Patent Publication Number: US-8120272-B2

Title: AC delay angle control for energizing a lamp

Description:
FIELD OF THE INVENTION 
     The present invention generally relates to a control circuit that provides a particular power to a load, and more specifically to a control circuit for a lamp that uses an alternating current (AC) input voltage to obtain a voltage suitable for lamp operation. 
     BACKGROUND OF THE INVENTION 
     High-intensity discharge (HID) lamps such as mercury vapor, metal halide, high-pressure sodium, low-pressure sodium lamps types are generally time consuming to ignite re-ignite. Typically, an ignition period of about twenty minutes may be needed in order for the lamp to sufficiently cool prior to attempting re-ignition. 
     Re-ignition may occur frequently, especially when the lamps are used with an unreliable power source. Generally, HID lamps will extinguish when power to the lamp is interrupted. Power interruptions of even a very short duration, e.g., tens of milliseconds, will often extinguish the lamp. 
     Since HID lamps are not illuminated during the lengthy ignition periods, they are often used in lamp system with an auxiliary lamp. The auxiliary lamp is responsive to the unlit HID lamp and accordingly provides light during the ignition period or whenever the HID lamp otherwise unavailable or unlit 
     These lamp systems generally include an alternating current (AC) power source which may have a variable amplitude. The auxiliary lamp, such as an incandescent lamp, generally requires a constant root mean squared (rms) voltage in order to operate properly. Accordingly, the lamp systems must include a control circuit for providing a constant root mean squared (rms) voltage. 
     SUMMARY OF THE INVENTION 
     Embodiments of the invention control an output voltage that is generated from a voltage input signal having a variable amplitude and/or frequency. In one embodiment, the invention provides a constant root mean squared (rms) voltage output for energizing a lamp from a voltage input signal having a variable amplitude and/or frequency. 
     In particular, embodiments of the invention estimate a delay angle as a linear function of a sensed input voltage. The delay angle is estimated so that when it is applied to the input voltage signal, a voltage output signal having a constant rms voltage is generated. 
     In one embodiment, the delay angle is estimated by a controller in a lamp system. By using a linear function to estimate the delay angle, the present invention provides an efficient method and device for controlling the voltage output signal. 
     Other objects and features will be in part apparent and in part pointed out hereinafter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1 and 2  are block diagrams illustrating lamp systems that have a control circuit according to an embodiment of the invention. 
         FIG. 3A  is graph illustrating an exemplary voltage input signal according to an embodiment of the invention. 
         FIG. 3B  is a graph illustrating an exemplary voltage output signal according to an embodiment of the invention. 
         FIG. 4  is a graph illustrating an actual and approximate relationship between maximum voltage of an input signal and delay angle for generating a 120 Volt rms output signal according to an embodiment of the invention. 
         FIG. 5  is a flow diagram of a method for determining a delay angle according to an embodiment of the invention. 
         FIG. 6  is a graph illustrating an actual and approximate relationship between slope of a linear region of an input signal and delay angle for generating a 120 Volt rms output signal according to an embodiment of the invention. 
         FIG. 7  is a graph illustrating an actual and approximate relationship between voltage difference in a linear region of an input signal and delay angle for generating a 120 Volt rms output signal according to an embodiment of the invention. 
         FIG. 8  is a flow diagram of a method for determining a delay angle according to an embodiment of the invention. 
         FIG. 9  is a graph illustrating an exemplary half-rectified voltage input signal according to an embodiment. 
         FIG. 10  is a three dimensional graph illustrating the relationship between maximum voltage of an input signal, frequency of an input signal, and delay angle. 
         FIG. 11  is flow diagram illustrating a method implemented by a control circuit in a lamp system having a variable voltage and variable frequency power source according to an embodiment of the invention. 
     
    
    
     Corresponding reference characters indicate corresponding parts throughout the drawings. 
     DESCRIPTION 
     Embodiments of the invention generally relate to a control circuit used with an input power source for energizing a load. In particular, embodiments of the invention determine a delay angle that is used to convert an alternating current (AC) input power signal from the input power source to an output power signal suitable for operating the load. 
     In an embodiment of the invention, the control circuit is used in a lamp system to generate a voltage signal suitable for energizing a lamp. For example, the control circuit may be used in a lamp system to account for variations, such as amplitude and/or frequency variations, in an input voltage signal supplied by the power source so that a constant voltage signal is provided to the lamp in the system. As generally known, some lamps, including incandescent lamps, operate most efficiently from a constant (broadly, substantially constant) voltage. Accordingly, the control circuit allows such lamps to efficiently operate in a lamp system that has a variable input voltage power source. 
       FIG. 1  illustrates an exemplary lamp system  100  according to an embodiment of the invention. The lamp system  100  includes an alternating current (AC) power source  102 , a lamp energizing circuit  104 , a primary lamp  106 , and an auxiliary lamp  108 . The illustrated lamp system  100  is configured for energizing the primary lamp  106  and the auxiliary lamp  108  wherein the primary lamp  106  includes one or more high-intensity discharge (HID) lamps (e.g., mercury vapor, metal halide, high-pressure sodium, low-pressure sodium lamps) and the auxiliary lamp  108  includes one or more incandescent lamps. The lamp system  100  may be configured for energizing other types of lamps, without departing from the scope of the invention. 
     The lamp energizing circuit  104  is adapted for receiving a variable voltage input signal from the power source  102  and generating a voltage output signal based on the received voltage input signal for energizing the primary lamp  106  and/or the auxiliary lamp  108 . There are numerous causes for variations in the input voltage. For example, in one embodiment, the power source  102  includes a first voltage source (e.g., 120 volts AC) and a second voltage source (e.g., 277 volts AC). Significant variations in the amplitude of the input voltage (e.g., amplitude may vary from between about 187 Volts to about 305 Volts) occur when the power source  102  changes between the first voltage source and the second voltage source. Additionally or alternatively, the first and second voltage sources may have different frequencies and thus significant variations in the frequency occur when the power source  102  changes between the first and second voltage sources. Additionally or alternatively, the voltage input signal may include smaller voltage variations due to signal distortion (e.g., harmonics/noise injected into the voltage input signal by other electrical devices). 
     The lamp energizing circuit  104  includes primary lamp energizing components  104  (e.g., rectifier  112 , smoothing capacitor  114 , power factor control circuit  116 , primary lamp driver  118 ) for generating a voltage output signal based on the received variable voltage input signal for selectively energizing the primary lamp  106 . The primary lamp energizing components  104  discussed herein are for energizing an HID lamp. For example, the primary lamp energizing components  104  may be included in an electronic ballast for energizing the HID primary lamp  106 . Additional or alternative components may be used for energizing other types of lamps without departing from the scope of the invention. The rectifier  112  (e.g., full wave rectifier) converts the AC voltage input signal to a direct current (DC) voltage signal. The smoothing capacitor  114  filters the rectified voltage input signal in order to minimize any AC ripple voltage present in the rectified voltage input signal. The power factor control circuit  116 , such as a boost converter, converts the filtered voltage input signal to a high DC voltage (e.g., 460 volts DC) signal. The primary lamp driver  118  (broadly, primary lamp driver and ignition circuit) includes an inverter circuit, such as a resonant converter, which converts the high DC voltage signal into a suitable AC voltage output signal for energizing the primary lamp  106 . 
     The lamp energizing circuit  104  includes a control circuit  120  for generating a substantially constant root mean square (rms) voltage output signal based on the received variable voltage input signal for selectively energizing the auxiliary lamp  108 . Voltage output (V o ) rms  is related to voltage input (V in ) rms  as follows 
     
       
         
           
             
               
                 
                   
                     
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                               n 
                             
                           
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                           sin 
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                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
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                     1 
                   
                   ) 
                 
               
             
           
         
       
     
     wherein θ represents a delay angle that is applied to the voltage input (V in ) rms . Additional details regarding the derivation of Equation 1 are given in Appendix A. 
     The control circuit  120  includes a voltage sensing component for sensing the voltage of the voltage input signal. According to the illustrated embodiment, the voltage sensing component comprises a voltage divider having two resistors (R 1 , R 2 ) for sensing the voltage of the half rectified voltage input signal. 
     The control circuit  120  includes a controller  122  (e.g., microcontroller, microprocessor) for receiving, via a controller input channel, the sensed voltage from the voltage sensing component. According to the illustrated embodiment, the sensed voltage V sense  is a voltage value of the input voltage signal at point A (V A ) which has been stepped down by the voltage divider R 1 , R 2 . Specifically, the sensed voltage V sense  is related to the input voltage signal V A  as follows
 
V sense =kv A   (Equation 2)
 
     Where k is given by 
     
       
         
           
             
               
                 
                   k 
                   = 
                   
                     
                       R 
                       2 
                     
                     
                       
                         R 
                         1 
                       
                       + 
                       
                         R 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     3 
                   
                   ) 
                 
               
             
           
         
       
     
     In one embodiment, the value of k is selected such that V sense  never exceeds a maximum voltage limit of the controller input channel. 
     The controller  122  is configured, based on the relationship set forth in Equation 1, to calculate a delay angle as a function of the sensed voltage V sense . More particularly, the controller  122  is configured to calculate a delay angle, which when applied to the voltage input signal, will generate an output signal having a particular rms voltage value (V o ) rms . The particular output rms voltage value (V o ) rms  is pre-selected based on the operating requirements of the auxiliary lamp  108 . For example, in the illustrated lamp system  100 , the particular rms voltage value (V o ) rms  may be 120 Volts, which is recommended for efficiently operating a standard incandescent lamp. 
     The control circuit  120  includes an AC converter  124  for modifying the voltage input signal according to the calculated delay angle to generate a constant rms AC voltage output signal. The generated AC voltage output signal is applied/provided to the auxiliary lamp  108  in order to energize the lamp  108 . 
     Referring to  FIG. 2 , in one embodiment, the AC converter  124  includes an AC chopper circuit  124  electrically connected to the power source  102  for receiving the voltage input signal and to the controller  122  via a controller output channel for receiving a control signal. The AC chopper circuit  124  generates the voltage output signal as a function of the voltage input signal and the control signal. The AC chopper circuit  124  is electrically connected to the auxiliary lamp  108  for providing the voltage output signal to the auxiliary lamp  108 . 
     For example, the AC chopper circuit  124  may include an AC bidirectional switch (e.g., triac  134 ) electrically connected between the power source  102  and the auxiliary lamp  108  that can be selectively operated in a conducting state or a non-conducting state. When the switch  134  is operated in the conducting state it conducts the voltage input signal and when the switch  134  is operated in the non-conducting state it does not conduct the voltage input signal. The triac  134  is operated in the non-conducting state during a delay period defined by the delay angle θ. The triac  134  is otherwise operated in the conducting state. As illustrated in  FIG. 2 , the AC chopper circuit  124  may also include a snubber circuit  132  for suppressing voltage transients in order to protect the switch  134 . 
       FIG. 3A  is a graph representing an exemplary voltage input signal as a function of time according to one embodiment of the invention. Each cycle of the voltage input signal includes a positive half cycle (portion of cycle during which voltage values are greater than zero) and a negative half cycle (portion of cycle during which voltage values are less than zero). As shown by the graph, the positive half cycle and the negative half cycle each include a portion in which the voltage values are increasing and a portion in which the voltage values are decreasing.  FIG. 3B  is a graph representing as a function of time an output voltage signal generated by blocking the voltage input signal by a delay angle θ. In particular, the voltage values of the voltage input signal are converted to zero during a period of time defined by the delay angle θ. As shown by the graphs, a delay angle θ is applied to the voltage input signal during the positive half cycles and the negative half cycles. 
     Referring generally to  FIGS. 3A and 3B , in one embodiment, the delay angle may be chosen so that the voltage output signal has a lower rms voltage than that of the input signal. In particular, when the delay angle θ defines a delay period which includes a maximum voltage value V max  of the voltage input signal, the voltage output signal will have a lower rms voltage than the voltage input signal. 
     As indicated by Equation 1, the relationship between the voltage input (V in ) rms  and the voltage output (V o ) rms  is non-linear. In order to simplify computations performed by the controller  122  for determining the delay angle, the controller  122  may be configured to estimate the delay angle based on an approximate linear relationship between the amplitude of the voltage input signal and the delay angle for a pre-selected output rms voltage value (V o ) rms . Thus, by simplifying the computations required by the controller  122 , the present invention provides an efficient an accurate method of determining a delay angle for a variable voltage input signal. 
       FIG. 4  is a graph illustrating the actual (non-linear) relationship between a maximum value of the voltage input signal V max  and the delay angle in order to generate a constant 120 V rms voltage output signal. The graph also illustrates a linear curve-fitting of the actual relationship plot. As shown by the graph, the linear curve fitting provides a relatively accurate estimate of the delay angle for each of the maximum voltage input values. 
     In one embodiment, the controller  122  is configured to identify a maximum voltage value for an AC cycle of the voltage input signal based on the voltage sensed by the voltage sensing component and to determine a delay angle for the AC cycle as a linear function of the identified maximum voltage value V max . More particularly, the controller  122  is configured to determine the delay angle used to modify the input signal, having a particular frequency f, according to the following linear formula:
 
Delay= A ( f )× V in max   +B ( f )  (Equation 4)
 
     Wherein A and B are constants pre-defined for generating an output signal having the pre-selected rms voltage (V o ) rms  from the input signal having the particular frequency f. 
     The controller  122  is configured to determine a delay angle for each positive half cycle and for each negative half cycle. In one embodiment, the controller  122  determines the delay angle for each positive half cycle using the linear formula of Equation 4 and determines the delay angle for each negative half cycle based on the determined delay angle for the corresponding positive half cycle. For example, the controller  122  may determine the delay angle for a negative half cycle included in a particular cycle by computing the sum of the determined delay angle for positive half cycle included in the particular cycle and a time period (T/2) corresponding to negative half of the particular cycle. 
       FIG. 5  is a flow chart illustrating a method  500  implemented by the controller  122  of a control circuit  120  used in a lamp system  100  to determine a delay angle for an auxiliary lamp  108  in the lamp system  100  according to an embodiment of the invention. In this lamp system  100 , the controller  122  is also used to control the operation of the primary lamp  106 . For example, the controller  122  is electrically connected to one or more of the primary lamp energizing components  104  for controlling the components  104 . 
     Once the method is initiated at  502 , the method at  504  determines whether the primary lamp  106  (i.e., main lamp) is lit (e.g, illuminated). If the primary lamp  106  is determined to be lit, the method at  506  initiates a set of instructions (e.g., software program) for controlling the operation of the primary lamp  106 . If the primary lamp  106  is determined not to be lit, the method at  508  identifies a maximum voltage value V max  for a particular cycle of the voltage input signal. For example, the controller  122  may receive a plurality of voltage values sensed by the voltage sensing component at a pre-defined/particular time interval during the cycle. The controller  122  may compare the plurality of received voltage values in order to identify the maximum voltage value (e.g., largest voltage values of the received voltage values). 
     The method at  510  includes identifying a zero crossing of the input signal for triggering the delay angle for the positive half cycle of the particular cycle. For example, the controller  122  may receive voltage values V sense  sensed by the voltage sensing component and identify the zero crossing based on the received voltage values V sense . The method at  512  determines the delay angle for the positive half cycle of the particular cycle using Equation 4 as discussed above. 
     The method at  514  determines the delay angle for the negative half cycle based on the determined delay angle for the positive half cycle. For example, the controller  122  may transmit a control signal to the AC converter  124  that causes the AC converter  124  to fire the delay angle during the negative half cycle after a time period corresponding to half of the particular cycle has elapsed since the firing of the delay angle for the positive half cycle. After the method at  514  determines the delay angle for the negative half cycle, the method returns to  502  and repeats the steps  502 - 514 . 
     Another embodiment of the invention contemplates that the amplitude of the voltage input signal for a particular cycle is proportional to the slope of voltage input signal near the zero-crossing (i.e., in the linear region) of the particular cycle.  FIG. 6  is a graph illustrating the actual (non-linear) relationship between slope of the linear region of the voltage input signal and a corresponding delay angle for generating a 120 V rms voltage output signal. The graph also illustrates a linear curve-fitting of the actual relationship plot. As shown by the graph, the linear curve fitting provides a relatively accurate estimate of the delay angle for each of the slope values. Similarly,  FIG. 7  illustrates an actual plot and a linear curve fitting plot of the relationship between voltage difference in the linear region of the voltage input signal and a corresponding delay angle for generating 120 V rms voltage output signal. As slope is directly proportional to voltage difference, a linear function may also be used to estimate a delay angle from a voltage difference in the linear region of the voltage input signal. Accordingly, in one embodiment, the controller  122  is configured to calculate voltage difference ΔV in  of voltage values near the zero crossing and to determine the delay angle based on the calculated voltage difference ΔV in . 
       FIG. 8  is a flow chart illustrating an example of such a method  800  which is implemented by the controller  122  of a control circuit  120  used in a lamp system  100  to determine a delay angle for an auxiliary lamp  108  in the lamp system  100 . In particular, the method includes steps  808 - 818  for determining a delay angle for a positive half cycle of the input signal. The method includes steps  820 - 830  for determining a delay angle for the negative half cycle of the input signal. In this lamp system  100 , the controller  122  is also used to control the operation of the primary lamp  106 . For example, the controller  122  is electrically connected to one or more of the primary lamp energizing components  104  for controlling the components  104 . 
     Once the method is initiated at  802 , the method at  804  determines whether the primary lamp  106  (i.e., main lamp) is lit (e.g, illuminated). If the primary lamp  106  is determined to be lit, the method at  806  initiates a set of instructions (e.g., software program) for controlling the operation of the primary lamp  106 . If the primary lamp  106  is determined not to be lit, the method receives at  808  a voltage value measured from the voltage input signal (e.g., near the zero crossing, in the linear region). For example, the controller  122  may receive the voltage value from the voltage sensing component. 
     The method at  810  determines whether the received voltage value is greater than or equal to a threshold voltage value (“ThresholdVoltage”). For example, the threshold value may represent a pre-defined voltage value that is far enough from the actual zero crossing that it is unlikely to include noise which may be present at the zero crossing and close enough to be in the linear region of the input signal so that it can be used to accurately estimate the delay angle using a linear function. If the received voltage value is not greater than or equal to the threshold voltage value, the method at  812  initiates a delay period (e.g., 120 microseconds) and then at  810  considers whether another received voltage value is greater than or equal to the threshold voltage value. For example, the controller  122  may cause the voltage sensing component to sense an initial voltage value. If the controller  122  determines that the initial voltage value is not greater than or equal to the threshold voltage value, the controller  122  may cause the voltage sensing component to, after the delay period, sense a subsequent voltage value. 
     Steps  810  and  812  are repeated until the method determines that the received voltage value is greater than or equal to the threshold voltage value. When the method determines that the received voltage value is greater than or equal to the threshold voltage value, the controller  122  initiates another delay period (e.g., 400 microseconds). When the delay period elapses, the method at  816  receives a voltage value (“first measured voltage value”, “MeasuredVoltage 1 ”) measured from the input signal. 
     The method at  818  determines the delay angle for the positive half cycle as a linear function of the threshold voltage value and the first measured voltage value. In particular, the method determines the delay angle for the positive half cycle according to the following linear formula:
 
Delay= A (MeasuredVoltage1−Threshold Voltage)+ B   (Equation 5)
 
     wherein A and B are constants pro-defined for generating an output signal having the pre-selected rms voltage (V o ) rms  from the input signal having the particular frequency f. 
     After determining the delay angle for the positive half cycle, the method determines the delay angle for the corresponding negative half cycle. In particular, the method receives at  820  a voltage value measured from the voltage input signal (e.g., near the zero crossing, in the linear region). For example, the controller  122  may receive the voltage value from the voltage sensing component. 
     The method at  822  determines whether the received voltage value is less than or equal to the first measured voltage value. It is to be noted that the received voltage may be compared to a different defined voltage value without departing from the scope of the invention. If the received voltage value is not less than or equal to the first measured voltage value, the method at  824  initiates a delay period (e.g., 120 microseconds) and then at  822  considers whether another received voltage value is less than or equal to the first measured voltage value. For example, the controller  122  may cause the voltage sensing component to operate in a similar manner as discussed above in conjunction with steps  810  and  812 . 
     Steps  822  and  824  are repeated until the method determines that the received voltage value is less than or equal to the first measured voltage value. When the method determines that the received voltage value is less than or equal to the first measured voltage value, the method initiates another delay period (e.g., 400 microseconds). When the delay period elapses, the method at  828  receives a voltage value (“second measured voltage value”, “MeasuredVoltage 2 ”) measured from the input signal. 
     The method at  830  determines the delay angle for the negative half cycle as a linear function of the first measured voltage value and the second measured voltage value. In particular, the method determines the delay angle for the negative half cycle according to the following linear formula:
 
Delay= A (MeasuredVoltage1−MeasuredVoltage2)+ C   (Equation 6)
 
     Wherein A and C are constants pre-defined for generating an output signal having the pre-selected rms voltage (V o ) rms  from the input signal having the particular frequency f. 
     After the method at  830  determines the delay angle for the negative half cycle, the method returns to  802  and repeats the steps  802 - 830 . 
       FIG. 9  is a graph of an exemplary input signal that has been rectified. The graph shows four exemplary voltage values (e.g., a first voltage value, a second voltage value, a third voltage value, and a fourth voltage value) with reference to the input signal which may be used to estimate a delay angle for the input signal. For purposes of consistency, the exemplary voltage values are labeled as they were described above with reference to method  800 . 
     In one embodiment, the frequency of the input voltage signal is pre-determined. Accordingly, the controller  122  may be configured to determine the delay angle using a linear formula in which the constants (e.g., A, B, C) are pre-defined for the pre-determined frequency. For example, as illustrated in  FIG. 4 , for an input signal having a frequency of 60 Hz, the controller  122  may be configured to use the linear formula:
 
Delay=0.0631(MeasuredVoltage1−ThresholdVoltage)+1.988
 
     in order to generate an output signal having a constant rms voltage of about 120 Volts. 
     In another embodiment, the frequency of input voltage signal may be variable and the controller  122  is configured to determine the frequency of the input voltage signal. For example, the controller  122  may determine the frequency by sampling the input voltage signal and measuring the time between zero crossings, maximum values, or other points consistently measured during the input voltage signal cycles. 
     The controller  122  may also include a storage memory for storing data associated with a plurality of frequencies. In particular, the storage memory may store sets of constants (A, B, C) each corresponding to one of the frequencies. The set of constants may be used in a linear formula in order to estimate the delay angle for the corresponding frequency. Accordingly, the controller  122  is configured to retrieve the set of constants which correspond to the determined frequency and determine the delay angle using a linear formula and the retrieved set of constants.  FIG. 10  is a three dimensional plot for determining a delay angle based on a maximum measured voltage value and a determined frequency. In one embodiment, the illustrated plot may be implemented by the controller  122  using the stored sets of constants. 
       FIG. 11  illustrates a flow diagram for a method implemented by the controller  122  in a lamp system  100  having a variable frequency voltage input signal. The controller  122  determines the frequency of the input voltage input signal and queries a look up table for a set of constants corresponding to the determined frequency. The controller  122  retrieves a set of constants from the look up table. The controller  122  also receives the maximum measured voltage value from the input signal. The controller  122  computes the delay angle using the retrieved set of constants and the received maximum voltage in Equation 4 discussed above to compute the delay angle. The controller  122  is then configured to transmit a control signal to the AC chopper circuit to fire the delay angle at about from the zero crossing of the positive half cycle and at about T/2 (where T is the period for the cycle) for negative half cycle from the triac firing location for the positive half. 
     Having described the invention in detail, it will be apparent that modifications and variations are possible without departing from the scope of the invention defined in the appended claims. 
     When introducing elements of the present invention or the preferred embodiments(s) thereof, the articles “a”, “an”, “the” and “said” are intended to mean that there are one or more of the elements. The terms “comprising”, “including” and “having” are intended to be inclusive and mean that there may be additional elements other than the listed elements. 
     In view of the above, it will be seen that the several objects of the invention are achieved and other advantageous results attained. 
     As various changes could be made in the above constructions, products, and methods without departing from the scope of the invention, it is intended that all matter contained in the above description and shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense. 
     APPENDIX A 
     
       
         
           
             
               
                 ( 
                 
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                   1 
                   T 
                 
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                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             β 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         β 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   v 
                   o 
                 
                 ) 
               
               rms 
             
             = 
             
               
                 
                   V 
                   m 
                 
                 
                   2 
                   ⁢ 
                   
                     π 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       ( 
                       
                         β 
                         - 
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             β 
                           
                           2 
                         
                       
                       ) 
                     
                     θ 
                     π 
                   
                   + 
                   
                     
                       ( 
                       
                         β 
                         - 
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             β 
                           
                           2 
                         
                       
                       ) 
                     
                     α 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       π 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   v 
                   o 
                 
                 ) 
               
               rms 
             
             = 
             
               
                 
                   V 
                   m 
                 
                 
                   2 
                   ⁢ 
                   
                     π 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       
                         π 
                         - 
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                           
                           2 
                         
                         - 
                         θ 
                         + 
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           2 
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                         - 
                       
                     
                   
                   
                     
                       
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             4 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                           
                           2 
                         
                         - 
                         α 
                         + 
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             α 
                           
                           2 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   v 
                   o 
                 
                 ) 
               
               rms 
             
             = 
             
               
                 
                   V 
                   m 
                 
                 
                   2 
                   ⁢ 
                   
                     π 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       
                         π 
                         - 
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                           
                           2 
                         
                         - 
                         θ 
                         + 
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           2 
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                         - 
                       
                     
                   
                   
                     
                       
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             4 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                           
                           2 
                         
                         - 
                         π 
                         - 
                         θ 
                         + 
                         
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   v 
                   o 
                 
                 ) 
               
               rms 
             
             = 
             
               
                 
                   V 
                   m 
                 
                 
                   2 
                   ⁢ 
                   
                     π 
                   
                 
               
               ⁢ 
               
                 
                   
                     
                       
                         π 
                         - 
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                           
                           2 
                         
                         - 
                         θ 
                         + 
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           2 
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                         - 
                       
                     
                   
                   
                     
                       
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             4 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                           
                           2 
                         
                         - 
                         π 
                         - 
                         θ 
                         + 
                         
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   v 
                   o 
                 
                 ) 
               
               rms 
             
             = 
             
               
                 
                   V 
                   m 
                 
                 
                   2 
                   ⁢ 
                   
                     π 
                   
                 
               
               ⁢ 
               
                 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                   
                   - 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                   + 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   v 
                   o 
                 
                 ) 
               
               rms 
             
             = 
             
               
                 
                   V 
                   m 
                 
                 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                   
                 
               
               ⁢ 
               
                 
                   π 
                   - 
                   θ 
                   + 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θcos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   v 
                   o 
                 
                 ) 
               
               rms 
             
             = 
             
               
                 
                   
                     ( 
                     
                       V 
                       
                         i 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                     
                     ) 
                   
                   rms 
                 
                 
                   π 
                 
               
               ⁢ 
               
                 
                   π 
                   - 
                   θ 
                   + 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θcos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ