Patent Publication Number: US-11397098-B2

Title: Method for detecting errors in a rotating position sensor system having sine and cosine signals

Description:
RELATED APPLICATION 
     This disclosure claims priority to U.S. Provisional Application Ser. No. 62/745,936, filed on Oct. 15, 2018, which is herein incorporated by reference in its entirety. 
    
    
     TECHNICAL FIELD 
     Embodiments of the present invention are related to position sensors and, in particular, to a method for detecting errors in a rotational position sensor system having sine and cosine signals. 
     DISCUSSION OF RELATED ART 
     Position sensors are used in various settings for measuring the position of one component with respect to another. Inductive position sensors can be used in automotive, industrial and consumer applications for absolute rotary and linear motion sensing. In many inductive positioning sensing systems, a transmitter coil is driven to induce eddy currents in a metallic target that is sliding or rotating above a set of receiver coils. Receiver coils receive the magnetic field generated from eddy currents and the transmitter coils and provide signals to a processor. The processor uses the signals from the receiver coils to determine the position of the metallic target above the set of receiver coils. The processor, transmitter coils, and receiver coils may all be formed on a printed circuit board (PCB). 
     However, these systems exhibit inaccuracies and are then calibrated to account for these inaccuracies. One major concern in a sensor system is the need for understanding the overall error of the sensor information provided by the signal chain in the sensor module (Sensor raw data+analog signal processing+digital signal processing). One effective way to investigate the error of a sensor system is to compare the sensor results with a known good reference. However, for cost and handling reasons, this method can be used only for product development and during the end-of-line calibration in a production line, but is not generally used on every unit produced during normal operation. 
     Another effective way is to use a redundant method where two independent sensors are measuring the same physical parameter (e.g. rotary position) and the Electric Control Unit (ECU) compares the two sensor results. A deviation of the two sensor output signals is considered a failure of the system, but it might not reveal which sensor is defective and which one is operating correctly. 
     Particularly in high speed sensor systems, as for example in a rotor position sensor of an electric motor, the sensor signal cannot be interrupted during operation, therefore it is not allowed, for example, to replace the sensor raw signal with a known reference signal for making a reference measurement. Therefore, the error analysis must be performed with the un-interrupted sine and cosine signals. 
     Therefore, there is a need to develop better methods of detecting and reducing position sensor errors. 
     SUMMARY 
     In some embodiments, a method of correcting for errors in a rotational position sensor having a sine signal and a cosine signal is presented. The method includes compiling data from the sine signal and the cosine signal over a period of rotation of a motor shaft; determining offset correction parameters from the data; correcting the data with the offset correction parameters; determining amplitude difference parameters from the data; correcting the data with the amplitude difference parameters; determining phase difference parameters from the data; correcting the data with the phase difference parameters; and using the offset correction parameters, the amplitude difference parameters, and the phase difference parameters to correct the sine signal and the cosine signal. 
     These and other embodiments are discussed below with respect to the following figures. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIGS. 1A, 1B, and 1C  illustrate an inductive position sensor. 
         FIG. 2A  illustrates raw signals of a position sensor. 
         FIG. 2B  illustrates a polar graph of an error-free sine and cosine signal. 
         FIG. 2C  illustrates rectangular coordinates of an error-free sine and cosine signal. 
         FIG. 3  illustrates a signal path of a position sensor system having sine and cosine signals. 
         FIGS. 4A and 4B  illustrate a process that can be executed on a processor of the position sensor system according to some embodiments. 
         FIGS. 5A, 5B, and 5C  illustrate a positive offset on the sine signal with an ideal cosine signal. 
         FIGS. 6A, 6B, and 6C  illustrate a negative offset on the sine signal with an ideal cosine signal. 
         FIGS. 7A, 7B, and 7C  illustrate a positive offset on the cosine signal with an ideal sine signal. 
         FIGS. 8A, 8B, and 8C  illustrates a negative offset on the cosine signal with an ideal sine signal. 
         FIGS. 9A, 9B, and 9C  illustrate a combination of offsets on the sine and the cosine signals. 
         FIGS. 10A, 10B, and 10C  illustrate an amplitude of the sine signal that is greater than the amplitude of the cosine signal. 
         FIGS. 11A, 11B, and 11C  illustrate an amplitude of the cosine signal that is greater than the amplitude of the sine signal. 
         FIGS. 12A, 12B, and 12C  illustrates a cosine to sine phase shift of less than 90 degrees. 
         FIGS. 13A, 13B, and 13C  illustrates a cosine to sine phase shift greater than 90 degrees. 
         FIGS. 14A, 14B, and 14C  illustrates a combination of gain and phase errors. 
         FIGS. 15A, 15B, and 15C  illustrates a combination of offset and gain errors. 
         FIGS. 16A, 16B, and 16C  illustrates a combination of offset and phase errors. 
         FIGS. 17A, 17B, and 17C  illustrates a combination of offset, gain, and phase errors. 
     
    
    
     These and other aspects of embodiments of the present invention are further discussed below. 
     DETAILED DESCRIPTION 
     In the following description, specific details are set forth describing some embodiments of the present invention. It will be apparent, however, to one skilled in the art that some embodiments may be practiced without some or all of these specific details. The specific embodiments disclosed herein are meant to be illustrative but not limiting. One skilled in the art may realize other elements that, although not specifically described here, are within the scope and the spirit of this disclosure. 
     This description illustrates inventive aspects and embodiments should not be taken as limiting—the claims define the protected invention. Various changes may be made without departing from the spirit and scope of this description and the claims. In some instances, well-known structures and techniques have not been shown or described in detail in order not to obscure the invention. 
     Some embodiments of this invention provide a simple method is described, which allows precise error determination from the sine and cosine signals without the need of a reference sensor and likewise without the need of a second, redundant sensor. The method is based on the fact that in a sensor system having offset errors, either on one or on both sensor signals (sine and/or cosine) there is a fixed relation between the normalized ripple on the magnitude of the signal and the normalized, full scale peak-to-peak error over a full period. Likewise, in a sensor system having gain and/or phase errors in the sensor signals (sine and/or cosine) there is the same fixed relation between the normalized ripple on the magnitude of the signal and the normalized peak-to-peak error over a full period. 
     Many types of position sensors, particularly position sensors for detecting rotary movement, provide the raw sensor data in the form of sinusoidal signal, where one period of the sensor signal may represent a full turn or fractions of a full turn of the item being measured (e.g. a rotor in an electric motor). In order to improve the signal quality and consequently the accuracy of the sensor system, a second signal is introduced, which is again sinusoidal in format, but phase shifted by a quarter of one phase, or 90 electrical degrees. While the first signal is typically designated as Sine-signal, the second signal is typically designated as Cosine signal. Such types of position sensor include, but are not limited to magnetic sensors, such as Hall, AMR (anisotropic magneto resistance), TMR (tunneling magneto resistance) or GMR (giant magneto resistance) sensors as well as inductive or eddy current sensors. 
       FIG. 1A  illustrates a block diagram of a position sensor  100 . As illustrated in  FIG. 1A , position sensor  100  includes a transmitter coil  104  and receiver coils  106  coupled to a controller  102 . Controller  102  drives transmitter coil  104  to generate a time-varying magnetic field, which induces eddy currents in a conductive target that is in a position relative to transmitter coil  104  and receiver coils  106 . Receiver coils  106  are positioned within transmitter coil  104  and provides a signal related to the spatial superposition of magnetic fields generated by the transmitter coil and the magnetic fields generated by the eddy currents induced in the conductive target positioned over the receiver coils  106 . In general, transmitter coil  104  and receiver coils  106  can be arranged to provide sensors of varying geometry, including linear and rotational positioning. 
       FIGS. 1B and 1C  illustrate a planar view and a cross-sectional view of a rotational inductive position sensor.  FIG. 1B  illustrates position sensor coils  110 . Position sensors coils  110  includes transmitter coil  104  and receiver coils  106  formed on a printed circuit board  112 . As is illustrated in  FIG. 1B , position sensor coils  110  are formed in a circular fashion. Transmitter coil  104  is formed in a circle around receiver coils  106 . Receiver coils  106  can typically be two coils, a sine-coil that generates a sine signal as a target is rotated over receiver coils  106  and a cosine-coil that generates a cosine signal as the target is rotated over receiver coils  106 . Leads  118  provide electrical connections between transmitter coil  104  and receiver coils  106  and controller  102  as illustrated in  FIG. 1A . 
       FIG. 1C  further illustrates operation of position sensor  100  using position sensor coils  110 . As shown in  FIG. 1C , transmitter coils  104  and receiver coils  106  are formed on PCB  112 . Transmitter coils  104  and receiver coils  106  can be formed on both sides of PCB  112 . A circular hole  108  is formed in the center of position sensor coils  110  through which a rotor shaft  114  can pass. A target  116 , which is positioned to rotate over receiver coils  106 , is coupled to rotor shaft  114 . 
       FIG. 2A  illustrates a graph of raw output signals from a position sensor  100  as illustrated in  FIGS. 1A, 1B , and IC. In this example, receiver coils  106  include a sine coil and a cosine coil. In  FIG. 2A , the horizontal axis of the graph represents the mechanical rotation (in this case 0-360 degrees), the vertical axis of the graph shows the output voltages from the sine coil and the cosine coil of receiver coils  106 . As the target is rotated over receiver coils through the full rotation, the sine coil provides a voltage given by:
 
 V   sine   =A  sin(α),
 
where A is the amplitude of the signal and a is the angular position. Similarly, the sine coil of the receiver coils  106  provides a voltage given by:
 
 V   cosine   =B  cos(α),
 
where B is the amplitude of the cosine coils of receiver coils  106  as target  116  is rotated. In the example illustrated in  FIG. 2A , the amplitude of the cosine signal B is slightly larger than the amplitude of the sine signal A. The angle α can be designated in degrees or radians. In an ideal environment, the amplitudes of the sine signal and the cosine signal would be the same (A=B).
 
     Having a sine and cosine signal that result from sine coils and cosine coils  106  available to provide position information simplifies the calculation of the actual absolute position of target  116 , which is the position of rotor  114 . In particular, the position being measured uses a simple arctangent function: 
             Position   =         arctan   ⁡     (         a     si   ⁢           ⁢   n       *     sin   ⁡     (   x   )             a     co   ⁢           ⁢   s       *     cos   ⁡     (   x   )           )       -&gt;   Position     =       arctan   ⁡     (       sin   ⁡     (   x   )         cos   ⁡     (   x   )         )       ⁢           ⁢   in   ⁢           ⁢   radians                   Position   =           arctan   ⁡     (         a     si   ⁢           ⁢   n       *     sin   ⁡     (   α   )             a   cos     *     cos   ⁡     (   α   )           )       *     180   π       -&gt;   Position     =       arctan   ⁡     (       sin   ⁡     (   α   )         cos   ⁡     (   α   )         )       ⁢           ⁢   in   ⁢           ⁢   degrees             
where: a sin : peak amplitude of sine signal,
         a cos : peak amplitude of cosine signal,   x: angular position in radians (0 . . . 2π), and   α: angular position in degrees (0 . . . 360).       

     The division of V sin/V cos eliminates the absolute level of the signal and thus provides a ratiometric measurement of sine divided by cosine, which is equal to the tangent of the angular position: tan(x)=sin(x)/cos(x). Consequently, the arctangent of sin(x)/cos (x) provides the angle (x): x=arctan(tan(x))=arctan(sin(x)/cos(x)). 
       FIG. 2B  illustrates another depiction, a polar coordinate system, of the sine and cosine output signals from receiver coils  106 . Another important parameter of the sensor system is the magnitude of the these signals, represented above as A and B. These magnitudes are representative of the signal level strength. In a polar coordinate system as shown in  FIG. 2B , the X-axis is represented by the cosine signal V cos  and the Y-axis is represented by the sine signal V sin . The value α in this graph depicts the angular vector of the signal, which is the rotational position of target  116  to be measured. The Magnitude (“MAG”) is the length of the vector, the peak level of the signals. In an error-free system, having undistorted sine and cosine signal (A=B), no offset, matching peak amplitude levels and exactly 90° phase shift between the sine and cosine functions, the magnitude is calculated in a simplified form as:
 
MAG=√{square root over ( V   sin   1   +V   cos   2 )}=√{square root over (( A  sin(α)) 2 +( A  cos(α)) 2 )}= A,  
 
where the magnitude (MAG)=the Magnitude level of the sine and cosine signals, α=the Angular position (in degrees, 0°≤α&lt;360°), and A is the amplitude of the sine function and the cosine function. In the polar coordinate example illustrated in  FIG. 2B , A=B=1.
 
     The polar representation of such signals, as shown in  FIG. 2B , which shows peak levels on sine and cosine signals, is a perfect circle, centered around the zero position, at the crossing of the X and Y axis. The cosine signal V cos =cos(α) is plotted on the horizontal axis X and the sine signal V sin =sin(α) is plotted on the vertical axis Y. Peak amplitude A for sine and A for the cosine signal is assumed to be 1 in the graphical representation depicted in the ideal situation illustrated in  FIG. 2B . The magnitude vector has a constant length across the full circle, and the positional error (i.e., the difference between the calculated position of the target and its actual position) over the full 360° range is zero. 
       FIG. 2C  illustrates the sine and cosine signals for an error free set of coordinates with magnitude  1 .  FIG. 2C  illustrates the ideal situation where the sine signal  202  represented as V sin (α) and the cosine signal  204  represented by V cos (α) with a resulting signal magnitude (MAG)  206  of 1 through all angles. This results in a full-scale (FS) percent error  208  of 0 through all angles. 
     In practice, the signal path (raw signals→analog processing→digital processing→accurate position) may contain errors. These can be categorized into (1) Offset errors: one or both signals are shifted in the voltage domain in either the positive or negative direction; (2) Gain mismatch errors: different gains of sine and cosine signal path resulting in a mismatch of the amplitudes; and (3) Phase errors resulting in a phase shift between sine signal and cosine signal that is not 90°. In a system having such errors, the magnitude vector length is no longer a constant value throughout one period. However, as is further illustrated below, these errors represent distinctive patterns over one period, based on the type of error. Embodiments of the present invention analyze the pattern of the magnitude vector length within a minimum of half a period of rotation, ideally one full period of rotation, and provides evidence about the type of errors and the size of the error in that system. This data can be used to correct the processing so that more accurate positional determinations can be made. 
     In a typical rotating position sensor system  100 , the input signals (sine and cosine signals) are constantly monitored and the angular position is calculated using the arctangent of the ratio of the sine and cosine signals as described above. In some embodiments, other appropriate methods, such as a look-up table, can be used. Likewise, the Magnitude of the signals using the square root of the square sum of sine and cosine signals, as described earlier is calculated. By monitoring the minimum and maximum signal levels (peak-to-peak magnitude) as well as the number of periods of that signal within one electrical period, the type of error along with the strength of that error can be estimated by the methods described further in this document. 
     The magnitude of a system  100  having offset, gain mismatch and phase errors is calculated as: 
             MAG   =             V     si   ⁢           ⁢   n     2     ⁡     (   α   )       +       V     c   ⁢           ⁢   os     2     ⁡     (   α   )           =           [       δ     s   ⁢           ⁢   i   ⁢           ⁢   n       +       a     si   ⁢           ⁢   n       ⁢     sin   ⁡     (   α   )           ]     2     +     [       δ     co   ⁢           ⁢   s       +       a     co   ⁢           ⁢   s       ⁢       cos   (     α   +   φ     ]     2                       
where MAG is the magnitude of a signal including offset, amplitude mismatch and phase errors, such as that depicted in the polar coordinate system illustrated in  FIG. 2B . In the above equation. δ sin  is the offset level on the sine signal, δ cos  is the offset level on the cosine signal, a sin  is the amplitude level of the sine signal, a cos  is the amplitude of the cosine signal, α is the angular position (0°≤α≤360 0 ), and φ is the phase error (difference from 90°) of the cosine signal relative to the sine signal.
 
       FIG. 3  shows a signal path diagram for system  100 . As is illustrated in  FIG. 3 , the sine signal and the cosine signal from position sensor receiver coils  106  is input to an analog circuit  320  and then to a digital electric control unit (ECU)  322 . Consequently, the raw sine and cosine signals from the position sensor receiver coils  106  is amplified, rectified, and filtered in analog section  320  before being sent to the receiving electrical control unit  322 , where the signals are converted to the digital domain and further processed. As illustrated in  FIG. 3 , the sine signal is input to an offset adder  302  where an AC offset voltage is added. The signal from offset adder  302  is then amplified in amplifier  306 , rectified in rectifier  324 , and filtered in filter  310 . As a result, an offset, an amplitude gain, and a phase are introduced to the rectified signal. Similarly, the cosine signal is input to an offset adder  304  where an offset voltage is added, an amplifier  308  that amplifies the signal, a rectifier  326 , and a filter  312  that filters the signal. Rectifiers  324  and  326  rectify the signals from amplifiers  306  and  308 , respectively, and output signals with DC average values proportional (x 2/π) to the amplitudes of the amplified sign and cosine receiver voltages, respectively. The remaining high frequency component of the signals is removed by filters  310  and  312 , respectively. The amplified and filtered sine signal and cosine signal are then input to ECU  322 . In ECU  322 , the sine signal is digitized in analog-to-digital converter (ADC)  314  to obtain a digitized signal V sin  and input to digital signal processing  318 . Similarly, the cosine signal is input to ADC  316  to obtain a digitized signal V cos  and input to digital signal processing  318 . 
     Digital signal processing  318  can include a microprocessor, memory (both volatile and non-volatile), and support circuitry sufficient to perform the functions as described here. Digital signal processing  318  receives the digitized signal V sin  from ADC  314  and the digitized signal V cos  from ADC  316  and determines the position of target  116 , for example by an arctangent conversion as described above. The calculated position is then available at the output of the digital signal processor  322  in the form of digital data. 
     In accordance with embodiments of the present invention, digital signal processor  322  also determines the type and magnitude of the errors in the received signals V sin  and V cos  by analyzing characteristic patterns plotted over a period of the rotation. Depending on the type of error (offset, gain, or phase error) the magnitude and error curves, plotted over the full phase, show characteristic patterns, which allows a conclusion of the error type and the magnitude of the error from analysis of the magnitude over the angle pattern. While offset related errors lead to a 1-periodic error signal and a 1-periodic magnitude over one mechanical phase, gain and phase related errors lead to a 2-periodic error signal. 
     A major requirement in a sensor system such as sensor system  100  is the need for understanding the overall error of the sensor information provided by the signal chain in the sensor module (sensor raw data→analog signal processing→digital signal processing). An effective way to investigate the error of a sensor system is to compare the sensor results with a known good reference. However, for cost and handling reasons, this method is used only for product development and during end of line calibration in a production line, but is not used on every unit during normal operation. Another effective way is to use a redundant method, where two independent sensors are measuring the same physical parameter (e.g. rotary position) and the Electric Control Unit (ECU) compares the two sensor results. A deviation of the two sensor output signals is considered a failure of the system, but it might not reveal which sensor is defective and which one is operating correctly. Particularly in a high-speed rotary sensor systems, as for example in a rotor position sensor of an electric motor, the sensor signal cannot be interrupted during operation, therefore it is not allowed for example to replace the sensor raw signal with a known reference signal for making a reference measurement. Therefore, the error analysis must be performed with the un-interrupted sine and cosine signals. 
       FIG. 4A  illustrates an example process  400  that can be executed on digital signal processing  318  as illustrated in  FIG. 3 . Process  400  receives the sine signal V sin  and the cosine signal V cos  in step  402 . This occurs as shaft  114  is rotated such that target  116  is rotated over receiver coils  106 . On each rotation of shaft  114 , a number of data points will be received, depending on the relationship between the rotational frequency of shaft  114  and the internal operating clock that determines data sampling of digital signal processing  318 . In step  404 , the data is compiled such that all of the V sin  and V cos  data over a rotational period of shaft  114 , for example data can be compiled over a full rotation period or over a number of full rotational periods to reduce the impact of noise by averaging, are stored. This compiled data can be continuously updated on each rotation of shaft  114 , or can be updated every few rotations of the shaft. 
     In step  406 , the error pattern is determined from the compiled V sin  and V cos . In step  408 , the offset, amplitude, and phase parameters are determined based on the error pattern recognition and the magnitude of the errors determined. Steps  406  and  408  are further discussed below in further detail. 
     In step  410 , corrections to the sine signal V sin  and the cosine signal V cos  can be determined. These corrections can be input to step  412 . Step  412  receives the originally received sine signal V sin  and cosine signal V cos  and corrects these signals according to the corrections determined by step  410  to provide step  414  with corrected signals V sin  and V cos . In step  414 , the position of target  116  over receiver coils  106  is determined. 
     The loop that includes steps  412  and  414  can be executed on each sampling cycle of the sine signal V sin  and the cosine signal V cos . Therefore, the position determination is accomplished on each sampled values V sin  and V cos . The loop of process  400  that includes steps  404  through  410  can be executed on each rotational period of shaft  114  (i.e., through each rotation through 360°) or on selected rotational periods of shaft  114 . 
     In some embodiments, steps  406 ,  408 , and  410  can be performed with a simple method that allows precise error determination from the sine and cosine signals V sin  and V cos  without the need of a reference sensor and likewise without the need of a second, redundant sensor. Some methods according to these embodiments are based on the observation that in a sensor system  100  having offset errors, either on one or on both sensor signals (sine and/or cosine), there is a fixed relation between the normalized ripple on the magnitude of the signals and the normalized, full scale peak-to-peak error over a full period of rotation. Likewise, in a sensor system  100  having gain and/or phase errors in the sensor signals (sine and/or cosine) there is the same fixed relation between the normalized ripple on the magnitude of the signal and the normalized peak-to-peak error over a full period. 
       FIG. 4B  illustrates a more detailed example of part of process  400 , in particular steps  402 - 410  illustrated in  FIG. 4B . As discussed above, in step  402  the sine signal V sin (α) and the cosine signal V cos (α) are received in digital signal processing  318 . Data is then compiled over one or more rotational periods (rotations of a through 360°) are accumulated in step  404 . Although in some embodiments all of the data points of V sin (α) and V cos (α) can be compiled and stored, in some embodiments data is extracted from the input signals V sin (α) and V cos (α) and stored instead. 
     As illustrated in  FIG. 4B , compilation step  404  can simultaneously compile multiple derived data over the rotational period or periods of the data acquisition. In step  416 , the signals V sin (α) and V cos (α) can be monitored and stored. The minimum and maximum values can be recorded over the rotational period or periods of the data acquisition in step  418 . 
     In some embodiments, in step  420  the magnitude is recorded as a function of angle MAG(α). As discussed above, the magnitude can be calculated from the signals V sin (α) and V cos (α) as discussed above and is given by:
 
MAG(α)=√{square root over ( V   sin   2 (α)+ V   cos   2 (α))}
 
In step  422 , the maximum and minimum values of MAG(α) over the data acquisition period can be recorded. In step  404 , some other values can be calculated and recorded for later use as well.
 
     In step  424 , an estimated error can be calculated from the maximum and minimum values of MAG(α). In that calculation, the normalized magnitude ripple can be defined as: 
               MAG   norm     =         MAG     ma   ⁢           ⁢   x       -     MAG     m   ⁢           ⁢   i   ⁢           ⁢   n             MAG     m   ⁢           ⁢   a   ⁢           ⁢   x       +     MAG     m   ⁢           ⁢   i   ⁢           ⁢   n                 
where MAG norm  is the normalized magnitude ripple of the sine and cosine signals over a full period and has a value between 0 and 1, MAG min  is the minimum magnitude level of MAG(α) over a full period, and MAG max  is the maximum magnitude level of MAG(α) over a full period. Consequently, MAG norm  is determined by determining the minimum MAG(α), MAG min , and the maximum MAG(α), MAG max , over a full period of compiled data from step  416  and calculating the normalized magnitude ripple MAG norm  as discussed above.
 
     Additionally, a normalized error can be estimated from the MAG norm . The normalized error err norm , which is also the peak-to-peak value of err(α), can be defined as: 
               err   norm     =         err     ma   ⁢           ⁢   x       -     err     m   ⁢           ⁢   i   ⁢           ⁢   n           360   ⁢   °             
where err norm  is the normalized full scale error over a full 360° rotation, err min  is the minimum positional error in degrees over the full 360° rotation, and err max  is the maximum positional error in degrees over the full 360° rotation. In some embodiments, err min  can be considered to be 0. Table 1 below illustrates the position of extremes of of MAG(α) and err(α) patterns with various combinations of error resulting in offset, amplitude mismatch, and phase. These patterns are further discussed below with respect to various Figures. In particular, error case 1.1 illustrates a positive offset error on the sine signal, such as that illustrated in  FIGS. 5A, 5B, and 5C . Error case 1.2 illustrates a negative offset on the sine signal, such as that illustrated in  FIGS. 6A, 6B , and  6 C. Error case 1.3 illustrates a positive offset on the cosine signal, such as that illustrated in  FIGS. 7A, 7B, and 7C . Error case 1.4 illustrates a negative offset on the cosine signal such as that illustrated in  FIGS. 8A, 8B, and 8C . Error case 1.5 illustrates a combination of offsets on the sine signal and the cosine signal as is illustrated in  FIGS. 9A, 9B, and 9C . As indicated in Table 1, these offset errors illustrate patterns of MAG(α) and err(α) that are sinusoidal and repeat every period of rotation (1-periodic sinusoidal). The error pattern description and magnitude pattern description columns indicate the positional angle where the maximum and minimum values of err(α) and MAG(α) are found in the pattern. The last column provides the ratio of the magnitude ripple MAG norm , and normalized error err norm .
 
     Error cases 2.1, 2.2, 3.1, 3.2, and 3.3 illustrate error cases where the patterns of MAG(α) and err(α) repeats twice every rotational period (2-periodic sinusoidal) (i.e., there are two maximum values and two minimum values of MAG(α) and err(α) for each rotational period). Error case 2.1 illustrates a case where the sine signal amplitude is greater than that of the cosine signal, which is illustrated in  FIGS. 10A, 10B, and 10C . Error case 2.2 illustrates a case where the amplitude of the cosine signal is greater than that of the sine signal, which is illustrated in  FIGS. 11A, 11B, and 11C . Error case 3.1 illustrates a case where the cosine signal phase shift from the sine signal is greater than 90°, which is illustrated in  FIGS. 12A, 12B, and 12C . Error case 3.2 illustrates a case where the cosine signal phase shift from the sine signal is less than 90°, which is illustrated in  FIGS. 13A, 13B, and 13C . 
     Error cases 4.1, 4.2, and 4.3 illustrate combinations of 1-periodic and 2-periodic patterned errors. Case 4.1 illustrates a combination of offset and gain errors, which is illustrated in  FIGS. 15A, 15B, and 15C . Case 4.2 illustrates a combination of offset and phase errors, which is illustrated in  FIGS. 16A, 16B, and 16C . Case 4.3 illustrates a combination of offset, gain, and phase errors, which is illustrated in  FIGS. 17A, 17B, and 17C . 
     As shown in Table 1 (below), for any combination of 1-periodic errors (offset errors on sine and/or cosine signal), the ratio between normalized Magnitude and normalized error is a fixed factor of 3.14:1. Likewise, for any combination of 2-periodic errors (gain or phase errors on sine and/or cosine signal), the ratio between normalized Magnitude and normalized error is a fixed factor of 3.14:1. Another indicator is that the waveform of the error err(α) resembles the 1st derivative of the magnitude MAG(α). To determine the error of a rotating sensor system, the maximum and minimum magnitude levels over a full period can be constantly monitored and the magnitude ripple calculated from these two values or the first derivative of MAG(α) can be determined to provide the shape of err(α). Dividing the magnitude ripple by the fixed factor of 3.14 leads to an estimation of the maximum normalized error of the system. 
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                   
                   
                 Magnitude 
                 Position of Error 
                 Position of Magnitude 
                   
               
               
                 Error 
                   
                 and Error 
                 pattern extremes 
                 pattern extremes 
                 Magnitude 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 case 
                 Type of error 
                 pattern 
                 Max pos 
                 Max neg 
                 Max pos 
                 Max neg 
                 vs. error 
               
               
                   
               
               
                 1.1 
                 Pos offset on Sin 
                 1-periodic 
                  0° 
                 180° 
                  90° 
                 270° 
                 3.14x 
               
               
                 1.2 
                 Neg. offset on Sin 
                 sinusoidal 
                 180° 
                  0° 
                 270° 
                  90° 
                 3.14x 
               
               
                 1.3 
                 Pos. offset on Cos 
                   
                 270° 
                  90° 
                  0° 
                 180° 
                 3.14x 
               
               
                 1.4 
                 Neg. offset on Cos 
                   
                  90° 
                 270° 
                 180° 
                  0° 
                 3.14x 
               
            
           
           
               
               
               
               
               
               
            
               
                 1.5 
                 Combinations 
                   
                 Variable 
                 Variable 
                 3.14x 
               
               
                   
                 of 1.1 to 1.4 
               
               
                   
                 offset errors 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 2.1 
                 Amplitude Sin &gt; Cos 
                 2-periodic 
                  45°, 225° 
                 135°, 315° 
                 90°, 270° 
                  0°, 180° 
                 3.14x 
               
               
                 2.2 
                 Amplitude Cos &gt; Sin 
                 sinusoidal 
                 135°, 315° 
                  45°, 225° 
                  0°, 180° 
                  90°, 270° 
                 3.14x 
               
               
                 3.1 
                 Cos phase shift &lt;90° 
                   
                  0°, 180° 
                  90°, 270° 
                 45°, 225° 
                 135°, 315° 
                 3.14x 
               
               
                 3.2 
                 Cos phase shift &gt;90° 
                   
                  90°, 270° 
                  0°, 180° 
                 135°, 315°  
                  45°, 225° 
                 3.14x 
               
            
           
           
               
               
               
               
               
               
            
               
                 3.3 
                 Combinations of gain 
                   
                 Variable 
                 Variable 
                 3.14x 
               
               
                   
                 and phase errors 
               
            
           
           
               
               
               
               
               
            
               
                 4.1 
                 Combinations of offset 
                 Combination of 1- and 2- periodic pattern, 
                 Combination of 1- and 2- 
                 Variable 
               
               
                   
                 and gain errors 
                 depending on error type 
                 periodic pattern, 
               
               
                   
                   
                   
                 depending on error type 
               
               
                 4.2 
                 Combinations of offset 
                 Combination of 1- and 2- periodic pattern, 
                 Combination of 1- and 2- 
                 Variable 
               
               
                   
                 and phase errors 
                 depending on error type 
                 periodic pattern, 
               
               
                   
                   
                   
                 depending on error type 
               
               
                 4.3 
                 Combinations of offset, 
                 Combination of 1- and 2- periodic pattern, 
                 Combination of 1-and 2- 
                 Variable 
               
               
                   
                 gain and phase errors 
                 depending on error type 
                 periodic pattern, 
               
               
                   
                   
                   
                 depending on error type 
               
               
                   
               
            
           
         
       
     
     It should be noted that this ratio of 3.14:1 is maintained only if the type or error is either 1-periodic (from offset errors) or 2-periodic (from gain or phase errors). Combinations of 1-periodic (offset) and 2-periodic (gain or phase) errors lead to non-sinusoidal magnitude and error patterns where there is no fixed ratio between magnitude ripple and maximum error. The ratio will be in a similar region (depending on the type of error), but not exactly 3.14:1. Therefore, in such cases, the magnitude ripple can still be used to determine the error over a full turn, but with less precision. This method also requires that the investigated sensor system is physically rotated by at least a movement that provides half of an electrical period. In practice, especially in constantly rotating systems, a full electrical period will be used as it simplifies the data collection to picking the maximum and minimum magnitude level. 
     The calculated error determined in step  424  as described above, is the maximum peak-to-peak error within one electrical period, it cannot be used to calculate the error at a given static position. However, since the error pattern resembles the 1 st  derivative of the magnitude waveform, the error at any position can be calculated from the 1 st  derivate of the magnitude signal. This could be performed by monitoring not only the maximum and minimum values of the magnitude but also by constantly measuring the magnitude level and calculating the rate of change through the delta of the magnitude over a given change of position in a moving system. 
     Further to the error monitoring based on the pattern of the magnitude signal, the analysis of the sine and cosine signals in a rotating position sensor system also provides inputs that allow an elimination of offset, amplitude mismatch, and phase errors. 
     In some embodiments, steps  420 ,  422 , and  424  may be omitted so that the magnitude and error are not calculated using the uncorrected input data V sin (α) and V cos (α). Instead, these calculations may be performed using corrected data so that the resulting error reflects the remaining error after corrections are made, as suggested in steps  442 - 446  discussed below. 
     In step  426  of  FIG. 4B , the offset errors can be determined. By constantly monitoring the sine and cosine input signals V sin (α) and V cos (α), the peak levels of each signals are recorded and analyzed as discussed above in step  420 . In step  426 , the offset of the sine signal can be calculated over 1 period of the maximum and minimum values of V sin (α) and V cos (α) determined in step  418  as: 
               Offset     SI   ⁢           ⁢   N       =           V   ⁢           ⁢     SIN     ma   ⁢           ⁢   x         +     V   ⁢           ⁢     SIN     m   ⁢           ⁢   i   ⁢           ⁢   n           2     -       V   REF     .             
The offset of the cosine signal is similarly calculated as:
 
               Offset     c   ⁢           ⁢   os       =           v   ⁢           ⁢     cos     ma   ⁢           ⁢   x         +     v   ⁢           ⁢     cos     m   ⁢           ⁢   i   ⁢           ⁢   n           2     -     V   REF             
In these equations, Offset SIN  is the offset on the sine signal V sin (α), Offset COS  is the offset on the cosine signal V cos (α), VSIN max  is the maximum signal level of the sine signal V sin (α) over a full period of rotational motion, VSIN min  is the minimum signal level of the sine signal V sin (α) over a full period of rotational motion, VCOS max  is the maximum signal level of the cosine signal V cos (α) over a full period of rotational motion, VCOS min  is the minimum signal level of the cosine signal V cos (α) over a full period of rotational motion, and VREF is a reference bias point of the sine and cosine signals.
 
     In step  428 , the calculated Offset SIN  can be subtracted from the sine signal V sin (α) to obtain an offset-free sine signal. Similarly, the calculated Offset COS  can be subtracted from the cosine signal V cos (α) to obtain an offset-free cosine signal. The corrected values of V sin (α) and V cos (α) can replace the values recorded in step  416 . Consequently, the values of Offset SIN  and Offset COS  can be provided to step  412  to use to correct the sine and cosine signals during operation. 
     In step  430 , the amplitude mismatch can be determined. By constantly monitoring the sine and cosine input signals and recording the peak levels, as described above in step  420 . The amplitude mismatch of sine and cosine signals is then calculated as: 
             Amplitude_mismatch   ⁢           =           V   ⁢           ⁢     SIN     ma   ⁢           ⁢   x         -     V   ⁢           ⁢     SIN     m   ⁢           ⁢   i   ⁢           ⁢   n               V   ⁢           ⁢     COS     ma   ⁢           ⁢   x         -     V   ⁢           ⁢     COS     m   ⁢           ⁢   i   ⁢           ⁢   n             .           
In this equation, the Amplitude_mismatch is the mismatch of sine and cosine signal levels where a value of 1 represents an ideal matching, VSINmax is the maximum signal level of the sine signal V sin (α), VSINmin is the minimum signal level of the sine signal V sin (α), VCOSmax is the maximum signal level of the cosine signal V cos (α), and VCOSmin is the minimum signal level of the cosine signal V cos (α). By multiplying V cos (α) by the calculated Amplitude_mismatch value, both the sine signals and the cosine signals will have the same amplitude levels. Note that the amplitude correction can be performed whether or not new minimum and maximum values are corrected based on the offset-corrected sine and cosine values.
 
     Consequently, in step  432 , the signals V sin (α) and V cos (α) can be corrected by multiplying the cosine signal V cos (α) by the Amplitude_mismatch. Again, in some embodiments, the recorded values of V sin (α) and V cos (α) can be replaced. At this point, the recorded values of V sin (α) and V cos (α) have been corrected for offset and for amplitude mismatch. 
     In steps  434  through  438 , the phase correction is determined. As illustrated in the ideal situation shown in  FIG. 2C , an ideal phase shift of 90° between sine and cosine signals is indicated by one signal crossing the vertical zero line while the opposite signal has either a minimum or a maximum at this position. In the case of a phase error between the sine and cosine signal (V sin (α) and V cos (α)), the zero-signal crossings no longer occur at one of the peaks of the opposite signal. 
     A simple method to correct phase errors has been described in patent application publication US20130268234A1 “Method and device for determining the absolute position of a movable body,” by a co-author if this application. As described therein, to correct a phase error, two new signals are generated, derived from the sum and the difference of the sine and cosine signals. The sum and the difference of a sine and cosine signal are always phase shifted by 90° independent of the initial phase shift error between the sine and cosine signal (except for 0° and 180° phase shifts where the difference or the sum becomes 0). After the sum and difference calculation, an initial phase shift error between the sine and cosine signals is represented as an amplitude mismatch of the sum and difference signals, which in a subsequent step can be corrected by the same method described for amplitude mismatch correction, or a phase correction can be determined. 
     In step  434 , the sum and difference signals can be determined based on the corrected values of V sin (α) and V cos (α). As such, the value of Sum(α) can be calculated as V sin (α)+V cos (α). The value of Diff(α) can be determined as V sin (α)−V cos (α). As discussed above, adding or subtracting sinusoidal waves of the same frequency but with a phase shift between them yields a sinusoidal wave of the same frequency, but different amplitude and phase relationships. As discussed above, the sum and the difference of two sinusoidal waves are phase shifted by 90° to each other, no matter what the phase shift of the original signals was (except for the special case of phase shifts of 0° and 180° where the sum or difference becomes 0). If both of the signals V sin (α) and V cos (α) have the same amplitude, but a phase relation different from 90°, this phase differences results in an amplitude mismatch between the sum and difference signals. This amplitude mismatch can easily be compensated by measuring the extremes of both signals (sum and difference) to obtain Sum max , Sum min , Diff max , and Diff min  and adjusting the amplitudes as discussed above with respect to the amplitude adjustment. Based on ideal original signals with a 90° phase relation and identical amplitudes, the sum and difference signals Sum(a) and Diff(a) are phase shifted by 45° relative to the original signals V sin (α) and V cos (α). Consequently, the position α can be calculated from the amplitude adjusted values of Sum(α) and Diff(α) while taking the positional phase shift of 45° into account by 
             position   =       arctan   ⁡     (     Diff   Sum     )       .           
In this case, the position of target  116  can be determined accurately while the phase correction is further taken into account. In these embodiments, from step  434 , the system proceeds to step  440  where the sum and difference values are then sued to calculate the angular position of target  116 .
 
     In some embodiments, step  434  proceeds to step  436 , where the maximum and minimum values of Sum(α) and Diff(α), Sum max , Sum min , Diff max , and Diff min  can be normalized. As discussed above, the sum and difference values can be normalized as: 
                 Sum   norm     =         Sum     ma   ⁢           ⁢   x       -     Sum     m   ⁢           ⁢   i   ⁢           ⁢   n             Sum     ma   ⁢           ⁢   x       +     Sum     m   ⁢           ⁢   i   ⁢           ⁢   n             ,     
     ⁢       Diff   norm     =           Diff     ma   ⁢           ⁢   x       -     Diff     m   ⁢           ⁢   i   ⁢           ⁢   n             Diff     m   ⁢           ⁢   a   ⁢           ⁢   x       +     Diff     m   ⁢           ⁢   i   ⁢           ⁢   n           .             
In these equations, Sum norm  is the normalized sum values, Sum max  is the maximum sum value as determined in step  418 , Sum min  is the minimum sum value as determined in step  418 , Diff norm  is the normalized difference value, Diff max  is the maximum difference value as determined in step  418 , and Diff min  is the minimum difference value as determined in step  434 .
 
     In step  436 , the phase difference of the cosine signal relative to 90° from the sine signal, α, can then be given by: 
             φ   =     2   ⁢           ⁢       arctan   ⁡     (       Diff   norm       Sum   norm       )       .             
Consequently, in step  438 , the sine and cosine signals can be corrected for phase differences based on the phase difference calculated in step  436 . In step  440 , the angular position can be calculated using the corrected values of V sin (α) and V cos (α) as described above, these values being corrected for the offset values Offset sin  and Offset cos , the amplitude mismatch, and the phase difference φ. As discussed above, in some embodiments step  440  uses the amplitude adjusted sum and difference values that uses V sin (α) and V cos (α) that have been corrected for the offset values Offset sin  and Offset cos  and the amplitude mismatch to directly calculate the position of target  116 . In step  446 , these correction parameters can be updated to step  412 , for continuous correction of the sine and cosine signals to provide a rotational position in step  414 . In some embodiments, the offset correction values, amplitude correction values, and phase correction can be updated while in other embodiments the offset correction values, amplitude correction values, and Sum/Diff amplitude correction values can be uploaded.
 
     In steps  444  and  446 , the remaining angular position error can be calculated. In step  442 , the values of MAG(α) can be calculated using the corrected values of V sin (α) and V cos (α) as discussed above. The first derivative of the corrected MAG signal can then be determined. In some embodiments, the first derivative of MAG(α) can be determined by monitoring its change as a function of positional angle through a particular range of positional angles. The first derivative of MAG(α) is indicative of the shape of the error err(α) and the peak-peak value of err(α) is related to the magnitude ripple value as discussed above. Alternatively, the MAG signals stored in step  420  can be calculated The magnitude pattern and the first derivative with respect to angular position over a half or full period can be analyzed to determine the remaining error as described above. 
     Therefore, a dynamic signal correction according to the present invention, and as described with  FIGS. 4A and 4B , may consist of the following actions. The method steps as described in  FIGS. 4A and 4B  may occur in any order, and may be executing simultaneously. Therefore, embodiments of the present invention can include (1) constantly monitor sine and cosine input signals while the position sensor system is rotating as described in step  404 ; (2) Record the minimum and maximum levels of the sine and cosine signals as described in step  416 ; (3) Determine the correct offset errors as described above with respect to step  426 ; (4) Apply offset correction to both sine and cosine signals as illustrated in step  428 ; (5) Correct amplitude mismatch as described above in step  430 ; (6) Apply the amplitude mismatch correction as described in step  432 ; (7) Apply phase correction described above, which includes calculating sum and difference of sine and cosine signals as described in step  418  and using amplitude adjusted sum and difference values to directly calculate the position or by normalizing the sum and difference values to calculate the phase difference as described in step  436  and  438 ; (8) Calculate the angular position, by means of arctangent or other methods as discussed with step  440  and step  414 ; (9) Calculate the magnitude of the sine and cosine signals by the square root of the square sums of sine and cosine signals as discussed in steps  442 ; (10) Record the minimum and maximum magnitude values along with their number of periods within one electrical period as described in step  442 ; (11) Determine the first derivative of the magnitude signal by monitoring the change with position within a fixed rotational window as illustrated in step  444  of  FIG. 4B ; and (12) Determine the remaining angular error by analyzing the magnitude pattern and rate of change over a half or full period as described in this document as described in step  446 . 
     As is illustrated in  FIGS. 4A and 4B , the correction of the sine and cosine signals and the calculation of the angular position can be performed throughout the process or, as illustrated in  FIG. 4A , as a separate step. In some embodiments, during calculation of the phase correction in step  434 , sine and cosine data that has been corrected for offsets and amplitude variation can be used. In those embodiments, the sine and cosine data itself is recorded in step  404  and the sum and difference signals are calculated during step  434 . 
     Calculation of the correction values as described in  FIGS. 4A and 4B  can be performed continuously while system  100  is operating. Alternatively, the process described in  FIGS. 4A and 4B  can be performed periodically and the correction parameters updated accordingly. In steps  412  and  414 , the latest updated correction parameters can then be used to continuously correct the values of V sin (α) and V cos (α) received in digital signal processing  318 . 
       FIGS. 5A, 5B, and 5C  illustrate an example with a positive offset on V sin (α) with no corresponding offset on V cos (α), which corresponds to case 1.1 of Table 1.  FIG. 5A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(α) and the error err(α).  FIG. 5B  is a plot of the values of MAG(α) and err(α) under the circumstance illustrated in  FIG. 5A .  FIG. 5C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude for comparison. As is illustrated, an offset on the sine signal V sin (o) results in a 1-periodic pattern in MAG(α) and err(α). This example illustrates a 5% positive offset on the sine signal V sin (α). As is illustrated, the pattern of err(α) is phase shifted by 90 electrical degrees (1 st derivative) relative to the pattern of MAG(α). As is illustrated, the positive maximum of err(α) (err max ) occurs at a position of 0° while the negative maximum (err min ) occurs at 180°. Similarly, the positive maximum of MAG(α) (MAGm ax ) occurs at a position of 90° while the negative maximum (MAG min ) occurs at 270°. The ratio between normalized magnitude ripple and peak-to-peak normalized error is fixed at a ratio of 3.14:1, based on the normalized magnitude ripple MAG norm  and the peak-to-peak normalized error err norm  as discussed above. The polar plot illustrated in  FIG. 5C  shows the ideal figure as dotted circle compared with MAG(α), the actual pattern is shifted up on the Y-axis, the sine axis. 
       FIGS. 6A, 6B, and 6C  illustrate an example with a negative offset on V sin (α) with no corresponding offset on V cos (α), which corresponds to error case 1.2 in Table 1.  FIG. 6A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(α) and the error err(α).  FIG. 6B  is a plot of the values of MAG(α) and err(α) under the circumstances illustrated in  FIG. 6A .  FIG. 6C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude for comparison.  FIGS. 6A, 6B, and 6C  illustrate a negative offset on the sine signal V sin (α) results in a 1-periodic pattern in MAG(α) and err(α). This example illustrates a 5% negative offset on the sine signal V sin (α). The pattern of err(α) is phase shifted by 90 electrical degrees (1st derivative) relative to the pattern of MAG(α). As is illustrated, the positive maximum of err(α) (err max ) occurs at a position of 180° while the negative maximum (err min ) occurs at 0°. Similarly, the positive maximum of MAG(α) (MAG max ) occurs at a position of 270° while the negative maximum (MAG min ) occurs at 90°. Normalized magnitude ripple and peak-to-peak normalized error is fixed at a ratio of 3.14:1, based on the normalized magnitude ripple MAG norm  and the peak-to-peak normalized error err norm  as discussed above. The polar plot illustrated in  FIG. 6C  shows the ideal figure as dotted circle, the actual pattern MAG(α) is shifted down on the Y-axis (sine). 
       FIGS. 7A, 7B, and 7C  illustrate an example with a positive offset on V cos (α) with no corresponding offset on V sin (α), which corresponds to error case 1.3 in Table 1.  FIG. 7A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(α) and the error err(α).  FIG. 7B  is a plot of the values of MAG(α) and err(α) under the circumstances illustrated in  FIG. 7A .  FIG. 7C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude for comparison.  FIGS. 7A, 7B, and 7C  illustrate the effects of the positive offset on a cosine signal. An offset on the cosine signal V cos (α) results in a 1-periodic pattern in MAG(α) and err(α).  FIGS. 7A, 7B, and 7C  illustrate an example having a 5% positive offset on the cosine signal V cos (α). The pattern in error err(α) is phase shifted by 90 electrical degrees (1st derivative) relative to the pattern in MAG(α). As is illustrated, the positive maximum of err(α) (err max ) occurs at a position of 270° while the negative maximum (err min ) occurs at 90°. Similarly, the positive maximum of MAG(α) (MAG max ) occurs at a position of 0° while the negative maximum (MAG min ) occurs at 180°. Normalized magnitude ripple and peak-to-peak normalized error is fixed at a ratio of 3.14:1, based on the normalized magnitude ripple MAG norm , and the peak-to-peak normalized error err norm  as discussed above. The polar plot of  FIG. 7C  shows the ideal figure as a dotted circle along with the actual pattern of MAG(α), which as illustrated is shifted right on the X-axis (cosine). 
       FIGS. 8A, 8B, and 8C  illustrate an example with a negative offset on V cos (α) with no corresponding offset on V sin (α), which corresponds to error case 1.4 of Table 1.  FIG. 8A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(α) and the error err(α).  FIG. 8B  is a plot of the values of MAG(α) and err(α) under the circumstances illustrated in  FIG. 8A .  FIG. 8C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude for comparison.  FIGS. 8A, 8B, and 8C  illustrate a negative offset on cosine signal, which results in a 1-periodic pattern in MAG(α) and err(α).  FIGS. 8A, 8B, and 8C  illustrate an example having a 5% negative offset on the cosine signal V cos (α). The pattern of err(α) is phase shifted by 90 electrical degrees (1st derivative) relative to the pattern of Magnitude MAG(α). As is illustrated, the positive maximum of err(α) (err max ) occurs at a position of 90° while the negative maximum (err min ) occurs at 270°. Similarly, the positive maximum of MAG(α) (MAG max ) occurs at a position of 180° while the negative maximum (MAG min ) occurs at 0°. Normalized magnitude ripple and peak-to-peak normalized error is fixed at a ratio of 3.14:1, based on the normalized magnitude ripple MAG norm  and the peak-to-peak normalized error err norm  as discussed above. The polar plot illustrated in  FIG. 8C  shows the ideal figure as dotted circle overlaid on the actual pattern of MAG(α), which is shifted left on the X-axis (cosine). 
       FIGS. 9A, 9B, and 9C  illustrate an example with a combination of offsets on both the cosine signal V cos (α) and the sine signal V sin (α), which corresponds to error case 1.5 in Table 1.  FIG. 9A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(α) and the error err(α).  FIG. 9B  is a plot of the values of MAG(α) and err(α) under the circumstances illustrated in  FIG. 9A .  FIG. 9C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude for comparison.  FIGS. 9A, 9B, and 9C  illustrate a combination of offsets on both the sine and cosine signals V cos (α) and V sin (α), which results in a 1-periodic pattern in both Magnitude MAG(α) and error err(α).  FIGS. 9A, 9B, and 9C  illustrate an example having a 5% positive offset on the sine signal V sin (α) and a 5% negative offset on the cosine signal V cos (α). The pattern in error err(α) is phase shifted by 90 electrical degrees (1st derivative) relative to the pattern in magnitude MAG(α). As is illustrated, the positive maximum of err(α) (err max ) and negative maximum (err min ) occurs at variable locations, but are characterized with a 1-periodic sinusoidal pattern. Similarly, the positive maximum of MAG(α) (MAG max ) and the negative maximum (MAG min ) are variable, but are characterized with a 1-periodic sinusoidal pattern that is shifted in position by 90° from that of the error maxima. Magnitude ripple and peak-to-peak normalized error is fixed at a ratio of 3.14:1, based on the normalized magnitude ripple MAG norm  and the peak-to-peak normalized error err norm  as discussed above. The polar plot illustrated in  FIG. 9C  shows the ideal figure as dotted circle the actual pattern of MAG(α), which is shifted both left on the X-axis (cosine) and up on the Y-axis (sine). 
       FIGS. 10A, 10B, and 10C  illustrate an example where the amplitude of the cosine signal V cos (α) is less than that of the sine signal V sin (α), which corresponds to error case 2.1 of Table 1.  FIG. 10A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(α) and the error err(α).  FIG. 10B  is a plot of the values of magnitude MAG(α) and error err(α) under the circumstances illustrated in  FIG. 10A .  FIG. 10C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude for comparison.  FIGS. 10A, 10B, and 10C  illustrate an amplitude of the sine signal V sin (α) that is greater than the amplitude of the cosine signal V cos (α). An amplitude mismatch between the sine and cosine signals V sin (α) and V cos (α) results in a 2-periodic pattern in magnitude MAG(α) and error err(α).  FIGS. 10A, 10B , and  10 C illustrate an example having a 5% larger sine signal V sin (α) relative to the cosine signal V cos (α). The pattern in error err(α) is phase shifted by 90 electrical degrees (1st derivative) relative to the pattern in magnitude MAG(α). As is illustrated, the positive maximum of err(α) (err max ) occurs at a positions 45° and 225° while the negative maximum (err min ) occurs at 135° and 315°. Similarly, the positive maximum of MAG(α) (MAG max ) occurs at a position of 90° and 270° while the negative maximum (MAG min ) occurs at 0° and 180 Magnitude ripple and peak-to-peak normalized error is fixed at a ratio of 3.14:1, based on the normalized magnitude ripple MAG norm  and the peak-to-peak normalized error err norm  as discussed above. The polar plot illustrated in  FIG. 10C  shows the ideal figure as dotted circle and the actual pattern MAG(α) in this example, which is stretched in the Y-axis (sine). 
       FIGS. 11A, 11B, and 11C  illustrate an example where the amplitude of the cosine signal V cos (α) is greater than that of the sine signal V sin (α), which corresponds to error case 2.2 of Table 1.  FIG. 11A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(α) and the error err(α).  FIG. 11B  is a plot of the values of magnitude MAG(α) and error err(α) under the circumstances illustrated in  FIG. 11A .  FIG. 11C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude for comparison for the example illustrated in  FIG. 11A .  FIGS. 11A, 11B, and 11C  illustrate an example where the amplitude of the cosine signal V cos (α) is greater than the amplitude of the sine signal V sin (α). An amplitude mismatch between the sine and cosine signals V sin (α) and V cos (α) results in a 2-periodic pattern in magnitude MAG(α) and error err(α).  FIGS. 11A, 11B, and 11C  show an example having a 5% larger cosine signal V cos (α) relative to the sine signal V sin (α). The pattern of error err(α) is phase shifted by 90 electrical degrees (1st derivative) relative to the pattern of the magnitude MAG(α). As is illustrated, the positive maximum of err(α) (err max ) occurs at a positions 135° and 315° while the negative maximum (err min ) occurs at 45° and 225°. Similarly, the positive maximum of MAG(α) (MAG max ) occurs at a position of 0° and 180° while the negative maximum (MAG min ) occurs at 90° and 270°. Magnitude ripple and peak-to-peak normalized error is fixed at a ratio of 3.14:1, based on the normalized magnitude ripple MAG norm  and the peak-to-peak normalized error err norm  as discussed above. The polar plot illustrated in  FIG. 11C  shows the ideal figure as dotted circle with the actual pattern of MAG(α), which is stretched in the X-axis (cosine). 
       FIGS. 12A, 12B, and 12C  illustrate an example where cosine signal V cos (α) to sine phase signal V sin (α) is less than 90 degrees, which corresponds to error case 3.1 of Table 1.  FIG. 12A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(α) and the error err(α).  FIG. 12B  is a plot of the values of magnitude MAG(α) and error err(α) under the circumstances illustrated in  FIG. 12A .  FIG. 12C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude for comparison for the example illustrated in  FIG. 12A .  FIGS. 12A, 12B, and 12C  illustrate a cosine to sine phase shift of less than 90 degrees. A phase mismatch between sine and cosine V sin (α) and V cos (α) signals of &lt;90° results in a 2-periodic pattern in magnitude MAG(α) and error err(α). Shown is an example having an 85° phase shift of the cosine signal V cos (α) relative to the sine signal V sin (α) (i.e., the phase shift discussed above φ is −5°). The pattern of error err(α) is phase shifted by 90 electrical degrees (1st derivative) relative to the pattern of magnitude MAG(α). As is illustrated, the positive maximum of err(α) (err max ) occurs at a positions 0° and 180° while the negative maximum (err min ) occurs at 90° and 270°. Similarly, the positive maximum of MAG(α) (MAG max ) occurs at a position of 45° and 225° while the negative maximum (MAG min ) occurs at 135° and 315°. Magnitude ripple and peak-to-peak normalized error is fixed at a ratio of 3.14:1, based on the normalized magnitude ripple MAG no and the peak-to-peak normalized error err norm  as discussed above. but the absolute error is always negative. The polar plot illustrated in  FIG. 12C  shows the ideal figure as dotted circle and the actual pattern of MAG(α), which is stretched in the 45° direction. 
       FIGS. 13A, 13B, and 13C  illustrate an example where cosine signal V cos (α) to sine phase signal V sin (α) is greater than 90 degrees, which corresponds to error case 3.2 of Table 1.  FIG. 13A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(α) and the error err(α).  FIG. 13B  is a plot of the values of magnitude MAG(α) and error err(α) under the circumstances illustrated in  FIG. 13A .  FIG. 13C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude MAG(α) for comparison in the example illustrated in  FIG. 13A .  FIGS. 13A, 13B, and 13C  illustrate a cosine to sine phase shift of greater than 90 degrees. A phase mismatch between the cosine signal V cos (α) and the sine signal V sin (α) of &gt;90° results in a 2-periodic pattern in the magnitude MAG(α) and error err(α).  FIGS. 13A, 13B, and 13C  show an example having a 95° phase shift of the cosine signal relative to the sine signal (i.e. the phase shift discussed above φ is 5°). The pattern in err(α) is phase shifted by 90 electrical degrees (1st derivative) relative to the pattern in magnitude MAG(α). As is illustrated, the positive maximum of err(α) (err max ) occurs at a positions 90° and 270° while the negative maximum (err min ) occurs at 0° and 180°. Similarly, the positive maximum of MAG(ca) (MAG max ) occurs at a position of 135° and 315° while the negative maximum (MAG min ) occurs at 45° and 225°. Magnitude ripple and peak-to-peak normalized error is fixed at a ratio of 3.14:1, based on the normalized magnitude ripple MAG norm  and the peak-to-peak normalized error err norm  as discussed above. However, the absolute error is always positive.  FIG. 13C  illustrates a polar plot that shows the ideal figure as dotted circle 
       FIGS. 14A, 14B, and 14C  illustrate an example with a combination of gain and phase errors in the cosine signal V cos (α) and the sine signal V sin (α), which corresponds to case 3.3 of Table 1.  FIG. 14A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(α) and the error err(α).  FIG. 14B  is a plot of the values of magnitude MAG(c) and error err(α) under the circumstances illustrated in  FIG. 14A .  FIG. 14C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude MAG(α) for comparison in the example illustrated in  FIG. 14A .  FIGS. 14A, 14B, and 14C  illustrate a combination of gain and phase errors. The combination of both amplitude and phase mismatch between sine and cosine signals V sin (α) and V cos (α) results in a 2-periodic pattern in magnitude MAG(α) and error err(α).  FIGS. 14A, 14B, and 14C  illustrate an example having a 93° phase shift of the cosine signal V cos (α) relative to the sine signal V sin (α) and a sine signal V sin (α) that is 5% larger relative to the cosine signal V cos (α). The pattern for err(α) is phase shifted by 90 electrical degrees (1st derivative) relative to the pattern of magnitude MAG(α). As is illustrated, the positive maximum of err(α) (err max ) and the negative maximum (err min ) occur at variable locations. Similarly, the positive maximum of MAG(α) (MAG max ) and the negative maximum (MAG min ) occur occur at variable locations. Magnitude ripple and peak-to-peak normalized error is fixed at a ratio of 3.14:1, based on the normalized magnitude ripple MAG norm  and the peak-to-peak normalized error err norm  as discussed above. However, the error is not symmetric to zero. The polar plot illustrated in  FIG. 14C  shows the ideal figure as dotted circle and the actual pattern of magnitude MAG(α), which is stretched in both the X-axis and in 135° direction. 
       FIGS. 15A, 15B, and 15C  illustrate an example with a combination of offset and gain errors in the cosine signal V cos (α) and the sine signal V sin (α), which corresponds to error case 4.1 of Table 1.  FIG. 15A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(a) and the error err(α).  FIG. 15B  is a plot of the values of magnitude MAG(α) and error err(α) under the circumstances illustrated in  FIG. 15A .  FIG. 15C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude MAG(α) for comparison in the example illustrated in  FIG. 15A .  FIGS. 15A, 15B, and 15C  illustrate a combination of offset and gain errors. The combination of both offset errors and amplitude mismatch between sine and cosine signals V sin (α) and V cos (α) results in a non-sinusoidal pattern in magnitude MAG(α) and error err(α), combining both 1-periodic and 2-periodic elements.  FIGS. 15A, 15B, and 15C  show an example having a 3% offset and 5% gain mismatch of the cosine signal V cos (α) relative to the sine signal V sin (α). The pattern in error err(α) does not resemble a sinusoidal waveform and is also no longer the direct 1st derivative of the pattern in magnitude MAG(α). The ratio between the ripple in magnitude MAG(α) and the peak-to-peak in error err(α) is not fixed at 3.14:1. In this particular case, the ratio is 2.75:1. The polar plot illustrated in  FIG. 15C  shows the ideal figure as dotted circle and the actual pattern of magnitude MAG(α), which is stretched and/or shifted in any direction depending on the signal having an offset and the amplitude mismatch (sine&gt;cosine or cosine&gt;sine). There is still a resemblance between the normalized magnitude ripple and peak-to-peak error, but since this ratio is no longer fixed at 3.14:1, the error estimation for such multi-point failures can only be made with less precision. 
       FIGS. 16A, 16B, and 16C  illustrate an example with a combination of offset and phase errors in the cosine signal V cos (α) and the sine signal V sin (α), which corresponds to error case 4.2 of Table 1.  FIG. 16A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(α) and the error err(ax).  FIG. 16B  is a plot of the values of magnitude MAG(c) and error err(α) under the circumstances illustrated in  FIG. 16A .  FIG. 16C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude MAG(α) for comparison in the example illustrated in  FIG. 16A .  FIGS. 16A, 16B, and 16C  illustrate a combination of offset and phase errors. The combination of both offset and phase errors between sine and cosine signals V sin (α) and V cos (α) results in a non-sinusoidal pattern in magnitude MAG(α) and error err(α), combining both 1-periodic and 2-periodic elements.  FIGS. 16A, 16B, and 16C  show an example having a 5% offset and 95° phase shift of the cosine signal V cos (at) relative to the sine signal V sin (α). The pattern of err(α) does not resemble a sinusoidal waveform and is also no longer the direct 1st derivative of the pattern of magnitude MAG(α) pattern. The ratio between the normalized magnitude ripple MAG norm  and the normalized peak-to-peak in error err norm  is not fixed at 3.14:1, in this particular case, the ratio is 3.53:1. The polar plot illustrated in  FIG. 16C  shows the ideal figure as dotted circle and the actual pattern MAG(α), which is stretched and/or shifted in the 45° or 135° direction depending on the signal having an offset and the type of phase shift (below or above 90°). There is still a resemblance between the ripple of magnitude MAG(α) and the peak-to-peak of error err(α), but since this ratio is no longer fixed at 3.14:1 the error estimation for such multi-point failures can only be made with less precision. 
       FIGS. 17A, 17B, and 17C  illustrate an example with a combination of offset, gain, and phase errors in the cosine signal V cos (α) and the sine signal V sin (α), which corresponds to error case 4.3 of Table 1.  FIG. 17A  illustrates plots of V sin (α) and V cos (α) along with the corresponding values of MAG(α) and the error err(α).  FIG. 17B  is a plot of the values of magnitude MAG(α) and error err(α) under the circumstances illustrated in  FIG. 17A .  FIG. 17C  illustrates a polar graph of MAG(α) overlaid on an ideal polar plot of magnitude MAG(α) for comparison in the example illustrated in  FIG. 17A .  FIGS. 17A, 17B, and 17C  illustrate a combination of offset, gain, and phase errors, which results in a non-sinusoidal pattern in magnitude MAG(α) and error err(α), combining both 1-periodic and 2-periodic elements. 
       FIGS. 17A, 17B, and 17C  show an example having a 3% offset of the sine signal V sin (α), a 5% larger cosine signal V cos (α) relative to the sine signal V sin (α), and an 87° phase shift between the cosine signal V cos (α) and the sine signal V sin (α). The pattern in error err(α) does not resemble a sinusoidal waveform and is also no longer the direct 1st derivative of the pattern in magnitude MAG(α). The ratio between in the normalized magnitude ripple MAG norm  and the normalized peak-to-peak error err norm  is not fixed at 3.14:1, in this particular case, the ratio is 3.15:1. The polar plot illustrated in  FIG. 17C  shows the ideal figure as dotted circle and the actual pattern of magnitude MAG(α), which is both shifted, stretched and rotated, depending on the signal having an offset and gain mismatch and the type of phase shift (below or above 90°). There is still a resemblance between the ripple of magnitude MAG(α) and the peak-to-peak in error err(α), but since this ratio is no longer fixed at 3.14:1, the error estimation for such multi-point failures can only be made with less precision. 
     For any single point failure resulting in an offset, gain, or phase error, the error estimation from the Magnitude ripple provides an error estimation with high precision. The method can also be applied to multi-point failures, as long as the failures result from the same error type, having either 1-periodic or 2-periodic patterns. For multi-point failures which are a combination of 1-periodic and 2-periodic failures, this method can only be applied with less precision. 
     The above detailed description is provided to illustrate specific embodiments of the present invention and is not intended to be limiting. Numerous variations and modifications within the scope of the present invention are possible. The present invention is set forth in the following claims.