PATENT DOCUMENT

Abstract:
A method for controlling a switching power converter provides an efficient algorithm for controlling the output voltage across loads that are relatively light with small transients. When the output voltage is at or below a predetermined first magnitude, a determination is made of the charge required for one or more pulses to increase the output voltage to a predetermined second magnitude which is greater than a target output voltage. Corrective action is taken to raise the output voltage to the second magnitude and the system takes no further corrective action until output voltage is determined to be at or below the first magnitude. The method is useful with synchronous or non-synchronous power converters of buck, boost, buck/boost or other topologies. The method further provides a simple means for determining the amount of charge removed from a battery.

Full Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This application is related to commonly assigned U.S. patent application Ser. No. 10/295,580 filed on Nov. 14, 2002, by Kent Kernahan, David F. Fraser and Jack Roan, entitled “Switching Power Converter” which is incorporated herein by reference in its entirety. Patent application Ser. No. 10/295,739, filed Nov. 14, 2002 by Kent Kernahan and John Carl Thomas entitled “Switching Power Converter”, now U.S. Pat. No. 6,825,644 issued Nov. 30, 2004 is also incorporated herein by reference in its entirety. 
   FIELD OF THE INVENTION 
   This invention relates to switching power converters, the operations of which are computer controlled. 
   BACKGROUND OF THE INVENTION 
   A digital switching power supply controller provides the ability to control a power supply through the use of algorithms executing logical decisions rather than by employing prior art techniques which use analog signals which are conditioned by various scale factors. In contrast to the prior art, the present invention relates to a method and circuitry for the regulation of the output voltage in a digital logic-based controller for a switching power supply. 
   Some products include a sleep, or low power, or standby mode. When a product is battery powered such modes are intended to extend battery life. Hence it is desirable to use a control method that is as efficient as possible. It is further desirable to provide a system and method for maintaining voltage regulation to relatively stable loads which maximizes supply efficiency. Another desirable advancement over the prior art is to provide a simple, low power method of battery energy management. 
   SUMMARY OF THE INVENTION 
   The present invention provides a control method which is applicable to a switching power converter and may be utilized with buck, boost and buck/boost topologies, any of which may be synchronous or non-synchronous. 
   In the method of the present invention, the magnitude of the output voltage of a switching power converter is sensed and compared to a predetermined lower limit voltage. No action is taken until the magnitude of the output voltage is equal to or less than the predetermined lower limit. If the magnitude of the output voltage is equal to or less than the predetermined lower limit, then corrective action is taken to cause the output voltage to increase to a magnitude which is greater than a target output voltage magnitude. The magnitude of the output voltage is monitored and no further corrective action is taken until the magnitude of the output voltage becomes equal to or less than the predetermined lower limit output voltage. The elapsed time between corrective action events is thus variable depending on, for example, the load being supplied by the switching power converter. This method is termed pulse frequency modulation control, or “PFM” control. In accordance with one embodiment of the present invention, PFM control is utilized by a controller in a switching power converter. PFM control is described herein for topologies including synchronous buck, non-synchronous boost, and four-FET buck/boost. One skilled in the art will recognize that the PFM control is applicable to other topologies, such as non-synchronous buck and synchronous boost. 
   The present invention is particularly advantageous when used in a system in which the load is relatively small and stable. The strategy of the method is summarized as:
         1. PFM control of conduction is utilized on an as needed basis.   2. Whenever the magnitude of the output voltage is determined to be equal to or less than a predetermined lower limit, PFM control is activated for corrective action.   3. PFM control utilizes the input voltage and calculates the amount of energy which must be injected into a driver circuit which will be sufficient to raise the output voltage to a predetermined upper voltage which is greater than a target output voltage.   4. The PFM system returns to an inactive mode, waiting until the magnitude of the output voltage is determined to be equal to or less than the predetermined lower limit.
 
The parameters used by the PFM control method are reused to provide a convenient, low power method for determining how much energy has been removed from a one or more batteries by counting the number of PFM events per second.
       

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a representative synchronous buck power converter. 
       FIG. 2  shows output voltage over time in relation to target, upper, and lower voltage limits. 
       FIG. 3  is a flow chart of the PFM method. 
       FIG. 4  shows the relationship between time and current during time period T where a single set of control signals are applied to the converter of  FIG. 1 . 
       FIG. 5  shows an embodiment wherein correction is achieved by using multiple shorter duration pulses during a correction period rather than a single set of pulses such as that shown in  FIG. 4 . 
       FIG. 6  is an example of values in a lookup table used in one embodiment of the invention. 
       FIG. 7  is a table of the voltages and circuit values for the devices in  FIG. 1  used to calculate the table entries of  FIG. 6 . 
       FIG. 8  illustrates a non-synchronous boost converter. 
       FIG. 9  illustrates a four transistor buck/boost converter. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   
     
       
             
           
             
             
             
           
         
             
                 
             
             
               Definitions, Acronyms, and Abbreviations 
             
             
                 
             
           
           
             
                 
             
           
        
         
             
                 
               CCM 
               Continuous Current Mode 
             
             
                 
               Coss 
               FET output capacitance 
             
             
                 
               Cycle 
               Procedure wherein no pulse is applied for one 
             
             
                 
               skipping 
               or more time periods 
             
             
                 
               DCM 
               Discontinuous Current Mode 
             
             
                 
               DPC 
               Digital Pulse Controller 
             
             
                 
               dX 
               Change in value of X, where X may be any 
             
             
                 
                 
               parameter such as I, V, Tp, Ts, etc. 
             
             
                 
               Icoil 
               Coil current 
             
             
                 
               L 
               Inductance value (h) 
             
             
                 
               REG 
               Regulation system, which includes REG_Eng, 
             
             
                 
                 
               REG_Sch, SC, SV, Variables and interfaces to 
             
             
                 
                 
               DPC, DAS, NFETDVR and SYS, as described in 
             
             
                 
                 
               aforementioned U.S. patent application No. 
             
             
                 
                 
               10/295,580 and U.S. Pat. No. 6,825,644 
             
             
                 
               SYS 
               Microcontroller-based system control as 
             
             
                 
                 
               described in aforementioned U.S. patent 
             
             
                 
                 
               application No. 10/295,580 and U.S. Pat. No. 
             
             
                 
                 
               6,825,644 
             
             
                 
               T 
               One time period 
             
             
                 
               Tp, t p   
               Time period during which a control FET is 
             
             
                 
                 
               turned on 
             
             
                 
               Ts, t s   
               Time period during which a synchronizing FET is 
             
             
                 
                 
               turned on, or during which a “low side” diode 
             
             
                 
                 
               conducts 
             
             
                 
               Vdbh 
               High dead band limit 
             
             
                 
               Vdbl 
               Low dead band limit 
             
             
                 
               Vdb 
               Dead band width; equals (Vdbh − Vbdl) 
             
             
                 
               Vo 
               Output voltage; voltage across the load 
             
             
                 
               Vtar 
               Target voltage 
             
             
                 
                 
             
           
        
       
     
   
   Referring to FIG. 12 of U.S. Pat. No. 6,825,464, according to one embodiment of the present invention, PFM control utilizes data from the ADC 1206 block and asserts control through the DPC 1201. The PFM control software resides within REG 1211 and interacts with SYS 1205. These functions are incorporated into the actions which are ascribed to the controller  112 ,  FIG. 1 , by the present invention. 
   Referring to  FIG. 1 , the basic operation of a buck switching power converter is the intermittent connection between an input voltage “Vin” at input terminal  100  and inductor L 1  by a control FET  114  for a time termed “Tp”. At the end of time Tp, control FET  114  turns off and synchronizing FET  116  is turned on for a time termed “Ts”. This is accomplished by a controller  112  supplying conduction control signals to the FETs via lines  128  and  130  which are connected to FETs  114  and  116 . This causes current to flow through inductor L 1  to the load  126 . Output voltage “Vo”, measured at output terminal  122 , is smoothed by a capacitor C 2   120 . FET  116  may be replaced by a diode to form a non-synchronous buck supply, in which case line  130  is not needed. Alternatively, as described in U.S. Pat. No. 6,825,644, FET  116  may not be turned on during time period T. In the  FIG. 1  topology, Ts is the time during which current from inductor L 1  continues to flow after FET  114  is turned off. Said differently, it is the time required for the current to return to zero after time Tp is completed. Those skilled in the art will recognize that the method of the present invention may be applied to any switching power converter topology, including but not limited to buck, boost, and buck/boost wherein any of them may be implemented as synchronous or non-synchronous designs. 
   The PFM control method is illustrated by  FIG. 2 , wherein Vo falls to a lower dead band limit “Vdbl”  202  at time T 1 . The controller  112  takes corrective action, raising Vo to approximately the upper dead band limit “Vdbh”  201 . This voltage swing is “Vdb” ( 203 )=(Vdbh−Vdbl). Vdbh and Vdbl are defined relative to the desired output voltage “Vtar”  200 . Vo will decline over time T 3  also indicated by reference character  205  until again reaching Vdbl  202  at time T 2 , when controller  112  again takes corrective action. 
   The method of the present invention is represented by the flow chart of  FIG. 3 . Following Start  300  the system samples Vo of the power converter being controlled. At Step  302  Vo is compared with Vdbl  202 . This process continues until Vo is equal to or less than Vdbl  202 . At that point the process moves to Step  304 . At Step  304  Vin is sampled, and the magnitude of Vin is used to calculate the pulse time Tp needed to raise Vo by Vdb  203 . Ts is also calculated. In the synchronous circuit of  FIG. 1 , FET  116  is driven for the time Ts. In a non-synchronous topology, for example, a circuit of the type shown in  FIG. 1  but without FET  116 , Ts represents the time during which current will continue to flow from the coil toward load  126  after the control FET  114  is turned off. In Step  306 , a conduction control signal of time length Tp is applied to control FET  114  by the controller  112  via line  128 . Thereafter, a conduction control signal of time duration Ts is applied to FET  116  via line  130 . The PFM control process then returns to Step  300  to resume comparing Vo to Vdbl  202  at Step  302 . 
   The PFM control method may be applied to various topologies. In each case there are alternative implementation strategies which may be selected, depending upon the end product&#39;s tolerance for ripple, computing power available, power budget and other factors. Three PFM control embodiments include: a) PFM control using a stored set of pre-calculated Tp and Ts pulse times, from which the pulse times are selected depending on input variables (“Method_ 1 ”); b) PFM control based on calculation of Tp and Ts each time corrective action is to be initiated using the present input variables and pre-stored system model parameters (“Method_ 2 ”); and c) PFM control based on calculation of Tp and Ts each time corrective action is to be initiated using the present input variables and an estimation of system model parameters, calculated as a function of system response (“Method_ 3 ”) 
     FIG. 4  shows the current through inductor L 1  over time. Coil current Icoil rises as indicated by reference character  402  with a slope of V 1 /L for the time Tp and falls with the slope V 2 /L for the time Ts, where V 1  is the voltage across the inductor L 1  during Tp, V 2  is the voltage across the inductor L 1  during Ts, and L is the inductance of inductor L 1 . The following relationship is derived from the principle of conservation of charge: 
   
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             Q 
           
           = 
           
             
               
                 
                   1 
                   2 
                 
                 · 
                 
                   ( 
                   
                     
                       T 
                       p 
                     
                     + 
                     
                       T 
                       s 
                     
                   
                   ) 
                 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               I 
             
             = 
             
               
                 
                   
                     1 
                     2 
                   
                   · 
                   Δ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   t 
                   · 
                   Δ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 I 
               
               = 
               
                 
                   C 
                   · 
                   Δ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   V 
                   . 
                 
               
             
           
         
       
     
   
   However, considering that the inductor L 1  current starts at zero and returns to zero, this provides: 
   
     
       
         
           EQ_ 
           ⁢ 
           1 
           ⁢ 
           
             : 
           
         
       
     
     
       
         
           
               
           
           ⁢ 
           
             
               
                 
                   V 
                   1 
                 
                 · 
                 
                   T 
                   p 
                 
               
               L 
             
             = 
             
               
                 
                   
                     
                       V 
                       2 
                     
                     · 
                     
                       T 
                       s 
                     
                   
                   L 
                 
                 ⇒ 
                 
                   
                     V 
                     1 
                   
                   · 
                   
                     T 
                     p 
                   
                 
               
               = 
               
                 
                   V 
                   2 
                 
                 · 
                 
                   
                     T 
                     s 
                   
                   . 
                 
               
             
           
         
       
     
   
   Expanding the equation, results in: 
   
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             t 
           
           = 
           
             
               
                 T 
                 p 
               
               + 
               
                 T 
                 s 
               
             
             = 
             
               
                 
                   T 
                   p 
                 
                 + 
                 
                   
                     
                       V 
                       1 
                     
                     
                       V 
                       2 
                     
                   
                   · 
                   
                     T 
                     p 
                   
                 
               
               = 
               
                 
                   T 
                   p 
                 
                 · 
                 
                   
                     ( 
                     
                       1 
                       + 
                       
                         
                           V 
                           1 
                         
                         
                           V 
                           2 
                         
                       
                     
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
   
   The peak coil current ΔI  400  is: 
   
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             I 
           
           = 
           
             
               
                 
                   V 
                   1 
                 
                 · 
                 
                   T 
                   p 
                 
               
               L 
             
             = 
             
               
                 
                   
                     
                       V 
                       1 
                     
                     · 
                     Δ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   t 
                 
                 
                   L 
                   · 
                   
                     ( 
                     
                       1 
                       + 
                       
                         
                           V 
                           1 
                         
                         
                           V 
                           2 
                         
                       
                     
                     ) 
                   
                 
               
               . 
             
           
         
       
     
   
   Therefore, using the relationship previously obtained: 
   
     
       
         
           
             
               C 
               · 
               Δ 
             
             ⁢ 
             
                 
             
             ⁢ 
             V 
           
           = 
           
             
               
                 
                   1 
                   2 
                 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 t 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               I 
             
             = 
             
               
                 
                   
                     1 
                     2 
                   
                   · 
                   Δ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   t 
                   · 
                   
                     
                       
                         
                           V 
                           1 
                         
                         · 
                         Δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                     
                     
                       L 
                       · 
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               V 
                               1 
                             
                             
                               V 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               = 
               
                 
                   
                     1 
                     
                       2 
                       · 
                       L 
                     
                   
                   · 
                   Δ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     t 
                     2 
                   
                   · 
                   
                     
                       
                         
                           V 
                           1 
                         
                         · 
                         
                           V 
                           2 
                         
                       
                       
                         
                           V 
                           1 
                         
                         + 
                         
                           V 
                           2 
                         
                       
                     
                     . 
                   
                 
               
             
           
         
       
     
   
   Finally, we have: 
   
     
       
         
           EQ_ 
           ⁢ 
           2 
           ⁢ 
           
             : 
           
         
       
     
     
       
         
           
               
           
           ⁢ 
           
             
               
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 V 
               
               = 
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       t 
                       2 
                     
                   
                   
                     ( 
                     
                       2 
                       · 
                       L 
                       · 
                       C 
                     
                     ) 
                   
                 
                 · 
                 
                   
                     ( 
                     
                       
                         V 
                         1 
                       
                       · 
                       
                         V 
                         2 
                       
                     
                     ) 
                   
                   
                     ( 
                     
                       
                         V 
                         1 
                       
                       + 
                       
                         V 
                         2 
                       
                     
                     ) 
                   
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
                 
             
             ⁢ 
             
               or 
               ⁢ 
               
                 : 
               
             
           
         
       
     
     
       
         
           
               
           
           ⁢ 
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               t 
             
             = 
             
               
                 
                   2 
                   · 
                   L 
                   · 
                   C 
                 
               
               · 
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       V 
                       · 
                       
                         
                           ( 
                           
                             
                               V 
                               1 
                             
                             + 
                             
                               V 
                               2 
                             
                           
                           ) 
                         
                         
                           ( 
                           
                             
                               V 
                               1 
                             
                             · 
                             
                               V 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
   
   For the topology of  FIG. 1 :
 
Δ V=V   dbh   −V   dbl  
 
 V   1   =V   in   −V   out   =V   in   −V   tar  
 
 V   2   =V   out   =V   tar  
 
   Therefore, combining EQ — 1 and EQ — 2, and solving for Tp and Ts provides: 
   
     
       
         
           EQ_ 
           ⁢ 
           3 
           ⁢ 
           
             : 
           
         
       
     
     
       
         
           
               
           
           ⁢ 
           
             
               T 
               p 
             
             = 
             
               
                 
                   2 
                   · 
                   L 
                   · 
                   C 
                 
               
               · 
               
                 
                   
                     
                       ( 
                       
                         
                           V 
                           dbh 
                         
                         - 
                         
                           V 
                           
                             db 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                       
                       ) 
                     
                     · 
                     
                       V 
                       tar 
                     
                   
                   
                     
                       V 
                       in 
                     
                     · 
                     
                       ( 
                       
                         
                           V 
                           in 
                         
                         - 
                         
                           V 
                           tar 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
               
           
           ⁢ 
           and 
         
       
     
     
       
         
           EQ_ 
           ⁢ 
           4 
           ⁢ 
           
             : 
           
         
       
     
     
       
         
           
               
           
           ⁢ 
           
             
               
                 T 
                 s 
               
               = 
               
                 
                   
                     
                       
                         V 
                         in 
                       
                       - 
                       
                         V 
                         tar 
                       
                     
                     
                       V 
                       tar 
                     
                   
                   · 
                   
                     T 
                     p 
                   
                 
                 - 
                 
                   GB 
                   1 
                 
                 - 
                 
                   GB 
                   2 
                 
               
             
             , 
             where 
           
         
       
     
   
   GB 1  is the time guard-band between the end of the conduction control signal provided to FET  114 , and the beginning of the conduction control signal provided to FET  116 , and GB 2  is the guard-band between the end of the conduction control signal provided to FET  116  and the beginning of the next conduction control pulse to be provided to FET  114 . It is of course desirable to avoid overlapping conduction of the control FET  114  and the synchronizing FET  116 . Guard band length is a function of the turn on and off times of the FETs used for FET  114  and FET  116  in the target system, as determined from their data sheet specifications. 
   In some designs, a single conduction control signal of length Tp could cause the maximum cur-rent in the associated inductor to exceed a maximum acceptable, such as the maximum current rating of the inductor. The associated inductor may have, for example, been selected for a small physical size or other requirements of the design.  FIG. 5  illustrates an embodiment wherein the time required for Tp is broken up into multiple, shorter time periods, separated by the appropriate Ts periods, to yield the required charge. In this example Tp is applied in time periods Tp 1 , Tp 2 , Tp 3 , and Ts in time periods Ts 1 , Ts 2 , and Ts 3 , resulting in a lower ΔI  500 . This embodiment may be applied to any of the PFM methods and topologies herein described. 
   Control Method_ 1 , is accomplished by using EQ — 3 and EQ — 4 to build a lookup table in which the PFM control utilizes at each activation, the input to the table being Vin. Vo is known to be Vdbl  202  at the time of activation. An example of such a table of values for Tp is shown in  FIG. 6 . The details of the values for the example are listed in  FIG. 7  in the block labeled “ FIG. 1  Circuit Details”. The exact values calculated are in column “Tp Calc” ( FIG. 6 ). 
   In the embodiment of Method_ 2 , Tp and Ts are calculated by PFM at each activation, again using EQ — 3 and EQ — 4. To speed up the calculation, a second order inverse polynomial approximation (a/(1+b*V IN +c*V IN   2 )) can be used. In this example the constants a, b, and c have been calculated using a Taylor Series to be: 
   a=6.658E-07 
   b=0.5349 
   c=0.0039 
   The results of this Method_ 2  are shown in  FIG. 6  column “Tp Approx”. Comparing the two columns of  FIG. 6 , it will be appreciated that the error using the approximation is very small. 
   Method_ 1  and Method_ 2  (actually, EQ_ 3  and EQ_ 4 ) ignore the effects of parasitic resistance in the physical circuit of the target application, which effects may amount to as much as a ΔV error of twenty percent. Parasitic resistance reduces the voltage across inductor L 1  during Tp while increasing the negative voltage across inductor L 1  during Ts. This results in the conduction permitted by the conduction control signals yielding less charge than expected using the calculations above. To appreciate these effects, reference is made to  FIG. 4  from which the following relationship is derived from the principle of conservation of charge: 
   
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             Q 
           
           = 
           
             
               
                 
                   1 
                   2 
                 
                 · 
                 
                   ( 
                   
                     
                       T 
                       p 
                     
                     + 
                     
                       T 
                       s 
                     
                   
                   ) 
                 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               I 
             
             = 
             
               
                 C 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               V 
             
           
         
       
     
     
       
         and 
       
     
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             
               Q 
               1 
             
           
           = 
           
             
               
                 
                   1 
                   2 
                 
                 · 
                 
                   T 
                   p 
                 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               I 
             
             = 
             
               
                 C 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 V 
                 1 
               
             
           
         
       
     
   
   However, considering that the inductor L 1  current starts at zero and returns to zero yields: 
   
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             I 
           
           = 
           
             
               
                 1 
                 L 
               
               · 
               
                 
                   ∫ 
                   0 
                   
                     T 
                     p 
                   
                 
                 ⁢ 
                 
                   
                     V 
                     1 
                   
                   · 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             = 
             
               
                 1 
                 L 
               
               · 
               
                 
                   ∫ 
                   0 
                   
                     T 
                     s 
                   
                 
                 ⁢ 
                 
                   
                     V 
                     2 
                   
                   · 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
           
         
       
     
   
   Assuming that input and output voltages do not change significantly over Tp and Ts results in:
 
 V   1   =V   in   −V   out   −R   p   ·I  
 
and
 
 V   2   =V   out   +R   s   ·I  
 
where R p  is the parasitic resistance in the current path during Tp and R s  is the parasitic resistance in the current path during Ts. Solving the integral provides:
 
   
     
       
         
           
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   I 
                 
                 = 
                 
                   
                     1 
                     L 
                   
                   · 
                   
                     
                       ∫ 
                       0 
                       
                         T 
                         p 
                       
                     
                     ⁢ 
                     
                       
                         V 
                         1 
                       
                       · 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     1 
                     L 
                   
                   · 
                   
                     
                       ∫ 
                       0 
                       
                         T 
                         p 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             V 
                             in 
                           
                           - 
                           
                             V 
                             out 
                           
                           - 
                           
                             
                               R 
                               p 
                             
                             · 
                             I 
                           
                         
                         ) 
                       
                       · 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       
                         T 
                         p 
                       
                       · 
                       
                         ( 
                         
                           
                             V 
                             in 
                           
                           - 
                           
                             V 
                             out 
                           
                         
                         ) 
                       
                     
                     L 
                   
                   - 
                   
                     
                       
                         R 
                         p 
                       
                       L 
                     
                     · 
                     
                       
                         ∫ 
                         0 
                         
                           T 
                           p 
                         
                       
                       ⁢ 
                       
                         I 
                         · 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       
                         T 
                         p 
                       
                       · 
                       
                         ( 
                         
                           
                             V 
                             in 
                           
                           - 
                           
                             V 
                             out 
                           
                         
                         ) 
                       
                     
                     L 
                   
                   - 
                   
                     
                       
                         
                           R 
                           p 
                         
                         L 
                       
                       · 
                       Δ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Q 
                       1 
                     
                   
                 
               
             
           
         
       
     
   
   However, assuming that: 
   
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             
               Q 
               1 
             
           
           ≅ 
           
             
               
                 
                   T 
                   p 
                 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               I 
             
             2 
           
         
       
     
   
   Which yields: 
   
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             I 
           
           = 
           
             
               
                 
                   T 
                   p 
                 
                 · 
                 
                   ( 
                   
                     
                       V 
                       in 
                     
                     - 
                     
                       V 
                       out 
                     
                   
                   ) 
                 
               
               L 
             
             - 
             
               
                 
                   R 
                   p 
                 
                 L 
               
               · 
               
                 
                   
                     
                       T 
                       p 
                     
                     · 
                     Δ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   I 
                 
                 2 
               
             
           
         
       
     
   
   Finally, 
   
     
       
         
           EQ_ 
           ⁢ 
           5 
           ⁢ 
           
             : 
           
         
       
     
     
       
         
           
               
           
           ⁢ 
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               I 
             
             = 
             
               
                 
                   
                     T 
                     p 
                   
                   · 
                   
                     ( 
                     
                       
                         V 
                         in 
                       
                       - 
                       
                         V 
                         out 
                       
                     
                     ) 
                   
                 
                 
                   L 
                   · 
                   
                     ( 
                     
                       1 
                       + 
                       
                         
                           
                             T 
                             p 
                           
                           · 
                           
                             R 
                             p 
                           
                         
                         
                           2 
                           · 
                           L 
                         
                       
                     
                     ) 
                   
                 
               
               ≅ 
               
                 
                   T 
                   p 
                 
                 · 
                 
                   ( 
                   
                     1 
                     - 
                     
                       
                         
                           T 
                           p 
                         
                         · 
                         
                           R 
                           p 
                         
                       
                       
                         2 
                         · 
                         L 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   
                     ( 
                     
                       
                         V 
                         in 
                       
                       - 
                       
                         V 
                         out 
                       
                     
                     ) 
                   
                   L 
                 
               
             
           
         
       
     
   
   Repeating the same derivation for the second slope of the inductor L 1  current (during Ts time) provides: 
   
     
       
         
           EQ_ 
           ⁢ 
           6 
           ⁢ 
           
             : 
           
         
       
     
     
       
         
           
               
           
           ⁢ 
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               I 
             
             = 
             
               
                 
                   
                     T 
                     s 
                   
                   · 
                   
                     V 
                     out 
                   
                 
                 
                   L 
                   · 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           
                             T 
                             s 
                           
                           · 
                           
                             R 
                             s 
                           
                         
                         
                           2 
                           · 
                           L 
                         
                       
                     
                     ) 
                   
                 
               
               ≅ 
               
                 
                   T 
                   s 
                 
                 · 
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         
                           T 
                           s 
                         
                         · 
                         
                           R 
                           s 
                         
                       
                       
                         2 
                         · 
                         L 
                       
                     
                   
                   ) 
                 
                 · 
                 
                   
                     V 
                     out 
                   
                   L 
                 
               
             
           
         
       
     
   
   From EQ — 5 and EQ — 6, the relationship between Tp and Ts is derived, specifically: 
   
     
       
         
           
             
               
                 T 
                 s 
               
               · 
               
                 V 
                 out 
               
             
             
               L 
               · 
               
                 ( 
                 
                   1 
                   - 
                   
                     
                       
                         T 
                         s 
                       
                       · 
                       
                         R 
                         s 
                       
                     
                     
                       2 
                       · 
                       L 
                     
                   
                 
                 ) 
               
             
           
           = 
           
             
               
                 
                   T 
                   p 
                 
                 · 
                 
                   ( 
                   
                     
                       V 
                       in 
                     
                     - 
                     
                       V 
                       out 
                     
                   
                   ) 
                 
               
               
                 L 
                 · 
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         
                           T 
                           p 
                         
                         · 
                         
                           R 
                           p 
                         
                       
                       
                         2 
                         · 
                         L 
                       
                     
                   
                   ) 
                 
               
             
             ⇒ 
             
               
 
             
             ⁢ 
             
               
                 
                   
                     
                       ⇒ 
                       
                         T 
                         s 
                       
                     
                     = 
                       
                     ⁢ 
                     
                       
                         T 
                         p 
                       
                       · 
                       
                         
                           ( 
                           
                             
                               V 
                               in 
                             
                             - 
                             
                               V 
                               out 
                             
                           
                           ) 
                         
                         
                           V 
                           out 
                         
                       
                       · 
                       
                         
                           ( 
                           
                             1 
                             - 
                             
                               
                                 
                                   T 
                                   s 
                                 
                                 · 
                                 
                                   R 
                                   s 
                                 
                               
                               
                                 2 
                                 · 
                                 L 
                               
                             
                           
                           ) 
                         
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 
                                   T 
                                   p 
                                 
                                 · 
                                 
                                   R 
                                   p 
                                 
                               
                               
                                 2 
                                 · 
                                 L 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     ≅ 
                       
                     ⁢ 
                     
                       
                         T 
                         p 
                       
                       · 
                       
                         
                           ( 
                           
                             
                               V 
                               in 
                             
                             - 
                             
                               V 
                               out 
                             
                           
                           ) 
                         
                         
                           V 
                           out 
                         
                       
                       · 
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               1 
                               
                                 2 
                                 · 
                                 L 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   
                                     T 
                                     s 
                                   
                                   · 
                                   
                                     R 
                                     s 
                                   
                                 
                                 + 
                                 
                                   
                                     T 
                                     p 
                                   
                                   · 
                                   
                                     R 
                                     p 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
             
           
         
       
     
   
   Ignoring second order effects: 
   
     
       
         
           
             T 
             s 
           
           ≅ 
           
             
               T 
               p 
             
             · 
             
               
                 ( 
                 
                   
                     V 
                     in 
                   
                   - 
                   
                     V 
                     out 
                   
                 
                 ) 
               
               
                 V 
                 out 
               
             
           
         
       
     
   
   Therefore: 
   
     
       
         
           
             T 
             s 
           
           ≅ 
           
             
               T 
               p 
             
             · 
             
               
                 ( 
                 
                   
                     V 
                     in 
                   
                   - 
                   
                     V 
                     out 
                   
                 
                 ) 
               
               
                 V 
                 out 
               
             
             · 
             
               [ 
               
                 1 
                 - 
                 
                   
                     
                       T 
                       p 
                     
                     
                       2 
                       · 
                       L 
                     
                   
                   · 
                   
                     ( 
                     
                       
                         R 
                         p 
                       
                       + 
                       
                         
                           
                             ( 
                             
                               
                                 V 
                                 in 
                               
                               - 
                               
                                 V 
                                 out 
                               
                             
                             ) 
                           
                           
                             V 
                             out 
                           
                         
                         · 
                         
                           R 
                           s 
                         
                       
                     
                     ) 
                   
                 
               
               ] 
             
           
         
       
     
   
   The total charge transferred to the output is found by 
   
     
       
         
           
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Q 
                 
                 = 
                 
                   
                     
                       C 
                       · 
                       Δ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     V 
                   
                   ≅ 
                   
                     
                       
                         
                           ( 
                           
                             
                               T 
                               p 
                             
                             + 
                             
                               T 
                               s 
                             
                           
                           ) 
                         
                         · 
                         Δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       I 
                     
                     2 
                   
                 
               
             
             
               
                 EQ_ 
                 ⁢ 
                 7 
               
             
           
         
       
     
   
   Now, assuming that R=R p ≅R s  and that Tp will only be adjusted over a narrow region (T p(min) ≦T p ≦T p(max) ) or 
   
     
       
         
           
             T 
             pa 
           
           = 
           
             
               
                 
                   T 
                   
                     p 
                     ⁡ 
                     
                       ( 
                       min 
                       ) 
                     
                   
                 
                 + 
                 
                   T 
                   
                     p 
                     ⁡ 
                     
                       ( 
                       max 
                       ) 
                     
                   
                 
               
               2 
             
             . 
           
         
       
     
   
   The above provides: 
   
     
       
         
           
             
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       I 
                     
                     ≅ 
                     
                       
                         T 
                         p 
                       
                       · 
                       
                         ( 
                         
                           1 
                           - 
                           
                             k 
                             p 
                           
                         
                         ) 
                       
                       · 
                       
                         
                           ( 
                           
                             
                               V 
                               in 
                             
                             - 
                             
                               V 
                               out 
                             
                           
                           ) 
                         
                         L 
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   where 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     k 
                     p 
                   
                   = 
                   
                     
                       
                         T 
                         pa 
                       
                       · 
                       R 
                     
                     
                       2 
                       · 
                       L 
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 and 
               
             
             
               
                 EQ_ 
                 ⁢ 
                 8 
               
             
           
           
             
               
                 
                   T 
                   s 
                 
                 ≅ 
                 
                   
                     T 
                     p 
                   
                   · 
                   
                     
                       ( 
                       
                         
                           V 
                           in 
                         
                         - 
                         
                           V 
                           out 
                         
                       
                       ) 
                     
                     
                       V 
                       out 
                     
                   
                   · 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           k 
                           p 
                         
                         · 
                         
                           ( 
                           
                             
                               V 
                               in 
                             
                             
                               V 
                               out 
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 EQ_ 
                 ⁢ 
                 9 
               
             
           
         
       
     
   
   Combining EQ — 7, EQ — 8, and EQ — 9 gives: 
   
     
       
         
           
             
               
                 
                   
                     T 
                     p 
                   
                   + 
                   
                     T 
                     s 
                   
                 
                 ≅ 
                   
                 ⁢ 
                 
                   
                     T 
                     p 
                   
                   · 
                   
                     { 
                     
                       1 
                       + 
                       
                         
                           
                             ( 
                             
                               
                                 V 
                                 in 
                               
                               - 
                               
                                 V 
                                 out 
                               
                             
                             ) 
                           
                           
                             V 
                             out 
                           
                         
                         · 
                         
                           [ 
                           
                             1 
                             - 
                             
                               
                                 k 
                                 p 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     V 
                                     in 
                                   
                                   
                                     V 
                                     out 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                     } 
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     T 
                     p 
                   
                   · 
                   
                     [ 
                     
                       
                         
                           V 
                           in 
                         
                         
                           V 
                           out 
                         
                       
                       - 
                       
                         
                           k 
                           p 
                         
                         · 
                         
                           
                             ( 
                             
                               
                                 V 
                                 in 
                               
                               - 
                               
                                 V 
                                 out 
                               
                             
                             ) 
                           
                           
                             V 
                             out 
                           
                         
                         · 
                         
                           ( 
                           
                             
                               V 
                               in 
                             
                             
                               V 
                               out 
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     T 
                     p 
                   
                   · 
                   
                     ( 
                     
                       
                         V 
                         in 
                       
                       
                         V 
                         out 
                       
                     
                     ) 
                   
                   · 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           k 
                           p 
                         
                         · 
                         
                           
                             
                               V 
                               in 
                             
                             - 
                             
                               V 
                               out 
                             
                           
                           
                             V 
                             out 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Q 
                 
                 = 
                 
                   
                     
                       C 
                       · 
                       Δ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     V 
                   
                   ≅ 
                     
                   ⁢ 
                   
                     
                       1 
                       2 
                     
                     · 
                     
                       T 
                       p 
                     
                     · 
                     
                       ( 
                       
                         
                           V 
                           in 
                         
                         
                           V 
                           out 
                         
                       
                       ) 
                     
                     · 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             k 
                             p 
                           
                           · 
                           
                             
                               
                                 V 
                                 in 
                               
                               - 
                               
                                 V 
                                 out 
                               
                             
                             
                               V 
                               out 
                             
                           
                         
                       
                       ) 
                     
                     · 
                     
                       T 
                       p 
                     
                     · 
                     
                       ( 
                       
                         1 
                         - 
                         
                           k 
                           p 
                         
                       
                       ) 
                     
                     · 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     ( 
                     
                       
                         V 
                         in 
                       
                       - 
                       
                         V 
                         out 
                       
                     
                     ) 
                   
                   L 
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     1 
                     
                       2 
                       · 
                       L 
                     
                   
                   · 
                   
                     T 
                     p 
                     2 
                   
                   · 
                   
                     ( 
                     
                       
                         V 
                         in 
                       
                       
                         V 
                         out 
                       
                     
                     ) 
                   
                   · 
                   
                     ( 
                     
                       
                         V 
                         in 
                       
                       - 
                       
                         V 
                         out 
                       
                     
                     ) 
                   
                   · 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           k 
                           p 
                         
                         · 
                         
                           
                             
                               V 
                               in 
                             
                             - 
                             
                               V 
                               out 
                             
                           
                           
                             V 
                             out 
                           
                         
                       
                     
                     ) 
                   
                   · 
                   
                     ( 
                     
                       1 
                       - 
                       
                         k 
                         p 
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                 ≅ 
                   
                 ⁢ 
                 
                   
                     1 
                     
                       2 
                       · 
                       L 
                     
                   
                   · 
                   
                     T 
                     p 
                     2 
                   
                   · 
                   
                     ( 
                     
                       
                         V 
                         in 
                       
                       
                         V 
                         out 
                       
                     
                     ) 
                   
                   · 
                   
                     ( 
                     
                       
                         V 
                         in 
                       
                       - 
                       
                         V 
                         out 
                       
                     
                     ) 
                   
                   · 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           k 
                           p 
                         
                         · 
                         
                           
                             V 
                             in 
                           
                           
                             V 
                             out 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
   
   Using the same name conventions as before, Tp and Ts can be solved by 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           T 
                           p 
                         
                         ≅ 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   
                                     k 
                                     p 
                                   
                                   2 
                                 
                                 · 
                                 
                                   
                                     V 
                                     in 
                                   
                                   
                                     V 
                                     tar 
                                   
                                 
                               
                             
                             ) 
                           
                           · 
                           
                             
                               2 
                               · 
                               L 
                               · 
                               C 
                             
                           
                           · 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               
                                 V 
                                 tar 
                               
                               
                                 
                                   V 
                                   in 
                                 
                                 - 
                                 
                                   V 
                                   tar 
                                 
                               
                             
                           
                           · 
                           
                             
                               
                                 
                                   V 
                                   dbh 
                                 
                                 - 
                                 
                                   V 
                                   dbl 
                                 
                               
                               
                                 V 
                                 in 
                               
                             
                           
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 and 
               
             
             
               
                 EQ_ 
                 ⁢ 
                 10 
               
             
           
           
             
               
                 
                   
                     T 
                     s 
                   
                   ≅ 
                   
                     
                       
                         T 
                         p 
                       
                       · 
                       
                         
                           
                             V 
                             in 
                           
                           - 
                           
                             V 
                             tar 
                           
                         
                         
                           V 
                           tar 
                         
                       
                       · 
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               k 
                               p 
                             
                             · 
                             
                               
                                 V 
                                 in 
                               
                               
                                 V 
                                 tar 
                               
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       GB 
                       1 
                     
                     - 
                     
                       GB 
                       2 
                     
                   
                 
                 , 
                 
                   
 
                 
                 ⁢ 
                 where 
               
             
             
               
                 EQ_ 
                 ⁢ 
                 11 
               
             
           
           
             
               
                 
                   k 
                   p 
                 
                 = 
                 
                   
                     R 
                     
                       4 
                       · 
                       L 
                     
                   
                   · 
                   
                     ( 
                     
                       
                         T 
                         
                           p 
                           ⁡ 
                           
                             ( 
                             min 
                             ) 
                           
                         
                       
                       + 
                       
                         T 
                         
                           p 
                           ⁡ 
                           
                             ( 
                             max 
                             ) 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 EQ_ 
                 ⁢ 
                 12 
               
             
           
         
       
     
   
   Equations EQ — 10, EQ — 11, and EQ — 12 represent an embodiment of PFM Method_ 3 , which is basically Method_ 2  with the refinement of adjustment for parasitics. However Method_ 3  utilizes static component values estimated by the designer. 
   The precise value of model parameters cannot be known in advance, causing significant variance. That is especially true for the values of L and C. They might vary as much 20% each, for a potential combined variation of 40% from assumed values. 
   A more accurate embodiment of Method_ 3  compares the actual variation of the output voltage (ΔV actual ), the amount by which Vo changed as the result of one pulse, in relation to the deployed pulse width of the conduction control signal applied to the control FET  114 , deriving the effective value of the L and C per: 
   
     
       
         
           
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Q 
                 
                 = 
                   
                 ⁢ 
                 
                   
                     C 
                     · 
                     Δ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     V 
                     actual 
                   
                 
               
             
           
           
             
               
                 ≅ 
                   
                 ⁢ 
                 
                   
                     1 
                     
                       2 
                       · 
                       L 
                     
                   
                   · 
                   
                     T 
                     p 
                     2 
                   
                   · 
                   
                     ( 
                     
                       
                         V 
                         in 
                       
                       
                         V 
                         out 
                       
                     
                     ) 
                   
                   · 
                   
                     ( 
                     
                       
                         V 
                         in 
                       
                       - 
                       
                         V 
                         out 
                       
                     
                     ) 
                   
                   · 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           k 
                           p 
                         
                         · 
                         
                           
                             V 
                             in 
                           
                           
                             V 
                             out 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     
       
         Therefore 
       
     
     
       
         
           
             
               2 
               · 
               L 
               · 
               C 
             
           
           ≅ 
           
             
               T 
               p 
             
             · 
             
               
                 
                   
                     
                       
                         V 
                         in 
                       
                       - 
                       
                         V 
                         
                           · 
                           tar 
                         
                       
                     
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         actual 
                       
                     
                   
                   · 
                   
                     ( 
                     
                       
                         
                           V 
                           in 
                         
                         
                           V 
                           tar 
                         
                       
                       - 
                       
                         k 
                         p 
                       
                     
                     ) 
                   
                 
               
               . 
             
           
         
       
     
   
   From this it will be appreciated that the specific values for L and C are not needed, only the product LC. The parameter √{square root over (2·L·C)} is estimated by measuring Vo after each pulse deployed and updating the model (EQ — 10). In other embodiments, EQ — 10 is slowly adapted using a rolling average technique, or periodically adjusted per a time schedule or other scheme. This approach adjusts for actual component values, including the effect of temperature during operation and longer term changes due to component aging. 
   In another embodiment, the PFM control method is utilized with a non-synchronous boost converter, such as shown in  FIG. 8 . Controller  800  provides a conduction control signal to FET Q 3  via line  810 , thus permitting current to flow in the inductor L 2  for a length of time Tp. Thereafter the conduction control signal is terminated and current of inductor L 2  flows through diode D 1  for a length of time Ts. 
   The width of the pulse of current (Tp+Ts) is calculated using the method of the present invention (PFM), and can be implemented in any of the following ways: a) PFM control using a stored set of pre-calculated pulses, from which it selects depending on its input variables; b) PFM control based on a calculation of a duration for Tp and Ts each time corrective action is to be initiated using the present input variables and pre-stored system model parameters; and c) PFM control based on calculation of Tp and Ts each time corrective action is to be initiated using present input variables and an estimation of system model parameters, calculated as a function of system response. The first embodiment is presented; one skilled in the art will understand other embodiments similar to those previously discussed as Method_ 2  and Method_ 3 . 
   Vin is assumed to be within a restricted range of voltages: V in(min) ≦V in ≦V in(max) . Looking to  FIG. 8 , we have the following relationship derived from the principle of conservation of charge: 
   
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             Q 
           
           = 
           
             
               
                 
                   1 
                   2 
                 
                 · 
                 
                   T 
                   s 
                 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               I 
             
             = 
             
               
                 C 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               V 
             
           
         
       
     
   
   However, considering that the current in inductor L 2  starts at 0 and returns to 0, this provides: 
                 V   1     ·     T   p       L     =             V   2     ·     T   s       L     ⇒       V   1     ·     T   p         =       V   2     ·     T   s               
where V 1  is the voltage across the inductor L 2  during time Tp and V 2  is the voltage across the inductor L 2  during time Ts.
 
   The peak current in inductor L 2  (ΔI) is: 
   
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             I 
           
           = 
           
             
               
                 
                   V 
                   2 
                 
                 · 
                 
                   T 
                   s 
                 
               
               L 
             
             = 
             
               
                 
                   V 
                   1 
                 
                 · 
                 
                   T 
                   p 
                 
               
               L 
             
           
         
       
     
   
   Therefore, using the relationship previously obtained: 
   
     
       
         
           
             
               C 
               · 
               Δ 
             
             ⁢ 
             
                 
             
             ⁢ 
             V 
           
           = 
           
             
               
                 
                   1 
                   2 
                 
                 · 
                 
                   T 
                   s 
                 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               I 
             
             = 
             
               
                 1 
                 2 
               
               · 
               
                 
                   
                     V 
                     1 
                   
                   · 
                   
                     T 
                     p 
                   
                 
                 
                   V 
                   2 
                 
               
               · 
               
                 
                   
                     V 
                     1 
                   
                   · 
                   
                     T 
                     p 
                   
                 
                 L 
               
             
           
         
       
     
     
       
         Now 
       
     
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               V 
             
             = 
             
               
                 
                   T 
                   p 
                   2 
                 
                 
                   2 
                   · 
                   L 
                   · 
                   C 
                 
               
               · 
               
                 
                   V 
                   1 
                   2 
                 
                 
                   V 
                   2 
                 
               
             
           
           , 
           
               
           
           ⁢ 
           or 
         
       
     
     
       
         
           
             T 
             p 
           
           = 
           
             
               1 
               
                 V 
                 1 
               
             
             · 
             
               
                 
                   
                     2 
                     · 
                     L 
                     · 
                     C 
                     · 
                     Δ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     V 
                     · 
                     
                       V 
                       2 
                     
                   
                 
               
               . 
             
           
         
       
     
   
   For the implementation of  FIG. 8 , this provides:
 
Δ V=V   dbh   −V   dbl ,
 
V 1 =V in ,
 
and
 
 V   2   =V   out   +V   D   −V   in   =V   tar   +V   D   −V   in ,
 
where V D  is the drop across diode  808 . Thus Tp, the only control parameter for a non-synchronous implementation, can be calculated as
 
   
     
       
         
           
             T 
             p 
           
           = 
           
             
               
                 
                   2 
                   · 
                   L 
                   · 
                   C 
                   · 
                   
                     ( 
                     
                       
                         V 
                         dbh 
                       
                       - 
                       
                         V 
                         dbl 
                       
                     
                     ) 
                   
                 
               
               
                 V 
                 in 
               
             
             · 
             
               
                 
                   
                     V 
                     tar 
                   
                   + 
                   
                     V 
                     D 
                   
                   - 
                   
                     V 
                     in 
                   
                 
               
               . 
             
           
         
       
     
   
   In another embodiment of the present invention PFM control is used to regulate a multiple transistor buck/boost power converter, such as that in  FIG. 9 , useful for producing an output voltage Vo which is either higher or lower than input voltage Vin. 
   The controller  900 , using times calculated by the PFM control methods, injects pulses of current into inductor L 3 , by closing switches Q 4  and Q 6 , for a length of time Tp; and thereafter opening them, letting the coil current flow thru MOSFETs Q 5 , and Q 7  for a length of time Ts. MOSFETs Q 5  and Q 7  may be replaced with diodes. 
   Looking again to  FIG. 9 , the following relationship is derived from the principle of conservation of charge: 
   
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             Q 
           
           = 
           
             
               
                 
                   1 
                   2 
                 
                 · 
                 
                   ( 
                   
                     
                       T 
                       p 
                     
                     + 
                     
                       T 
                       s 
                     
                   
                   ) 
                 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               I 
             
             = 
             
               
                 C 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               V 
             
           
         
       
     
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             
               Q 
               1 
             
           
           = 
           
             
               
                 
                   1 
                   2 
                 
                 · 
                 
                   T 
                   p 
                 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               I 
             
             = 
             
               
                 C 
                 · 
                 Δ 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 V 
                 1 
               
             
           
         
       
     
   
   However, considering that the inductor L 3  current starts at zero and returns to zero, this provides: 
   
     
       
         
           
             Δ 
             ⁢ 
             
                 
             
             ⁢ 
             I 
           
           = 
           
             
               
                 1 
                 L 
               
               · 
               
                 
                   ∫ 
                   0 
                   
                     T 
                     p 
                   
                 
                 ⁢ 
                 
                   
                     V 
                     1 
                   
                   · 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
             = 
             
               
                 1 
                 L 
               
               · 
               
                 
                   ∫ 
                   0 
                   
                     T 
                     s 
                   
                 
                 ⁢ 
                 
                   
                     V 
                     2 
                   
                   · 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     t 
                   
                 
               
             
           
         
       
     
   
   Assuming that input and output voltages do not change significantly over Tp and Ts, results in:
 
 V   1   =V   in   −R   p   ·I  
 
 V   2   =V   out   +R   s   ·I  
 
   Where: 
   R p  is the parasitic resistance in the current path during Tp and R s  is the parasitic resistance in the current path during Ts. 
   Solving the integral, provides: 
   
     
       
         
           
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   I 
                 
                 = 
                   
                 ⁢ 
                 
                   
                     1 
                     L 
                   
                   · 
                   
                     
                       ∫ 
                       0 
                       
                         T 
                         p 
                       
                     
                     ⁢ 
                     
                       
                         V 
                         1 
                       
                       · 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     1 
                     L 
                   
                   · 
                   
                     
                       ∫ 
                       0 
                       
                         T 
                         p 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             V 
                             in 
                           
                           - 
                           
                             
                               R 
                               p 
                             
                             · 
                             I 
                           
                         
                         ) 
                       
                       · 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       
                         T 
                         p 
                       
                       · 
                       
                         V 
                         in 
                       
                     
                     L 
                   
                   - 
                   
                     
                       
                         R 
                         p 
                       
                       L 
                     
                     · 
                     
                       
                         ∫ 
                         0 
                         
                           T 
                           p 
                         
                       
                       ⁢ 
                       
                         I 
                         · 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       
                         T 
                         p 
                       
                       · 
                       
                         V 
                         in 
                       
                     
                     L 
                   
                   - 
                   
                     
                       
                         
                           R 
                           p 
                         
                         L 
                       
                       · 
                       Δ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Q 
                       1 
                     
                   
                 
               
             
           
         
       
     
   
   However, assuming that: 
   
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Q 
                       1 
                     
                   
                   ≅ 
                   
                     
                       
                         
                           
                             T 
                             p 
                           
                           · 
                           Δ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         I 
                       
                       2 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     we 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     find 
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     I 
                   
                   = 
                   
                     
                       
                         
                           T 
                           p 
                         
                         · 
                         
                           V 
                           in 
                         
                       
                       L 
                     
                     - 
                     
                       
                         
                           R 
                           p 
                         
                         L 
                       
                       · 
                       
                         
                           
                             
                               T 
                               p 
                             
                             · 
                             Δ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           I 
                         
                         2 
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   Finally 
                   , 
                 
               
             
             
               
                   
               
             
           
           
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   I 
                 
                 = 
                 
                   
                     
                       
                         T 
                         p 
                       
                       · 
                       
                         V 
                         
                           i 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           n 
                         
                       
                     
                     
                       L 
                       · 
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               
                                 T 
                                 p 
                               
                               · 
                               
                                 R 
                                 p 
                               
                             
                             
                               2 
                               · 
                               L 
                             
                           
                         
                         ) 
                       
                     
                   
                   ≅ 
                   
                     
                       T 
                       p 
                     
                     · 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             
                               T 
                               p 
                             
                             · 
                             
                               R 
                               p 
                             
                           
                           
                             2 
                             · 
                             L 
                           
                         
                       
                       ) 
                     
                     · 
                     
                       
                         V 
                         
                           i 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           n 
                         
                       
                       L 
                     
                   
                 
               
             
             
               
                 EQ_ 
                 ⁢ 
                 13 
               
             
           
         
       
     
   
   Repeating the same reasoning for the slope of the inductor L 3  current during the Ts time we have: 
   
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     I 
                   
                   = 
                   
                     
                       
                         
                           T 
                           s 
                         
                         · 
                         
                           V 
                           out 
                         
                       
                       
                         L 
                         · 
                         
                           ( 
                           
                             1 
                             - 
                             
                               
                                 
                                   T 
                                   s 
                                 
                                 · 
                                 
                                   R 
                                   s 
                                 
                               
                               
                                 2 
                                 · 
                                 L 
                               
                             
                           
                           ) 
                         
                       
                     
                     ≅ 
                     
                       
                         t 
                         s 
                       
                       · 
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               
                                 T 
                                 s 
                               
                               · 
                               
                                 R 
                                 s 
                               
                             
                             
                               2 
                               · 
                               L 
                             
                           
                         
                         ) 
                       
                       · 
                       
                         
                           V 
                           out 
                         
                         L 
                       
                     
                   
                 
                 ⁢ 
                 
                     
                 
               
             
             
               
                 EQ_ 
                 ⁢ 
                 14 
               
             
           
         
       
     
   
   From EQ — 13 and EQ — 14, the relationship between Tp and Ts is: 
   
     
       
         
           
             
               
                 T 
                 s 
               
               · 
               
                 V 
                 out 
               
             
             
               L 
               · 
               
                 ( 
                 
                   1 
                   - 
                   
                     
                       
                         T 
                         s 
                       
                       · 
                       
                         R 
                         s 
                       
                     
                     
                       2 
                       · 
                       L 
                     
                   
                 
                 ) 
               
             
           
           = 
           
             
               
                 
                   
                     T 
                     p 
                   
                   · 
                   
                     V 
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                   
                 
                 
                   L 
                   · 
                   
                     ( 
                     
                       1 
                       + 
                       
                         
                           
                             T 
                             p 
                           
                           · 
                           
                             R 
                             p 
                           
                         
                         
                           2 
                           · 
                           L 
                         
                       
                     
                     ) 
                   
                 
               
               ⇒ 
               
                 
 
               
               ⇒ 
               
                 T 
                 s 
               
             
             = 
             
               
                 
                   T 
                   p 
                 
                 · 
                 
                   
                     V 
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                   
                   
                     V 
                     out 
                   
                 
                 · 
                 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           
                             T 
                             s 
                           
                           · 
                           
                             R 
                             s 
                           
                         
                         
                           2 
                           · 
                           L 
                         
                       
                     
                     ) 
                   
                   
                     ( 
                     
                       1 
                       + 
                       
                         
                           
                             T 
                             p 
                           
                           · 
                           
                             R 
                             p 
                           
                         
                         
                           2 
                           · 
                           L 
                         
                       
                     
                     ) 
                   
                 
               
               ⁢ 
               
                 
 
               
               ≅ 
               
                 
                   t 
                   p 
                 
                 · 
                 
                   
                     V 
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                   
                   
                     V 
                     out 
                   
                 
                 · 
                 
                   [ 
                   
                     1 
                     - 
                     
                       
                         1 
                         
                           2 
                           · 
                           L 
                         
                       
                       · 
                       
                         ( 
                         
                           
                             
                               T 
                               s 
                             
                             · 
                             
                               R 
                               s 
                             
                           
                           + 
                           
                             
                               T 
                               p 
                             
                             · 
                             
                               R 
                               p 
                             
                           
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
   
   Ignoring second order effects: 
   
     
       
         
           
             T 
             s 
           
           ≅ 
           
             
               T 
               p 
             
             · 
             
               
                 V 
                 
                   i 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   n 
                 
               
               
                 V 
                 out 
               
             
           
         
       
     
   
   Therefore, 
   
     
       
         
           
             T 
             s 
           
           ≅ 
           
             
               T 
               p 
             
             · 
             
               
                 V 
                 
                   i 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   n 
                 
               
               
                 V 
                 out 
               
             
             · 
             
               
                 [ 
                 
                   1 
                   - 
                   
                     
                       
                         T 
                         p 
                       
                       
                         2 
                         · 
                         L 
                       
                     
                     · 
                     
                       ( 
                       
                         
                           R 
                           p 
                         
                         + 
                         
                           
                             
                               V 
                               
                                 i 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 n 
                               
                             
                             
                               V 
                               out 
                             
                           
                           · 
                           
                             R 
                             s 
                           
                         
                       
                       ) 
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
   
   The total charge transferred to the output is: 
   
     
       
         
           
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Q 
                 
                 = 
                 
                   
                     
                       C 
                       · 
                       Δ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     V 
                   
                   ≅ 
                   
                     
                       
                         
                           ( 
                           
                             
                               T 
                               p 
                             
                             + 
                             
                               T 
                               s 
                             
                           
                           ) 
                         
                         · 
                         Δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       I 
                     
                     2 
                   
                 
               
             
             
               
                 EQ_ 
                 ⁢ 
                 15 
               
             
           
         
       
     
   
   Now, assuming the following conditions:
 
R=R p ≅R s ,
 
and
 
( T   p(min)   ≦T   p   ≦T   p(max) )
 
will only be adjusted over a narrow region, given as
 
               T   pa     =         T     p   ⁡     (   min   )         +     T     p   ⁡     (   max   )           2       ,         
yields:
 
   
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     I 
                   
                   ≅ 
                   
                     
                       T 
                       p 
                     
                     · 
                     
                       ( 
                       
                         1 
                         - 
                         
                           k 
                           p 
                         
                       
                       ) 
                     
                     · 
                     
                       
                         V 
                         
                           i 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           n 
                         
                       
                       L 
                     
                   
                 
                 , 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     where 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       k 
                       p 
                     
                   
                   = 
                   
                     
                       
                         T 
                         pa 
                       
                       · 
                       R 
                     
                     
                       2 
                       · 
                       L 
                     
                   
                 
                 , 
                 and 
               
             
             
               
                 EQ_ 
                 ⁢ 
                 16 
               
             
           
           
             
               
                 
                   T 
                   s 
                 
                 ≅ 
                 
                   
                     T 
                     p 
                   
                   · 
                   
                     
                       V 
                       
                         i 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                     
                     
                       V 
                       out 
                     
                   
                   · 
                   
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             k 
                             p 
                           
                           · 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   V 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     n 
                                   
                                 
                                 
                                   V 
                                   out 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
             
             
               
                 EQ_ 
                 ⁢ 
                 17 
               
             
           
         
       
     
   
   Combining EQ — 15, EQ — 16 and EQ — 17 yields: 
   
     
       
         
           
             
               
                 
                   
                     T 
                     p 
                   
                   + 
                   
                     T 
                     s 
                   
                 
                 ≅ 
                   
                 ⁢ 
                 
                   
                     T 
                     p 
                   
                   · 
                   
                     { 
                     
                       1 
                       + 
                       
                         
                           
                             V 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                           
                           
                             V 
                             out 
                           
                         
                         · 
                         
                           [ 
                           
                             1 
                             - 
                             
                               
                                 k 
                                 p 
                               
                               · 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       V 
                                       
                                         i 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         n 
                                       
                                     
                                     
                                       V 
                                       out 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                     } 
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     T 
                     p 
                   
                   · 
                   
                     [ 
                     
                       1 
                       + 
                       
                         
                           V 
                           in 
                         
                         
                           V 
                           out 
                         
                       
                       - 
                       
                         
                           k 
                           p 
                         
                         · 
                         
                           
                             V 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                           
                           
                             V 
                             out 
                           
                         
                         · 
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 V 
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                 
                               
                               
                                 V 
                                 out 
                               
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     T 
                     p 
                   
                   · 
                   
                     ( 
                     
                       1 
                       + 
                       
                         
                           V 
                           
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             n 
                           
                         
                         
                           V 
                           out 
                         
                       
                     
                     ) 
                   
                   · 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           k 
                           p 
                         
                         · 
                         
                           
                             V 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                           
                           
                             V 
                             out 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Q 
                 
                 = 
                   
                 ⁢ 
                 
                   
                     C 
                     · 
                     Δ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   V 
                 
               
             
           
           
             
               
                 ≅ 
                   
                 ⁢ 
                 
                   
                     1 
                     2 
                   
                   · 
                   
                     T 
                     p 
                   
                   · 
                   
                     ( 
                     
                       1 
                       + 
                       
                         
                           V 
                           
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             n 
                           
                         
                         
                           V 
                           out 
                         
                       
                     
                     ) 
                   
                   · 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           k 
                           p 
                         
                         · 
                         
                           
                             V 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                           
                           
                             V 
                             out 
                           
                         
                       
                     
                     ) 
                   
                   · 
                   
                     T 
                     p 
                   
                   · 
                   
                     ( 
                     
                       1 
                       - 
                       
                         k 
                         p 
                       
                     
                     ) 
                   
                   · 
                   
                     
                       V 
                       
                         i 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                     
                     L 
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     1 
                     2 
                   
                   · 
                   
                     T 
                     p 
                     2 
                   
                   · 
                   
                     
                       V 
                       
                         i 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                     
                     L 
                   
                   · 
                   
                     ( 
                     
                       1 
                       + 
                       
                         
                           V 
                           
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             n 
                           
                         
                         
                           V 
                           out 
                         
                       
                     
                     ) 
                   
                   · 
                   
                     [ 
                     
                       1 
                       - 
                       
                         
                           k 
                           p 
                         
                         · 
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 V 
                                 
                                   i 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                 
                               
                               
                                 V 
                                 out 
                               
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
   
   Then, using the same name conventions used above in Method  1 , the solution for the circuit of  FIG. 9  is: 
   
     
       
         
           
             T 
             p 
           
           ≅ 
           
             
               
                 [ 
                 
                   1 
                   + 
                   
                     
                       
                         k 
                         p 
                       
                       2 
                     
                     · 
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             V 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                           
                           
                             V 
                             tar 
                           
                         
                       
                       ) 
                     
                   
                 
                 ] 
               
               · 
               
                 
                   2 
                   · 
                   L 
                   · 
                   C 
                   · 
                 
               
             
             ⁢ 
             
               
                 
                   
                     V 
                     tar 
                   
                   
                     V 
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                   
                 
               
               · 
               
                 
                   
                     
                       V 
                       dbh 
                     
                     - 
                     
                       V 
                       dbl 
                     
                   
                   
                     
                       V 
                       
                         i 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                     
                     + 
                     
                       V 
                       tar 
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             T 
             s 
           
           ≅ 
           
             
               
                 T 
                 p 
               
               · 
               
                 
                   V 
                   
                     i 
                     ⁢ 
                     
                         
                     
                     ⁢ 
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   A further benefit of the PFM control method is the ability to make use of the information used in PFM control to derive an estimate of the current delivered to the power converter. For a battery-powered device this is useful in “fuel gauging”, i.e., keeping track of how much energy remains in a one or more batteries used for power. 
   The method used is to accumulate voltage changes. A count (PR) is kept of how many pulses get deployed per second. The amount of charge in each pulse is
 
Δ Q=C·ΔV   actual   =C· ( V   dbh   −V   dbl ).
 
   Therefore an estimate of the load current is given by
 
 I   load   =PR·ΔQ=PR·C· ( V   dbh   −V   dbl ).
 
   For the circuit of  FIG. 1 , the accuracy of this calculation depends upon the tolerance of the capacitor C 2 . Since all terms are relatively fixed, it is possible to keep track of current, hence energy removed from the battery, by simply counting the number of pulses.