FEATURES OF FREQUENCY ANALYSIS OF TORQUE CONTROL LOOP IN AC ELECTRIC DRIVES

In the article, the structural scheme of the synchronous motor torque control channel is shown and mathe-matically described. Passing of a sine signal in the torque control loop is considered, equations for describing the dynamic properties of the current control loop are given. Specific features relating to the frequency characteristics of the channel for controlling of the electromagnetic torque in the adjustable variable speed drive with synchronous electric machines are considered. The control channel is regarded as a linear system with amplitude modulation. A comparative analysis of the experimental and calculated Bode plots of the current and torque control loops is carried out.


Introduction
AC variable speed drives are operated by close loop scheme with the inner loop of the indirect control of the electromagnetic torque and outer loop -speed. In addition, in case there is a direct dependence between the armature current of the motor and its torque in typical DC drive, that simplifies the inner loop setup, in AC electric drives there is no such explicit dependence. The oscillate features of the description of dynamic properties of the electromagnetic torque actuating path in the variable-frequency electric drive with the synchronous and synchronous reluctance machine lower with use of the device.

Theoretical
The control loop of the torque (CLT) in the variable-frequency synchronous electric drive should be carried to multivariate the same systems of regulation with the amplitude modulation of the U in input signal. The sequence of mathematical operations, by means of which signal transmission processes through links of one phase of the synchronous motor stator are described, is illustrated by the block scheme. Here links are connected sequentially: PCGB -phase current generated block; CLC -the control loop of the stator phase current; SM -the synchronous motor [1].
In a mathematical model of PCGB after the operation of multiplication (modulating action) of the U in input signal on a sine function is the U ref reference signal on the motor input CLC: where ω 1 -angular frequency of a modulating signal equal to the voltage set frequency to the frequency converter by the angular position sensor of the motor rotor; t -time.
The modulating signal can be generated by the angular position sensor of the motor rotor directly that is seen in diagrams of the torque vector control formation [2] or via coordinate conversions of variables [3]. In the first case the spatial resultant vector of current of the stator is created in polar coordinates, in the second -in orthogonal. Then the signal U ref will be transformed by a control loop of the stator phase current (CLC) to the stator current I [4]. Connection between these values can be described by the CLC transfer function: The link SM considers interaction of the stator phase current i with the excitation rotor. The electromagnetic torque of the motor M i resulting from this interaction is considered as an outcome of the current (i) secondary multiplication (demodulation) by the sinusoidal value of the same frequency, as to PCGB, but with the γ -degree phase shift: M i = L m I r i sin (ω 1 t + γ) , where L m -the maximum value of the mutual induction coefficient between the rotor winding and one of the stator phases; I r -rotor current of the synchronous motor; γ -the angle of shift between the sinusoidal quantity given on inputs of PCG Band SM [5].

Practical
For application to the analysis of processes in the variable-frequency AC electric drive of the standard device of the Bode plot and transfer functions will consider passing of a sine signal through links of one phase of the control loop of the torque [6].
Let give to the control loop input of the torque CLT a sine signal of the frequency ω, i.e. U in = U m sin ω 1 t, Then to the output of the first unit of multiplication (a link PCGB) We see that the first harmonic of frequency ω is lost, but two summary harmonicas appear (ω + ω 1 ) together with the difference ω -ω 1 frequencies.
Here A CLC1 (ω -ω 1 ) and A CLC2 (ω + ω 1 ) -values of the Bode plot of the gain of the CLC link with frequencies (ω -ω 1 ) and (ω + ω 1 ); ϕ 1 (ω -ω 1 ) and ϕ 2 (ω + ω 1 ) -the values of the phase-shift-frequency of the CLC link with the same frequencies. Generally, these harmonicas differ among themselves on the amplitudes and the phase angle only in case when CLC is inertia less, the amplitudes and phases at them identical. After secondary multiplication in the SM link each of harmonicas will again break up into two items, then It follows from the last expression that at output of the single-phase torque control loop the monosine signal U in of the frequency ω will be transformed to the signal containing four harmonicas: two with the basic frequency ω, but with different phase angles, and two harmonicas with the side frequencies (ω − 2ω 1 ) and (ω + 2ω 1 ). In the three-phase variablefrequency electric drive the electromagnetic torque of the synchronous motor is provided as a result of the combined action of three identical single-phase control loops connected in parallel. These control loops correspond to phases A, B and C of the stator and have the general input signal and the adder on the output [7]. Feature of this structure is that signals are multiplied in the links of PCGB and SM by the sinusoidal values forming a three-phase symmetric system among themselves. It leads to the fact that at the system output adder (which shall contain twelve items) two or three of sine curves are mutually equilibrated. In expression for the electromagnetic torque there are two trebled harmonicas of the basic frequency ω. The amplitudes of these harmonicas and the value of their phase shifts are defined by the frequency responses of the CLC link of CLC on the side frequencies (ω-ω 1 ) and (ω + ω 1 ): Let's consider the expression given above for the electromagnetic torque of the synchronous motor in more detail. In the elementary case we read CLC as an inertialess link. In electric drives with a supply of windings of the stator from the gated transformers working in the current source mode, the bandwidth range of frequencies of CLC reaches several thousand radianper second, so, the accepted assumption not only simplifies a pattern of processes, but also slightly distinguishes it from the true one. In case of the inertialess CLC its Bode plot (A CLC = const) and the phase frequency characteristics (φ CLC = 0) [8]. The vector of the resultant torque (M) turns out as the amount of vectors, equal on length (M 1 and M 2 ). At the same time, the vector (M 1 ) advances the vector (U in ) by the angle (γ), and the vector (M 2 ) shifts behind on the same angle. The expression for the torque takes a form: M = 1,5U in L m I r A CLC cosγ, i.e., with a constant amplitude of the sine input signal of U in = const, we have M = const overall the frequency band. If γ = 0, then the directions of a vector of U in and composed by M 1 and M 2 match [9].
The frequency characteristics of the torque control loop, in case of the different fixed values ω 1 . Let's simultaneously compare the values ω 1 and ω c . For descriptive reasons, we will provide items (M 1 and M 2 ) in the vectorial form. Let us remind that these are sinusoidal values of the frequency ω, amplitudes which phase shifts depend on the values of side frequencies (ω-ω 1 ) and (ω + ω 1 ). Let us address the CLT vectorial charts (Fig. 1) when U in vector is accepted to the basic (input) vector, and for a day off -the vector (M) represented as the amount of items (M 1 and M 2 ) [10]. At first, we will consider a case when (ω 1 <<ω c ) that is traced by operation of the electric drive on emphasis, and also in the mode of "creeping" speeds; then (ω ± ω c ≈ ω), therefore A CLC1 (ω -ω 1 ) ≈ A CLC2 (ω + ω 1 ) ≈ A CLC (ω); ϕ 1 (ω -ω 1 ) ≈ ϕ 2 (ω + ω 1 ) ≈ ϕ(ω). The modules of items M 1 (ω-ω 1 ) and M 2 (ω + ω 1 ) are equal, and in case of the frequency change, the amplitude and phase shift equally change (Fig. 1, a).
The frequency responses of CLC and CLT in the relative units match, the expression for the torque takes a form: When ω 1 and ω T are comparable in their values and (ω -ω 1 ) and (ω + ω 1 ) differ significantly, and because of influence of inertialess in the CLC links, it is necessary to consider a limited bandwidth range of the frequencies of the CLC items (M 1 and M 2 ) changing differently as a frequency. Most considerably this difference is shown in the field of the cutoff frequency (ω c ); then in case of frequency rate of tension on the statorclose to the value of this frequency in case of ω 1 ≈ ω c , the difference of frequencies (ω -ω 1 ) lies within the working bandwidth range of CLC frequencies where the vector (M 1 ) changes slightly. The other side frequency (ω + ω 1 ) quits for the right boundary of the uniform passage of frequencies where range of a vector (M 2 ) significantly decreases [11].
As a result, there is |M 1 | > |M 2 |. Therefore, the value and the direction of a resultant vector (M = M 1 + M 2 ) are defined largely by the vector of M 1 which changes a little (Fig. 1, b) [12]. At the resultant range of frequency characteristics, some broad banding of uniform passage of frequencies in case of the reduced values of the Bode plot of CLT is observed, and the phase of frequency characteristic CLT passes higher than the phase of the CLC frequency characteristics. The described phenomenon is shown stronger than frequency rates ω 1 and the angular speed of the motor are closer to the frequency rate of a cutoff (ω c ) than CLC. In the range of very high frequencies of the U in test signal of when ω → ∞, in case of any finite frequency rate (ω 1 ), the frequency responses of CLC and CLT match. Around mid frequencies, when values (ω 1 and ω c ) are close, these characteristics differ, and for their computation, it is necessary to use the above-stated expression [13].

Analysis results
Calculation was executed for CLT in which the CLT closed-loop of each phase of the stator was approximated by an oscillatory link of the second order with the transfer function W CLC (p) = 1/(1 + 2ζT p + T 2 p 2 ).
Design bode plots corresponding to this transfer function are received by substituting of = jω and for convenience are provided to functions of the dimensionless value (ω c ). The bode plot of the gain of CLC and CLT were provided in the relative units. Their values were taken for basic values of the current amplitude and the torque in case of ω = 0 and ω = 0 [14]. The coefficient of damping was accepted equal ζ = 0,5 that corresponds to a standard setup of the CLC closed loop with the relative cutoff frequency ω = 1 and the phase margin ∆ϕ ≈ 50°. The design bode plot of CLC and CLT in case of ω = 0 matches (Fig. 2,  curves 1) and corresponds to the current transformation model of i A , i B , i C in the torque of M.
In case of the increased values ω when the angular speed of the electric drive approaches cutoff frequency (ω ≈ ω ), the Bode plot of CLT changes (Fig. 2,  curves 2). The gain decreases, but the Bode plot of the phase for about a decade around the frequency of the CLC cutoff of passes on 40-50° above that is explained by the dominating impact of the first item in the above-stated expression for the torque. On the Bode plot of CLT in comparison with the Bode plot of the phase of CLC in case of values ω > ω , the increased rise of the phase not only remains, but also increases a little. However, the practical effect from this doesn't manage to be derived as receiving flat gain of CLC on a section with the steep gradient of value of gain requires unfairly big voltage-forcing of power supplies of the stator circuit [15].
The described comparison of the Bode plot of CLC and CLT through all the range of frequencies shows the following. First, it is necessary to set a system of the electric drive at small speeds or even in case of the slowed-down motor as these modes are characterized by the smallest marginal stability. Secondly, at the stage of an approximate choice of structure and parameters of the correcting links in the electric drive it is possible to replace internal CLT with CLC. This result seems very paradoxical but is full of decency: on the one hand, it simplifies calculation procedures, on the other -the inaccuracy of measurements only increases the "margin of safety" of calculations. At last, it is necessary to observe the principle of division of movements and to set up CLC of phase currents of the stator it is isolated from other actuating paths [16]. The experimental Bode plots were defined for specification of the accepted mathematical model of CLC, the accounting of all significant factors influencing the dynamic characteristics of the electric drive, the conclusion confirmation was obtained during calculations.
Bode plots were defined for CLC of a phase and CLT of the electric drive. The block diagram of the experimental assembly and the characteristic are given in Fig. 3. For convenience of comparing of the options, the relative values of amplification gain in case of low frequencies are accepted as identical [17].
The trial sine signal of the different fixed frequencies in the range from 1 to 10000 rad/s was given from the output terminal of the measurement unit "Vector" [6] on an input of PCGB and further arrived on three parallel operating CLC of phase currents of the stator. Each CLC was executed on the basis of the AI single-phase transistor autonomous inverters (Maxi Maestro 25/127 type). As the SM, the synchro-nous generator BMZ 4,5 was used (4,5 kV•a; 1500 rpm; U ph = 127 V; I ph = 10 A). The phase currents were measured by current sensors (CS) (Lem HY-05-P). In case of determination of the experimental characteristics of each of the local CLC of the stator phase of the SM for an output variable accepted voltage of one of CS, and the input voltage of the measurement unit "Vector" [18].
In case of the experimental determination of Bode plots CLT were used in series in the PLC programmable controller (Atmega 8535) which executed the computing operations corresponding to the SM link, the rotor-position encoder of the RPE (Omron E6C3 AJ5C type) and the digital-analog converter DAC (is an Atmega 8535 part). The value of assessment of the torque M obtained on a DAC output was accepted to an output signal of CLT [19].
The experimental Bode plots of CLC and CLT (Fig. 4) was removed in the diagram with a proportional regulator of the current, and in case of the amplification gain (K CLC = 5) the open-loop circuit for-  med by links of RC, AI, CS (Fig. 3) has quite big (to (2-4)•10 -3 rad/s) band of the uniform passage of frequencies. It allows almost complete weakening of the influence of cross coupling on the nature of the processes caused by existence of mutual induction between stator windings [20]. The experiment also showed the legitimacy of approximation of the CLC of a phase of the stator an oscillatory link of the second order in the range of frequencies at least to (2-4)•10 -3 rad/s. A little smaller band of uniform passage of frequencies observed in CLT in comparison with CLC should be explained with existence of some inertia in the procedure of the digital conversions, which are realized in a PLC [21]. The experimental bode plots of CLC of the stator phase (and, respectively, CLT) in the electric drive with the filed regulated reluctance machine [7] have a band of uniform passage of frequencies about 2-3 times wider (Fig. 4, curves 3). It can be explained by the fact that with increase in number of phases of the winding of the stator (the experiment was made in case of m = 6) and an invariable bore diameter width of a phase zone, number of a phase winding and leakage inductance decreases [22].

Conclusion
In case of the frequency analysis of the dynamic properties of variable speed drive of an alternating current with the synchronous motors, it is convenient to consider CLT as the linear system with amplitude shift keying [2]. Using the principle of division of movements, it is possible to consider independent processes in the phase CLC and CLT. Application of the experimental characteristics allows, without complicating the mathematical description of system of the electric drive, to automatically consider influence of the cross couplings caused by existence of reciprocal magnetic coupling between windings in the motor. The modern inverters executed on threephase converters allow receiving in phase circuits of regulation of current a band of uniform passage of several thousand a radian in a second. The rough estimate of the dynamic properties of the torque control loop of the can be received using only bode plots of control loop of the phase currents of the stator.