Patent Publication Number: US-11051369-B2

Title: Radio frequency heating apparatus using direct-digital radio frequency power control and fine-tune power control

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation-in-part of U.S. application Ser. No. 14/873,920 filed on Oct. 2, 2015, which claims the benefit of U.S. Provisional Application No. 62/066,465, filed Oct. 21, 2014. Each of the foregoing applications is incorporated by reference herein in its entirety, for all purposes. This application further incorporates herein by reference in its entirety, International Application No. PCT/US2015/056669, filed on Oct. 21, 2015. 
    
    
     BACKGROUND 
     Field 
     The disclosed embodiments relate to radio frequency (RF) industrial power systems. More particularly, the disclosed embodiments relate to RF heating power systems, which are used in RF induction heating and melting applications, as well as plasma generation, corona discharge, and RF heating of hydrocarbons. 
     SUMMARY 
     The embodiments disclosed herein include a method to directly control radio frequency (RF) power of an induction heating or industrial RF power system, which enables control over output power by utilizing direct-digital RF power control combined with one or more techniques of fine-tune power control. Combining direct-digital RF power control with fine-tune power control allows full control of the RF power without requiring an intermediate full-power control stage in the system, such as a silicon-controlled rectifier (SCR) controller or DC-to-DC conversion stages, reduces the quantity of RF devices used in the system, and eliminates efficiency issues found in methods of directly controlling RF power. 
     A radio frequency inductive heating apparatus in accordance with the embodiments disclosed herein includes a control device, a plurality of radio frequency devices, a plurality of transformers, a resonant tank circuit, a heating element, and a first power supply, and in certain embodiments, a second power supply. Each of the plurality of radio frequency devices is selectively activated by the control device. In certain embodiments, each of the plurality of transformers includes a primary winding and a secondary winding, and each of the plurality of radio frequency devices is operatively coupled to the primary winding of one of the plurality of transformers. The secondary winding of each of the plurality of transformers is operatively coupled to the resonant tank circuit, and the heating element is operatively coupled to the resonant tank circuit. 
     In yet another embodiment, each of the plurality of radio frequency devices may be operatively coupled to each one of the plurality of transformers to generate an output current, and at least one of the plurality of radio frequency devices are selectively activated by the control device based on a power set signal and a feedback signal, the power set signal representing a desired output current associated with the resonant tank circuit, the feedback signal representing an electrical control parameter associated with the resonant tank circuit. 
     In yet another embodiment, the plurality of radio frequency devices may include a first radio frequency device and a second radio frequency device. The first radio frequency device is operatively coupled to the first power supply, and the second radio frequency device is operatively coupled to the second power supply. 
     The plurality of radio frequency devices may receive a radio frequency signal from the control device. The radio frequency signal may provide an operating frequency associated with the plurality of radio frequency devices, and the operating frequency may be based on a resonant frequency of the resonant tank circuit. In certain embodiments, the second power supply may be a variable power supply, and an output voltage associated with the first power supply may be greater than an output voltage associated with the second power supply. 
     In yet another embodiment, the at least one of the plurality of radio frequency devices may be selectively activated by the control device based on a power set signal and a feedback signal. The power set signal may represent a desired voltage associated with the resonant tank circuit, and the feedback signal may represent an electrical parameter associated with the resonant tank circuit. 
     In certain embodiments at least one of the plurality of radio frequency devices may include a full bridge inverter that receives a first phase signal and a second phase signal. A phase difference between the first phase signal and the second phase signal may be selectively adjusted to modify an output voltage associated with the at least one of the plurality of radio frequency devices. 
     The plurality of transformers may include a first transformer and a second transformer. The first transformer may include a first turns ratio, the second transformer may include a second turns ratio, and the first turns ratio may be different from the second turns ratio. The plurality of radio frequency devices may include a first radio frequency device and a plurality of second radio frequency devices. The plurality of transformers may include a first transformer and a plurality of second transformers. The first radio frequency device may be operatively coupled to the primary winding of the first transformer, and each of the plurality of second radio frequency devices may be operatively coupled to the primary winding of one of the plurality of second transformers. The plurality of second transformers may include a turns ratio equal to one of n:1 and (n−1):1, where n represents a quantity of second radio frequency devices. Each of the plurality of second transformers may include a turns ratio equal to 2 n :1, where n represents an increasing positive integer. 
     The radio frequency inductive heating apparatus may include a first power supply and a second power supply, and the plurality of second radio frequency devices may include a first radio frequency device and a second radio frequency device. The first radio frequency device may be operatively coupled to the first power supply, and the second radio frequency device may be operatively coupled to the second power supply. The second power supply may be a variable power supply, and an output voltage associated with the first power supply may be greater than an output voltage associated with the second power supply. An output voltage associated with the second power supply may be twice an output voltage associated with the first power supply. At least one of the first power supply and the second power supply may include a rectifier. 
     A maximum number of radio frequency devices that are used may be based on a value of the output current associated with one of the plurality of radio frequency devices and a maximum allowable operating current of one of the plurality of radio frequency devices. 
     A method of providing radio frequency inductive heating in accordance with the embodiments disclosed herein include activating a plurality of radio frequency devices selectively by a control device, coupling each of the plurality of radio frequency devices operatively to a primary winding of one of a plurality of transformers, coupling a secondary winding of each of the plurality of transformers operatively to a resonant tank circuit, coupling a heating element operatively to the resonant tank circuit, and coupling the plurality of radio frequency devices operatively to a first power supply, wherein each plurality of the radio frequency devices generate an output current, and at least one of the plurality of radio frequency devices is selectively activated by the control device based on a power set signal and a feedback signal, the power set signal representing a desired current associated with the resonant tank circuit, the feedback signal representing an electrical control parameter associated with the resonant tank circuit. 
     In certain embodiments, the method further includes coupling a first radio frequency device operatively to a first power supply, and coupling a second radio frequency device operatively to a second power supply. 
     The method may further include receiving a radio frequency signal from the control device by the plurality of radio frequency devices. The radio frequency signal may provide an operating frequency associated with the plurality of radio frequency devices, and the operating frequency may be based on a resonant frequency of the resonant tank circuit. The second radio frequency device is operatively coupled to the second power supply. The second power supply may be a variable power supply, and an output voltage associated with the first power supply may be greater than an output voltage associated with the second power supply. 
     The method may include activating the at least one of the plurality of radio frequency devices selectively by the control device based on a power set signal and a feedback signal. The power set signal may represent a desired voltage associated with the resonant tank circuit, and the feedback signal may represent an electrical parameter associated with the resonant tank circuit. At least one of the plurality of radio frequency devices may include a full bridge inverter, and the method may include receiving, by the full bridge inverter, a first phase signal and a second phase signal, and modifying a phase difference between the first phase signal and the second phase signal selectively to modify an output voltage associated with the at least one of the plurality of radio frequency devices. 
     The plurality of transformers may include a first transformer and a second transformer, and the first transformer may include a first turns ratio that is different from the second turns ratio of the second transformer. The plurality of radio frequency devices may include a first radio frequency device and a plurality of second radio frequency devices, and the plurality of transformers may include a first transformer and a plurality of second transformers. The method may include coupling the first radio frequency device operatively to a primary winding of the first transformer, and coupling each of the plurality of second radio frequency devices operatively to a primary winding of one of the plurality of second transformers. 
     Each of the plurality of second transformers may include a turns ratio equal to one of n:1 and (n−1):1, where n represents a quantity of second radio frequency devices. Each of the plurality of second transformers may include a turns ratio equal to 2 n :1, where n represents an increasing positive integer. The plurality of second radio frequency devices may include a first radio frequency device and a second radio frequency device, and the method may include coupling the first radio frequency device operatively to a first power supply, and coupling the second radio frequency device operatively to a second power supply. 
     The second power supply may be a variable power supply, and an output voltage associated with the first power supply may be greater than an output voltage associated with the second power supply. The output voltage associated with the second power supply may be twice an output voltage associated with the first power supply. At least one of the first power supply and the second power supply may include a rectifier. 
     A maximum number of radio frequency devices that are used may be based on a value of the output current associated with one of the plurality of radio frequency devices and a maximum allowable operating current of one of the plurality of radio frequency devices. 
     Other embodiments will become apparent from the following detailed description considered in conjunction with the accompanying drawings. It is to be understood, however, that the drawings are designed as an illustration only and not as a definition of the limits of any of the embodiments. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The following drawings are provided by way of example only and without limitation, wherein like reference numerals (when used) indicate corresponding elements throughout the several views, and wherein: 
         FIG. 1A  is a block diagram showing a first embodiment of a radio frequency (RF) heating system using direct-digital RF control and fine-tune power control; 
         FIG. 1B  is a block diagram of yet another embodiment of the radio frequency (RF) induction heating system using direct-digital RF control and fine-tune power control as shown in  FIG. 1A ; 
         FIG. 1C  is a block diagram of yet another embodiment of the radio frequency (RF) induction heating system using direct-digital RF control and fine-tune power control as shown in  FIG. 1A ; 
         FIG. 2  is a graph of output voltage as a function of a quantity of activated RF devices using direct-digital RF control and fine-tune power control in the RF heating system; 
         FIG. 3  is a graph of output power as a function of the quantity of activated RF devices using direct-digital RF control and fine-tune power control in the RF heating system; 
         FIG. 4A  is a diagram of a phase-shift circuit that provides fine-tune power control in the RF heating system; 
         FIG. 4B  shows phase control signals and an output power signal associated with the phase-shift circuit shown in  FIG. 4A ; 
         FIG. 5  is a graph of phase angle as a function of the quantity of activated RF devices in the RF heating system; 
         FIG. 6  is a graph of output voltage as a function of the quantity of activated RF devices in the RF heating system; 
         FIG. 7A  is a block diagram of a circuit utilizing a variable power supply (VPS) to provide fine-tune power control in the RF heating system; 
         FIG. 7B  is a graph of output voltage from the VPS as a function of a control signal; 
         FIG. 8  is a graph of output voltage from the VPS as a function of the quantity of activated RF devices in relation to use of the circuit shown in  FIG. 7A  in the RF heating system; 
         FIG. 9  is a graph of output voltage as a function of the quantity of activated RF devices in relation to use of the circuit shown in  FIG. 7A  in the RF heating system; 
         FIG. 10A  is a block diagram of a portion of the RF heating system showing a last RF device and transformer; 
         FIG. 10B  is a block diagram of trim RF devices and transformers that are substituted for the last RF device and transformer shown in  FIG. 10A  in the RF heating system; 
         FIG. 10C  is a graph of output voltage as a function of the quantity of activated RF devices in relation to use of the circuit shown in  FIG. 10B  in the RF heating system; 
         FIG. 11A  is a block diagram of trim RF devices with binary weighting substituted for the last RF device shown in  FIG. 10A ; 
         FIG. 11B  is a graph of output voltage as a function of the quantity of activated RF devices in relation to use of the circuit shown in  FIG. 11A  in the RF heating system; 
         FIG. 12  is a block diagram of a circuit utilizing a VPS to control the trim RF devices with binary weighting; 
         FIGS. 13A-B  are block diagrams of two embodiments that couple a current transformer (CT) sensor in the RF heating system; 
         FIG. 14  is a block diagram of a control device used in the RF heating system; 
         FIG. 15A  is a block diagram of trim RF devices with binary weighting substituted for the last RF device shown in  FIG. 10A ; 
         FIG. 15B  is a graph of output voltage as a function of the quantity and identity of activated RF devices in relation to utilization of the circuit shown in  FIG. 15A  in the RF heating system; 
         FIG. 16  is a block diagram showing another embodiment of the radio frequency (RF) heating system using direct-digital RF control and fine-tune power control; 
         FIG. 17  is a graph of output voltage of one of the RF devices as a function of time; 
         FIG. 18  is a graph of output voltage as a function of the quantity of activated RF devices; 
         FIG. 19  is a graph of the quantity of activated RF devices, output power, output voltage, and duty cycle as a function of time; and 
         FIG. 20  is a block diagram of at least a portion of an exemplary machine in the form of a computing system that performs methods according to one or more embodiments disclosed herein. 
     
    
    
     It is to be appreciated that elements in the figures are illustrated for simplicity and clarity. Common but well-understood elements that are useful or necessary in a commercially feasible embodiment are not shown in order to facilitate a less hindered view of the illustrated embodiments. 
     DETAILED DESCRIPTION 
     Methods to control radio frequency (RF) power are used in the field of RF induction heating and related RF heating fields. These methods use several power conversion steps to provide RF power, which include rectifying alternating current (AC) line power to a direct current (DC) voltage. These methods include monitoring the DC voltage or current to control power delivered to an RF stage, which is referred to as a gain stage or pre-regulator. Most of these methods include an intermediate gain stage that uses a full-power rated stage to vary the DC voltage or current to control the RF stage. 
     The embodiments disclosed herein are directed to a radio frequency induction heating apparatus  10  shown in  FIG. 1A  that uses direct-digital radio frequency (RF) power control and fine-tune power control. An alternating current (AC) power source  12  is applied to a direct current (DC) central rectifier  14 , which results in a DC voltage V+ applied to a plurality of RF devices ( 16 ,  34 ). As an alternative to the central rectifier  14  for all of the RF devices ( 16 ,  34 ), a dedicated rectifier  20  may be coupled to each of the RF devices ( 16 ,  34 ), as shown by a dashed line  23  to the RF device  16 . 
     Each of the plurality of RF devices  16 ,  34  is coupled to a transformer  18 ,  40 , which has a secondary winding  33 ,  41  connected in series to generate an output voltage Vs. The output voltage Vs is used to drive a resonant tank circuit  22  that includes an induction heating coil  24  or corona discharge capacitor  26 . A feedback signal  28  is used to monitor the output voltage Vs, current, and/or frequency at the input of the resonant tank circuit  22 . 
     A control device  30  provides on/off control signals  11 ,  13 ,  15  to each RF device  16 ,  34 , as well as RF frequency signals  17 ,  19 ,  21  to drive the RF devices  16 ,  34 . The RF frequency signals  17 ,  19 ,  21  include two phases that are phase shifted with respect to each other. The frequency of the RF frequency signals are determined by a resonant frequency of the resonant tank circuit  22 . The control device  30  provides closed-loop power control based on a power setting signal  32  and the feedback signal  28 . A variable power supply (VPS)  36  optionally provides a variable controlled DC voltage  37  to the last RF device  34 . In some embodiments, the VPS  36  is not used or removed from the apparatus  10 , in which case the voltage is provided by the output voltage Vs to each of the RF devices  16 ,  34 , as shown by bypass connection  38 . 
     The last RF device  34  may be implemented differently than the remaining RF devices  16  and may have a unique transformer  40  with a different turns ratio than the remaining transformers  18 . The last RF device  34  can include a plurality of lower power trim RF devices, each of which may have a transformer with a ratio that increases an effective RF device number associated with the apparatus  10 . 
     The control device  30  controls the output power by using a combination of direct-digital RF power control for each RF device  16 ,  34  combined with one or more fine-tune power control methods including, but not limited to, phase-shift control and a variable power supply applied to the last power device  34 . This combination of direct-digital power control and fine-tune power control offers an advantageous solution to controlling the power of the resulting induction heating system. 
     The embodiments disclosed herein directly control one or more RF devices to implement closed-loop power control rather than amplification or modulation of an external signal. In addition, the disclosed embodiments use fine-tune power control with direct-digital power control in induction heating applications. These features greatly reduce the overall number of RF devices used to achieve a desired power, which reduces system cost and improves overall efficiency. 
     The disclosed embodiments apply a direct-digital RF power control scheme combined with one or more fine-tune power control techniques to manage RF power without requiring additional gain stages. The disclosed embodiments allow operation with a variable RF frequency that tracks a resonance frequency of the resonant tank circuit  22 . The feedback signal  28  provides a sample of the current at the resonance frequency of the resonant tank circuit  22 . The voltage of the current sample is estimated by the control device  30  or can be sampled directly from the resonant tank circuit  22 . 
     Tracking variable RF frequency uses the voltage of the resonant tank circuit  22 , which is derived by the control device  30 , and/or the current derived from the feedback signal  28  using a phased-locked loop (PLL) circuit, which may be implemented digitally, to track the resonance frequency of the resonant tank circuit  22 . Thus, the disclosed embodiments achieve improved power control with fewer power stages and higher efficiency than conventional techniques. 
     As discussed above,  FIG. 1A  shows a radio frequency (RF) heating system  10  using a direct-digital RF power control method combined with one or more fine-tune control methods disclosed herein. The RF heating system  10  includes an AC power source  12  coupled to a DC rectifier  14 , which provides a DC voltage V+ to the RF devices  16 ,  34 . As an alternative to using one central DC rectifier  14  for each of the RF devices  16 ,  34 , dedicated rectifiers  20 , each of which may be coupled to one of the RF devices  16 ,  34  as shown by a dashed line  23  connected to the first RF device  16 , may be substituted for the central rectifier  14 . 
     Each of the plurality of RF devices  16 ,  34  is coupled to a primary winding  31 ,  39  of one of a plurality of transformers  18 ,  40 . Secondary windings  33 ,  41  of the transformers  18 ,  40  are connected in series to provide the output voltage Vs. The output voltage Vs is used to drive a resonant tank circuit  22 , which includes a work component, an induction heating coil  24 , corona discharge capacitor  26 , and/or the like. The resonant tank circuit  22  includes a series combination, parallel combination, and/or an alternative combination of resonant components. The feedback signal  28  is used to provide an alternating current sample to the control device  30 , from which the voltage Vs, current amplitude, phase, and/or frequency is estimated by the control device  30 . 
     A current transformer (CT) sensor may be used to obtain the feedback signal  28 . CT sensors are used for measuring alternating electric current. CT sensors, together with voltage (or potential) transformers, are known as instrument transformers. If the current in a circuit is too high to apply directly to measurement instrumentation, the CT sensor provides a reduced current accurately proportional to the current in the circuit, which can be conveniently connected to measuring and recording instruments, such as the control device  30 . The CT sensor isolates measurement instruments from a potentially high voltage in the circuit to be monitored. 
     Like any other transformer, the CT sensor has a primary winding, a magnetic core, and a secondary winding. Alternating current in the primary winding generates an alternating magnetic field in the core, which then induces an alternating current in the secondary winding. An essential objective of current transformer design is to ensure that the primary and secondary circuits are efficiently coupled so that the secondary current is linearly proportional to the primary current. The CT sensor includes a length of wire wrapped around a silicon steel ring applied to the circuit being measured. The primary winding includes, for example, a single turn of a conductor, with a secondary winding having multiple turns. 
       FIGS. 13A and 13B  show two embodiments of the resonant tank circuit  226 ,  228 . In  FIG. 13A , the resonant tank circuit  226  is coupled to approximately twenty (20) feet of flexible lines  230  since the resonant tank circuit  226  is mobile. The flexible lines  230  are then coupled to the secondary windings  33 ,  41  of the transformers  18 ,  40 . If the flexible lines  230  are configured to handle  4000 A, the flexible lines  230  may become bulky, in which case water cooling is used.  FIG. 13B  shows another embodiment in which the flexible lines  230  are coupled to a transformer  232 , which is then coupled to the secondary windings  33 ,  41  of the transformers  18 ,  40 . In this embodiment, the flexible lines  230  do not carry as much current as in the embodiment shown in  FIG. 13A . 
     In  FIG. 1A , the control device  30  provides a control signal  11 ,  13 ,  15  and an RF frequency signal  17 ,  19 ,  21  to each RF device  16 ,  34 . The RF frequency signals  17 ,  19 ,  21  include two (2) channels, phase A and phase B, which are phase shifted with respect to each other. The frequency of the RF frequency signals  17 ,  19 ,  21  is determined by the resonance frequency of the resonant tank circuit  22  utilizing a variety of techniques, such as analog, digital, and/or hybrid PLL circuits. The control device  30  provides closed-loop power control based on the power setting signal  32  and the feedback signal  28 . The power setting signal  32  is obtained from a user interface that allows setting of the power level. The power setting signal  32  can be an analog signal or a digital signal, which is provided to the control device  30  to provide a power reference value. 
     An embodiment that provides a variable controlled DC voltage to the last RF device  34  uses a variable power supply  36  as shown in  FIG. 1A . In some embodiments, the variable power supply  36  is not used, which is represented by the bypass connection  38 . 
     The last RF device  34  may be implemented differently than the remaining RF devices  16  and can have a different transformer  40  with a different turns ratio than the remaining transformers  18 . The last RF device  34  can also include lower power or trim devices, each of which has a different transformer ratio to further increase an effective RF device number. For multiple trim RF devices with different ratios, the effective RF number is equal to Neff=2 n *(N+1)−1, where N represents the number of RF devices  16  (i.e., N full steps) and n is the number of trim RF devices included in the last RF device  34  (i.e., n binary steps). An embodiment in which the last RF device  34  is substantially the same size as the remaining RF devices  16 , and the VPS  36  is set to half of Vs, would result in Neff=2N. 
     The control device  30  controls output power of the RF heating system  10  by utilizing a combination of independent, direct-digital RF power control to each RF device  16 ,  34  combined with one or more fine-tune power control methods including, but not limited to, phase-shift control and trim variable power supply control of the last power device  34 . This combination of direct-digital power control and fine-tune power control provides a substantially improved solution for controlling the power of the RF heating system. 
     The RF devices or amplifiers  16 ,  34  may include a plurality of solid-state technologies, such as silicon controlled rectifiers (SCR), insulated-gate bipolar transistors (IGBT), and metal-oxide semiconductor field effect transistors (MOSFET) configured in a variety of topologies including, but not limited to, a full bridge amplifier, half bridge amplifier, class A amplifier, class AB amplifier, class C amplifier, and/or class D amplifier circuit. The RF devices  16 ,  34  are typically implemented as a class D, full bridge amplifier, but other topologies can also be used. The transistors used can vary widely from metal oxide semiconductor field effect transistor (MOSFET) or insulated gate bipolar transistor (IGBT) designs to newer switching semiconductors. A typical device used is the HiperFET power MOSFET Q3-Class, part number IXFB44N100Q3, which is available from IXYS Corporation, 1590 Buckeye Drive, Milpitas, Calif. 95035, 408 457 9000, www.ixys.com. 
     The transformers  18 ,  40  are typically implemented with a M:1 or M:0.5 ratio having multiple primary windings and a single-turn or half-turn secondary winding. Ratios can vary to allow for proper impedance matching of the RF device  16 ,  34  to the output circuit or resonant tank circuit  22 . Typical ratios include 2:1 to 40:1. 
     The control device  30  provides the control signals  11 ,  13 ,  15  and RF frequency signals  17 ,  19 ,  21  independently to each RF device  16 ,  34 , each of which then generates an RF drive voltage that is applied to the primary windings  31 ,  39  of the corresponding transformer  18 ,  40 . The control device  30  provides a closed-loop power control system by comparing the feedback signal  28  with the power setting signal  32  to determine the level of power required. The control device  30  operates by selectively activating power devices  16 ,  34  using the control signals  11 ,  13 ,  15  as needed. The signals  17 ,  19 ,  21  can be the same or different signals. However, the RF devices  16 ,  34  are controlled independently with the on/off control signals  11 ,  13 ,  15 . 
     A block diagram of the control device  30  as shown in  FIGS. 1A, 1B, and 1C , is further illustrated in  FIG. 14 . The control device  30  includes a control panel  234 , external interface  236 , non-volatile random access memory (NV RAM)  238 , microprocessors  240 , flash memory  242 , light emitting diodes  244 , field programmable gate arrays (FPGA)  246 , oscillators  248 , power-on-reset circuits (POR)  250 , buffers  252 , capacitors  254 , resistors  256 , low pass filters  258 , external connectors  260 , buffers  262 , drivers  264 , analog-to-digital converter  266 , transformer  268 , operational amplifiers  270 , simple switcher  272 , and inverters  274 . 
     Another embodiment of the radio frequency induction heating apparatus  10  shown in  FIG. 1A , is presented in  FIG. 1B . The topology in  FIG. 1B  is similar to the radio frequency induction heating apparatus  10  of  FIG. 1A , but differs with respect to the transformers secondary coils connections. Specifically, each of the plurality of RF devices  16  is coupled to a transformer  18 , which has a secondary winding  33  connected in parallel to generate an output current Is. The output current Is is used to drive a resonant tank circuit  22  that includes an induction heating coil  24  or corona discharge capacitor  26 . A feedback signal Cs  28  is used to monitor the output current Is, voltage, and/or frequency at the input of the resonant tank circuit  22 . 
     Yet another embodiment of the radio frequency induction heating apparatus shown in  FIG. 1B  is shown in  FIG. 1C . In  FIG. 1C , each of the outputs of RF devices  16  are connected directly in parallel to the common load  22 . In such embodiment shown in  FIG. 1C , each of the transformers  18  shown in  FIG. 1B  are omitted. 
     Referring back to  FIG. 1B  and the embodiment shown in  FIG. 1C , the output current Is, is in both of the embodiments, a result of the sum of each operating RF device  16  output current (or alternatively, the respective transformers secondary windings  33 ,  41  currents if present, as shown in the embodiment of  FIG. 1B ). Thus, current control can be applied in either embodiment  FIG. 1B  or  FIG. 1C . The number N of operating RF devices  16  is determined by the value of the current Is in accordance with equation (1) as defined hereinbelow:
 
 Is/N≤I 1max  (1)
 
wherein I1max is the maximum allowable operating current of one RF device  16 .
 
     The current control algorithm is similar to the voltage control algorithm defined hereinbelow (for example, indicated in equations (4)-(8)).  FIG. 2 ,  FIG. 3 ,  FIG. 6  and  FIG. 9  can be used for visualization of this current control algorithm, using Is instead of Vs in the Y-axes labels where applicable. The embodiments presented in  FIG. 13A  and  FIG. 13B  of tank configurations are both compatible for operation with the embodiment of  FIGS. 1B  and C. The embodiments shown in  FIG. 17  and  FIG. 18  control methods are also applicable in the embodiments of  FIGS. 1B and 1C . 
       FIG. 2  shows the output voltage Vs, which is an AC voltage, of the RF heating system  10  as a function of the quantity of RF power devices  16 ,  34  that have been activated. Each of the RF devices  16  provides a substantially equivalent output voltage step. By selectively activating individual RF devices  16 ,  34 , the output voltage Vs is stepwise modified. The output power, as shown in  FIG. 3 , varies in proportion to the square of the output voltage Vs. When a modification in power is required as determined by the control device  30 , the correct identity and/or quantity of RF devices  16 ,  34  is activated. 
     Power control using direct-digital RF control of the RF devices  16 ,  34  is implemented with the plurality of RF devices  16 ,  34  to produce a sufficiently fine power control. As shown in  FIG. 3 , using five (5) RF devices, power steps  44  are relatively coarse. However, using a larger quantity of RF devices may be too costly for many applications. Accordingly, the disclosed embodiments use a variety of techniques to combine direct-digital RF power control of RF devices with one or more methods of fine-tune power control to achieve a finer resolution of power control. 
     One embodiment for fine-tune power control is shown in  FIGS. 4A and 4B . By applying phase shift control of the RF devices  16 ,  34 , fine-tune power control can be achieved without requiring additional components. A full bridge inverter  46  shown in  FIG. 4A  is implemented as the RF device topology in this embodiment, which could also be implemented using alternative inverter topologies. The full bridge inverter  46 , which can be implemented as the last RF device  34  shown in  FIG. 1A , includes field effect transistors  45 ,  47  connected in series between a positive voltage V+ and ground, and field effect transistors  49 ,  51  connected in series between the positive voltage V+ and ground. The primary winding  59  of a transformer  53 , which can be the last transformer  40  shown in  FIG. 1A , is connected in series across a node disposed between field effect transistors  45 ,  47  and a node disposed between field effect transistors  49 ,  51 . The voltage Vs is obtained across the secondary winding  61  of the transformer  53 .  FIG. 4B  shows waveforms  48  including a phase A signal  53  from the control device  30 , which is coupled to gate terminals of field effect transistors  45 ,  51 , a phase B signal  55  from the control device  30 , which is coupled to gate terminals of field effect transistors  47 ,  49 , and a power signal  57 , which is provided across the secondary winding  59  of the transformer  53 . The phase A signal  53  and phase B signal  55  are combined to provide a magnitude of the power signal  57 . The phase A signal  53  and phase B signal  55  are modulated in phase with respect to each other to control the transfer of power from the output of the RF device  46 . 
     When combining direct-digital RF control with phase-shift control, as an implementation of fine-tune power control, the value of the phase shift depends on the quantity of RF devices being activated. Power is a function of the square of the output voltage Vs 2 . In a system having five (5) RF devices, if one (1) RF device is activated, the total power is approximately 4%. With one RF device activated, 100% of the output of that RF device is controlled, so that the phase is shifted by 0-180 degrees. If a second RF device is added, the total power is increased to 18%, but only the incremental power difference, which is approximately 12%, is controlled with the phase control. That is, approximately 70% of the total power is controlled with the phase control or about 0-120 degrees. Once 120 degrees, or 4% of the total power, is obtained, a single RF device can be used. 
     As additional RF devices are used, the change in power that is controlled decreases as a ratio of the total power at a given number of RF devices, and thus the phase angle used to control those RF devices decreases as a result. As five (5) RF devices are activated, 45% of the power range is controlled with phase control, or only 78 degrees of phase shift. 
     As additional RF devices are activated, this difference decreases with each incremental step.  FIG. 5  shows a relative phase shift angle as a function of the quantity of RF devices activated. The phase angle range  50  decreases as additional RF devices are activated. This feature allows for a significant reduction in power loss by reducing the phase angle range at higher power, which represents a substantial improvement over conventional techniques. 
       FIG. 6  shows a smoothed output voltage Vs achieved by using the direct-digital RF control with phase-shift control. As shown in  FIG. 6 , the large voltage steps shown in  FIG. 3  have been eliminated without using additional RF devices. Thus, this embodiment enables the use of fewer RF devices to achieve finer power control when compared with embodiments in which only direct-digital RF control is used. Accordingly, in this embodiment, losses experienced when switching large amounts of RF power with high phase angles is avoided by using phase-shift control. 
     Another fine-tune power control method is shown in  FIGS. 7A and 7B . This method utilizes the variable power supply (VPS)  54  to control a level of DC voltage applied to the RF device  56 , which can be implemented as the last RF device  40  shown in  FIG. 1A . The VPS  54  can be implemented using any topology that allows for control of the output DC voltage Vf. The control device  30  shown in  FIG. 1A  controls the VPS  54  directly, thereby controlling the output voltage of the RF device  56 , as shown in a graph  58  of variable voltage Vf as a function of fine-tune control voltage  59  provided to the VPS  54  in  FIG. 7B . 
       FIG. 8  shows a graph  69  of the variable voltage Vf applied to the RF device  56  shown in  FIG. 7A  as a function of the quantity of RF devices activated. This embodiment provides a smooth output voltage Vs ramp shown in  FIG. 9  as more or fewer RF devices are activated. This embodiment also provides an improvement over other techniques by requiring fewer RF devices to achieve greater resolution in power control. The VPS  54  is rated as 1/N times the overall power rating of the RF devices, where N represents a total quantity of RF devices, which provides a significant reduction when compared with other implementations that do not combine direct-digital RF control with the variable power supply control. Reducing the size and power of the VPS  54  improves overall system efficiency, reduces system cost, and reduces the quantity of stored energy components in the VPS  54 . 
       FIG. 10B  shows an embodiment that can be used alone or in combination with the disclosed embodiments. The RF device  62  shown in  FIG. 10A , which can be the last RF device  40  shown in  FIG. 1A , is replaced with a plurality of n trim RF devices  66  shown in  FIG. 10B . Each RF device  64  is coupled to a corresponding transformer  68 , which provides 1/n or 1/(n−1) times the voltage of the RF device  62  shown in  FIG. 10A . The RF devices  66  are referred to as trim devices and shown as four devices in  FIG. 10B , but can be implemented using any quantity of trim devices. The trim devices  66  are used with larger RF devices  16  shown in  FIG. 1A  to multiply the effective quantity of RF devices. The effective quantity of direct-digital RF devices Neff is provided by Neff=(N+1)*n, where N represents the quantity of RF devices  16  (i.e., N full steps) and n represents the quantity of trim RF devices in the last RF device  34  (i.e., n steps). Thus, for three (3) large total steps with four (4) trim steps, the effective quantity of RF devices Neff is equal to sixteen (16). This is shown in  FIG. 10C , which combines smaller and larger steps to increase the effective number of steps  70 . This embodiment provides the advantage of improving power control by reducing the quantity of full-size steps and increasing the quantity of smaller steps. If the step size is adequate for power control, the step can be utilized by itself or combined with other fine-tune power control methods disclosed herein. 
       FIG. 11A  shows another embodiment, in which the trim RF devices  74  have a transformer ratio that is binary-weighted, as shown in a four-device configuration  72 . Thus, the transformer ratios are provided as 2:1, 4:1, 8:1, and 16:1 in this embodiment or, in general, 2 n :1, where n is an increasing positive integer. The trim RF devices  74  are combined in all possible combinations to further increase the effective quantity of RF devices as shown along the horizontal axis in  FIG. 11B . For the same three (3) RF devices and four (4) trim RF devices configuration, an effective quantity of RF devices of forty-five (45) is achieved without requiring additional RF devices. More generally, Neff=2 n *(N+1)−1. Another embodiment includes the binary-weighted configuration depending on tradeoffs between RF device cost and power control resolution. When combined with phase-shift control shown in  FIG. 4A , this embodiment has the effect of reducing the phase angle range by the effective quantity of RF devices Neff, thereby further improving system efficiency. When combined with the trim variable power supply embodiment shown in  FIG. 7A , this embodiment enables the use of a lower power VPS, thus further reducing the cost of implementation. The binary-weighted embodiment also uses lower-power, and thus smaller, stored energy components. 
       FIG. 12  shows another embodiment that utilizes a VPS  78  to provide a fixed voltage equal to half the voltage V+, which feeds one of the trim RF devices  80  with half of the voltage V+. This embodiment enables both transformer ratios associated with the trim RF devices  80  to be the same, which increases RF device reuse yet maintains the same effective quantity of RF devices (Neff=2 n *(N+1)−1) as in the binary-weighted embodiment. 
       FIG. 16  shows a block diagram of another embodiment  286  of the radio frequency (RF) heating system using direct-digital RF control and fine-tune power control. In this embodiment  286 , a single trim RF device  288  is used that is substantially equivalent to the remaining RF devices  290  with the same transformer ratios for each of the transformers  292 . However, the trim RF device  288  is provided with half of the input voltage that is provided to the remaining RF devices  290 . This embodiment effectively doubles the RF device count. For example, using this embodiment with five (5) RF devices, the equivalent resolution of a nine (9) RF device system is achieved. This embodiment includes a dedicated rectifier  294  that provides ac-to-dc conversion for each RF device  290 , which can also be combined with phase-shift control, as discussed hereinabove. 
     These embodiments can be combined even further by adding additional devices with weighted transformer ratios and/or additional voltages to increase the effective device number. The input voltage can be adjusted to a binary ratio of the input voltage in a 1:1 ratio, which is the same ratio as that used with the RF devices  290 , or the transformer ratio can be adjusted to a binary weighted ratio of the RF devices  290 . The output voltage of each is equal to the input voltage divided by the ratio, so either the input voltage or the transformer ratio can be adjusted to obtain the same net effect, as shown in the configuration  280  of  FIG. 15A  and the corresponding graph  282  of output voltage Vs as a function of the quantity of activated RF devices shown in  FIG. 15B . This embodiment can also be combined with phase-shift control to achieve greater control resolution by adding the phase-shift control to the RF signal as discussed above. 
     Another embodiment, which is also based on  FIG. 16 , is directed to fine-tune control with direct digital control and includes pulse controlling the on/off control signals, and thus the RF duty cycle of the RF output, for one or more RF devices  290 . This embodiment includes modifying the duty cycle of the last RF device  288  for fine-tune control, however, the duty cycle of the on/off control signals can be modified prior to application to any one or more of the RF devices  288 ,  290 . 
       FIG. 17  is a graph of the output voltage of one of the RF devices  288 ,  290 , in which the output voltage is a pulsed RF signal  300  due to the duty cycle of the on/off control signal being applied to one of the RF devices  288 ,  290 . The duty cycle is equal to an on-time T of the pulsed RF signal  300  divided by a duration of a period P of the pulsed RF signal  300  or
 
Duty Cycle= T/P.   (2)
 
     A minimum on-time of the RF device  288 ,  290  is provided by
 
 T min=1/ f,   (3)
 
where f is the operating frequency. This embodiment enables on/off control of each RF device  288 ,  290  on a cycle-by-cycle basis, which allows for control of the duty cycle of any or all RF devices  288 ,  290  as long as Tmin is equal to or greater than one period of the pulsed RF signal  300 . If Tmin is less than one period of the pulsed RF signal  300 , switching losses of the RF device  288 ,  290  are increased. The period P should be greater than or equal to ten times the on-time T to enable reasonable resolution of power control. Increasing the period P to one hundred times the on-time or greater would enable more accurate control of the voltage.
 
     There is a tradeoff between resolution of the voltage control and response time of the power control. The control bandwidth BW is limited by the following equation and should be equal to or greater than the overall control bandwidth of the control system as provided by the following equation:
 
BW= f*T min/ P.   (4)
 
     In order to control the output voltage Vs, the duty cycle and quantity of activated RF devices  288 ,  290  are controlled. In the graph of  FIG. 18 , which shows the output voltage as a function of the quantity of activated RF devices, the number of steps and the duty cycle of the last RF device are controlled. The output voltage Vs will have some ripple based on the duty cycle of the pulsed RF signal  300 . In heating applications, a key parameter is the average power transferred to an item to be heated. The impulse effect on the output voltage Vs is averaged, and thus will not affect heating to a great degree. 
     The output voltage Vs and output power Pout are determined from the following equations:
 
 Vs =( N− 1)* V +duty cycle* V,   (5)
 
 P out=(( N− 1)* V +duty cycle* V ) 2   *R , and  (6)
 
 P out= Vs   2   *R,   (7)
 
where N represents the quantity of activated RF devices (N=1 to Neff), V represents a secondary voltage associated with the lowest power RF device  288 ,  290 , and R represents a resistance of the heating element  24 ,  26 . As discussed above, the effective RF device number Neff is given by the following equation:
 
 N eff=2 n *( N+ 1)−1,  (8)
 
where n represents a quantity of binary steps, and N represents a quantity of full steps.
 
       FIG. 19  shows graphs of the quantity of activated RF devices  310 , output power Pout  312 , output voltage Vs  314 , and duty cycle  316  as a function of time. This embodiment includes four (4) full RF devices, one (1) trim RF device, and an effective RF device number Neff equal to nine (9). 
     Any and all control methods disclosed herein may be combined together and implemented using the RF heating system. For different applications and power sizes, different configurations of RF devices and fine-tune power control methods may be used. 
     One or more embodiments, or elements thereof, can be implemented in the form of an apparatus including a storage device or memory, and at least one processing device or processor that is coupled to the memory and operative to perform a method according to one or more embodiments. 
     One or more embodiments disclosed herein, or a portion thereof, make use of software running on a computer or workstation. By way of example, only and without limitation,  FIG. 20  is a block diagram of an embodiment of a machine in the form of a computing system  200 , within which is a set of instructions  202  that, when executed, cause the machine to perform any one or more of the methodologies according to embodiments disclosed herein. In one or more embodiments, the machine operates as a standalone device; in one or more other embodiments, the machine is connected (e.g., via a network  222 ) to other machines. In a networked implementation, the machine operates in the capacity of a server or a client user machine in a server-client user network environment. Exemplary implementations of the machine as contemplated by embodiments disclosed herein include, but are not limited to, a server computer, client user computer, personal computer (PC), tablet PC, personal digital assistant (PDA), cellular telephone, mobile device, palmtop computer, laptop computer, desktop computer, communication device, personal trusted device, web appliance, network router, switch or bridge, or any machine capable of executing a set of instructions (sequential or otherwise) that specify actions to be taken by that machine. 
     The computing system  200  includes a processing device(s)  204  (e.g., a central processing unit (CPU), a graphics processing unit (GPU), or both), program memory device(s)  206 , and data memory device(s)  208 , which communicate with each other via a bus  210 . The computing system  200  further includes display device(s)  212  (e.g., liquid crystal display (LCD), flat panel, solid state display, or cathode ray tube (CRT)). The computing system  200  includes input device(s)  214  (e.g., a keyboard), cursor control device(s)  216  (e.g., a mouse), disk drive unit(s)  218 , signal generation device(s)  220  (e.g., a speaker or remote control), and network interface device(s)  224 , operatively coupled together, and/or with other functional blocks, via bus  210 . 
     The disk drive unit(s)  218  includes machine-readable medium(s)  226 , on which is stored one or more sets of instructions  202  (e.g., software) embodying any one or more of the methodologies or functions herein, including those methods illustrated herein. The instructions  202  may also reside, completely or at least partially, within the program memory device(s)  206 , the data memory device(s)  208 , and/or the processing device(s)  204  during execution thereof by the computing system  200 . The program memory device(s)  206  and the processing device(s)  204  also constitute machine-readable media. Dedicated hardware implementations, such as but not limited to ASICs, programmable logic arrays, and other hardware devices can likewise be constructed to implement methods described herein. Applications that include the apparatus and systems of various embodiments broadly comprise a variety of electronic and computer systems. Some embodiments implement functions in two or more specific interconnected hardware modules or devices with related control and data signals communicated between and through the modules, or as portions of an ASIC. Thus, the example system is applicable to software, firmware, and/or hardware implementations. 
     The term “processing device” as used herein is intended to include any processor, such as, for example, one that includes a CPU (central processing unit) and/or other forms of processing circuitry. Further, the term “processing device” may refer to more than one individual processor. The term “memory” is intended to include memory associated with a processor or CPU, such as, for example, RAM (random access memory), ROM (read only memory), a fixed memory device (for example, hard drive), a removable memory device (for example, diskette), a flash memory and the like. In addition, the display device(s)  212 , input device(s)  214 , cursor control device(s)  216 , signal generation device(s)  220 , etc., can be collectively referred to as an “input/output interface,” and is intended to include one or more mechanisms for inputting data to the processing device(s)  204 , and one or more mechanisms for providing results associated with the processing device(s). Input/output or I/O devices (including but not limited to keyboards (e.g., alpha-numeric input device(s)  214 , display device(s)  212 , and the like) can be coupled to the system either directly (such as via bus  210 ) or through intervening input/output controllers (omitted for clarity). 
     In an integrated circuit implementation of one or more embodiments, multiple identical die are typically fabricated in a repeated pattern on a surface of a semiconductor wafer. Each such die may include a device described herein, and may include other structures and/or circuits. The individual dies are cut or diced from the wafer, then packaged as integrated circuits. One skilled in the art would know how to dice wafers and package die to produce integrated circuits. Any of the exemplary circuits or method illustrated in the accompanying figures, or portions thereof, may be part of an integrated circuit. Integrated circuits so manufactured are considered part of the disclosed embodiments. 
     An integrated circuit in accordance with the disclosed embodiments can be employed in essentially any application and/or electronic system in which buffers are utilized. Suitable systems for implementing one or more embodiments include, but are not limited, to personal computers, interface devices (e.g., interface networks, high-speed memory interfaces (e.g., DDR3, DDR4), etc.), data storage systems (e.g., RAID system), data servers, etc. Systems incorporating such integrated circuits are considered part of the disclosed embodiments. Given the teachings provided herein, one of ordinary skill in the art will be able to contemplate other implementations and applications. 
     In accordance with various embodiments, the methods, functions or logic described herein is implemented as one or more software programs running on a computer processor. Dedicated hardware implementations including, but not limited to, application specific integrated circuits, programmable logic arrays and other hardware devices can likewise be constructed to implement the methods described herein. Further, alternative software implementations including, but not limited to, distributed processing or component/object distributed processing, parallel processing, or virtual machine processing can also be constructed to implement the methods, functions or logic described herein. 
     The embodiment contemplates a machine-readable medium or computer-readable medium containing instructions  202 , or that which receives and executes instructions  202  from a propagated signal so that a device connected to a network environment  222  can send or receive voice, video or data, and to communicate over the network  222  using the instructions  202 . The instructions  202  are further transmitted or received over the network  222  via the network interface device(s)  224 . The machine-readable medium also contains a data structure for storing data useful in providing a functional relationship between the data and a machine or computer in an illustrative embodiment of the systems and methods herein. 
     While the machine-readable medium  202  is shown in an example embodiment to be a single medium, the term “machine-readable medium” should be taken to include a single medium or multiple media (e.g., a centralized or distributed database, and/or associated caches and servers) that store the one or more sets of instructions. The term “machine-readable medium” shall also be taken to include any medium that is capable of storing, encoding, or carrying a set of instructions for execution by the machine and that cause the machine to perform anyone or more of the methodologies of the embodiment. The term “machine-readable medium” shall accordingly be taken to include, but not be limited to: solid-state memory (e.g., solid-state drive (SSD), flash memory, etc.); read-only memory (ROM), or other non-volatile memory; random access memory (RAM), or other re-writable (volatile) memory; magneto-optical or optical medium, such as a disk or tape; and/or a digital file attachment to e-mail or other self-contained information archive or set of archives is considered a distribution medium equivalent to a tangible storage medium. Accordingly, the embodiment is considered to include any one or more of a tangible machine-readable medium or a tangible distribution medium, as listed herein and including art-recognized equivalents and successor media, in which the software implementations herein are stored. 
     It should also be noted that software, which implements the methods, functions and/or logic herein, are optionally stored on a tangible storage medium, such as: a magnetic medium, such as a disk or tape; a magneto-optical or optical medium, such as a disk; or a solid state medium, such as a memory automobile or other package that houses one or more read-only (non-volatile) memories, random access memories, or other re-writable (volatile) memories. A digital file attachment to e-mail or other self-contained information archive or set of archives is considered a distribution medium equivalent to a tangible storage medium. Accordingly, the disclosure is considered to include a tangible storage medium or distribution medium as listed herein and other equivalents and successor media, in which the software implementations herein are stored. 
     Although the specification describes components and functions implemented in the embodiments with reference to particular standards and protocols, the embodiment are not limited to such standards and protocols. 
     The illustrations of embodiments described herein are intended to provide a general understanding of the structure of various embodiments, and they are not intended to serve as a complete description of all the elements and features of apparatus and systems that might make use of the structures described herein. Many other embodiments will be apparent to those of skill in the art upon reviewing the above description. Other embodiments are utilized and derived therefrom, such that structural and logical substitutions and changes are made without departing from the scope of this disclosure. Figures are also merely representational and are not drawn to scale. Certain proportions thereof are exaggerated, while others are decreased. Accordingly, the specification and drawings are to be regarded in an illustrative rather than a restrictive sense. 
     Such embodiments are referred to herein, individually and/or collectively, by the term “embodiment” merely for convenience and without intending to voluntarily limit the scope of this application to any single embodiment or inventive concept if more than one is in fact shown. Thus, although specific embodiments have been illustrated and described herein, it should be appreciated that any arrangement calculated to achieve the same purpose are substituted for the specific embodiments shown. This disclosure is intended to cover any and all adaptations or variations of various embodiments. Combinations of the above embodiments, and other embodiments not specifically described herein, will be apparent to those of skill in the art upon reviewing the above description. 
     In the foregoing description of the embodiments, various features are grouped together in a single embodiment for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting that the claimed embodiments have more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive subject matter lies in less than all features of a single embodiment. Thus the following claims are hereby incorporated into the detailed description, with each claim standing on its own as a separate example embodiment. 
     The abstract is provided to comply with 37 C.F.R. § 1.72(b), which requires an abstract that will allow the reader to quickly ascertain the nature of the technical disclosure. It is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the claims. In addition, in the foregoing Detailed Description, it can be seen that various features are grouped together in a single embodiment for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that the claimed embodiments require more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive subject matter lies in less than all features of a single embodiment. Thus the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as separately claimed subject matter. 
     Although specific example embodiments have been described, it will be evident that various modifications and changes are made to these embodiments without departing from the broader scope of the inventive subject matter described herein. Accordingly, the specification and drawings are to be regarded in an illustrative rather than a restrictive sense. The accompanying drawings that form a part hereof, show by way of illustration, and without limitation, specific embodiments in which the subject matter are practiced. The embodiments illustrated are described in sufficient detail to enable those skilled in the art to practice the teachings herein. Other embodiments are utilized and derived therefrom, such that structural and logical substitutions and changes are made without departing from the scope of this disclosure. This Detailed Description, therefore, is not to be taken in a limiting sense, and the scope of various embodiments is defined only by the appended claims, along with the full range of equivalents to which such claims are entitled. 
     Given the teachings provided herein, one of ordinary skill in the art will be able to contemplate other implementations and applications of the techniques of the disclosed embodiments. Although illustrative embodiments have been described herein with reference to the accompanying drawings, it is to be understood that these embodiments are not limited to the disclosed embodiments, and that various other changes and modifications are made therein by one skilled in the art without departing from the scope of the appended claims.