Patent Publication Number: US-10333444-B2

Title: System and method for stability control in adjustable speed drive with DC link thin film capacitor

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates generally to adjustable speed drives (ASDs) and, more particularly, to a system and method for stabilizing the output current of ASDs when using small DC link thin film capacitors. 
     In a conventional ASD, an alternating current (AC) power input is converted to a direct current (DC) power on a DC link by a rectifier and then to an AC power output by an inverter. The AC power output has the desired characteristics for operating an AC motor or other AC driven load. Often, a large electrolytic capacitor bank measuring between a few hundred and thousands of microfarads (μF) is used on the DC link to stabilize the DC link voltage and decouple the inverter side motor control of the ASD from the input rectifier operation. These electrolytic capacitor banks have a large capacity for energy storage and keep the DC link voltage fairly steady. Although electrolytic capacitors are effective to stabilize the DC link voltage, they have several drawbacks. 
     As one example, when a front end diode rectifier is used in the ASD, the AC input source current becomes severely distorted by the electrolytic capacitor bank such that low order harmonics pollute the utility grid. These harmonics can cause higher root mean square currents through connected transformers and feeder equipment. Sensitive equipment like instrumentation, computers, and communications systems may fail to function correctly or suffer damage. 
     As another example, electrolytic capacitors dry out and have a limited lifetime, which is a reliability concern. Then, upon replacement of an electrolytic capacitor bank, the capacitors need to be reformed or conditioned if they have been in storage for an extended period of time. This results in an inefficient installation process. The electrolytic capacitors will lose their charge in storage, so they need to be pre-charged. If the electrolytic capacitor banks are not pre-charged before energizing the ASD and the ASD does not have a pre-charge circuit, a high inrush current can flow through the rectifier and into the electrolytic DC link capacitor bank. 
     Because of the drawbacks of using electrolytic capacitors, smaller thin film capacitors are being used as a replacement. Thin film capacitors provide enhanced reliability, improved input current harmonic performance, reduced system size and cost, and out of the box installation with unlimited shelf life. However, thin film capacitors generally cannot provide the stability that electrolytic capacitors can. A significant amount of total harmonic distortion (THD) and DC link voltage ripple may thus be present in the system when thin film capacitors are employed. 
     In order to overcome the instability of the thin film capacitors, stability control strategies have been developed for ASDs. However, these control strategies generally include complicated algorithms that are not intuitive for users. In addition, while the control strategies may make the system more stable, they still do not provide enough stability for the ASD to produce a sine wave current, which is the ASD output for operating motors. Instead, the ASD output may approximate an oscillating wave, for example. 
     It would therefore be desirable to provide a system and method for analyzing and controlling the stability of an ASD with a small DC link thin film capacitor bank such that the ASD is stable under all normal operating conditions and the ASD output approximates a sine wave with low THD and DC link voltage ripple. 
     BRIEF DESCRIPTION OF THE INVENTION 
     Embodiments of the present invention provide a system and method for controlling the stability of an ASD incorporating a small DC Link thin film capacitor. 
     In accordance with one aspect of the invention, a control system for actively damping an output of an ASD having a DC link thin film capacitor is programmed to calculate a d-axis damping coefficient and a q-axis damping coefficient for stabilizing an output of the ASD based at least on a voltage across the DC link thin film capacitor at a steady operating point. The control system is further programmed to extract d-axis and q-axis perturbations in d-axis and q-axis output currents of the ASD using a high pass filter, damp the d-axis perturbation and the q-axis perturbation with the d-axis damping coefficient and the q-axis damping coefficient, respectively, and calculate a damping frequency based on the damped d-axis perturbation and the damped q-axis perturbation. The control system is also programmed to damp an angle of rotation of a reference motor speed command for controlling the ASD using the damping frequency. 
     In accordance with another aspect of the invention, a method of stabilizing an ASD having a DC link thin film capacitor incorporated therein includes receiving, at a controller, input parameters comprising a current output by the ASD from a current sensor and a reference motor speed command and computing, with the controller, a d-axis stabilization factor and a q-axis stabilization factor for stabilizing the ASD based on at least one steady state voltage across the DC link thin film capacitor. The method additionally includes transforming the current output by the ASD into a d-axis current and a q-axis current with the controller, obtaining d-axis and q-axis perturbations in the d-axis and q-axis currents, respectively, with a high pass filter in the controller, and compensating the d-axis perturbation and the q-axis perturbation with the controller using the d-axis stabilization factor and the q-axis stabilization factor, respectively. Furthermore, the method includes computing a feedback frequency with the controller based on the compensated d-axis perturbation and the compensated q-axis perturbation and compensating a reference motor speed command with the feedback frequency to change an angle of rotation for controlling an inverter of the ASD. 
     In accordance with yet another aspect of the invention, an ASD includes an input connectable to an AC source, a rectifier connected to the input to convert an AC power input to a DC power at an output thereof, and a DC link coupled to the rectifier output to receive the DC power therefrom. The DC link has a thin film capacitor and an inductor positioned thereon and is coupled to a DC side of an inverter having an AC side connectable to a motor. The ASD includes at least one current sensor configured to measure currents output from the AC side of the inverter and a control system coupled to the inverter as well. The control system is programmed to determine a d-axis compensation coefficient and a q-axis compensation coefficient designed to stabilize the ASD output currents under all operating conditions based on at least one steady operating point for a set of operating parameters including voltage across the DC link thin film capacitor and a current through the inductor. In addition, the control system is programmed to extract a perturbation in the inverter output currents from the at least one current sensor using a high pass filter, compensate a d-axis component of the perturbation using the d-axis compensation coefficient and a q-axis component of the perturbation using the q-axis compensation coefficient, and determine a compensation frequency by combining the compensated d-axis and q-axis perturbations. Moreover, the control system is programmed to add the compensation frequency to a reference motor speed command to adjust an angle of rotation for controlling the inverter and control the inverter based on the amplitude of the reference motor speed command and the adjusted angle of rotation. 
     Various other features and advantages of the present invention will be made apparent from the following detailed description and the drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawings illustrate preferred embodiments presently contemplated for carrying out the invention. 
       In the drawings: 
         FIG. 1  is a schematic of an ASD system including an ASD having a small DC link thin film capacitor, according to an embodiment of the invention. 
         FIG. 2  is a block diagram of the ASD system of  FIG. 1  including a more detailed view of a controller of the ASD having an active damping module, according to an embodiment of the invention. 
         FIG. 3  is a pair of graphs comparing DC link voltage, u dc , voltage ripple, and phase a and b output currents, i a , i b , at 50 Hz with and without the control scheme provided by the controller of the ASD of  FIGS. 1-2 . 
         FIG. 4  is a coefficient matrix, A, for a state space model of the ASD system of  FIGS. 1-2 , according to an embodiment of the invention, according to an embodiment of the invention. 
         FIG. 5  is a coefficient matrix, A 0 , resulting from entering steady operating values into the coefficient matrix, A, of  FIG. 5 , according to an embodiment of the invention. 
         FIG. 6  is a coefficient matrix, A 1 , obtained from the linearization of the coefficient matrix, A, of  FIG. 5 , according to an embodiment of the invention. 
         FIG. 7  is a coefficient matrix, A d , obtained by incorporating the active damping module of the controller of the ASD of  FIGS. 1-2  into the coefficient matrix, A, of  FIG. 4 , according to an embodiment of the invention 
         FIG. 8  is a coefficient matrix, A 9d , obtained by expanding the coefficient matrix, A d , from a 7×7 matrix to a 9×9 matrix, according to an embodiment of the invention. 
         FIG. 9  is a coefficient matrix, A 90 , resulting from entering steady operating values into the coefficient matrix, A 9d , of  FIG. 8 , according to an embodiment of the invention. 
         FIG. 10  is a coefficient matrix, A 9l , obtained from the linearization of the coefficient matrix, A 9d , of  FIG. 8 , according to an embodiment of the invention. 
         FIGS. 11A-11B  are a flowchart illustrating a technique for determining d-axis and q-axis compensation coefficients and a cutoff frequency for stabilizing the ASD system of  FIGS. 1-2 , according to an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments of the invention relate to a system and method for controlling an ASD that includes a small DC link thin film capacitor such that the ASD is stable under all normal operating conditions. The ASD includes a control system that is programmed or configured to analyze the stability of the ASD output and actively stabilize the output of the ASD. The control system calculates compensation, stabilizing, or damping coefficients, constants, or factors and a corner frequency for a high pass filter for equilibrium or steady operating points using Lyapunov&#39;s First Method. The control system extracts perturbations in current output by the ASD, compensates the perturbations by multiplying them with the compensation coefficients, and adds the compensated perturbations together to obtain a feedback, compensation, or damping frequency or speed. The control system then adds the compensation frequency to a reference motor speed command in order to modify, alter, or adjust an angle of rotation used to control an inverter of the ASD. This active damping control scheme significantly reduces the ASD output current THD and DC link voltage ripple. 
     Referring to  FIG. 1 , an ASD system or circuit  10  including an ASD  12  is shown, according to an embodiment of the invention. ASD  12  includes an input  14  connected to an AC source  16  and an output  18  connected to a motor or induction machine  20 . ASD  12  further includes a rectifier  22  connected between input  14  and an inductor, L; a DC link  24  having a small DC link thin film capacitor, C, thereacross; and an inverter  26  having a plurality of switch-diode pairs  28  coupled between DC link  24  and output  18 . Small DC link thin film capacitor, C, may also be in the form of a capacitor bank having multiple small DC link thin film capacitors. ASD  12  also includes a control system or controller  30  for controlling inverter  26 . ASD  12  may additionally include a filter, such as, for example, an electromagnetic interference filter or LC filter, at input  14  and/or output  18 . 
     Referring now to  FIG. 2 , ASD system  10  of  FIG. 1  including ASD  12  is shown with a more detailed view of controller  30 , according to an embodiment of the invention. As shown, ASD  12  additionally includes a set of sensors  32  including at least one voltage and current sensor for sensing or measuring a DC link voltage, u dc , across the DC link thin film capacitor, C, and two phases of the output current, i a , i b , flowing from inverter  26  to induction machine  20 . While phase a and b output currents, i a  i b , of inverter  26  are being measured in  FIG. 2 , the output currents of phases b and c or phases a and c may be measured instead. At a dq/abc block  34 , the sampled phase a and b output currents, {tilde over (ι)} a , {tilde over (ι)} b , measured by the current sensors are then transformed from an abc reference frame in which the a, b, and c axes are in the same plane 120° apart from each other to a direct-quadrature (dq) reference frame in which the d and q axes are 90° apart from each other and the d axis is rotated away from the a axis toward the b axis by an angle of rotation, θ, using a Clarke Transform and then a Park Transform with the angle of rotation, θ, from a previous iteration of the control scheme. The output of dq/abc block  34  is a sampled current, {tilde over (ι)} s , in the dq reference frame that is input into an active damping module  36 . 
     Active damping module  36  includes a high pass filter (HPF)  38  that extracts a perturbation or variation in the sampled current, {tilde over (ι)} s , according to: 
                   {               i     sd   ⁢           ⁢   2       =       s     s   +     ω   c         ⁢       i   ~     sd                     i     sq   ⁢           ⁢   2       =       s     s   +     ω   c         ⁢       i   ~     sq               ,             [     Eqn   .           ⁢   1     ]               
where {tilde over (ι)} sd  is the d-axis component of the sampled current, {tilde over (ι)} s ; {tilde over (ι)} sq  is the q-axis component of the sampled current, {tilde over (ι)} s ; ω c  is the corner or cutoff frequency of HPF  38 ; i sd2  is the d-axis component of the perturbation; and i sq2  is the q-axis component of the perturbation. After the perturbation has been extracted by HPF  38 , active damping module  36  damps, compensates, or stabilizes the perturbation. More specifically, active damping module  36  multiplies the d-axis component of the perturbation, i sd2 , and the q-axis component of the perturbation by respective d-axis and q-axis damping, compensation, or stabilization coefficient, constant, or factors, k d , k q , at damping, compensation, or stabilization blocks  40 ,  42 . The selection of the damping coefficients, k d , k q , and the cutoff frequency, ω c , will be discussed further below with respect to  FIG. 11 .
 
     Once the perturbation has been damped, active damping module  36  obtains a feedback, compensation, or damping frequency, ω o , at the output of summation block  44  according to:
 
ω o   =k   d   i   sd2   +k   q   i   sq2   [Eqn. 2].
 
     Compensation frequency, ω o , is the output of active damping module  36  and is fed into summation block  46  along with a reference motor speed command, ω r *, which is an input for controller  30 . Reference motor speed command, ω r *, is a scalar command such as, for example, 100*π radians/second in one non-limiting embodiment. Summation block  46  adds compensation frequency, ω o , with reference motor speed command, ω r *, to obtain a damped, compensated, or stabilized speed, ω d . The damped speed, ω d , is then integrated at integration block  48  to obtain a damped, compensated, or stabilized angle of rotation, θ, for the dq reference frame. The angle of rotation, θ, is adjusted, modified, or altered from what the angle of rotation would have been if reference motor speed command, ω r *, were integrated without adding compensation frequency, ω o . The above-described adjustment made via compensation frequency, ω o , is how controller  30  stabilizes the output of inverter  26  of ASD  12 . 
     Reference motor speed command, ω r *, is also input into a Volts per Hertz (V/Hz) block  50  simultaneously with summation block  46 . Based on the reference motor speed command, ω r *, V/Hz block  50  outputs an inverter output voltage reference, u f , according to: 
                       u   f     =       u   sq     =       kU   b     ⁢       ω   e       ω   b             ,           [     Eqn   .           ⁢   3     ]               
where u sq  is the q-axis component of the inverter output voltage reference, u f ; k is a constant that can be set to a desired value to control the output of V/Hz block  50 ; U b  is the base voltage of induction machine  20 ; ω e  is the synchronous speed of induction machine  20 ; and ω b  is the rated speed of induction machine  20 . In a non-limiting embodiment in which ASD  12  is equipped with a small 3% DC link thin film capacitor (where 100% would be when an electrolytic capacitor is used), k is set to a value of 0.9. The d-axis component, u sd , of the inverter output voltage reference, u f , is set to 0. Since the compensation frequency, ω o , is not added to the reference motor speed command, ω r *, before being input into V/Hz block  50 , the compensation frequency, ω o , does not affect the inverter output voltage reference, u f .
 
     The inverter output voltage reference, u f , along with the adjusted angle of rotation, θ, is input into dq/αβ block  52 . dq/αβ block  52  performs an inverse Park Transform on the inverter output voltage reference, u f , using the adjusted angle of rotation, θ, to transform the inverter output voltage reference, u f , from the dq reference frame to an αβ in which an α axis is aligned with the a axis and a β axis is perpendicular to the α axis. dq/αβ block  52  outputs the result of the transformation to a space vector pulse width modulation (SVPWM) block  54 . SVPWM block  54  controls inverter  26  according to the output of dq/αβ block  52  and the sampled DC link voltage, ũ dc , from sensors  32 . 
     Referring now to  FIG. 3 , a pair of graphs  56 ,  58  comparing DC link voltage, u dc , voltage ripple, and phase a and b output currents, i a , i b , at 50 Hz are shown with and without the control scheme provided by controller  30  of  FIGS. 1-2 . As shown in graph  58 , without the use of the control scheme provided by controller  30 , there is a 264V DC link voltage, u dc , ripple, and the phase a and b output currents, i a , i b , are distorted with a 58% THD and do not approximate the desired sine wave output. However, as shown in graph  58 , when using the control scheme provided by controller  30 , the DC link voltage, u dc , ripple is reduced to 100V and the THD on the phase a and b output currents, i a , i b , is reduced to 16%. In addition, the phase a and b output currents, i a , i b , more closely approximate a sine wave for operating motor  20 . Thus, the control scheme implemented by controller  30  provides a significant improvement in the DC link voltage, u dc , ripple and the THD and waveform shape in the current output by inverter  26 . In other words, the control scheme of controller  30  is effective to stabilize the output of ASD  12  of  FIGS. 1-2 . 
     In order to analyze the stability of ASD system  10 , ASD system  10  may be described or modeled as a state space equation including the V/Hz control of ASD  12  and taking into account the voltage fluctuation on small DC link thin film capacitor, C, using a 7×7 matrix. The state space equation for ASD system  10  not including active damping module  36  is given by: 
                       dx   dt     =     Ax   +   Bu       ,           [     Eqn   .           ⁢   4     ]               
where x is the state vector including the system state variables; u is the input vector including the system inputs; and A and B are system coefficient matrices including coefficients describing ASD system  10 . State vector, x, is a 7×1 matrix, the transpose of which is given by:
 
 x =[ i   sd   i   sq ψ sd ψ sq ω r   i   L   u   C ] T   [Eqn. 5],
 
where i sd , i sq  are the stator current on the d and q axis; ψ sd , ψ sq  are the stator flux on the d and q axis; ω r  is the rotor speed; i L  is the current flowing through inductor, L; and u C  is the voltage on small DC link thin film capacitor, C.
 
     Input vector, u, is a 2×1 matrix, the transpose of which is given by:
 
 u =[ T   L   E ] T   [Eqn. 6],
 
where T L , is the load torque and E is given by:
 
                     E   =         3   ⁢     2       π     ⁢     V   LL         ,           [     Eqn   .           ⁢   7     ]               
where V LL  is the three-phase source line voltage.
 
     Coefficient matrix, B, is a 7×2 matrix, the transpose of which is given by: 
                     B   =       [         0       0       0       0         -       n   p     J           0       0           0       0       0       0       0         1   L         0         ]     T       ,           [     Eqn   .           ⁢   8     ]               
where n p  is the number of poles and J is the inertia of motor  20 .
 
     The coefficient matrix, A, is a 7×7 matrix given by Eqn. 9, which is shown in  FIG. 4 . In the coefficient matrix, A, L m  is the mutual inductance; L s  is the stator inductance; L r  is the rotor inductance; R s  is the stator resistance; and R r  is the rotor resistance. In addition, σ, τ r , and R eq  (the equivalent resistance caused by the dead-time, t d ) are given by: 
                     σ   =     1   -       L   m   2         L   s     ⁢     L   r             ;           [     Eqn   .           ⁢   10     ]                   τ   r     =       L   r       R   r         ;           [     Eqn   .           ⁢   11     ]                   R   eq     =       4   ⁢           ⁢     t   d     ⁢     f   sw     ⁢     u   C         π   ⁢         i   sd   2     +     i   sq   2               ,           [     Eqn   .           ⁢   12     ]               
where f sw  is the switching frequency.
 
     Eqn. 4 shows that ASD system  10  is a coupled nonlinear system. In order to analyze the stability of the nonlinear system, Lyapunov&#39;s First Method is adopted. An equilibrium, steady state, or steady operating point solution for Eqn. 4 is given by:
 
0 7,1   =A   0   x   0   +Bu   0   [Eqn. 13],
 
where 0 7,1  is a 7×1 matrix in which every element is “0”; x 0  is the state vector, x, at a steady operating point, the transpose of which is given by:
 
 x   0 =[ i   sd0   i   sq0 ψ sd0 ψ sq0 ω r0   i   L0   u   C0 ] T   [Eqn. 14];
 
u 0  is the input vector, u, at a steady operating point, the transpose of which is given by:
 
 u   0 =[ T   L0   E   0 ] T   [Eqn. 15];
 
and A 0  is coefficient matrix, A, at a steady operating point given by Eqn. 16, which is shown in  FIG. 5 . In coefficient matrix, A 0 , ω b ′ is given by:
 
                     ω   b   ′     =         ω   b     k     .             [     Eqn   .           ⁢   17     ]               
The subscript “0” at the end of system variables in Eqns. 13-15 represents system variables at the steady operating point. Controller  30  may be used to obtain the steady state solution for Eqn. 13.
 
     The state equation of the small-signal system is obtained by linearization and is given by: 
                       dx   dt     =         A   l     ⁢   x     +   Bu       ,           [     Eqn   .           ⁢   18     ]               
where A l  is coefficient matrix, A, after the linearization and is given by Eqn. 19, which is shown in  FIG. 6 . In Eqn. 18, R eq0,d , R eq0,q , X eq0,dq , and i eq0  are given by:
 
     
       
         
           
             
               
                 
                   
                     
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     Controller  30  may be used to calculate the eigenvalues of Eqn. 18 by substituting the parameters of ASD system  10  and the steady state solution from Eqn. 13. These eigenvalues give the pole distribution for ASD system  10 . Any eigenvalues/poles located in the right half of the s-plane will make the system unstable. Thus, if controller  30  determines that any poles are located in the right half of the s-plane, ASD system  10  is unstable. Analyzing ASD system  10  using Eqns. 4, 13, and 18 provide a benefit over previous systems by incorporating the current, i L  through inductor, L, and the voltage, u C , across DC link thin film capacitor, C, such that state vector, x, includes 7 variables and coefficient matrix, A, is a 7×7 matrix. 
     In order to stabilize any instability found by analyzing ASD system  10  using Eqns. 4, 13, and 18, the damping coefficients, k d , k q , and the cutoff frequency, ω c , are incorporated into the system model. To incorporate the damping coefficients, k d , k q , and the cutoff frequency, ω c , into the model given by Eqn. 4, the model given by Eqn. 4 is modified to include active damping module  36  according to: 
                       dx   dt     =         A   d     ⁢   x     +   Bu       ,           [     Eqn   .           ⁢   24     ]               
where A d  is the coefficient matrix, A, modified by active damping module  36  according to Eqn. 25, which is shown in  FIG. 7 . However, because the d-axis and q-axis perturbations, i sd2 , i sq2 , are not represented by state variables in coefficient matrix, A d , intermediate d-axis and q-axis perturbations, i sd1 , i sq1 , are introduced according to:
 
                   {               i     sd   ⁢           ⁢   1       =         ω   c       s   +     ω   c         ⁢       i   ~     sd                     i     sq   ⁢           ⁢   1       =         ω   c       s   +     ω   c         ⁢       i   ~     sq               .             [     Eqn   .           ⁢   26     ]               
As such, the d-axis and q-axis perturbations, i sd2 , i sq2 , are now given by:
 
     
       
         
           
             
               
                 
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     Hence, the complete model for ASD system  10  incorporating active damping module  36  is given by: 
                         dx   9     dt     =         A     9   ⁢           ⁢   d       ⁢     x   9       +       B   9     ⁢   u         ,           [     Eqn   .           ⁢   28     ]               
where x 9  is a state vector having 9 state variables, the transpose of which is given by:
 
 x   9 =[ i   sd   i   sq   i   sd1   i   sq1 ψ sd ψ sq ω r   i   L   u   C ] T   [Eqn. 29];
 
B 9  is a 9×2 coefficient matrix, the transpose of which is given by:
 
                     B   9     =         [         0       0       0       0         -       n   p     J           0       0           0       0       0       0       0         1   L         0         ]     T     .             [     Eqn   .           ⁢   30     ]               
and A 9d  is coefficient matrix, A d , modified to a 9×9 coefficient matrix given by Eqn. 31, which is shown in  FIG. 8 .
 
     In order to solve for the damping coefficients, k d , k q , and the cutoff frequency, ω c , using Lyapunov&#39;s First Method, the steady state and small-signal system equations must be found. The steady state equation is given by:
 
0 9,1   =A   90   x   90   +B   9   u   90   [Eqn. 32],
 
where 0 9,1  is a 9×1 matrix in which every element is “0”; x 90  is the state vector, x 9 , at a steady operating point, the transpose of which is given by:
 
 x   90 =[ i   sd0   i   sq0   i   sd10   i   sq10 ψ sd0 ψ sq0 ω r0   i   L0   u   C0 ] T   [Eqn. 33];
 
u 0  is the input vector, u, at a steady operating point, the transpose of which is given by:
 
 u =[ T   L0   E   0 ] T   [Eqn. 34];
 
and A 90  is the coefficient matrix, A 9d , at a steady operating point given by Eqn. 35, which is shown in  FIG. 9 . The state equation of the small-signal system obtained by linearization of Eqn. 28 is given by:
 
                         dx   9     dt     =         A     9   ⁢           ⁢   l       ⁢     x   9       +       B   9     ⁢   u         ,           [     Eqn   .           ⁢   36     ]               
where A 9l  is the coefficient matrix, A 9 , after the linearization and is given by Eqn. 37, which is shown in  FIG. 10 .
 
     Referring now to  FIGS. 11A-11B , a technique or process  60  is shown for determining the appropriate values for the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c , used by active damping module  36  in controller  30  of ASD  12  shown in  FIGS. 1-2 . Process  60  may be performed by controller  30  of ASD  12  or by another controller. However, process  60  will be described as though controller  30  is performing process  60 . The d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c , must be calculated by controller  30  before controller  30  can effectively stabilize the output of inverter  26  of ASD  12 . Process  60  is described below in conjunction with Eqns. 35 and 39 that are used by controller  30  in process  60  to solve for the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c . 
     Process  60  starts at STEP  62  when controller  30  is activated. At STEP  64 , a wide variety of operating parameters for motor  20  are input into controller  30 . The operating parameters are used in Eqn. 35. Controller  30  solves Eqn. 35 for x 90  multiple times by varying the rotor speed from low frequency to rated frequency, such as, for example, from 5 Hz to 50 Hz, and by varying the load from no load to full load. At STEP  68 , if a solution for x 90  exists at any steady operating point, controller  30  saves that solution in a matrix at STEP  70 . Because x 90  is a 9×1 matrix and all of the solutions are stored in the same matrix, the resulting solution matrix is a 9×N matrix, where N is the number of solutions. After controller  30  saves a solution at STEP  70  or determines that a solution did not exist at STEP  68 , controller  30  determines whether all operating conditions have been considered at STEP  72 . If some operating conditions have not been considered, process  60  moves back to STEP  66  to vary the operating conditions. 
     Once controller  30  has considered all operating conditions, process  60  moves to STEP  74 . At STEP  74 , a set of the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c , are chosen or selected from a given range. At STEP  76 , controller  30  enters the selected values for the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c , into Eqn. 39. After controller  30  has solved for the eigenvalues (poles and zeros) of Eqn. 39 for the selected values for the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c , controller  30  determines if any of the resulting eigenvalues are positioned in the right half of the s-plane at STEP  78 . Eigenvalues located in the right half of the s-plane will make the system unstable. Thus, if controller  30  determines that any eigenvalues are in the right half of the s-plane, controller  30  discards the currently selected values for the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c , and process  60  moves to STEP  80 . At STEP  80 , controller  30  determines whether all possible values for the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c , have been considered. If so, process  60  ends at STEP  82 . If not, process  60  moves back to STEP  74 , where controller  30  selects another set of the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c , in the given range. 
     Going back to STEP  78 , if none of the eigenvalues are located in the right half of the s-plane, process  60  moves to STEP  84 . At STEP  84 , controller  30  identifies the dominant poles and then the worst damped dominant pole of the eigenvalues under all operating conditions. At STEP  86 , controller  30  determines whether all operating conditions have been considered for solving Eqn. 39. If not, process  60  moves back to STEP  76  to solve for the eigenvalues. If so, process  60  moves to STEP  88 , where controller  30  compares the worst damped dominant pole with the worst damped dominant pole of any previously identified optimal parameters for the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c . 
     If controller  30  determines at STEP  90  that the worst damped dominant pole of the currently selected d-axis and q-axis compensation coefficients, k d , k q , and cutoff frequency, ω c , is better than the previously identified optimal parameters or if no optimal parameters have been selected yet, controller  30  sets the current values for the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c , as the optimal parameters at STEP  92 . If not, controller  30  discards the currently selected d-axis and q-axis compensation coefficients, k d , k q , and cutoff frequency, ω c , and process  60  moves to STEP  80 . Again, at STEP  80 , if controller  30  determines that all possible values for the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c , have not been considered, process  60  moves to STEP  76 . If all possible values for the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c , have been considered, process  60  ends, and controller  30  will use the values for the d-axis and q-axis compensation coefficients, k d , k q , and the cutoff frequency, ω c , that were set as the optimal parameters in STEP  92  to stabilize the output of ASD  12  of  FIGS. 1-2 . 
     Beneficially, embodiments of the invention thus provide a system and method for stabilizing an output of an ASD having a small DC link thin film capacitor and controlling a motor. The ASD includes a controller that receives two phases of the output current of the ASD and a DC link voltage across the DC link thin film capacitor or capacitor bank from a plurality of sensors and a reference motor speed command. The sampled output currents of the ASD are input into an active damping module, which outputs a compensation frequency based on d-axis and q-axis compensation coefficients previously determined by the controller. The compensation frequency is added to the reference motor speed command to form a damped speed that is integrated to find a damped angle of rotation. An SVPWM block controls the inverter based on the reference motor speed command and the damped angle of rotation, which stabilizes the output of the inverter by reducing the ASD output current THD and the DC link voltage ripple on the DC link thin film capacitor. Thus, the use of the compensation frequency to damp the reference motor speed command results in a more stable output for ASDs that include DC link thin film capacitor(s). 
     According to one embodiment of the present invention, a control system for actively damping an output of an ASD having a DC link thin film capacitor is programmed to calculate a d-axis damping coefficient and a q-axis damping coefficient for stabilizing an output of the ASD based at least on a voltage across the DC link thin film capacitor at a steady operating point. The control system is further programmed to extract d-axis and q-axis perturbations in d-axis and q-axis output currents of the ASD using a high pass filter, damp the d-axis perturbation and the q-axis perturbation with the d-axis damping coefficient and the q-axis damping coefficient, respectively, and calculate a damping frequency based on the damped d-axis perturbation and the damped q-axis perturbation. The control system is also programmed to damp an angle of rotation of a reference motor speed command for controlling the ASD using the damping frequency. 
     According to another embodiment of the present invention, a method of stabilizing an ASD having a DC link thin film capacitor incorporated therein includes receiving, at a controller, input parameters comprising a current output by the ASD from a current sensor and a reference motor speed command and computing, with the controller, a d-axis stabilization factor and a q-axis stabilization factor for stabilizing the ASD based on at least one steady state voltage across the DC link thin film capacitor. The method additionally includes transforming the current output by the ASD into a d-axis current and a q-axis current with the controller, obtaining d-axis and q-axis perturbations in the d-axis and q-axis currents, respectively, with a high pass filter in the controller, and compensating the d-axis perturbation and the q-axis perturbation with the controller using the d-axis stabilization factor and the q-axis stabilization factor, respectively. Furthermore, the method includes computing a feedback frequency with the controller based on the compensated d-axis perturbation and the compensated q-axis perturbation and compensating a reference motor speed command with the feedback frequency to change an angle of rotation for controlling an inverter of the ASD. 
     According to yet another embodiment of the present invention, an ASD includes an input connectable to an AC source, a rectifier connected to the input to convert an AC power input to a DC power at an output thereof, and a DC link coupled to the rectifier output to receive the DC power therefrom. The DC link has a thin film capacitor and an inductor positioned thereon and is coupled to a DC side of an inverter having an AC side connectable to a motor. The ASD includes at least one current sensor configured to measure currents output from the AC side of the inverter and a control system coupled to the inverter as well. The control system is programmed to determine a d-axis compensation coefficient and a q-axis compensation coefficient designed to stabilize the ASD output currents under all operating conditions based on at least one steady operating point for a set of operating parameters including voltage across the DC link thin film capacitor and a current through the inductor. In addition, the control system is programmed to extract a perturbation in the inverter output currents from the at least one current sensor using a high pass filter, compensate a d-axis component of the perturbation using the d-axis compensation coefficient and a q-axis component of the perturbation using the q-axis compensation coefficient, and determine a compensation frequency by combining the compensated d-axis and q-axis perturbations. Moreover, the control system is programmed to add the compensation frequency to a reference motor speed command to adjust an angle of rotation for controlling the inverter and control the inverter based on the amplitude of the reference motor speed command and the adjusted angle of rotation. 
     The present invention has been described in terms of the preferred embodiment, and it is recognized that equivalents, alternatives, and modifications, aside from those expressly stated, are possible and within the scope of the appending claims.