Induction motor vector control

Abstract: An induction motor vector control apparatus employing a rotating Cartesian coordinate system (d,q) having a d-axis held in coincidence with a secondary flux of the induction motor. Another rotating Cartesian coordinate system (γ, δ) having a γ-axis held in coincidence with the induction motor primary current to detects a δ-axis induction motor primary voltage change caused by a change in the induction motor secondary resistance. The δ-axis primary voltage change contains no component related to a primary resistance change and it is used to compensate the secondary resistance for its change.

Description: BACKGROUND OF THE INVENTION This invention relates to an apparatus for controlling an adjustable speed electric motor and, more particularly, to an apparatus for vector control of an induction motor. Electric power converters or inverters have been employed for the application of adjustable speed drives using alternating current motors. A typical converter includes a direct current (DC) rectifier for rectifying three-phase AC input voltage and for supplying the resulting direct current (DC) bus potential to an inverter. The inverter comprises a plurality of pairs of series-connected switching elements to generate an adjustable frequency output. In many applications, such a frequency adjustment is effected through a control circuit which employs a pulse width modulation (PWM) control technique in producing variable frequency gating pulses to periodically switch the respective switching elements so as to operate the motor at a variable speed. The motor can be propelled (motoring mode) or retarded (braking mode) as desired by appropriately varying the frequency and the amplitude of the excitation that the inverter applies to the motor (see for example the 3 rd European Conference on Power electronics and applications Vol. 3,12 October 1989, pages 1079-1085). The actual motor speed is sensed and compared with a commanded motor speed. A speed error signal, which depends on the difference between the actual and desired values of motor speed, is derived and applied to a proportional plus integral control circuit which converts it into a torque command signal. The control circuit responds to the torque command signal by controlling the operation of the inverter so as to vary, as a function of the torque command signal, the amplitude of the voltages supplied from the inverter to the motor. In order to provide more accurate motor control and linear motor torque control for variations in commanded torque, vector control has been proposed and employed. Such vector control utilizes a secondary flux rotational speed together with the torque command signal to control the momentary values of the frequency and amplitude of the stator current of the motor. It has been proposed to compensate for the influence of changes in the primary and secondary resistances of the induction motor on the vector control. The compensation has been made on an assumption that the excitation current is constant. However, this assumption is not satisfied when the secondary flux changes. A secondary resistance compensation is disclosed in JP-A-61147788. SUMMARY OF THE INVENTION Therefore, a main object of the invention is to provide an improved vector control apparatus which can provide more accurate motor control. There is provided, in accordance with a first embodiment of the invention, an apparatus employing a rotating Cartesian coordinate system (d, q) having a d-axis and q-axis, the d-axis being held in coincidence with a secondary flux of the induction motor, for vector control of an adjustable-speed induction motor. The apparatus comprises means for applying a primary current and voltage to drive the induction motor, means sensitive to an induction motor angular velocity for producing an induction motor angular velocity value ωr, means for producing a d-axis secondary flux command value λ2d*, means for calculating a d-axis primary current command value i1d* as a function of the d-axis secondary flux command value λ2d* and a secondary time constant command value L2*/R2*, means for producing a q-axis primary current command value i1q*, means employing a rotating Cartesian coordinate system (γ, δ) having a γ-axis and δ-axis, the γ-axis being held in coincidence with the primary current I1, for calculating a γ-axis primary current command value i1γ* and a phase angle ψ of the γ-axis with respect to the d-axis as a function of the primary current command values i1d* and i1q*, means for calculating the γ- and δ-axis primary voltage command values v1γ* and v1δ* as a function of the γ-axis primary current command value i1γ*, the phase angle ψ, the d-axis secondary flux command value λ2d*, and a primary voltage angular frequency ω0, means for sensing the primary current and converting the sensed primary current into γ- and δ-axis primary current values i1γ and i1δ, means for calculating a γ-axis primary voltage change Δv1γ as a function of the γ-axis primary current value i1γ and the γ-axis primary current command value i1γ*, means for calculating a δ-axis primary voltage change Δv1δ as a function of the δ-axis primary current value i1δ and a δ-axis primary current command value i1δ*, means for adding the γ-axis primary voltage change Δv1γ to the γ-axis primary voltage command value v1γ* to produce a γ-axis primary voltage command signal v1γ, means for adding the δ-axis primary voltage change Δv1δ to the δ-axis primary voltage command value v1δ* to produce a δ-axis primary voltage command signal v1δ, means for calculating a secondary resistance change as a function of the d-axis primary current command value i1d*, the q-axis primary current command value i1q*, the γ-axis primary current command value i1γ*, the d-axis secondary flux command value λ2d*, the primary voltage angular frequency ω0 and the δ-axis primary voltage change Δv1δ, means for correcting the secondary time constant command value L2*/R2* based upon the secondary resistance change, means for calculating a slip frequency ωs as a function of the q-axis primary current command value i1q*, the d-axis secondary flux command value λ2d*, and the corrected secondary time constant command value, means for adding the slip frequency ωs to the induction motor angular velocity value ωr to produce the primary voltage angular frequency ω0, and means for controlling the motor driving means to adjust the primary voltage based upon the primary voltage command signals v1γ and v1δ, and the primary voltage angular frequency ω0. In a second embodiment of the invention, the induction motor vector control apparatus of the first embodiment further comprises means for subtracting the δ-axis primary current i1δ from the δ-axis primary current command value i1δ* to produce a difference (i1δ* - i1δ), means for multiplying the difference (i1δ* - i1δ) by a leakage inductance Lσ to produce a multiplied value (i1δ* - i1δ)·Lσ, means for integrating the difference (i1δ* - i1δ) to produce an integrated value Δv1δI, means for adding the multiplied value to the integrated value to produce a δ-axis primary voltage change Δv1δ, and wherein the δ-axis primary voltage change used by the means for calculating a secondary resistance change is the integral value Δv1δI. In a third embodiment of the invention, the induction motor vector control apparatus comprises means for applying a primary current and voltage to drive the induction motor, means sensitive to an induction motor angular velocity for producing an induction motor angular velocity value ωr, means for producing a d-axis secondary flux command value λ2d*, means for calculating a d-axis primary current command value i1d* as a function of the d-axis secondary flux command value λ2d*, and a secondary time constant command value L2*/R2*, means for producing a q-axis primary current command value i1q*, means employing a rotating Cartesian coordinate system (γ, δ) having a γ-axis and δ-axis, the γ-axis being held in coincidence with the primary current I1, for calculating a γ-axis primary current command value i1γ* and a phase angle ψ of the γ-axis with respect to the d-axis as a function of the primary current command values i1d* and i1q*, means for calculating the γ- and δ-axis primary voltage command values v1γ* and v1δ* as a function of the γ-axis primary current command value i1γ*, the phase angle ψ, the d-axis secondary flux command value λ2d*, and a primary voltage angular frequency ω0, means for sensing the primary current and converting the sensed primary current into γ- and δ-axis primary current values i1γ and i1δ, means for calculating a γ-axis primary voltage change Δv1γ as a function of the γ-axis primary current value i1γ and the γ-axis primary current command value i1γ*, means for calculating a δ-axis primary voltage change Δv1δ as a function of the δ-axis primary current value i1δ and a δ-axis primary current command value i1δ*, means for adding the γ-axis primary voltage change Δv1γ to the γ-axis primary voltage command value v1γ* to produce a γ-axis primary voltage command signal v1γ, means for adding the δ-axis primary voltage change Δv1δ to the δ-axis primary voltage command value v1δ* to produce a δ-axis primary voltage command signal v1δ, means for calculating a slip frequency command value ωs* as a function of the q-axis primary current command value i1q* and the d-axis secondary flux command value λ2d*, means for calculating a difference of the δ-axis primary voltage change Δv1δ from a δ-axis primary voltage change command value, means for calculating a slip frequency change Δωs as a function of the calculated difference, means for adding the slip frequency change Δωs to the slip frequency command value ωs* to produce a slip frequency ωs, means for adding the slip frequency ωs to the induction motor angular velocity value ωr to produce the primary voltage angular frequency ω0, and means for controlling the motor driving means to adjust the primary voltage based upon the primary voltage command signals v1γ and v1δ, and the primary voltage angular frequency ω0. In a fourth embodiment of the invention, the induction motor vector control apparatus of the third embodiment further comprises means for subtracting the δ-axis primary current i1δ from the δ-axis primary current command value i1δ* to produce a difference (i1δ* - i1δ), means for multiplying the difference (i1δ* - i1δ) by a leakage inductance Lσ to produce a multiplied value (i1δ* - i1δ)·Lσ, means for integrating the difference (i1δ* - i1δ) to produce an integrated value Δv1δI, means for adding the multiplied value to the integrated value to produce a δ-axis primary voltage change Δv1δ, means for calculating a secondary resistance change K as a function of the calculated difference, means for multiplying the calculated difference by the slip frequency command value ωs* to produce a slip frequency change Δωs. BRIEF DESCRIPTION OF THE DRAWINGS The invention will be described in greater detail by reference to the following description taken in connection with the accompanying drawings, in which like reference numerals identify like elements in the several figures and in which: Fig. 1 is a schematic diagram showing a prior art induction motor vector control apparatus; Fig. 2 is a diagram showing the relationship between the primary and secondary currents and voltages expressed in coordinate systems (d, q) and (γ, δ) ; Fig. 3 is a diagram showing the relationships between the primary voltage changes expressed in the coordinate systems (d, q) and (γ, δ); Fig. 4 is a circuit diagram showing an equivalent circuit for an induction motor; Fig. 5 is a diagram showing the vectors of the currents produced in the equivalent circuit of Fig. 4; Fig. 6 is a diagram showing the vectors of the voltages produced in an equivalent circuit for an idling induction motor; Fig. 7 is a schematic diagram showing a first embodiment of an induction motor vector control apparatus made in accordance with the invention; Fig. 8 is a schematic diagram showing a second embodiment of the induction motor vector control apparatus of the invention; Fig. 9 is a schematic diagram showing a modified form of the second embodiment of the invention; Fig. 10 is a schematic diagram showing a third embodiment of the induction motor vector control apparatus of the invention; Fig. 11 is a schematic diagram showing a modified form of the third embodiment of the invention; Fig. 12 is a schematic diagram showing a fourth embodiment of the induction motor vector control apparatus of the invention; Fig. 13 is a schematic diagram showing a modified form of the fourth embodiment of the invention; Fig. 14 is a schematic diagram showing another modified form of the fourth embodiment of the invention; Fig. 15 is a flow diagram showing the programming of the digital computer used in the induction motor vector control apparatus of the invention; and Fig. 16 is a flow diagram showing the programming of the digital computer used in the induction motor vector control apparatus of the invention. DETAILED DESCRIPTION OF THE INVENTION Prior to the description of the preferred embodiments of the present invention, the prior art vector control apparatus of Fig. 1 is briefly described. The principles on which the conventional vector control apparatus is based are as follows: An induction motor voltage equation expressed in a two-dimensional Cartesian coordinate system (d, q) rotating at the same angular velocity as the angular velocity of the primary voltage to the induction motor is given as where ωs = ω o - ωr, Lσ = (L1 · L2 - M²)/L2, v1d is the primary voltage in the d-axis, v1q is the primary voltage in the q-axis, i1d is the primary current in the d-axis, i1q is the primary current in the q-axis, λ2d is the secondary flux in the d-axis, λ2q is the secondary flux in the q-axis, P is the differentiating operator, R1 is the primary resistance, R2 is the secondary resistance, L1 is the primary inductance, L2 is the secondary inductance, M is the excitation inductance, Lσ is the equivalent leakage inductance, ω o is the primary voltage angular frequency, ωr is the rotor angular frequency, and ωs is the slip frequency. Assuming now that the d-axis is held in coincidence with the direction of the secondary flux, the q-axis secondary flux λ2q is zero, the d-axis secondary flux λ2d is a constant value 2, the d-axis secondary current i2d is zero, and the q-axis secondary current i2q is equal to the secondary current i2. It is, therefore, possible to control the induction motor like DC motors. The d- and q-axis secondary flux λ2d and λ2q are λ2d = M · i1d + L2 · i2d λ2q = M · i1q + L2 · i2q Since i2d = 0 and i2q = 0 under the vector control, the d-axis secondary flux is equal to M·i1d and the q-axis primary current i1q is equal to - L2/M·i2q. Thus, the q-axis primary current i1q is proportional to the torque current. From the fourth row of Equation (1), the following equation is derived: - R2 L2 · M · i1q + ωs · λ2d = 0 This equation is solved for the slip frequency as ωs = R2 L2 · M · i1q λ2d = R2 L2 · M · i1q M · i1d = R2 L2 · i1q i1d Thus, the vector control is made by getting the d-axis primary current i1d at λ2d/M and the slip frequency ωs at a value calculated from Equation (5). As can be seen from Equation (5), the secondary resistance R2 used in calculating the slip frequency ωs changes with changes in ambient and rotor temperatures. It is, therefore, required to estimate the secondary resistance change based upon inverter output voltage and to correct the slip frequency command value ωs* based upon the secondary resistance change estimated from the inverter voltage. If the secondary resistance changes are ignored, the vector control is degraded for torque control accuracy and torque response time. If the inverter output voltage is used as it is to estimate the secondary resistance change, however, a component related to the primary resistance change will be introduced into the estimated secondary resistance change. For this reason, it is desirable to estimate the secondary resistance change based upon a signal which is independent from the primary resistance change. The vector control apparatus is arranged to control excitation currents iu, iv and iw that an inverter 10 applies to an induction motor IM by utilizing motor-flux and motor-torque command signals i1d* and i1q*. The induction motor IM has three-phase stator windings which are energized by the output of the inverter 10 and a rotor coupled to drive a mechanical load. The inverter 10 includes a plurality of parallel pairs of series-connected switching element arranged and controlled to convert DC input power into AC output power having adjustable frequency and voltage magnitude. The inverter 10 is controlled by a pulse-width-modulation (PWM) circuit 11 which includes a PWM waveform generator, a triangle waveform generator and a gating circuit. The PWM waveform generator receives a triangle wave signal from the triangle waveform generator and controls the gating circuit to produce gating pulses so as to periodically switch the respective switching elements of the inverter in a predetermined sequence and at a desired frequency. The vector control apparatus includes a rotor angular velocity sensor 12 connected to a pulse pickup transducer PP. This transducer is associated with the rotor of the induction motor IM for producing a series of electric pulses of a repetition rate directly proportional to the speed of rotation of the rotor. The rotor angular velocity sensor 12 receives the electric pulses and produces an actual rotor angular velocity signal ωr indicative of the sensed angular velocity ωr of the rotor. The actual rotor angular velocity signal ωr is fed to a secondary flux command generator 13 which receives a command signal λ2d*/M*. The secondary flux command generator 13 produces a command signal i1d* which remains at a value λ2d*/M* when the actual rotor angular velocity ωr is less than a predetermined value and decreases as the actual rotor angular velocity ωr increases when the actual rotor angular velocity ωr exceeds the predetermined value. The secondary flux command signal i1d* is fed from the secondary flux command generator 13 to a calculation circuit 16. The actual rotor angular velocity signal ωr is also fed to a summing circuit 14 which subtracts it from a rotor angular velocity command signal ωr* to provide an error output signal. In order to enhance the induction motor control for stability and response time, a proportional plus integral operation is provided on this error signal at a proportional plus integral circuit 15 to produce a torque command signal i1q*. The torque command signal i1q* is applied to the calculation circuit 16. The calculation circuit 16 utilizes the command signals i1d* and i1q* to calculate command values v1d* and v1q* for the d- and q-axis components of the primary voltage applied to the induction motor IM. The conventional vector control apparatus also includes a coordinate converter 17 which senses two excitation currents iu and iv to the induction motor IM and converts them into d- and q-axis primary current signals i1d and i1q. The d-axis primary current signal i1d is applied to a summing circuit 18 which subtracts it from the secondary flux command signal i1d* fed thereto from the secondary flux command generator 13 to provide an error signal. A proportional plus integral operation is provided on this error signal at a proportional plus integral circuit 19 to produce a signal Δv1d. The signal Δv1d is then applied to a summing circuit 22 which adds it to the command signal v1d* to correct the command signal v1d*. The corrected signal vid is applied to a coordinate transformer 24. Similarly, the q-axis primary current signal i1q is applied to a summing circuit 20 which subtracts it from the torque command signal i1q* fed thereto from the proportional plus integral circuit 15 to provide an error signal. A proportional plus integral operation is provided on this error signal in a proportional plus integral circuit 20 to produce a signal Δv1q. The signal Δv1q is applied to a summing circuit 23 which adds it to the signal v1q* to correct the command signal v1q*. The corrected signal v1q is applied to the coordinate transformer 24. The signals Δv1d and Δv1q contain components related to changes in the primary and secondary resistances R1 and R2. Fig. 2 is a diagram showing the relationship between the primary and secondary currents and voltages expressed in coordinate systems (d, q) and (γ, δ), and Fig. 3 is a diagram showing the relationship between the primary voltage changes expressed in the coordinate systems (d, q) and (γ, δ). In these figures, the character V indicates the primary voltage, the character E is the secondary voltage, the character Δv1 is the primary voltage change, the character Δv1γ is the γ-axis primary voltage change, the character Δv1δ is the δ-axis primary voltage change, the character ψ is the phase angle of the γ-axis with respect to the d-axis, the character I0 is the excitation current, and the character I2 is the torque current. The δ-axis primary voltage change Δv1δ is given as Δv1δ = -Δv1d · sinψ + Δv1q · cosψ where sinψ = I2/I1 = i1q/i1γ and cosψ = I0/I1 = i1d/i1γ. The d- and q-axis primary voltage changes contains components related to first and second resistance changes. In the conventional apparatus, however. the vector control is made on an assumption that i1d = λ2d/M, that is, the d-axis excitation current i1d is constant without regard to secondary flux control. For this reason, the slip frequency cannot be calculated with high accuracy. The relationship between the d-axis secondary flux λ2d and the d-axis primary current ild may be derived from the third row of Equation (1) as - R2 L2 · M · i1d + ( R2 L2 + P) · λ2d - ωs · λ2q = 0 i1d = L2 R2 · λ2d M · ( R2 L2 + P) = λ2d M · (1 + L2 R2 · P) As can be seen from Equation (8), the d-axis primary current i1d should be controlled in a fashion of time advance of first order with respect to changes in the d-axis secondary flux λ2d. In other words, the condition λ2d = M·i1d cannot be established when the secondary flux command λ2d* changes. The principles of the invention will be described with reference to Figs. 4 and 5. Fig. 4 shows an asymmetrical T-I type equivalent circuit for an induction motor. Fig. 5 is a diagram showing the vectors of the currents produced in the equivalent circuit of Fig. 4. Assuming now that the γ-axis is held in coincidence with the direction of the primary current I1, the γ-axis primary current i1γ is equal to the primary current I1 and the δ-axis primary current i1δ is zero. The induction motor voltage equation expressed in the two-dimensional Cartesian coordinate system (γ, δ) rotating at the same angular velocity as the angular velocity of the primary voltage to the induction motor IM is given as From the third and fourth rows of Equation (9), the following equations are derived: - R2 L2 · M · i1γ + ( R2 L2 + P) · λ2γ - ωs · λ2δ = 0 ωs · λ2γ + ( R2 L2 + P) · λ2δ = 0 From Equation (11), the following equation is derived: R2/L2 + P = - λ2γ · ωs/λ2δ Substituting Equation (12) into Equation (10) gives the following equation: - R2 L2 · M · i1γ - λ2γ² λ2δ · ωs - λ2δ · ωs = 0 - R2 L2 · M · i1γ - ( λ2γ² + λ2δ² λ2δ ) · ωs = 0 Thus, ωs = - λ2δ λ2γ² + λ2δ² · R2 L2 · M · i1γ The d- and q-axis secondary fluxes λ2γ and λ2δ are λ2γ = λ2d cosψ λ2δ = - λ2dsinψ λ2γ² + λ2δ² = λ2d² Substituting Equations (14), (15) and (16) into Equation (13) gives the following equation: ωs = R2 L2 · M · i1γ · sinψ λ2d = R2 L2 · M · i1q M · i1d = R2 L2 · i1q i1d It can be seen from Equation (17) that the slip frequency ωs can be calculated from the same equation as Equation (5) used to calculate the slip frequency in the conventional vector control apparatus if the γ- and δ-axis primary currents i1γ and i1δ are controlled to establish conditions of i1γ = I1 and i1δ = 0 with the γ-axis being held in coincidence with the direction of the primary current I1. If the vector control is made with regard to secondary flux control, that is, the d-axis secondary flux λ2d is not equal to M·i1d, the slip frequency ωs is ωs = R2 L2 · i1q (λ2d/M) Considerations are made to the voltage commands v1γ* and v1δ*. Since the vector control is made on the coordinate system (γ, δ) with i1γ* = I1 and i1δ* = 0, the following equation can be derived from Equation (9): Equation (19) may be modified to remove the terms affixed to the differentiating operator P. When vector control conditions are satisfied, the following equations are given λ2γ* = λ2d* · cosψ =M · λ2d* M · cosψ λ2δ* = - λ2d* · sinψ = - M · λ2d* M · sinψ Thus, multiplying out (20) v1γ* = R1 · i1γ* + M² L2 · ω₀ · λ2d* M · sinψ v1δ* = Lσ · ω₀ · i1γ* + M² L2 · ω₀ · λ2d* M · cosψ The term P·i1γ affixed to Lσ in Equation (19) cannot be ignored when a rapid change occurs in the torque command i1q* and/or when the excitation current command i1d* changes. With regard to the term P·i1γ, the voltage commands v1γ* and v1δ* are v1γ* = R1 · i1γ* + Lσ · P · i1γ* + M² L2 · ω₀ · λ2d* M · sinψ = R1(1 + Lσ R1 · P)i1γ* + M² L2 · ω₀ · λ2d* M · sinψ v1δ* = Lσ · ω₀ · i1γ* + M² L2 · ω₀ · λ2d* M cosψ Considerations are made to a secondary flux change with a secondary resistance change. The following equations are derived from the third and fourth rows of Equation (9): - R2 L2 · M · i1γ + ( R2 L2 + P) · λ2γ - ωs · λ2δ = 0 ωs · λ2γ + ( R2 L2 + P) · λ2δ = 0 Multiplying Equations (27) and (28) by L2/R2 gives -M · i1γ + (1 + L2 R2 · P) · λ2γ - L2 R2 · ωs · λ2δ = 0 L2 R2 · ωs · λ2γ + (1 + L2 R2 · P) · λ2δ = 0 Multiplying Equation (29) by (1 + L2 · P/R2) gives -M · (1 + L2 R2 · P) · i1γ + (1 + L2 R2 · P)² · λ2γ - L2 R2 · ωs · (1 + L2 R2 · P) · λ2δ = 0 Multiplying Equation (30) by L2·ωs/R2 gives ( L2 R2 · ωs)² · λ2γ + L2 R2 · ωs · (1 + L2 R2 · P) · λ2δ = 0 Adding Equation (31) to Equation (32) gives λ2γ = M·(1 + L2 R2 ·P)·i1γ (1 + L2 R2 ·P)² + ( L2 R2 ·ωs)² Multiplying Equation (29) by L2 · ωs/R2 gives -M · L2 R2 · ωs · i1γ + L2 R2 · ωs · (1 + L2 R2 · P) · λ2γ - ( L2 R2 · ωs)² · λ2δ = 0 Multiplying Equation (30) by (1 + L2 · P/R2) gives L2 R2 · ωs · (1 + L2 R2 · P) · λ2γ + (1 + L2 R2 P)² · λ2δ = 0 Subtracting Equation (34) from Equation (35) gives λ2δ = - M · L2 R2 · ωs · i1γ (1 + L2 R2 · P)² + ( L2 R2 · ωs)² It is now assumed that the currents are controlled to respective command values and thus i1γ* = i1γ, i1δ* = i1δ = 0, i1d* = i1d, and i1q* = i1q. Since R2 = (1 + K)·R2* where K is the secondary resistance change, the term L2·ωs/R2 used in Equations (33) and (36) may be expressed as the following, with the use of (18) L2 R2 · ωs = L2* (1 + K) · R2* · R2* L2* · i1q* λ2d* M* = 1 (1 + K) · i1q* λ2d* M* Since the excitation current is controlled as shown in Equation (8), the following equation is obtained: i1d* = λ2d* M* · (1 + L2* R2* · P) Assuming now that the secondary time constant L2/R2 of the term (1 + L2·P/R2) is equal to L2*/R2*, that is, the secondary resistance R2 is unchanged in a short time, the following equation is given: 1 + L2 R2 · P = i1d* λ2d* M* Substituting Equations (39) and (37) into Equations (33) and (36) gives λ2γ = λ2d* · (1 + K)² · i1d* · i1γ* (1 + k)² · i1d*² + i1q*² λ2δ = -λ2d* · (1 + K) · i1q* · i1γ* (1 + k)² · i1d*² + i1q*² Thus, the secondary flux command values are given as λ2γ* = λ2d* · cosψ= λ2d* · i1d* i1γ* λ2δ* = -λ2d* · sinψ = -λ2d* · i1q* i1γ* ψ = tan⁻¹ i1q* i1d* Using Equations (40) and (42), the following equation is given: Δλ2γ = λ2γ - λ2γ* = λ2γ* · { (1 + K)² · i1γ*² (1 - k)² · i1d*² + i1q*² - 1} Using Equations (41) and (43), the following equation is given Δλ2δ = λ2δ - λ2δ* = λ2δ* · { (1 + K) · i1γ*² (1 + k)² · i1d*² + i1q*² - 1} When the secondary flux changes, the primary current is from Equation (20) v1γ = R1 · i1γ* - M* L2* · ω₀(λ2δ* + Δλ2δ) v1δ = Lσ* · ω₀ · i1γ* + M* L2* · ω₀(λ2γ* + Δλ2γ) Since the primary voltage command value is expressed by Equations (23) and (24), the voltage changes are derived from Equations (23), (24), (47) and (48) along with Equations (45) and (46). First (47) and (23) Δv1γ = v1γ - v1γ* = - M* L2* · ω₀ · Δλ2δ substitute (46) = - M* L2* · ω₀ · λ2δ* { (1 + K) · i1γ*² (1 + k)² · i1d*² + i1q*² - 1} = M*² L2* · ω₀ · λ2d* M* · i1q* i1γ* { (1 + K)² · i1γ*² (1 + k)² · i1d*² + i1q*² - 1} First (48) and (24) Δv1δ = v1δ - v1δ* = + M* L2* · ω₀ · Δλ2γ substitute (45) = + M* L2* · ω₀ · λ2γ* { (1 + K)² · i1γ*² (1 + k)² · i1d*² + i1q*² - 1} = M*² L2* · ω₀ · λ2d* M* · i1d* i1γ* { (1 + K)² · i1γ*² (1 + k)² · i1d*² + i1q*² - 1} where it has been assumed that M/L2 is equal to M*/L2*, that is the excitation inductance is unchanged in a short time. Since the γ-axis primary voltage v1γ contains a component related to the term R1·i1γ*, it contains a component related to a voltage change caused by a change in the primary resistance R1. With regard to a change in the primary resistance R1, Equation (47) becomes, using R1 = (1+A1)·R1* v1γ = (1 + A1)R1* · i1γ* - M L2 · ω₀(λ2δ* + Δλ2δ) = R1* · i1γ* - M L2 ω₀ · λ2δ* + λ1 · R1* i1γ* - M L2 · ω₀ · Δλ2δ Thus, Δv1γ = v1γ - v1γ* = A1 · R1* · i1γ* + M2 L2* · ω₀ · λ2d* M* i1q* i1γ* { (1 + K) · i1γ*² (1 + k)² · i1d*² + i1q*² - 1} where A1 is a primary resistance change. Although the term Δv1, which contains a component related to a change in the primary resistance R1, is not suitable for use in correcting for a change in the secondary resistance R2, the δ-axis primary voltage change Δv1δ contains no component related to the primary resistance R1 and it may be considered as a voltage change component caused by a secondary resistance change. It is, therefore, possible to eliminate the influence of the primary resistance R1 by detecting the change Δv1δ of the δ-axis primary voltage v1δ and using the detected change Δv1δ to correct the secondary resistance change. As a result, the secondary resistance correction can be free from the influence of the primary resistance R1 which changes with changes in temperature. Normally, the influence of voltage drops caused by the primary resistance R1 increases at slow induction motor speeds. Since the δ-axis primary voltage v1δ contains no component related to a voltage drop caused by the primary resistance R1, however, it is possible to provide an accurate secondary resistance correction at slow induction motor speeds. In addition, it is possible to estimate the primary resistance R1 by detecting the γ-axis primary voltage change Δv1γ which is a voltage component caused only by a change in the primary resistance R1 when the secondary resistance correction is made based upon the δ-axis primary voltage change Δv1δ The secondary resistance change K is given by modifying Equation (50) as 1 + K = M*² L2* · ω₀ · λ2d* M* · i1d* · i1q*² + i1γ* · i1q*² · Δv1δ M*² L2* · ω₀ · λ2d* M* · i1d* · i1q*² - i1γ* · i1d*² · Δv1δ Thus, if the δ-primary voltage change Δv1δ is detected, the secondary resistance change K can be calculated from Equation (53). Considerations are made to identification of the primary resistance R1 and the excitation inductance M during induction motor idling operation. The ratio of the excitation current to the torque current changes to change the primary voltage with changes in the excitation inductance M. Since the primary voltage changes with changes in the secondary resistance R2, it is impossible to distinguish a primary voltage change caused by an excitation inductance change from a primary voltage change caused by a secondary resistance change. Since the torque current i1q is zero during induction motor idling operation, however, the primary voltage change is free from the influence of the secondary resistance change. It is, therefore, possible to make an excitation inductance correction using the primary voltage change detected during induction motor idling operation. Fig. 6 is a diagram showing the vectors of the voltage produced in a T-I type equivalent circuit for an idling induction motor. Since the torque current i1q is zero during induction motor idling operation, the d-axis and q-axis of the coordinate system (d,q) are in coincidence with the -axis and δ-axis of the coordinate system (γ,δ), respectively. Thus, considerations are made to the coordinate system (d, q). During induction motor idling operation where the q-axis primary current i1q is zero, the primary current is derived from Equation (19) with the term P being ignored as v1d = R1 · i1d viq = Lσ · ω₀ · i1d + M² L2 · ω₀ · λ2d* M It is now assumed that i1d* = i1d since the d-axis primary current i1d is controlled to the command value i1d*, AM is the excitation inductance change, and the change in the equivalent leakage inductance Lσ is small and can be ignored. The command voltage value during induction motor idling operation is expressed as v1d* = R1* · i1d* viq* = Lσ · *ω₀ · i1d* + M*² L2* · ω₀ · λ2d* M* Using the primary resistance change A1 and the excitation inductance change AM. the primary voltage can be expressed as v1d = (1 + A1) · R1* · i1d* viq = Lσ · *ω₀ · i1d* + (1 + AM) · M*² L2* · ω₀ · λ2d* M* The d- and q-axis primary voltage changes Δv1d and Δv1q are expressed in Equations (60) and (61) derived from Equations (56), (57), and (58) and (59), and the primary resistance change A1 and the excitation inductance change AM are expressed in Equations (62) and (63) derived from Equations (60) and (61). Δv1d = v1d - v1d* = A1 · R1* · i1d* Δv1q = v1q - v1q* =AM · M*2 L2* · ω₀ λ2d* M* A1 = Δv1d R1* i1d* AM = Δv1q M*² L2* · ω₀ · λ2d* M* Since λ2d* = M* · i1d* when the excitation command is unchanged, the excitation inductance change AM is AM = Δv1q M*² L2* · ω₀ · i1d* As can be seen from the foregoing, it is possible to identify the primary resistance R1 and the excitation inductance M using the primary voltage change during induction motor idling operation. That is, the primary resistance change A1 can be derived from the d-axis primary voltage change Δv1d, and the excitation inductance change AM can be derived from the q-axis primary voltage change Δv1q. If the secondary resistance command R2* is in coincidence with the actual secondary resistance R2, the slip frequency command ωs* may be calculated from Equation (18). However, the actual secondary resistance R2 changes with changes in temperature. According to the invention, the δ-axis primary voltage change Δv1δ is utilized to calculate the secondary resistance change K and the calculated secondary resistance change K is utilized to correct the secondary resistance command R2*. The corrected secondary resistance command is used in calculating the slip frequency command ωs*. That is, the slip frequency command ωs* is calculated from the following equation: ωs* = (1 + K) · R2* L2* · i1q* λ2d*/M* The primary resistance R1 changes with changes in temperature. However, the secondary resistance correction is free from the influence of the primary resistance change since the δ-axis primary voltage change Δv1δ contains no component related to the primary resistance, as can be seen from Equation (50). In this respect, the invention is similar to the vector control circuit of Fig. 1. However, the conventional vector control apparatus is quite different from the invention in that the vector control is made only on a single coordinate system (d, q). Referring to Fig. 7, there is shown a schematic block diagram of a vector control apparatus embodying the invention. The vector control apparatus is arranged to control excitation currents iu, iv and iw that an inverter 40 applies to an induction motor IM by utilizing motor-flux and motor-torque command current signals i1d* and i1q*. The induction motor IM has three-phase stator windings which are energized by the output of the inverter 40 and a rotor coupled to drive a mechanical load. The inverter 40 includes a plurality of parallel pairs of series-connected switching element arranged and controlled to convert DC input power into AC output power having adjustable frequency and voltage magnitude. The inverter 40 is controlled by a pulse-width-modulation (PWM) circuit 41 which includes a PWM waveform generator, a triangle waveform generator and a gating circuit. The PWM waveform generator receives a triangle wave signal from the triangle waveform generator and controls the gating circuit to produce gating pulses so as to periodically switch the respective switching elements of the inverter in a predetermined sequence and at a desired frequency. The vector control apparatus includes a rotor speed sensor 42 connected to a pulse pickup transducer PP. This transducer is associated with the rotor of the induction motor IM for producing a series of electric pulses of a repetition rate directly proportional to the speed of rotation of the rotor. The rotor speed sensor 42 receives the electric pulses and produces an actual rotor angular velocity signal ωr indicative of the sensed rotor angular velocity. The actual rotor angular velocity signal ωr is fed to a calculation circuit 43 which receives a command signal λ2d*/M*. The calculation circuit 43 produces a command signal which remains at a value λ2d*/M* when the actual speed signal ωr is less than a predetermined value and decreases as the actual speed signal ωr increases when the actual speed signal ωr exceeds the predetermined value. The command signal is applied from the calculation circuit 43 to a calculation circuit 44 where a secondary flux command signal in terms of the primary current command value i1d* is calculated from Equation (8) as i1d* = λ2d M (1 + L2 R2 P) The actual speed signal ωr is also applied to a summing circuit 45 which subtracts it from a speed command signal ωr* to provide an error output signal. In order to enhance the induction motor control for stability and response time, a proportional plus integral operation is provided on this error signal in a proportional plus integral circuit 46 to produce a torque command signal i1q*. These command signals i1d* and i1q* are applied to a first coordinate converter 47. The first coordinate converter 47 utilizes the command signals i1d* and i1q * to calculate a γ-axis primary current command value i1γ* and a phase angle ψ of the γ-axis of the coordinate system (γ, δ) held in coincidence with the direction of the primary current I1 with respect to the d-axis of the coordinate system (d, q) held in coincidence with the direction of the secondary flux as ψ = tan⁻¹(i1q*/i1d*), I1 = i1d*² + i1q*² The calculated phase angle ψ is fed to a calculation circuit 48 which utilizes the calculated phase angle in the form of sinψ and cosψ, along with the calculated value λ2d*/M* fed from the calculation circuit 44 and the primary voltage angular frequency 0, to calculate γ- and δ-axis primary voltage command values v1γ* and v1δ* from Equations (23) and (24). The vector control apparatus also includes a second coordinate converter 50 which senses two excitation currents iu and iv supplied from the inverter 40 to the induction motor IM and converts them into γ- and δ-axis primary current signals i1γ and i1δ. The γ-axis primary current i1γ is supplied to a summing circuit 51 which subtracts it from the γ-axis primary current command signal i1γ* to provide an error signal. A proportional plus integral operation is provided on this error signal in a proportional plus integral circuit 52 to produce a γ-axis primary voltage change signal Δv1γ. The signal Δv1γ is applied to a summing circuit 56 which adds it to the command signal v1γ* to correct the command signal viγ*. The corrected signal viγ is applied to a coordinate transformer 58. Similarly, the δ -axis primary current signal i1δ is fed to a summing circuit 53 which subtracts it from a δ-axis primary current command signal i1δ* to provide an error signal. A proportional plus integral operation is provided on this error signal in a proportional plus integral circuit 54 to produce a δ-axis primary voltage change signal Δv1δ. The signal Δv1δ is then applied to a summing circuit 57 which adds it to the δ-axis primary voltage command signal v1δ* to correct the command signal v1δ*. The corrected signal v1δ is applied to the coordinate transformer 58. The coordinate transformer 58 receives the corrected signals v1γ and v1δ and produces a signal |V1| indicative of the magnitude of the primary voltage vector V1 and a value  indicative of the phase angle of the primary voltage vector V1 with respect to the γ-axis. The value |V1| is fed directly to the PWM circuit 41, whereas the value  is fed to the PWM circuit 41 through a summing circuit 59 where it is added to an value ϑ (=ω0·t) to be described later. The PWM circuit 51 converts the values |V1| and  + ϑ into primary voltage command signals causing the inverter 40 to produces three-phase excitation currents iu, iv and iw to the induction motor IM. The numeral 60 designates a secondary resistance change calculation circuit which utilizes the command signal λ2d*/M* fed thereto from the calculation circuit 43, the command signal i1d* fed thereto from the calculation circuit 44, the command signal i1q* fed thereto form the proportion plus integral circuit 46, the primary voltage angular velocity ω0 fed thereto from a summing circuit to be described later, the command signal i1γ* fed thereto from the first coordinate converter 47 and the command signal Δi1δ fed thereto from the proportional plus integral circuit 54 to calculate a secondary resistance change K from Equation (53). The calculated secondary resistance change K is fed to a slip angle frequency calculation circuit 61 which utilizes it, along with the command signals λ2d*/M* and i1q*, to calculate a slip frequency ωs from Equation (65). If a digital computer is used to make calculations at the circuits of Fig. 7, the slip frequency ωs is calculated as follows: The secondary resistance change K and the slip frequency ωs are calculated in synchronism with a series of clock pulses. The slip factor calculation circuit 61 uses the secondary resistance value R2n-1 calculated in the last or (n-1)th cycle of execution of the program to calculate the secondary resistance value S2n in the present or nth cycle of execution of the program. Assuming now that Kn is the secondary resistance change value calculated in the nth cycle of execution of the program, R2n is the secondary resistance value calculated in the nth cycle of execution of the program, and a predetermined value R2* is assigned to an initial value R20 for the secondary resistance value R2n, successive calculations are performed as follows: First calculation: R21 = (1 + K1) · R20 =(1 + K1) · R2* Second calculation: R22 = (1 + K2) · R21 = (1 + K2) · (1+K1) · R2* · · · Nth calculation: R2n = (1 + Kn) · R2(n - 1) = (1 + Kn) · (1 + Kn - 1) ··· (1 + K1) · R2* Assuming that ωsn is the slip frequency value calculated in the nth cycle of execution of the program, the value ωsn is given as ωsn = (1 + Kn) · ωs(n - 1) The value ωs(n-1) is calculated and stored in the (n-1)th cycle of execution of the program and it is used, along with the value Kn, to calculate the value ωsn from Equation (66). The initial value ωs1 for the slip frequency ωs is given as ωs1 = (1 + K1) · R2*(1/L2*) · i1q*/(λ2d*/M*) The calculated slip frequency ωs* is fed to the summing circuit 62 which adds it to the sensed rotor angular velocity ωr to provide a primary voltage angular frequency command signal ω0 which is then fed to an integrator circuit 63. The integrator circuit converts it into an angular position signal ϑ indicative of the angular position of the secondary flux. The angular position signal ϑ is then fed to the summing circuit 64. The numeral 65 designates an identification circuit operable in response to an output from a comparator 66. The comparator 66 compares the torque command signal i1q* with a predetermined value, for example, equal to 5% of the rated torque current. When the torque command signal i1q* is less than the predetermined value, the induction motor IM is idling and the comparator 66 produces a command signal causing the identification circuit 65 to operate. The command signal is also applied to disable the secondary resistance change calculation circuit 60 since the vector control is subject to no influence of the secondary resistance change while the induction motor IM is idling. The identification circuit 65 measures the γ-axis primary voltage change signal Δv1γ and the d-axis primary current command signal i1d* during induction motor idling operation and utilizes them to calculate a primary resistance change A1 from Equation (62). The calculated primary resistance change A1 is utilized to identify the primary resistance R1. The identification circuit 65 also measures the δ-axis primary voltage signal Δv1δ, the primary voltage angular frequency ω0 and the signal λ2d*/M* and utilizes them to calculate an excitation inductance change AM from Equation (63). The calculated excitation inductance change AM is used to identify the excitation inductance M²/L2. In order to calculate the γ-axis primary voltage command v1γ* , from Equation (25), with regard to the differentiating operator P, the calculation circuit 48 may be arranged to use the term R1*·(1 + Lσ/R1·P) in place of the term R1*. This is effective to enhance the vector control for control accuracy and response time. Referring to Fig. 8, there is shown a second embodiment of the vector control apparatus of the invention. The second embodiment is similar to that of the first embodiment except for the arrangement of the proportional plus integral circuits 70 and 74. Like reference numerals have been applied to Fig. 8 with respect to the equivalent components shown in Fig. 7. With the secondary flux change being ignored, the following equations can be derived from Equation (19): v1γ = R1 · i1γ + Lσ · P · i1γ + M²/L2 · ω₀ · (λ2d/M) · sinψ v1δ = Lσ · ω₀ · i1γ + Lσ · P · i1δ + M²/L2 · ω₀ · (λ2d/M) · cosψ As can be seen from Equations (67) and (68), each of the v1γ and v1δ changes by a value corresponding to the time rate of change of the corresponding one of the v1γ and v1δ with a rapid change in the primary current. That is, the δ-axis primary voltage change contains a component related to the secondary resistance change and also a component related to the time rate of change of the primary current, whereas the γ-axis primary voltage change contains a component related to the primary resistance and excitation inductance changes and also a component related to the time rate of change of the primary current. In this embodiment, both of (1) the primary voltage changes (Δv1γ and Δv1δ) which contain the terms Lσ·P·i1γ and Lσ·P·i1δ, and (2) the primary voltage changes (Δv1 I and Δv1δI) which do not contain the terms Lσ·P·i1 and Lσ·P·i1δ are calculated. The primary voltage changes Δv1γ and Δv1δ are used to control the primary voltage. The primary voltage changes Δv1γI and Δv1δI are used to correct the secondary resistance change and identify the primary resistance. The proportional plus integral circuit 70 includes a proportional element 71, an integral element 72 and a summing circuit 73. The proportional element 71 calculates a value (i1γ* - i1γ) x Lσ/Ts corresponding to Lσ·P·i1γ where Ts is the calculation time period. The integral element 72 integrates the value (i1γ* - i1γ). The integrated value is added to the calculated value (i1γ* - i1γ) x Lσ/Ts in the summing circuit 73. The added value Δv1γ is fed to the summing circuit 56. Similarly, the proportional plus integral circuit 74 includes a proportional element 75, an integral element 76 and a summing circuit 77. The proportional element 75 calculates a value (i1δ* -i1δ) x Lσ/Ts corresponding to Lσ·P·i1δ. The integral element 76 integrates the value (i1δ* - i1δ). The integrated value is added to the calculated value (i1δ* - i1δ) x Lσ/Ts in the summing circuit 77. The added value Δv1δ is fed to the summing circuit 57. The values (i1γ* - i1γ)/Ts and (i1δ* - i1δ)/Ts are calculated by differentiating elements. With this arrangement, the values Δv1γI and Δv1δ1 are free from the influence of a sudden primary current change. It is, therefore, possible to improve the accuracy with which the secondary resistance is corrected and the primary current is identified. Referring to Fig. 9, there is shown a modified form of the second embodiment where the δ-axis primary voltage change signal Δv1δI is fed to the secondary resistance change calculation circuit 60 through a filter 78 which provides a time lag of first order to the δ-axis primary voltage change signal Δv1δI. Since the δ-axis voltage change signal Δv1δI contains ripples particularly at low frequencies, it is desirable to remove the ripples to provide a stable secondary resistance change correction. The time constant of the filter 78A is varied in inverse proportion to the primary voltage angular frequency ω0. The filter transfer function G(S) is represented as G(S) = 1/1 + ST1 where T1 = 1/f0 = 2π/ω0, T1 is the time constant of the filter, f0 is the output frequency of the inverter 40, and ω0 is the primary angular frequency. For this purpose, a time constant setting circuit 79 receives a primary voltage angular frequency signal ω0 from the summing circuit 62 to vary the time constant of the filter 78 in inverse proportion to the primary voltage angular frequency ω0. It is desirable to provide a limiter 79a between the filter 78 and the time constant setting circuit 79 for varying the filtering effect according to the induction motor speed. Referring to Fig. 10, there is shown a third embodiment of the vector control apparatus of the invention. The third embodiment is similar to that of the first embodiment except that the secondary resistance change calculation circuit 60 is removed and replaced with a proportional plus integral circuit 80. Like reference numerals have been applied to Fig. 10 with respect to the equivalent components shown in Fig. 7. The proportional plus integral circuit 80 receives an input from a summing circuit 81 which subtracts the δ-axis primary voltage change Δv1δ fed from the proportional plus integral circuit 54 from a δ-axis primary voltage change command value Δv1δ* (= 0). The proportional plus integral circuit 80 calculates a slip frequency change Δωs from the present slip frequency command value ωs*. The calculated slip frequency change Δωs is fed from the proportional plus integral circuit 80 through a switch 82 to a summing circuit 83 where it is added to the slip frequency command value ωs* fed from the slip frequency calculation circuit 61. The switch 82 opens to interrupt the signal from the proportional plus integral circuit 80 to the summing circuit 83 in response to the command signal from the comparator 66, that is, when the induction motor IM is idling. The slip frequency calculation circuit 61 calculates the slip frequency command value ωs* from the following equation with the second resistance R2 being assumed to be unchanged from its command value: ωs = R2* L2* · i1q* λ2d*/M* The slip frequency command value ωs* is fed from the summing circuit 83 to the summing circuit 62. With this arrangement, the slip frequency command value is automatically corrected according to the secondary resistance. It is to be understood that the proportion plus integral circuits 52 and 54 may be replaced with the proportional plus integral circuit 70 and 74 of Fig. 8 in order to improve the accuracy with which the secondary resistance is corrected and the primary current is identified. Referring to Fig. 11, there is shown a modified form of the third embodiment. The δ-axis primary voltage change Δv1δ caused by a secondary resistance change is given by Equation (50). As can be seen from Equation (50), the δ-axis primary voltage change Δv1δ changes in direct proportion to the primary angular frequency ω0. For this reason, the primary angular frequency ω0 and thus the δ-axis primary voltage change Δv1δ are very small in a low frequency region or a motor locked condition. Therefore, the secondary resistance correction response is slow since the values Δv1δ and Δv1δI are very small. This modification improves the response time by providing a gain control circuit 84 which varies the gain Kp of the proportional plus integral circuit 80 in inverse proportion to the primary angular frequency ω0 as Kp = Kp* x ωOTRQ/ω0 where Kp* is the gain of the proportional plus integral circuit 80 at ω0TRQ, and 0TRQ is the ground angular frequency. It is preferable to enhance the operation stability of the proportional plus integral circuit 80 by providing a limiter 85 which limits the gain Kp between lower and upper limits Kp1 and Kp2. The primary angular frequency ω0 is set at ω0OTRQ and the gain Kp is set at Kp* if the primary angular frequency ω0 is in a steady output region. Referring to Fig. 12, there is shown a fourth embodiment of the vector control apparatus of the invention. The fourth embodiment is similar to that of the third embodiment except that a multiplying circuit 90 is provided. Like reference numerals have been applied to Fig. 12 with respect to the equivalent components shown in Fig. 10. When a sudden change occurs in the torque current command signal i1q* or the excitation current command signal λ2d*/M*, the slip frequency ωs will change. With the use of a proportional plus integral circuit arranged to produce a slip frequency change signal Δωs, the slip frequency change signal should change with a change in the command signal i1q* or λ2d*/M*. For this reason, the correction for secondary resistance changes has a slow response to a change in the torque current command signal i1q* or the excitation current command signal λ2d*/M*. This problem can be eliminated by arranging the secondary resistance correcting circuit in a manner to directly output the secondary resistance change K. Using the secondary resistance change K, the slip frequency is expressed as ωs = (1 + K) · R2* L2* · i1q* λ2d*/M* = R2* L2* · i1q* λ2d*/M + K · R2* L2* · i1q* λ2d*/M = ωs* + Δωs Assuming that the secondary resistance change K is a constant, the slip frequency change Δωs changes with a change in the command signal i1q* or λ2d*/M*, as can be seen from Equation (70). The proportional plus integral circuit 80 receives an input from the summing circuit 81 which subtracts the δ-axis primary voltage change Δv1δ fed from the proportional plus integral circuit 54 from a δ-axis primary voltage change command value Δv1δ* (= 0). The proportional plus integral circuit 80 calculates a secondary resistance change K. The calculated secondary resistance change K is fed from the proportional plus integral circuit 80 through the switch 82 to the multiplying circuit 90 where it is multiplied by the slip frequency command signal ωs* fed from the slip frequency calculation circuit 61. The product Δωs = K x ωs* is added to the slip frequency command signal ωs* fed from the slip frequency calculation circuit 61 in the summing circuit 83. The output of the summing circuit 83 is coupled to the summing circuit 62. The switch 82 opens to interrupt the signal from the proportional plus integral circuit 80 to the multiplying circuit 90 in response to the command signal from the comparator 66, that is, when the induction motor IM is idling. Referring to Fig. 13, there is shown a modification of the fourth embodiment. In this modification, the proportion plus integral circuits 52 and 54 are replaced with the proportional plus integral circuit 70 and 74 of Fig. 8 in order to improve the accuracy with which the secondary resistance is corrected and the primary current is identified. Referring to Fig. 14, there is shown another modified form of the fourth embodiment. The δ-axis primary voltage change Δv1δ caused by a secondary resistance change is given by Equation (50). As can be seen from Equation (50), the δ-axis primary voltage change Δv1δ changes in direct proportion to the primary angular frequency ω0. For this reason, the primary angular frequency ω0 and thus the δ-axis primary voltage change Δv1δ are very small in a low frequency region or a motor locked condition. Therefore, the secondary resistance correction response is slow since the values Δv1δ and Δv1δI are very small. This modification improves the response time by providing a gain control circuit 91 which varies the gain Kp of the proportional plus integral circuit 80 in inverse proportion to the primary angular frequency ω0 as Kp = Kp* x ωOTRQ/ω0 where Kp* is the gain of the proportional plus integral circuit 80 at ω0TRQ, and ω0TRQ is the ground angular frequency. It is preferable to enhance the operation stability of the proportional plus integral circuit 80 by providing a limiter 92 which limits the gain Kp between lower and upper limits Kp1 and Kp2. The primary angular frequency ω0 is set at ω0TRQ and the gain Kp is set at Kp* if the primary angular frequency ω0 is in a steady output region. As described in connection with Figs. 11 and 14, The δ-axis primary voltage change Δv1δ caused by a secondary resistance change is given by Equation (50). As can be seen from Equation (50), the δ-axis primary voltage change Δv1δ changes in direct proportion to the primary angular frequency ω0. For this reason, the primary angular frequency ω0 and thus the δ-axis primary voltage change Δv1δ are very small in a low frequency region or a motor locked condition. Thus, the secondary resistance correction response is slow since the values Δv1δ and Δv1δI are very small. It is, therefore, desired to increase the accuracy with which the secondary resistance change correction is made and to shorten the identification time. As can be seen from Equation (50), the δ-axis primary voltage change Δv1δ is in direct proportion to the primary voltage angular frequency ω0. Since the primary voltage angular frequency ω0 is equal to the slip frequency s in a motor clocked condition (ωr = 0), the δ-axis primary voltage change Δv1δ is in direct proportion to the slip frequency ωs. The slip frequency command value ωs* is given as ωs* = R2* L2* · i1q* λ2d*/M* When the torque current command value i1q* is small (at low load conditions), the slip frequency command value ωs* and thus the δ-axis primary voltage change Δv1δ are small. A mechanical brake, which is normally used with the induction motor IM, permits the induction motor IM to drive in a motor locked condition. If the torque current command value i1q* is set at a great value (for example, 50% to 100% of its maximum value) with the application of braking to drive the induction motor in a motor locked condition, the slip frequency command value ωs* and thus the δ-axis primary voltage change Δv1δ will be great. The great δ-axis primary voltage change Δv1δ is used to calculate the secondary resistance change from Equation (53) or operate the proportional plus integral circuit 80 so as to increase the accuracy of the secondary resistance change correction and to shorten the identification time. Fig. 15 is a flow diagram of the programming of the digital computer used in the induction motor vector control apparatus to calculate the secondary resistance change. The computer program is entered at the point 102. At the point 104 in the program, braking is applied to the induction motor IM. At the point 106 in the program, the torque current command value i1q* is set at a great value (for example, 50% to 100% of its maximum value). At the point 108 in the program, the value (1 + K) is calculated as a function of the δ-axis primary voltage change Δv1δ from Equation (53). At the point 110 in the program, the calculated value (1 + K) is held. The held value is used as an initial value of the secondary resistance change when the induction motor IM is returned into a normal operating condition. The program then proceeds to the end point 112. Fig. 16 is a flow diagram of the programming of the digital computer used in the induction motor vector control apparatus to operate the proportional plus integral circuit 80. The computer program is entered at the point 202. At the point 204 in the program, braking is applied to the induction motor IM. At the point 206 in the program, the torque current command value i1q* is set at a great value (for example, 50% to 100% of its maximum value). At the point 208 in the program, the proportional plus integral circuit 80 is operated with the δ-axis primary voltage change Δv1δ is applied to the input of the proportional plus integral circuit 80. At the point 210 in the program, a counter is set. At the point 212 in the program, a determination is made as to whether or not the counter accumulates a count corresponding to a time required for the identification. If the answer to this question is "yes", then the program proceeds to the point 214. Otherwise, the program is returned to the point 212. At the point 214 in the program, the output of the proportional plus integral circuit 80 is held. The held secondary resistance change is used as an initial value of the secondary resistance change when the induction motor IM is returned into a normal operating condition. Following this, the program proceeds to the end point 216.

Claims: An apparatus employing a rotating Cartesian coordinate system (d, q) having a d-axis and q-axis, the d-axis being held in coincidence with a secondary flux of the induction motor (IM), for vector control of an adjustable-speed induction motor, comprising: means (40) for applying a primary current and voltage to drive the induction motor; means (PP, 42) sensitive to an induction motor angular velocity for producing an induction motor angular velocity value ωr; means (43) for producing a d-axis secondary flux command value λ2d*; means (44) for calculating a d-axis primary current command value i1d* as a function of the d-axis secondary flux command value λ2d* and a secondary time constant command value L2*/R2*; means (46) for producing a q-axis primary current command value i1q*; means (47) employing a rotating Cartesian coordinate system (γ, δ) having a γ-axis and δ-axis, the γ-axis being held in coincidence with the primary current I1 for calculating a γ-axis primary current command value i1γ* and a phase angle ψ of the γ-axis with respect to the d-axis as a function of the primary current command values i1d* and i1q*; means (48) for calculating the γ- and δ-axis primary voltage command values v1γ* and v1δ* as a function of the γ-axis primary current command value i1γ*, the phase angle ψ, the d-axis secondary flux command value λ2d*, and a primary voltage angular frequency ω0; means (50) for sensing the primary current and converting the sensed primary current into γ- and δ-axis primary current values i1γ and i1δ; means (51, 52) for caluclating a γ-axis primary voltage change Δv1γ as a function of the γ-axis primary current value i1γ and the γ-axis primary current command value i1γ*; means (53, 54) for calculating a δ-axis primary voltage change Δv1δ as a function of the δ-axis primary current value i1δ and a δ-axis primary current command value i1δ*; means (56) for adding the γ-axis primary voltage change Δv1γ to the γ-axis primary voltage command value v1γ* to produce a γ-axis primary voltage command signal v1γ; means (57) for adding the δ-axis primary voltage change Δv1δ to the δ-axis primary voltage command value v1δ* to produce a δ-axis primary voltage command signal v1δ; means (60) for calculating a secondary resistance change (K) as a function of the d-axis primary current command value i1d*, the q-axis primary current command value i1q*, the γ-axis primary current command value i1γ*, the d-axis secondary flux command value λ2d*, the primary voltage angular frequency ω0, the excitation inductance command value M*, the secondary inductance command value L2* and the δ-axis primary voltage change Δv1δ according to the equation 1 + K = M*² L2* · ω₀ · λ2d* M* · i1d* · i1q*² + i1γ* · i1q*² · Δv1δ M*² L2* · ω₀ · λ2d* M* · i1d* · i1q*² - i1γ* · i1d*² · Δv1δ means (61) for correcting the secondary time constant command value L2*/R2* based upon the secondary resistance change; means (61) for calculating a slip frequency ωs as a function of the q-axis primary current command value i1q*, the d-axis secondary flux command value λ2d*, and the corrected secondary time constant command value; means (62) for adding the slip frequency ωs to the induction motor angular velocity value ωr to produce the primary voltage angular frequency ω0; and means (58, 41) for controlling the motor driving means to adjust the primary voltage based upon the primary voltage command signals v1γ and v1δ, and the primary voltage angular frequency ω0. The induction motor vector control apparatus as claimed in claim 1, wherein the δ-axis primary voltage change command value Δv1δ is zero. The induction motor vector control apparatus as claimed in claim 1, wherein the γ- and δ-axis primary voltage command value calculating means calculates the γ- and δ-axis primary voltage command values v1γ* and v1δ* as v1γ* = R1 · i1γ* + M² L2 · ω₀ · λ2d* M · sin ψ v1δ* = Lσ · ω₀ · i1γ* + M² L2 · ω₀ · λ2d* M · cos ψ where R1 is the primary resistance of the induction motor, M is the excitation inductance of the induction motor, and L2 is the secondary inductance of the induction motor, and wherein the induction motor vector control apparatus preferably further includes means for calculating a change A1 in the primary resistance R1 as a function of the primary resistance R1, the γ-axis primary voltage change Δv1γ, and the d-axis primary current command value i1d* when the induction motor is idling, means for correcting the primary resistance R1 based upon the calculated primary resistance change A1, means for calculating a change AM in the excitation inductance M as a function of the excitation inductance M, the δ-axis primary voltage change Δv1δ, the primary voltage angular frequency ω0, and the d-axis secondary flux command value λ2d* when the induction motor is idling, and means for correcting the excitation inductance M based upon the calculated excitation inductance change AM. The induction motor vector control apparatus as claimed in claim 1, wherein the γ- and δ-axis primary voltage command value calculating means calculates the γ- and δ-axis primary voltage command values v1γ* and v1δ* as v1γ* = R1(1 + Lσ R1 · P)i1γ* + M² L2 · ω₀ · λ2d* M · sin ψ v1δ* = Lσ · ω₀ · i1γ* + M² L2 · ω₀ · λ2d* M · cos ψ where R1 is the primary resistance of the induction motor, M is the excitation inductance of the induction motor, L2 is the secondary inductance of the induction motor, and P is a differentiating operator, and wherein the induction motor vector control apparatus preferably further includes means for calculating a change A1 in the primary resistance R1 as a function of the primary resistance R1, the γ-axis primary voltage change Δv1γ, and the d-axis primary current command value i1d* when the induction motor is idling, means for correcting the primary resistance R1 based upon the calculated primary resistance change A1, means for calculating a change AM in the excitation inductance M as a function of the excitation inductance M, the δ-axis primary voltage change Δv1δ, the primary voltage angular frequency ω0, and the d-axis secondary flux command value λ2d* when the induction motor is idling, and means for correcting the excitation inductance M based upon the calculated excitation inductance change AM. The induction motor vector control apparatus as claimed in claim 1, comprising: means (53) for subtracting the δ-axis primary current i1δ from the δ-axis primary current command value i1δ* to produce a difference (i1δ* - i1δ); means (75) for multiplying the difference (i1δ* - i1δ) by a leakage inductance Lσ to produce a multiplied value (i1δ - i1δ)·Lσ; means (76) for integrating the difference (i1δ* - i1δ) to produce an integrated value Δv1δI; means (77) for adding the multiplied value to the integrated value to produce a δ-axis primary voltage change Δv1δ; and wherein the δ-axis primary voltage change used by the means (60) for calculating a secondary resistance change is the integrated value Δv1δI. The induction motor vector control apparatus as claimed in claim 5, wherein the δ-axis primary voltage change command value Δv1δ is zero. The induction motor vector control apparatus as claimed in claim 5, further including filter means (78) having a time constant to provide a time lag of first order to the integrated value Δv1δI fed to the secondary resistance change calculating means, and means (79) for controlling the time constant of the filter means in inverse proportion to the primary voltage angular frequency ω0. The induction motor vector control apparatus as claimed in claim 5, wherein the γ-axis primary voltage change calculating means (70) includes: means (51) for subtracting the γ-axis primary current i1γ from the γ-axis primary current command value i1γ* to produce a difference (i1γ* - i1γ); means (71) for multiplying the difference (i1γ* - i1γ) by a leakage inductance Lσ to produce a multiplied value (i1γ* - i1γ)·Lσ; means (72) for integrating the difference (i1γ* - i1γ) to produce an integrated value Δv1γI; and means (73) for adding the multiplied value to the integrated value to produce the γ-axis primary voltage change Δv1γ, and wherein the γ- and δ-axis primary voltage command value calculating means (70, 74) preferably calculates the γ- and δ-axis primary voltage command values v1γ* and v1δ* as v1γ* = R1 · i1γ* + M² L2 · ω₀ · λ2d* M · sin ψ v1δ* = Lσ · ω₀ · i1γ* + M² L2 · ω₀ · λ2d* M · cos ψ where R1 is the primary resistance of the induction motor, M is the excitation inductance of the induction motor, and L2 is the secondary inductance of the induction motor. The induction motor vector control apparatus as claimed in claim 8, further including means for calculating a change A1 in the primary resistance R1 as a function of the primary resistance R1, the integrated value Δv1γI, and the d-axis primary current command value i1d* when the induction motor is idling, means for correcting the primary resistance R1 based upon the calculated primary resistance change A1, means for calculating a change AM in the excitation inductance M as a function of the excitation inductance M, the integrated value Δv1δI, the primary voltage angular frequency ω0, and the d-axis secondary flux command value λ2d* when the induction motor is idling, and means for correcting the excitation inductance M based upon the calculated excitation inductance change AM, wherein the induction motor vector control apparatus preferably further includes filter means having a time constant to provide a time lag of first order to the integrated value Δv1δI fed to the secondary resistance change calculating means, and means for controlling the time constant of the filter means in inverse proportion to the primary voltage angular frequency ω0. The induction motor vector control apparatus as claimed in claim 8, wherein the γ- and δ-axis primary voltage command value calculating means calculates the γ- and δ-axis primary voltage command values v1γ* and v1δ* as v1γ* = R1(1 + Lσ R1 · P)i1γ* + M² L2 · ω₀ · λ2d* M · sin ψ v1δ* = Lσ · ω₀ · i1γ* + M² L2 · ω₀ · λ2d* M · cos ψ where R1 is the primary resistance of the induction motor, M is the excitation inductance of the induction motor, L2 is the secondary inductance of the induction motor, and P is a differentiating operator, wherein the induction motor vector control apparatus preferably further includes means for calculating a change A1 in the primary resistance R1 as a function of the primary resistance R1, the integrated value Δv1γI and the d-axis primary current command value i1d* when the induction motor is idling, means for correcting the primary resistance R1 based upon the calculated primary resistance change A1, means for calculating a change AM in the excitation inductance M as a function of the excitation inductance M, the integrated value Δv1δI, the primary voltage angular frequency ω0, and the d-axis secondary flux command value λ2d* when the induction motor is idling, and means for correcting the excitation inductance M based upon the calculated excitation inductance change AM, and wherein the induction motor vector control apparatus preferably further includes filter means having a time constant to provide a time lag of first order to the integrated value Δv1δI fed to the secondary resistance change calculating means, and means for controlling the time constant of the filter means in inverse proportion to the primary voltage angular frequency ω0. An apparatus employing a rotating Cartesian coordinate system (d, q) having a d-axis and q-axis, the d-axis being held in coincidence with a secondary flux of the induction motor (IM), for vector control of an adjustable-speed induction motor, comprising: means (40) for applying a primary current and voltage to drive the induction motor; means (PP, 42) sensitive to an induction motor angular velocity for producing an induction motor angular velocity value ωr; means (43) for producing a d-axis secondary flux command value λ2d*; means (44) for calculating a d-axis primary current command value i1d* as a function of the d-axis secondary flux command value λ2d* and a secondary time constant command value L2*/R2*; means (46) for producing a q-axis primary current command value i1q*; means (47) employing a rotating Cartesian coordinate system (γ, δ) having a γ-axis and δ-axis, the γ-axis being held in coincidence with the primary current I1 for calculating a γ-axis primary current command value i1γ* and a phase angle ψ of the γ-axis with respect to the d-axis as a function of the primary current command values i1d* and i1q*; means (48) for calculating the γ- and δ-axis primary voltage command values v1γ* and v1δ* as a function of the γ-axis primary current command value i1γ*, the phase angle ψ, the d-axis secondary flux command value λ2d*, and a primary voltage angular frequency ω0; means (50) for sensing the primary current and converting the sensed primary current into γ- and δ-axis primary current values i1γ and i1δ; means (51, 52) for caluclating a γ-axis primary voltage change Δv1γ as a function of the γ-axis primary current value i1γ and the γ-axis primary current command value i1γ*; means (53, 54) for calculating a δ-axis primary voltage change Δv1δ as a function of the δ-axis primary current value i1δ and a δ-axis primary current command value i1δ*; means (56) for adding the γ-axis primary voltage change Δv1γ to the γ-axis primary voltage command value v1γ* to produce a γ-axis primary voltage command signal v1γ; means (57) for adding the δ-axis primary voltage change Δv1δ to the δ-axis primary voltage command value v1δ* to produce a δ-axis primary voltage command signal v1δ; means (61) for calculating a slip frequency command value ωs* as a function of the q-axis primary current command value i1q* and the d-axis secondary flux command value λ2d*; means (81) for calculating a difference of the δ-axis primary voltage change Δv1δ from a δ-axis primary voltage change command value; means (80) for calculating a slip frequency change Δωs as a function of the calculated difference; means (83) for adding the slip frequency change Δωs to the slip frequency command value ωs* to produce a slip frequency ωs; means (62) for adding the slip frequency ωs to the induction motor angular velocity value ωr to produce the primary voltage angular frequency ω0; and means (58, 41) for controlling the motor driving means to adjust the primary voltage based upon the primary voltage command signals v1γ and v1δ, and the primary voltage angular frequency ω0. The induction motor vector control apparatus as claimed in claim 11, wherein the δ-axis primary voltage change command value Δv1δ is zero, and/or wherein the slip frequency change calculating means is a proportional plus integral amplifier having a variable gain, and/or wherein the induction motor vector control apparatus further includes means for controlling the gain of the proportional plus integral amplifier in inverse proportion to the primary voltage angular frequency ω0, and/or wherein the γ- and δ-axis primary voltage command value calculating means calculates the γ- and δ-axis primary voltage command values v1γ* and v1δ* as v1γ* = R1 · i1γ* + M² L2 · ω₀ · λ2d* M · sin ψ v1δ* = Lσ · ω₀ · i1γ* + M² L2 · ω₀ · λ2d* M · cos ψ where R1 is the primary resistance of the induction motor, M is the excitation inductance of the induction motor, and L2 is the secondary inductance of the induction motor, wherein the induction motor vector control apparatus optionally further includes means for calculating a change A1 in the primary resistance R1 as a function of the primary resistance R1, the γ-axis primary voltage change Δv1γ, and the d-axis primary current command value i1d* when the induction motor is idling, means for correcting the primary resistance R1 based upon the calculated primary resistance change A1, means for calculating a change AM in the excitation inductance M as a function of the excitation inductance M, the δ-axis primary voltage change Δv1δ, the primary voltage angular frequency ω0, and the d-axis secondary flux command value λ2d* when the induction motor is idling, and means for correcting the excitation inductance M based upon the calculated excitation inductance change AM. The induction motor vector control apparatus as claimed in claim 11, wherein the γ- and δ-axis primary voltage command value calculating means calculates the γ- and δ-axis primary voltage command values v1γ* and v1δ* as v1γ* = R1(1 + Lσ R1 · P)i1γ* + M² L2 · ω₀ · λ2d* M · sin ψ v1δ* = Lσ · ω₀ · i1γ* + M² L2 · ω₀ · λ2d* M · cos ψ where R1 is the primary resistance of the induction motor, M is the excitation inductance of the induction motor, L2 is the secondary inductance of the induction motor, and P is a differentiating operator, wherein the induction motor vector control apparatus preferably further includes means for calculating a change A1 in the primary resistance R1 as a function of the primary resistance R1, the γ-axis primary voltage change Δv1γI and the d-axis primary current command value i1d* when the induction motor is idling, means for correcting the primary resistance R1 based upon the calculated primary resistance change A1, means for calculating a change AM in the excitation inductance M as a function of the excitation inductance M, the δ-axis primary voltage change Δv1δ, the primary voltage angular frequency ω0, and the d-axis secondary flux command value λ2d* when the induction motor is idling, and means for correcting the excitation inductance M based upon the calculated excitation inductance change AM. The induction motor vector control apparatus as claimed in claim 11, wherein the slip frequency change calculating means includes means for calculating a secondary resistance change K as a function of the calculated difference, and means for multiplying the secondary resistance change K by the slip frequency command ωs* to produce the slip frequency change Δωs. The induction motor vector control apparatus as claimed in claim 11, comprising: means (53) for subtracting the δ-axis primary current i1δ from the δ-axis primary current command value i1δ* to produce a difference (i1δ* - i1δ); means (75) for multiplying the difference (i1δ* - i1δ) by a leakage inductance Lσ to produce a multiplied value (i1δ* - i1δ)·Lσ; means (76) for integrating the difference (i1δ* - i1δ) to produce an integrated value Δv1δI; means (77) for adding the multiplied value to the integrated value to produce a δ-axis primary voltage change Δv1δ; means (80) for calculating a secondary resistance change K as a function of the calculated difference; means (90) for multiplying the calculated difference by the slip frequency command value ωs* to produce a slip frequency change Δωs; and/or wherein the δ-axis primary voltage change command value Δv1δ is zero, and/or wherein the slip frequency change calculating means is a proportional plus integral amplifier having a variable gain, and wherein the induction motor vector control apparatus preferably further includes means for controlling the gain of the proportional plus integral amplifier in inverse proportion to the primary voltage angular frequency ω0.

Inventors: YAMADA TETUO, YAMAMOTO YASUHIRO, YAMADA, TETUO, YAMAMOTO, YASUHIRO
Applicants: MEIDENSHA ELECTRIC MFG CO LTD, KABUSHIKI KAISHA MEIDENSHA
Application year: 1991
Main classification: H02P5
IPC codes: H02P21