Abstract:
An apparatus and method for determining an angle ψ an estimate of a true angle θ from a resolver excitation signal V ex (t)=A sin(ωt) and modulated resolver output signals V s (t,θ)=A×sin(θ)×sin(ωt) and V C (t,θ)=A×cos(θ)×sin(ωt), where ω is an angular frequency and t is time, is provided. The apparatus may include a converter comprising a closed loop phase-locked loop (PLL) system configured to produce an angle ψ and two digital signals BitØ(θ) and Bit 1 (θ) for estimating the true angle θ using V ex (t), V s (t,θ) and V C (t,θ) signals.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    The present application is related to and claims the benefit and priority of U.S. Provisional Patent Application No. 62/063,535, filed Oct. 14, 2014, the entirety of which is hereby incorporated herein by reference. 
     
    
     FIELD OF THE INVENTION 
       [0002]    Embodiments of the invention may generally relate to a resolver to converter device and, in particular, some embodiments may specifically relate to a phase-locked loop (PLL)-type resolver/converter apparatus and method. 
       BACKGROUND 
       [0003]    Resolvers have so far been widely used as reliable angle transducers especially suited to hostile environments. These devices are used in various positioning applications including robots, machine tools, aircrafts, radars and satellite antennas. Resolvers resemble small electric motors and are essentially rotary transformers designed so the coefficient of coupling between rotor and stator windings varies with the shaft angle (see the catalog of Admotec “ Understanding Resolvers and Resolver - to - Digital Conversion ” http://www.admotec.com/TT02.pdf, 1998 (non-patent document 1)). According to this document, the rotor winding of the resolver is used as primary and is supplied with a sinusoidal excitation voltage, 
         [0000]        V   ex ( t )= A   ex  sin ω t   (1)
 
         [0000]    As a result, the two windings located at right angles in the stator produce amplitude-modulated AC signals, one with the sine and the other with the cosine of shaft angle, θ. If the angular velocity of the rotor is much smaller than co, a condition usually fulfilled because of the relatively elevated excitation or carrier frequency (typically a few kHz), the stator waveforms of the resolver are given by: 
         [0000]        V   C ( t ,θ)= A  cos θ sin ω t   (2)
 
         [0000]        V   s ( t ,θ)= A  sin θ sin ω t   (3)
 
         [0000]    where A is a constant determined by the amplitude of the excitation signal and the transformation ratio, α, between stator and rotor windings (A=αA ex ). Hence, the resolver converts the mechanical angle θ to analog modulated signals V s (t,θ) and V C (t,θ), having sin(θ) and cos(θ) components as modulating signals respectively. The signals and modules illustrated in  FIG. 1  are some of the elements for the processing of information out of the resolver. 
         [0004]    Non-Patent Background Documents:
   1. Catalog of Admotec “ Understanding Resolvers and Resolver - to - Digital Conversion ” http://www.admotec.com/TT02.pdf, 1998   2. L. Ben-Brahim and M. Benammar, “A new PLL method for resolvers,” 2010  International Power Electronics Conference  (IPEC), Sapporo, Japan, 21-24 Jun. 2010, pp. 299-305   3. N. Al-Emadi, L. Ben-Brahim, and M. Benammar, “A New Tracking Technique for Mechanical Angle Measurement,” Measurement, Volume 54, August 2014, Pages 58-64   4. R. G. Meyer, W. M. C. Sansen, S. Lui and S. Peeters, “The Differential Pair as a Triangle-Sine Wave Converter,”  IEEE Journal of Solid - State Circuits, SC -11, June 1976, pp. 418-420   5. R. C. Gupta, “Bhaskara I&#39;s Approximation to Sine,”  Indian Journal of History of Science , Vol. 2, no. 2, 1967, pp. 121-136   
 
         [0010]    Patent Background Documents:
   1. PLL Type Resolver/Digital Converter JP60079221 (May 7, 1985)   2. Resolver To Digital Converter U.S. Pat. No. 5,162,798 (Nov. 10, 1992)   3. Resolver/Digital Converter And Control Apparatus U.S. Pat. No. 7,009,535 (Mar. 7, 2006)   4. Resolver To 360 degrees Linear Analog Converter And Method WO2000033466A 1 (Jun. 8, 2000)   5. PLL type resolver to 360 degrees linear converter apparatus GB2448350B (Jan. 6, 2011)   
 
       SUMMARY OF THE INVENTION 
       [0016]    One embodiment is directed to an apparatus for determining an angle vi an estimate of a true angle θ from a resolver excitation signal V ex (t)=A sin(ωt) and modulated resolver output signals V s (t,θ)=A×sin(e)×sin(ωt) and V C (t,θ)=A×cos(θ)×sin(ωt), where ω is an angular frequency and t is time. The apparatus comprises a converter comprising a closed loop phase-locked loop (PLL) system configured to produce an angle ψ and two digital signals BitØ(θ) and Bit 1 (θ) for estimating the true angle θ using V ex (t), V s (t,θ) and V C (t,θ) signals. 
         [0017]    In an embodiment, the high frequency signal V ex (t) is an excitation signal generated by an external generator, and the modulated signals V s (t,θ) and V C (t,θ) are response signals received from an external device in response to the excitation signal V ex (t) and to a mechanical angle θ. 
         [0018]    According to one embodiment, the closed-loop PLL system comprises a phase detector configured to produce a |sin(ωt)|×(|sin(θ)|cos(ψ)−|cos(θ)|sin(ψ)) signal from the signals V s (t,θ), V C (t,θ), sin(ψ) and cos(ψ), a controller and integrator configured to ensure that ψ tracks changes of the true angle θ, and a quadrant identification circuit configured to use the signals V ex (t), V s (t,θ) and V C (t,θ) to produce two binary signals BitØ(θ) and Bit 1 (θ) for identifying the four quadrants of the true angle θ. 
         [0019]    In an embodiment, the apparatus may further include a sin(ψ) and cos(ψ) extraction circuit comprising means for converting triangular waveform ψ into sin(ψ) and cos(ψ) using signal shaping or approximation methods. 
         [0020]    Another embodiment is directed to a method including determining an angle ψ an estimate of a true angle θ from a resolver excitation signal V ex (t)=A sin(ωt) and modulated resolver output signals V s (t,θ)=A×sin(θ)×sin(ωt) and V C (t,θ)=A×cos(θ)×sin(ωt), where ω is an angular frequency and t is time. The determining may comprise producing, by a converter comprising a closed loop phase-locked loop (PLL) system, an angle ψ and two digital signals BitØ(θ) and Bit 1 (θ) for estimating the true angle θ using V ex (t), V s (t,θ) and V C (t,θ) signals. 
         [0021]    According to an embodiment, the method may further include producing, by a phase detector of the closed-loop PLL system, a |sin(ωt)|×(|sin(θ)|cos(ψ)−|cos(θ)|sin(ψ)) signal from the signals V s (t,θ), V C (t,θ), sin(ψ) and cos(ψ). The method may also include ensuring, by a controller and integrator, that ψ tracks changes of the true angle θ, and using, by a quadrant identification circuit, the signals V ex (t), V s (t,θ) and V C (t,θ) to produce two binary signals BitØ(θ) and Bit 1 (θ) for identifying the four quadrants of the true angle θ. 
         [0022]    In one embodiment, the method may also include converting, by a sin(ψ) and cos(ψ) extraction circuit, triangular waveform ψ into sin(ψ) and cos(ψ) using signal shaping or approximation methods. 
         [0023]    Another embodiment is directed to an apparatus including determining means for determining an angle ψ an estimate of a true angle θ from a resolver excitation signal V ex (t)=A sin(ωt) and modulated resolver output signals V s (t,θ)=A×sin(θ)×sin(ωt) and V C (t,θ)=A×cos(θ)×sin(ωt), where ω is an angular frequency and t is time. The determining means may comprise means for producing, by a converter comprising a closed loop phase-locked loop (PLL) system, an angle ψ and two digital signals BitØ(θ) and Bit 1 (θ) for estimating the true angle θ using V ex (t), V s (t,θ) and V C (t,θ) signals. 
         [0024]    In an embodiment, the apparatus may further include means for producing, by a phase detector of the closed-loop PLL system, a |sin(ωt)|×(|sin(θ)|cos(ψ)−|cos(θ)|sin(ψ)) signal from the signals V s (t,θ), V C (t,θ), sin(ψ) and cos(ψ). The apparatus may also include means for ensuring, by a controller and integrator, that ψ tracks changes of the true angle θ, and means for using, by a quadrant identification circuit, the signals V ex (t), V s (t,θ) and V C (t,θ) to produce two binary signals BitØ(θ) and Bit 1 (θ) for identifying the four quadrants of the true angle θ. 
         [0025]    According to one embodiment, the apparatus may further include means for converting, by a sin(ψ) and cos(ψ) extraction circuit, triangular waveform ψ into sin(ψ) and cos(ψ) using signal shaping or approximation methods. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0026]    For proper understanding of the invention, reference should be made to the accompanying drawings, wherein: 
           [0027]      FIG. 1  illustrates a diagram depicting the principle of operation of a resolver with its input (excitation) and output signals; 
           [0028]      FIG. 2  illustrates a conventional PLL converter with VCO, LUT and DACs; 
           [0029]      FIG. 3  illustrates the PLL converter scheme of GB2448350B (Patent document 5); 
           [0030]      FIG. 4  illustrates a block diagram of an apparatus, according to an embodiment of the present invention; 
           [0031]      FIG. 5  illustrates input and output waveforms of a converter for one full revolution of the resolver shaft with a resolver excitation frequency of 200 Hz, according to one embodiment; 
           [0032]      FIG. 6  illustrates a block Diagram of an extraction circuit of sin(ψ) and cos(ψ) from ψ, according to an embodiment; 
           [0033]      FIG. 7  illustrates an example analog implementation of conversion of ψ voltage into sin(ψ) voltage using signal shaping technique, according to an embodiment; 
           [0034]      FIG. 8  illustrates an example analog implementation of conversion of ψ voltage into sin(ψ) voltage using rational approximation technique, according to an embodiment; 
           [0035]      FIG. 9  illustrates a block diagram of a quadrant identification circuit, according to an embodiment; and 
           [0036]      FIG. 10  illustrates input and output waveforms of the proposed converter for one full revolution of the resolver shaft with a resolver excitation frequency of 4 kHz, according to an embodiment. 
       
    
    
     DETAILED DESCRIPTION 
       [0037]    It will be readily understood that the components of the invention, as generally described and illustrated in the figures herein, may be arranged and designed in a wide variety of different configurations. Thus, the following detailed description of embodiments of a PLL-type resolver/converter apparatus and method, as represented in the attached figures, is not intended to limit the scope of the invention, but is merely representative of selected embodiments of the invention. 
         [0038]    The features, structures, or characteristics of the invention described throughout this specification may be combined in any suitable manner in one or more embodiments. For example, the usage of the phrases “certain embodiments,” “some embodiments,” or other similar language, throughout this specification refers to the fact that a particular feature, structure, or characteristic described in connection with the embodiment may be included in at least one embodiment of the present invention. Thus, appearances of the phrases “in certain embodiments,” “in some embodiments,” “in other embodiments,” or other similar language, throughout this specification do not necessarily all refer to the same group of embodiments, and the described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments. 
         [0039]    Additionally, if desired, the different functions discussed below may be performed in a different order and/or concurrently with each other. Furthermore, if desired, one or more of the described functions may be optional or may be combined. As such, the following description should be considered as merely illustrative of the principles, teachings and embodiments of this invention, and not in limitation thereof. 
         [0040]    Commercial resolver converters are built mainly around the successful PLL tracking method. This PLL system is based upon the operation in a closed loop in which an estimated angle, ψ, tracks the shaft angle, θ, as illustrated in  FIG. 2 . The system includes a demodulator for removing the carrier (excitation signal) from the resolver signals and means of computation of the sine and cosine of the estimated angle, usually from a look-up table addressed with an up/down counter. The phase detector may be composed of two multipliers and one subtractor to calculate the error signal sin(ε−ψ). The PI controller guarantees at steady state the convergence of the estimated angle ψ towards the true angle, θ by making sin(θ−ψ)≈θ≈θ−ψ. This PLL system may include: 1. Demodulator, 2. Phase Detector (PD), 3. Voltage Controlled Oscillator (VCO), 3. Up-Down counter, 4. Digital-to-Analog Converters (DACs), and 5. Look-Up Table (LUT) for the digital measurement of the sine and cosine of the estimated angle. 
         [0041]    A modified implementation of the classical PLL converter has been presented in patent document 5, as illustrated in  FIG. 3 . Implementation of this method has been reported in non-patent documents 2 and 3. Although the aim of this was to simplify implementation of the PLL converter, the method is still complicated and requires availability of three precise and synchronized reference signals (sine, cosine and sawtooth waveforms). A scheme of comparator and sample and hold devices is used to determine the required sine and cosine of the estimated angle ψ. Another drawback of this method is that the scheme produces a single sample of sin(ψ) and cos(ψ) in each period of the excitation signal as the sampling frequency (f=ω/2π). This means that the converter can only work at low resolvers&#39; speed of rotation (i.e., slow variation of θ and therefore of ψ). Additionally, the method requires an additional way of resetting the integrator at the end of each revolution of the shaft of the resolver, which introduces additional complexity. 
         [0042]    In view of the above, embodiments of the present invention relate to a resolver to linear analog/digital converter apparatus and method for determining the unknown resolver rotor angle θ from the modulated resolver signals: sin(ωt)×sin(θ) and sin(ωt)×cos(θ). More specifically, one embodiment includes a PLL closed-loop method for converting the amplitudes of the modulated transducer signals into a measure of the input angle without using synchronized reference signals, demodulators, VCOs, low-pass filters (LPFs), DACs and LUTs. A PLL method, according to an embodiment, is based on a closed loop system which provides an estimated value of the input angle and which includes (i) computation of the sine and cosine values of this estimated angle, and (ii) the demodulation of the resolver signals. 
         [0043]    Thus, a method of conversion, according to certain embodiments of the invention, may be applied to a resolver without using demodulators, synchronized reference signals, sample and hold devices, LUT, DAC, VCO, and LPF. Instead, an embodiment (i) derives the required sine and cosine values of the estimated angle from the converter output using simple signal shaping arrangement, and (ii) does not require demodulation of the resolver signals. 
         [0044]    In classical PLL converters, (i) the determination of the sine and cosine values of the estimated angle requires complicated schemes that involve either ADC and DAC conversion or sampling synchronized reference signals, and (ii) the demodulation is achieved using either sampling or multiplication and filtering techniques. In contrast, embodiments of the invention provide several advantages, which may include: (1) embodiments derive the required sine and cosine values of the estimated angle from the converter output using simple arrangement without LUTs and without the need for synchronized reference signals, and (2) embodiments do not require demodulation of the resolver signals. In addition, embodiments of the invention may be implemented easily using digital or basic analog electronic circuitry. 
         [0045]    One embodiment is directed to an apparatus, for example in the form of a resolver to analog or digital converter, for converting the amplitude-modulated input sine and cosine signals into signals representative of position and/or speed. For instance, an embodiment includes a new scheme applied to a resolver to converter device. Conventional PLL methods of resolver-to-digital (R/D) converters are based on a closed loop system which provides an estimated angle value that tracks the input angle. As mentioned above, the implementation of conventional PLL converters requires (i) computation of the sine and cosine values of this estimated angle, and (ii) the demodulation of the resolver signals. 
         [0046]    Embodiments of the invention, however, simplify the derivation of the angle without compromising the required dynamics and precision of the conversion process, and may include a R/D converter circuit applied to a resolver or sinusoidal quadrature encoders. 
         [0047]    As discussed above, conventional PLL methods of R/D converters use arrangements that are complicated to implement, particularly in hardware. An embodiment of the present invention, based on the PLL technique and applied for example to a resolver, does not require complicated hardware for its implementation. Embodiments of the invention can simplify the derivation of the angle without compromising required dynamics and precision of these techniques. 
         [0048]    Although the prior art system, PLL, thus described is capable of high performance both at transient and at steady state, there are a number of aspects that can be improved. The major difficulty with prior art systems is the implementation which needs a mixed analog and digital circuit which presents a complex and difficult task in the construction of the converter. Furthermore, as mentioned above, many components are used such as a demodulator, a Look-up table (addressable memory), an up-down counter, DACs and a VCO. VCO and its associated components have a maximum operating frequency which, when combined with the resolution of the converter, defines a maximum resolver angular speed that the converter can track. Further, VCOs operate linearly only over a limited range (see patent document 2). 
         [0049]    A modified implementation of the classical PLL converter has been presented in patent document 5. However, the method described therein is still complicated and requires the availability of three precise and synchronized reference signals (sine, cosine and sawtooth waveforms) which is not a trivial task. It uses a resettable integrator and a scheme of comparator and sample and hold devices that slow down the operation of the feedback loop of the converter which can only work at low resolvers&#39; speed of rotation. 
         [0050]    Therefore, an objective of the present invention is to simplify the PLL implementation by reducing the number of components used in the circuit, and by avoiding the need for three synchronized reference signals.  FIG. 4  illustrates a detailed block diagram of an embodiment of the present invention for the resolver to linear converter using PLL method. As illustrated in  FIG. 4 , a converter  10  produces a linear analog signal that is linearly proportional to the shaft angle of resolver  20  in the four identifiable quadrants of the full 360 degrees angle range. A generator  30  produces a sinusoidal carrier excitation signal to the resolver. 
         [0051]    The resolver  20  converts the mechanical angle θ to analog modulated signals V s (t,θ) and V C (t,θ), having sin(θ) and cos(θ) components as modulating signals, respectively. The V s (t,θ) and V C (t,θ) output signals from the resolver  20  and the excitation signal are processed by the converter  10  to provide a measure of the input mechanical angle θ. 
         [0052]    The converter  10  may incorporate two absolute value circuits  40  that are used to provide the absolute values of the modulated output signals V s (t,θ) and V C (t,θ) (equations 2 and 3) of the resolver. The absolute values of V s (t,θ) and V C (t,θ) are inputs to the phase detector (PD)  50 . The additional inputs to the PD  50  are sin(ψ) and cos(ψ) derived from the extraction circuit  80  as discussed in detail below. The PD  50  produces an error signal related to the sine of a combination of ψ and θ. This error signal is input to a controller  60  which may be a P controller. The output of the controller  60  is integrated by an integrator  70 . The closed loop will work towards an error signal (output of PD  50 ) close to zero. At steady state, the error signal converges to zero and the integrator  70  produces an output ψ which is an estimate of the true angle θ. 
         [0053]    The phase detector PD circuit  50  may be composed of two multipliers and one subtractor to calculate the error signal ε. Ignoring the actual amplitudes of signals, and using normalized inputs to the PD, it can be written: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
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         [0000]    The average (dc) error is: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0000]    The average error may be determined in the four quadrants of input angle θ as shown in Table I below. At steady state, the error signal converges to zero and the integrator  70  produces an output ψ which is an estimate of the true angle θ, as shown in Table I.  FIG. 5  illustrates the converter input and output waveforms (V ex (t), V s (t,θ), V C (t,θ) and ψ(θ)) for a full one revolution of the resolver shaft at a fixed speed of 200 rpm. For clarity reasons and in order to show details, the waveforms are shown for a low resolver excitation frequency f=200 Hz (or ω=400π rad/s); in reality, however, this is typically 4 kHz. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE I 
               
             
             
               
                   
               
               
                 Loop error signal and relationship between input angle θ and converter output ψ. 
               
             
          
           
               
                 Quadrant of 
                   
                 Converter 
               
               
                 input angle 
                   
                 output ψ at 
               
               
                 θ 
                 Loop error signal 
                 steady state 
               
               
                   
               
               
                 θ ε [0, π/2] 
                 
                   
                     
                       
                         
                           
                             
                               
                                 
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                                            
                                           
                                             ( 
                                             θ 
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           ψ 
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           θ 
                                           ) 
                                         
                                       
                                        
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           ψ 
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               = 
                               
                                 
                                   2 
                                   π 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             θ 
                                             - 
                                             π 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           ψ 
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             θ 
                                             - 
                                             π 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           ψ 
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               = 
                               
                                 
                                   2 
                                   π 
                                 
                                 × 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       θ 
                                       - 
                                       π 
                                       - 
                                       ψ 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 ψ = θ − π 
               
               
                   
               
               
                 θ ε [3π/2, 2π] 
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   ɛ 
                                   _ 
                                 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                               = 
                               
                                 
                                   2 
                                   π 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     
                                       
                                         - 
                                         
                                           sin 
                                            
                                           
                                             ( 
                                             θ 
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           ψ 
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           θ 
                                           ) 
                                         
                                       
                                        
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           ψ 
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               = 
                               
                                 
                                   2 
                                   π 
                                 
                                 × 
                                 
                                   ( 
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             
                                               2 
                                                
                                               π 
                                             
                                             - 
                                             θ 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           ψ 
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               2 
                                                
                                               π 
                                             
                                             - 
                                             θ 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           ψ 
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               = 
                               
                                 
                                   2 
                                   π 
                                 
                                 × 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         2 
                                          
                                         π 
                                       
                                       - 
                                       θ 
                                       - 
                                       ψ 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 ψ = 2π − θ 
               
               
                   
               
             
          
         
       
     
         [0054]      FIG. 6  illustrates a block diagram of the extraction circuit  80  of sin(ψ) and cos(ψ) from ψ, according to one embodiment. Given that the signal ψ(θ) is of triangular nature, its conversion to an approximation of sin(ψ) can be achieved using various methods including signal shaping approximation techniques. The same technique can be applied to generate an approximation of cos(ψ) by first inverting ψ(θ) and then adding an offset equivalent to π/2. 
         [0055]      FIG. 7  illustrates one possible way for implementing the signal shaping technique. The circuit illustrated in  FIG. 7  has been tuned for a ψ(θ) signal with a maximum amplitude of 10V (equivalent to π/2).  FIG. 8  illustrates an example of analog implementation for the approximation of sin(ψ) from ψ(θ) using approximation techniques (see non-patent document 5). In  FIG. 8 , an analog multiplier is used to implement the following rational approximation: 
         [0000]    
       
         
           
             
               
                 
                   
                     10 
                      
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         θ 
                         ) 
                       
                     
                   
                   ≈ 
                   
                     
                       ψ 
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               9 
                                
                               ψ 
                             
                             140 
                           
                         
                         ) 
                       
                     
                     
                       
                         7.3 
                         12 
                       
                       - 
                       
                         ψ 
                         40 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    It is noted that the analog multiplier used may be a single quadrant multiplier as both input and outputs are positive. 
         [0056]      FIG. 9  illustrates a block diagram of the quadrant identification circuit  90  producing the two digital outputs BitØ(θ) and Bit 1 (θ), according to an embodiment. Given the triangular shape of ψ(θ), this quadrant identification is used in order to have an unambiguous measure of θ. The circuit  90  may include three comparators, two D-flip-flops and one XOR gate. The principle of operation is based on determining the phases of V s (t,θ) and V C (t,θ) relative to V ex (t); the phases depend on the signs of sin(θ) and cos(θ).  FIG. 10  illustrates all input and output waveforms of the proposed converter for one full revolution of the resolver shaft with a typical resolver excitation frequency of 4 kHz. 
         [0057]    In view of the above, embodiments of the present invention provide a resolver converter that uses the resolver output signals without the need for demodulation. Additionally, the required determination of the sine and cosine of the estimated angle is carried out in a straightforward way without the need for complicated mixed analog/digital circuitry. These improvements when compared to conventional PLL method reduce cost and simplify design without compromising performance. 
         [0058]    In an embodiment, the converter output ψ is equivalent to angle varying between 0 and π/2, and hence sin(ψ) and cos(ψ) are always positive. This means all inputs and outputs of the two multipliers of the PD ( 60 ) can be single quadrant multipliers, as opposed to a conventional PLL method which require expensive four-quadrant multipliers. 
         [0059]    The fact that ψ varies between 0 and π/2 simplifies greatly the determination of sin(ψ) and cos(ψ) as simple signal shaping networks or simple approximation formulas may be used to convert ψ into sin(ψ) and cos(ψ) using the extraction circuit  80 . In conventional methods, the output of the integrator represents an angle varying between 0 and π or 2π, rendering extraction of sin(ψ) and cos(ψ) from ψ more challenging and complicated. 
         [0060]    Additionally, given that ψ varies between 0 and π/2, an analog implementation of the converter would provide an analog voltage output with a full scale representing one quadrant (90 degrees) of input angle as opposed to the four quadrants (360 degrees) for the conventional converter. This means better sensitivity of the present invention compared to the conventional converter. 
         [0061]    The converter, according to an embodiment, works with modulated resolver signals without the need for demodulators as is the case in conventional methods. This is made possible, in part, by using the absolute values circuits  40  which result in an error signal (output of PD  50 ) with a non-zero average value. In conventional methods without absolute units, the error signal would have a zero average value if no demodulators are used; this would mean that the control loop would not work and the converter would fail to provide a measure of θ. 
         [0062]    One having ordinary skill in the art will readily understand that the invention as discussed above may be practiced with steps in a different order, and/or with hardware elements in configurations which are different than those which are disclosed. Therefore, although the invention has been described based upon these preferred embodiments, it would be apparent to those of skill in the art that certain modifications, variations, and alternative constructions would be apparent, while remaining within the spirit and scope of the invention. In order to determine the metes and bounds of the invention, therefore, reference should be made to the appended claims.