing and ~egulation can be omitted if the rotor frequency is limited to the maximum permissible value and the machine operates at constant flux. This can be obtained by a positive feedbach system shown on figure 9 which operates as follows: The speed reference is as normally compared with the real speed, and an error signal is fed to the speed regulator SR. However,the output of this regulator is strictly limited to a value proportional to the allowable speed drop determined by the maximum rotor frequency. The output signal of the regulator is added to the ceal speed signal thus giving the synchronous speed signal n~. The firing and sequence unit FSU determines the rotat~onal direction by the sign of ns and controls the output frequency of the semiconductor power converter which is proportional to the input value Insl. When a speed drop is commanded by the regulator the output frequency of SPC falls, making ASM working as a generator. If regenerative braking is provided there are no problems, the energy is automatically fed into the battery or by the antiparallel inverter to the mains supply. If braking resistors are used, a logic circuit must operate the switches. This can be done by sensing on the internal dcvoltage of the converter, point b of figure 8. A voltage rise indicates generator mode, and consequently the switches must be operated. While both positive and negative slip are limited by the speed difference ~n the stator current is under control in generator mode also. The voltage control is done by the flux regulator as previously explained. CONCLUSIONS It is possible to realize an asynchronous motor variable speed drive with performance compatible to a qualified dc-motor drive. Options to the standard are necessary, or motor parameters must be well known and complex regulator functions must be provided. The advantages of the squirrelcage motor is the ruggedness, free from brushes and commutators, and the much lower price compared with the dc motor. On the other side the converter and regulator system is much more complex and expensive than the rectifier for the dc-motor. Calculations indicate that the asynchronous motor drives are able to compete from the range of about ,0 kW and up when the development costs are written off. In addition they often provide system advantages such as less maintenance, less space and no positioning problems.
726
H. Raphael, A. Skjellnes
References /1/
Schonnung, A & Stemmler, H.: Geregelter Drehstrom Umkehrantrieb mit gesteuertem Umrichter nach dem Unterschwingungsverfahren. Brown Boveri Mitteilungen. Bd.51 Nr.8/9 1964.
/2/
Lipo, Th. & Krause, P.C.: Stability Analysis of a Rect1fier Inverter Induction Motor Drive. IEEE Trans. vol. PAS 88 Nr.l 1969.
/3/
Lipo Th. & Nelson, R.: Stability Analysis of a Symmetrical Induction Machine. IEEE Trans. vol. PAS 88 Nr.ll 1969.
/4/
Raphael, H., Fylling, H.B., Skjellnes, A.E., Undeland, T.M.: Str~mretterstyrte Asynkronmotorer. EFl teknisk rapport 1742 1973 (Norwegian).
/5/
Murphy. J.: Thyristor Control of A.C.Motors. Pergamon Press 1973 P. 104/122.
/6/
Barton, T.H. & Stefanovic, V.R.: Static Torque Characteristics of Induction Motor with a Variable Frequency Supply. Conference Paper C 73 199-7. IEEE 1973 Winter Power Meeting, New York Jan.28. - Feb.2.1973
727