DE4032056C1 - Determining slip frequency for control of asynchronous motor - calculating cross product of stator current voltage and frequency vectors to increase stator frequency bandwidth towards DC - Google Patents

Abstract

The asynchronous motor slip frequency is derived from the internal torque divided by the square of the rotor flux. The inner torque is determined as a scalar product of the stator current and the rotor EMF vectors, multiplied by the stator frequency. The square of the rotor flux multiplied by the stator frequency is defined by a given differential equation in terms of the rotor time constant, the rotor reactance and the differential quotient of the stator frequency. USE - For induction motor drive with high dynamic performance.

Description

The invention relates to a method according to the preamble of the claim 1 and an arrangement for performing the method.

Drives with converter-fed asynchronous motors (ASM) for smaller ones Services and / or with not very high demands on the dynamic Behaviors are usually carried out without a speed sensor. The known method used for this is the control of the stator voltage depending on the stator frequency (U / f characteristic) with a constant or current-dependent basic value to compensate for the ohmic voltage drop. Using this method comes in rough approximation the stator flux was kept constant.

Much better dynamic properties can be achieved with a constant rotor flux. At constant rotor flux, torque and Flow formation decoupled in the ASM. However, this requires the stator voltage (or the stator current) as a function of magnitude and phase angle a constant component, the stator frequency and the slip frequency be specified. If the stator frequency is additionally determined by a Control loop formed with subordinate slip frequency control, receives you have a dynamically good and tilt-proof drive.

Knowing the slip frequency is therefore reliable Drive control indispensable.

The method explained has become known through the publication based on the preamble of claims 1 and 5: "CONTROL METHODS FOR GOOD DYNAMIC PERFORMANCE INDUCTION MOTOR DRIVES BASED ON CURRENT AND VOLTAGE AS MEASURED QUANTITIES " by R. Joetten, G. Maeder, May 1982 Orlando / Florida / USA.

The slip frequency determination method described therein is based on the known relationship: "The slip frequency is proportional to the internal torque divided by the square of the rotor flux linkage, Eq. (8, 3) ".

Provided that the rotor flux linkage is kept constant will or only that slowly changes, e.g. B. in field weakening, can the components of the rotor flux chaining by components of the Rotor EMF can be expressed, Eq. (10, 12). This makes integrators avoided with their drift problems.

From this the Eq. (11) the slip frequency ω₂:

Fig. 4 of the publication shows an equivalent circuit for calculating the slip frequency.

The Eq. However, (11) has two main practical implications Disadvantage:

• 1. Since the components of the rotor EMF e _{2 α} , e _{2 β} are proportional to the stator frequency ω ₁ , the numerator and denominator of the quotient change quadratically with the stator frequency. In the case of a large stator frequency range, this leads to very large number ranges, which at least can no longer be processed with the required accuracy by an analog divider.
• 2. The components of the rotor EMF are according to Eq. (12) formed by subtracting the ohmic voltage drop R ₁ i _{1 α, β} and the inductive (1 / K ₁₁ ) di _{1 α, β} / dt from the stator terminal voltage U _{1 α, β} . The stator resistor R₁ in the arithmetic circuit Fig. 4 is a constant parameter, during which is thermally variable in the ASM. The measured rotor EMF will therefore deviate from the true rotor EMF in the ASM. This leads to an increasing error when the stator frequency is reduced. There is therefore a stator frequency of +/- 3 in the publication cited on page 400 bottom left. . . 5% of their nominal value as the lower limit of the usability of the method.
  Additional measures are proposed on page 402 of the publication for the area between this limit and the stator frequency zero, which is traversed when starting, stopping and reversing. The output of the arithmetic circuit is then suppressed in the frequency range around zero and the adjustment speed of the stator frequency leaves this range. When reversing through the frequency range around zero, the adjustment speed can be kept at the value shortly before reaching this range in a refined method.

It is the aim of the present invention, with a further development the procedure described in the cited publication to bring the lower limit of the stator frequency so close to zero that additional measures are not necessary and the drive in the entire stator frequency range with the exception of "zero" slip frequency is.

Behavior is used to explain the extended procedure the ASM by means of an equation system related to the stator-fixed α, β axes to which the measured values are assigned.

This is based on the statements of the books by: H. Kleinrath / converter-fed induction machines, Springer-Verlag 1980; H. Bühler / Introduction to the theory of regulated three-phase drives Volume 1 basics and Volume 2 applications. Birkhäuser-Verlag, 1st edition 1977

The sizes and constants are standardized. The x, y coordinate system rotating with stator angular frequency f _S ω _N is aligned with the space vector rotor flux linkage Ψ _R - Figure 1.

The ASM equations in the stator-fixed α, β system are:

From Eq. (20, 22, 23) will:

Here is the rotor EMF:

e _R = e _A + each _B

With the approach according to Figure 1 (diagram of the room convectors):

Can the left side of Eq. (25) into a transformative and a rotational voltage can be broken down:

With the neglect:

you get the one from the cited publication without integration known simple relationship of rotor flux chaining:

Ψ _R = Ψ _A + jΨ _B

This in Eq. (24) inserted results in the product "Stator frequency times internal torque":

f _s m _i ≈ i _a e _A + i _β e _B (33)

Now a relationship in the form f _s Ψ ² _R must also be found for the rotor flux chaining in order to cover the number range of the quotient in Eq. (45) restrict.

For this purpose, from Eq. (21, 23, 29) further relationships for the components the rotor flux linkage formed:

The change in the rotor flux linkage cannot here as in Eq. (30) are neglected because of T _R »1 / ω _N.

Eq. (34) expanded with Ψ _A , Eq. (35) expanded and added with Ψ _B results in:

With Gln. (31, 32) and the following approach:

Since it is an integration with fixed negative feedback, exist no drift problems.

From US-PS 44 42 393 the use of the dot product (inner product) according to equation (33) and the crossing product (outer product) according to equation (38) from voltage and current vector is known. However, the voltage vector is the undiminished stator voltage, which is why the results according to the description in column 6, above, of US Pat. No. 4,442,393 mean the active power W _y and reactive power W _x supplied to the motor. The determination of the slip frequency indicated in the abstract, line 4 from below in this patent specification gives no indication of the acquisition of universal functions. In contrast, according to the invention, universal functions are developed which can be processed in a simpler manner, analog or digital.

There is a simple solution for:

Ψ _R = const. ⇒ f _S Ψ² _R = x _R (i _α e _B - i _β e _A ) (39)

For the cross product of stator current and rotor EMF can also another relationship can be used because the ohmic Voltage drop cancels out, as the following calculation shows:

This allows a relationship to be established between the model Rotor flux chaining and the real rotor flux chaining in the ASM derive - the following labeling is introduced:

AMS size index 1
Model sizes index 2

Provided that the stator sizes of terminal voltage, current and frequency are measured with negligible errors, Eq. (40, 41, 42) and the approach according to Figure 1:

The cross product determined by the model and thus also f _S1 Ψ² _R2 is therefore independent of the thermally variable stator resistance r _S1 . An error only arises when the scattering reactance x _{σ 2} is mismatched, which, however, is frequency-proportional and therefore uncritical, at least in stationary operation.

This is a product "stator frequency independent of the stator resistance times square of the rotor flux linkage ".

The known relationship for the slip frequency f _{R can be} obtained from Eq. (34) with -Ψ _B , Eq. (35) expanded and added with Ψ _A , as well as Eq. (24) is inserted:

The slip frequency is thus determined without delay if according to Fig. 1 the control or regulation of the drive in a field-oriented manner Rotor flux is established.

The quotient on the right in Eq. (45) is calculated using Eq. (33, 38) realized by a motor model according to Figure 2, whereby a small decaying disturbance function in f _R / r _R can occur in transient operation at a very low stator frequency, since a possible rotor flux change in Eq. (30) was neglected.

However, it is the ohmic stator voltage drop in Eq. (33) for the inside Torque still included. To solve this problem, a further publication used: U. Baader / Dissertation: The direct self-regulation (DSR). A process for highly dynamic Control of induction machines. Progress reports VDI, row 21 No. 35 Düsseldorf / VDI publishing house 1988.

In the dissertation, chapter 4.1.1 "The engine model", is for one model the ASM for stator flow determination a method for synchronization described by model and ASM, especially in the dashed framed blocks in Figure 4.1.1 / 2.

The components of the stator current i ′ _{s α} , i ′ _{s β} calculated by the model are compared with the true ASM i _{s α} , i _{s β} and the difference is fed to PI controllers. The manipulated variables correct the model in such a way that the calculated components of the stator _{flux chaining} Ψ _{µα, β} follow the true values of the ASM almost true to the amount and phase.

This procedure should also be used here. For the product stator frequency f _S1 times model stator current i _S2 , Eq. (21, 23, 30) derived the relationship, under the prerequisite for the further calculation:

Ψ _R = const.

The difference is formed from the ASM stator current f _S1 i _S1 as the setpoint and the actual value f _S1 i _S2 and broken down into components, supplied to two similar controllers - as shown in Figure 2. Their common controlled system has proportional behavior in the low frequency range, so that integral controllers can be used. The x, y system, which is fixed in the rotating field, is selected for examining the control processes, and the space vectors transformed into this are identified by a superscript k. The manipulated variables of the controllers are combined into a complex room vector y ^k _R , which reads:

In stationary operation:

the I-controllers have a finite gain 1 / (ω _N T _i ) because they have to process alternating variables in the original stator-fixed α, β system. Now the integration time constant T _i can be made very small (which will be explained later), so that the control deviation is practically zero - now shown again in the stator-fixed α, β system:

ω _N T _i 0.01 ⇒ i _S1 - i _S2 → 0 (49)

From Eq. (46) the components of the model stator current are derived:

After insertion of the Eq. (33, 39, 42, 45, 49) two relationships are obtained, according to which the two regulated components e _A2 , e _{B2 of} the rotor EMF are determined:

In the case of corrective adaptation of the model to the ASM parameters:

x ′ _h2 = x ′ _h1

x _{σ 2} = x _{σ 1} ⇒ i _{α 1} e _{β 2} - i _{β 1} e _{α 2} = i _{α 1} e _B1 - i _{β 1} -e _A1

according to Eq. (44) the solutions result from Eq. (52, 53):

e _A2 ≈ e _A1 ; e _B2 ≈ e _B1 (54)

This means that the model components of the rotor EMF match the real ones ASM and the thermally variable ohmic stator voltage drop is eliminated at least with correct adjustment.

The control loops are dimensioned for idling f _R2 = 0. Then every regulator takes effect according to Eq. (50, 51) only derives its actual value from the voltage of the other channel, so that two integral controllers connected in series form the transfer function, which would mean instability. To remedy this, the manipulated variables of the controllers y _A , y _{B are} permanently coupled as shown in Figure 2, each weighted with the factor K. This dampens and stabilizes the control loops, like the transfer function G _o (S) of the open control loop in the x, y- System shows, where proportional behavior of the controlled system is required in the low frequency range:

f _R2 = 0

The crossing frequency ω _d and the associated phase angle ϕ _o (ω _d ) can be calculated from this:

In stationary operation, the closed control loop has the Transfer function:

In order to meet this condition, ω _N T ′ _i must be made very small, as already in Eq. (49) stipulates what according to Eq. (56) means a high crossover frequency. If the motor model is built in analog technology as shown in Figure 2, the high-frequency behavior of the operational amplifiers and function blocks used determines the permissible level of the crossover frequency, since they must not have any significant phase lag at this point.

The crossover frequency will surely be above 2π5 kHz, which according to the Condition of Eq. (49) a maximum stator frequency of greater than 130 Hz means.

The phase distance in Eq. (57) depends almost exclusively on the coupling factor K from. A value of K = 0.5 results in 48 ° phase separation, which in part can be consumed by the elements of the controlled system. With K = 0 is also the phase distance zero and the previously described arises Instability.

Description of the arrangement for performing the method

Figure 2 shows the structural diagram of the engine model. The measured values of the stator sizes u _{α 1} , u _{β 1} , i _{α 1} , i _{b 1} , f _{s1 are} applied on the input side. The model sizes determined from this are marked with index 2. It is assumed that measurement errors are negligible and calculation errors of the model are due to mismatching of its parameters.

The cross product of stator current and rotor EMF for flow calculation according to Eq. (38) is formed in the multipliers 3 a, 3 b and the summing point 6 with the voltages e _{α 2} , e _{β 2} according to the right side of Eq. (42). However, its left side can also be used with e * _A2 , e * _{B2 of} the summing points 5 a, 5 b, since the ohmic voltage drops from 4 a, 4 b cancel each other out.

In block 8 , according to Eq. (38) realized the last summand on the right. This only has a significant influence on rapid changes in the stator frequency, so that the rotor _{flux linkage} Ψ _{R2 is} set to the nominal value Ψ _R2N for its short duration in the operating mode without field weakening.

If the motor model is built in analog technology, only two-quadrant dividers are available commercially, whereby the denominator cannot be variable and, for reasons of accuracy, must not fall below a minimum value. Therefore, the product f _S1 R2² _R2 delivered by block 10 is divided into amount and sign in block 18 . The amount is fed to the denominator input of the divider 21 via the minimum value limitation 20 , N _o . With N _o , the value of zero can be assigned to the output of the divider 21 if it is undefined at a stator frequency close to zero. The sign from block 13 is added again to the output of the divider 21 via the polarity switch 22 , so that the correct slip frequency f _R2 / r _R2 , which is based on the rotor resistance, appears at the main output 27 . The sign signal for actuating the polarity switch 22 can also be derived from the stator frequency f _s1 , since it has to be switched when the rotating field direction is changed.

When executing the engine model in digital technology, functions 19, 20, 22 are not required, and the output of block 10 directly forms the denominator of quotient 21 . Its undetermined value is then limited to zero stator frequency, and a certain value can be assigned to it via a conditional branching.

The model stator currents f _S1 i _{α 2} , f _S1 i _{β 2} are calculated according to Eq. (50, 51) formed in the summing points 24 a, 24 b and provided with the model parameter in 23 a, 23 b in the controller inputs 12 a, 12 b of the ASM stator currents f generated in the multipliers 11 a, 11 b _S1 i _{α 1} , f _S1 i _{β 1} subtracted. The differences influence the components of the rotor EMF e _A2 , e _B2 in the summing points 15 a, 15 b via the manipulated variables y _A , y _{B of} the controllers 13 a, 13 b in the sense of a correction according to Eq. (54) of the default values e * _A2 , e * _B2 coming from the summation points 5 a, 5 b. These will be severely faulty, in particular at a low stator frequency, due to the model stator resistance r _S2 that is permanently set in FIG. 4 a, 4 b, regardless of the connected ASM, depending on the exemplary scatter or the thermal state of the ASM.

At higher stator frequency, e.g. B. in the field weakening range, this error becomes meaningless, so that the effect of the controller 13 a, 13 b may decrease depending on frequency or in some other way. The arrangement is therefore also suitable for operation with stator frequencies above the 130 Hz mentioned.

In the multipliers 17 a, 17 b and the summing point 18 , the scalar product of stator current i _{α 1} , i _{b 1} and rotor EMF e _A2 , e _{B2 is} formed, which is equal to the product stator frequency times internal torque f _S1 m _i2 and that Counter input of the divider 21 is supplied.

The manipulated variables of the controllers 13 a, 13 b are y via the paths _A, 14 a, 16 a and 14 14 b adding and y _B, b, 16 b to 14 a subtracting mutually rigidly coupled, each evaluated by a factor of K is preferably in the Range 0 <K1, to stabilize the control loops.

Figure 3 shows the structure of an additional circuit for the motor model for operation with field weakening.

It is assumed that the default value of the rotor flux linkage Ψ * _{R2 is present} in the rest of the control or regulation of the drive and can be used, as also shown in the publication by R. Joetten and G. Maeder in Fig. 7 bottom left.

With the additional circuit, according to Eq. (38) on the right the last summand is realized with variable rotor flux linkage Ψ _R2 , the default value Ψ * _{R2 being} used to avoid positive feedback. It is squared in the multiplier 8.1 and multiplied in a further multiplier 8.2 by the differential quotient of the stator frequency sf _S1 from block 8 , which is weighted with half the rotor time constant T _R2 . The result is added at summing point 9 and processed according to Figure 2.

Claims (9)

1. A method for determining the slip frequency of an asynchronous machine according to the relationship internal torque divided by the square of the rotor flux linkage, the internal torque being multiplied approximately as a scalar product of the space vectors stator current and rotor EMF by the stator frequency, characterized in that the square multiplied by the stator frequency the rotor flux linkage from the cross product of the space vectors stator current and the stator voltage e _{α 2} , e _{β 2} reduced by the voltage drop across the entire leakage reactance and the stator frequency f _S1 according to the equation: is determined with the rotor time constant T _R2 and the rotor reactance X _R2 , the equation Ψ _{R2N being} the nominal value of the rotor _{flux linkage} and e _{α 2} , e _{β 2 being} the components of the stator voltage vector reduced by the voltage drop across the entire leakage reactance. 2. The method according to claim 1, characterized in that the square of the rotor flux linkage multiplied by the stator frequency from the cross product of the space vectors stator current and the reduced by the voltage drop across the entire leakage reactance and ohmic stator resistance stator voltage e * _A2 , e * _{B2 is} determined. 3. The method according to claims 1 or 2, characterized in that when operating with field weakening the differential quotient of the stator frequency in the integrand of the equation of claim 1 is evaluated with a constant T _R2 / 2 and multiplied by a variable Ψ * ² _R2 , the Ψ * _{R2 is} the default value of the rotor flux linkage. 4. The method according to claim 1, characterized in
that from the determined slip frequency and the EMF vector according to equations ( 50, 51 ) a model current vector is formed, that the model current vector is compared component by component with the measured current vector in controllers, and
that the controllers correct the components of the EMF vector. 5. Arrangement for performing the method according to claim 1,
with summation points ( 2, 5 ), which form an EMF vector from the measured voltage and current vector by subtracting the ohmic and inductive voltage drops,
with a circuit ( 17, 18 ) which forms the inner product of the current vector and the EMF vector as a measure of the inner torque,
with a divider which divides the internal torque by the square of the rotor flux formed from the current and voltage vector and thus determines the slip frequency, characterized by
a circuit which forms the cross product of the current vector multiplied by the rotor inductance X _R2 and the voltage vector e _{α 2} , e _{β 2} reduced by the voltage drop across the entire current reactance and feeds it to an adder ( 9 ),
a differentiator ( 8 ) that differentiates and with the factor supplies weighted stator frequency signal f _{s to} the adder,
an integrator connected downstream of the adder with the integration constant T _R2 / 2 and a direct feedback to the adder, the output signal of which is fed to the divider ( 21 ) as a square of the rotor flux multiplied by the stator frequency f _s . 6. Arrangement according to claim 5, characterized in that the denominator input of the divider ( 21 ) only the amount of the product stator frequency times the square of the rotor flux linkage is supplied via an adjustable minimum limit ( 20 ) and the sign of the product by means of a polarity switch ( 22 ) the output value the divider ( 21 ) is added again. 7. Arrangement according to claim 6, characterized in that the signal for actuating the polarity switch ( 22 ) is derived from the sign of the stator frequency. 8. Arrangement according to claim 5, characterized by a unit consisting of the determined slip frequency and the EMF vector according to the equations ( 50, 51 ) forms a model current vector, and controllers, which compare the model current vector with the actual current vector component by component and whose manipulated variables are added to the EMF components. 9. Arrangement according to claim 8, characterized in that for the stabilization of the control loops, the manipulated variables of the controllers ( 13 ) are mutually coupled, such that the manipulated variable of a controller multiplies by a fixed factor K1, added to the manipulated variable of the other controller and the The manipulated variable of the other controller multiplied by the factor K is subtracted from the manipulated variable of one controller.