High voltage power supply allowing transformers to be run in parallel for higher output power

[0002] This application is related to U.S. patent application Ser. No. 08/962,762, filed Nov. 3, 1997, now U.S. Pat. No. 6,067,236 the contents of which are incorporated by reference herein.

[0003] The present invention relates to a power supply for providing high voltage power from a low voltage source. The present invention can be manufactured so as to have a thin profile (less than 0.25 inch in width) and therefore is particularly well-suited for use as a power supply for a flat screen CRT (cathode ray tube).

[0004] Although the general concept of providing power conversion is known, the recently developed technology of flat screen CRTs has presented a greater need for power conversion circuitry capable of providing high voltage power from a low voltage source (e.g., 8,000 volts from a 5 volt source). In order to maximize the compactness of such flat screen CRTs, it is important that their power supplies also be compact. Further, in the highly competitive market for consumer electronics, it is important that the power supply be constructed at the lowest cost possible and therefore in a manner which permits automated manufacturing.

[0005] As is generally known by those skilled in the art, zero voltage switching of switches in electrical circuitry has been used to reduce the losses associated with opening and closing the switches. Thus, the switches have been operated in such a manner as to improve the efficiency of the electrical circuitry which incorporates the switches.

[0006] To increase the output power of conventional power supplies, two or more converters, including two or more transformers, two or more constant current sources and associated circuitry for each transformer, can be connected together. Alternatively, a power supply with increased output power has been provided by incorporating a larger transformer design. Thus, an increase in output power necessitates a much larger power supply design because of the increased number of components and/or a large transformer. The cost of manufacture is therefore increased and possibility of automated construction becomes limited. Moreover, if two or more converters are connected together to obtain an increased maximum output power, the multiple converters will often detrimentally interact with one another so that one or more of the transformer cores saturates.

[0007] It is therefore a primary object of the present invention to provide a power supply device capable of providing high voltage power from a low voltage source.

[0008] Another object of the present invention is to provide such a power supply device using components which facilitate automated manufacturing of the power supply device.

[0009] Another object of the present invention is to provide a power supply device having a high power output which can be manufactured to provide a thin overall profile of no more than 0.25 inches in width.

[0010] Another object of the present invention is to provide a power supply device having a high output-to-input voltage ratio (e.g., at least 500 times), compact size, and a relatively steady output voltage which remains within a close range of (e.g., at least within 10% or less) a desired voltage level, even if the load on the power supply varies significantly.

[0011] Another object of the present invention is to provide a power supply device including a resonant oscillator which is capable of maintaining a cycle to cycle flux balance of the transformer of the resonant oscillator and thus a cycle to cycle energy balance of the transformer. Due to the maintenance of the energy balance of the resonant oscillator's capacitive and inductive elements, the present invention allows multiple transformers to be connected together to form the resonant oscillator to thereby increase the maximum output power capability of the power supply device relative to a resonant oscillator having only a single transformer. The maximum output power capability thus increases as the number of connected transformers increases.

[0012] Another object of the present invention is to operate multiple transformers of a resonant oscillator at the natural resonant frequency of the equivalent inductive and capacitive elements forming the individual transformers so that the transformer cores can be prevented from saturating. Any detrimental interaction between the transformers is thus eliminated. The power supply of the present invention is therefore capable of providing power conversion in a highly efficient manner.

[0013] Another object of the present invention is to provide a power supply device in which multiple transformers of a resonant oscillator are connected directly in parallel to increase the output power capability with a lower number of components than what is conventionally required. Accordingly, the costs for constructing the power supply is relatively low since the same transformer design is used in the power supply device and the construction can be implemented by an automated process.

[0014] A power supply device according to the present invention therefore includes a current regulator, a resonant oscillator and a zero voltage level detector. The current regulator is connectable to a low voltage source (e.g., a 5 volt battery) and is arranged to provide a constant amount of current regardless of the changing load that is imposed on it.

[0015] The resonant oscillator preferably includes a first transformer, with its associated distributed capacitance and inductance, switches coupled to the primary winding and a switch driver for opening and closing the switches. The resonant oscillator receives a signal from the current regulator and transfers energy in a resonant fashion at the natural resonant frequency of its capacitive and inductive elements. The transfer is provided from the source to the capacitive and inductive elements of the transformer and to the load connected to the power supply.

[0016] The zero voltage level detector detects when the voltage level of a signal output from the current regulator reaches a zero level. In response to such a detection, the zero voltage level detector outputs an enable signal which triggers the switch driver to open and close the switches of the resonant oscillator so that a cycle to cycle energy energy balancebalance of the transformer of the resonant oscillator is maintained. That is, the resonance of the resonant oscillator is maintained through the operation of the zero voltage level detector to achieve a cycle to cycle balance of energy between the magnetic and capacitive elements of the resonant oscillator.

[0017] According to a preferred exemplary embodiment, the switches of the resonant oscillator are connected to opposite ends of the primary winding of the resonant oscillator and the current regulator is connected to a center tap of the primary winding. The switch driver receives the enable signal from the zero voltage level detector when the voltage level of the signal output from the current regulator reaches a zero level and opens and closes the switches in a push-pull mode in response to receiving the enable signal.

[0018] According to a preferred exemplary embodiment, the resonant oscillator can further include a second transformer which is connected in parallel to the primary winding of the first transformer. Through these two transformers, the output power capability of the power supply device can be increased (at least double) relative to a power supply device having only a single transformer. The transformers can be operated in a resonant mode at the natural resonant frequency of the combined capacitive and inductive elements of the individual transformers so that the transformer cores can be prevented from saturating and thus achieve a high conversion efficiency. More than two transformers can be connected together to further increase the maximum output power capability of the power supply device.

[0019] Another object of the present invention is to provide a power supply device having a multiplier which is connected to a secondary winding of the transformer. The multiplier is arranged so as to provide an output signal which has a DC voltage level which is significantly higher than the source voltage level. Preferably, the multiplier includes an assembly of interconnected capacitors and diodes to provide voltage multiplication.

[0020] Preferably, the maximum DC voltage level output from the multiplier is at least 500 times the source voltage level when a single transformer is used in the power supply device. Moreover, the maximum power level output from the multiplier can be at least two times the level with a single transformer when multiple transformers are connected in parallel in the power supply device. The maximum power level output from the multiplier thus increases as additional transformers are connected in parallel to a first transformer.

[0021] The power supply device may also include sampling circuitry connected electrically between the multiplier and the current regulator. The sampling circuitry provides a voltage indicative of current without drawing more than 0.003% of the power flowing to the electrical load.

[0022] Preferably, the current regulator, resonant oscillator, zero voltage level detector, multiplier and sampling circuitry are packaged together such that a profile of the power supply device is less than 0.25 inch in thickness. The power supply device of the present invention can thus assist to maximize the compactness of the device (e.g., a flat screen CRT) in which it is incorporated.

[0023] As will be appreciated by those skilled in the art, further embodiments and modifications can be made to the power supply device described herein without departing from the scope and spirit of the present invention.

[0024] FIG. 1 is a block diagram illustrating a power supply device in accordance with a preferred embodiment of the present invention.

[0025] FIG. 2 is a block diagram illustrating an equivalent transformer model of a power supply device in accordance with a preferred embodiment of the present invention.

[0026] FIG. 3 is a circuit diagram illustrating the transformer of FIG. 2.

[0027] FIG. 4A is a circuit diagram illustrating a power supply device in accordance with the preferred embodiment of FIGS. 1-3.

[0028] FIG. 4B is a voltage waveform diagram illustrating the operation of the circuit of FIG. 4A.

[0029] FIG. 5 is a circuit diagram of a transformer in accordance with a preferred embodiment of the present invention.

[0030] FIG. 6 is a circuit diagram of a transformer in accordance with a preferred embodiment of the present invention.

[0031] FIGS. 7A-7B are circuit diagrams of a transformer in accordance with a preferred embodiment of the present invention.

[0032] FIG. 8 is a circuit diagram illustrating an alternative preferred embodiment of a transformer circuit of the present invention.

[0033] FIGS. 9A-9B are circuit diagrams illustrating another alternative preferred embodiment of a transformer circuit of the present invention.

[0034] FIG. 10A is a circuit diagram illustrating a power supply device in accordance with an alternative preferred embodiment of the present invention.

[0035] FIG. 10B is a voltage waveform diagram illustrating the operation of the circuit of FIG. 10A.

[0036] FIG. 11A is a circuit diagram illustrating a power supply device according to a preferred embodiment of the present invention.

[0037] FIG. 11B is a voltage waveform diagram illustrating the operation of the circuit of FIG. 11A.

[0038] FIG. 12 is a circuit diagram illustrating an equivalent transformer model of a power supply device of the present invention.

[0039] With reference to FIG. 1, a power supply device according to a preferred embodiment of the present invention will now be described.

[0040] The illustrated power supply device includes a current regulator 10, a resonant oscillator 100 and a zero voltage level detector 50. The resonant oscillator 100 preferably includes a transformer 20, switches SW1, SW2 and switch commutator driver 60, although it will be understood that suitable alternatives and equivalents can be contemplated by those skilled in the art.

[0041] The current regulator 10 receives a DC voltage V_DCfrom a voltage source such as a 5 volt battery to provide an output signal having a voltage V1 and a current I1. The current I1 has a controlled wave shape. The current regulator 10 thus limits the rate of current I1 flowing to the resonant oscillator 100 so that the current I1 is at a constant current source level regardless of the load imposed on it.

[0042] As illustrated in FIGS. 2-3, the transformer 20 of the resonant oscillator 100 includes a primary winding and a secondary winding. The primary winding of the transformer 20 has distributed capacitance C_pand the secondary winding of the transformer 20 has a distributed capacitance C_S. The primary winding of transformer 20 has winding inductive L_Pand the secondary winding has winding inductance L_S. The primary winding has a center tap which receives the signal from the current regulator 10 and which divides the primary winding into primary sub-windings P1, P2. One end of primary sub-winding P1 is connected to switch SW1 and one end of sub-winding P2 is connected to switch SW2. Switches SW1, SW2 are connected to switch commutator driver 60 to receive controlling signals therefrom.

[0043] The zero voltage level detector 50 detects when the voltage V1 of the signal from the current regulator 10 reaches a zero level. Upon detecting the voltage V1 reaching a zero level, the zero voltage level detector 50 provides an enable signal to trigger the switch commutator driver 60. Upon receiving the enable signal from the zero voltage level detector 50, the switch commutator driver 60 changes the operative states of switches SW1, SW2. Specifically, the switch commutator driver 60 simultaneously operates switches SW1, SW2 in a push-pull mode so that when switch SW1 is closed, switch SW2 is opened, and when SW1 is opened, SW2 is closed.

[0044] A multiplier 30 is electrically connected to the secondary winding of the transformer 20. The multiplier 30 converts the voltage V4 (see, e.g., see FIG. 4A) across the secondary winding into a high voltage output (for example, a voltage output having a voltage level of 4000 volts to 8000 volts) which, in turn, can be applied to the load. The multiplier can be formed by an interconnected array of diodes and capacitors as illustrated in FIG. 10A. Alternative circuitry forming the multiplier 30 is described in related application 08/962,762, the contents of which are incorporated by reference herein.

[0045] The current drawn by the load is sampled by sampling circuitry 40. The sampling circuitry 40, in response to the current being drawn, provides a voltage V_PROGRAMwhich is indicative of the current being drawn by the load. The voltage V_PROGRAMis applied to the current regulator 10 which is capable of adjusting the current I1 dependant upon the voltage V_PROGRAM. The current regulator 10 thus provides dynamic current regulation in a manner dependant upon V_PROGRAM. Such dynamic control prevents significant deviations in the high voltage output HV OUT which might otherwise occur when the current being drawn by the load changes.

[0046] If, for example, the current being drawn by the load increases, this increase incurred is detected by the sampling circuitry 40. Upon detecting the increase, the current sampling network sends an appropriate voltage V_PROGRAMto the current regulator 10. The current regulator 10 will chop the voltage VDC by switching on and off a switch provided in the current regulator 10 at a specific rate to keep the current I1 flowing to the center tap of the transformer 20 constant. This results in an increase in V₁to maintain HV out at a constant level.

[0047] increases?]Likewise, the current being drawn by the load decreases, this decrease incurred is detected by the sampling circuitry 40. Upon detecting the decrease, the current sampling network sends and appropriate voltage V_programto the current regulator 10. The current regulator 10 will chop the voltage VDC by switching on and off a switch provided in the current regulator 10 at a specific rate to keep the current I1 flowing to the increases?]center tap of the transformer 20 constant. This results in a decrease in V₁to maintain HV out at a constant level.

[0048] FIGS. 4A-4B illustrate the operation of the power supply device according to a preferred embodiment of the present invention.

[0049] At time t_o, switch commutator driver 60 simultaneously closes switch SW1 and opens switch SW2. Voltage V3 is thus clamped to zero volts while voltage V1 of the signal provided to the center tap of the primary winding and voltage V2 rise (and later fall) sinusoidally at the resonant frequency of the resonant oscillator 100. The resonant frequency is determined by the value of the effective inductance l_pand capacitance C_Pof the primary winding. As illustrated in FIG. 4B, the peak voltage of V2 is twice the peak voltage of V1.

[0050] At time t₁, the voltage V1 falls to a zero level. This level is detected by the zero voltage level detector 50 which in turn sends an enable signal to the switch commutator driver 60. The switch commutator driver 60 simultaneously opens switch SW1 and closes SW2 in response to receiving the enable signal. Voltage V2 is thus clamped to a zero level, while voltage V3 rises (and later falls) sinusoidally with voltage V1 at the resonant frequency of the resonant oscillator 100. The peak voltage of V3 is twice that of voltage V1. The elapsed time from time t₀to time t₁is equal to half the resonant period (1/(2·resonant frequency)).

[0051] At time t₂, voltage V1 again resonates back to a zero level. This is detected by the voltage level detector 50 which in turn sends another enable signal to switch commutator driver 60. Switch commutator driver 60 simultaneously reopens switch SW2 and recloses switch SW1. The voltage waveform of voltages V1, V2 and V3 from time t₂to time t₄then repeats the cycle illustrated between time t₀to time t₂. The elapsed time from t₀to t₂and from t₂to t₄is equal to the resonant period (1/resonant frequency).

[0052] As illustrated in FIG. 4B, voltage V4 across the secondary winding oscillates between +V_{sec pk}(+V2×N) to −V_{sec pk}(V3×N), where N is the turn ratio of the transformer. Voltage V4 is then increased by multiplier 30 to provide the necessary high voltage output HV OUT to the load.

[0053] As illustrated by FIGS. 4A-4B, the zero voltage level detector 50 enables the resonant oscillator 100 to maintain its resonance. Accordingly, the cycle to cycle energy balance of the resonant oscillator 100 can be maintained. By maintaining the cycle to cycle energy balance of the resonant oscillator 100, additional transformers may be connected in parallel to increase the output power.

[0054] FIGS. 5-7 illustrate an exemplary transformer model for the power supply device in accordance with the present invention wherein the transformer is a Coiltronics CTX21065 transformer.

[0055] Specifically, FIG. 5 illustrates a transformer model of the Coiltronics transformer having nodes 1-5. The Coiltronics transformer has the following characteristics:

[0056] Transformer open circuit resonant frequency F_oc=198 KHz $\begin{array}{ccc}{N}_{\mathrm{p1}}=10\mathrm{turns}& L_{\mathrm{p1}}=10.80\mathrm{\mu H}& \mathrm{Nodes}1\&3\\ N_{\mathrm{p2}}=10\mathrm{turns}& L_{\mathrm{p2}}=11.10\mathrm{\mu H}& \mathrm{Nodes}1\&2\\ N_{\sec }=1340\mathrm{turns}& L_{\sec }=155.7\mathrm{mH}& \mathrm{Nodes}4\&5\end{array}$

[0057] The calculation for the A_Lnumber for the core to derive the total inductance of both primary sub-windings is: $\begin{array}{c}{A}_L=\frac{\left(\frac{L_{\mathrm{p1}}}{{10}^{-6}}\right)}{N_{\mathrm{p1}}^2}·{10}^4=\frac{\left(\frac{10.8\times {10}^{-6}}{{10}^{-6}}\right)}{{10}^2}·{10}^4=\frac{10.8}{100}·{10}^4=1080\\ L_{\mathrm{ptotal}}=\left(\frac{N_{\mathrm{ptotal}}^2{A}_L}{{10}^{-6}}\right)·{10}^{-6}=\left(\frac{{20}^2·1080}{104}\right)·{10}^{-6}=43.2·{10}^{-6}\\ \left(\mathrm{nodes}2\&3\right)43.2\mathrm{\mu H}\end{array}$

[0058] (nodes 2 & 3) 43.2 μH.

[0059] The turns ratio from nodes 3 & 2 to 4 & 5 is: $\frac{1340}{20}=67.$

[0060] As illustrated is FIG. 6, the secondary distributed capacitance is: $\begin{array}{c}{C}_{\sec }=\frac{1}{{\left(2\pi f_{\mathrm{oc}}\right)}^2{L}_{\sec }}=\frac{1}{{\left(2\mathrm{\pi 198}\mathrm{Khz}\right)}^2·155.7\mathrm{mH}}=4.15·{10}^{-12}\\ =4.15\mathrm{pf}.\end{array}$

[0061] The resonant frequency with multiplier 30 connected is: $F_{\mathrm{mc}}=82\mathrm{KHz}$ $C_{\mathrm{mc}}=\frac{1}{{\left(2\pi f_{\mathrm{mc}}\right)}^2·L_{\sec }}=\frac{1}{{\left(2\pi 82\mathrm{KHz}\right)}^2·155.7\mathrm{mH}}=24.19\mathrm{pf}.$

[0062] Thus, the multiplier effective capacitance is:

[0063] C_MULT=C_MC−C_SEC=24.19pf−4.14pf=20.05pf.

[0064] To minimize transformer loss the conversion frequency goal is 61 kHz. Thus, the effective C_SEC: $\mathrm{Effective}{C}_{\mathrm{SEC}}=\frac{1}{{\left(2\pi 61\mathrm{KHz}\right)}^2·155.7\mathrm{mH}}=43.72\mathrm{pf}.$

[0065] The additional capacitance to be added is:

[0066] 43.72pf−24.19pf=19.53pf.

[0067] As illustrated in FIG. 7a, it is preferable to reflect the secondary side capacitance to the primary side and add the additional capacitance to the primary winding. Accordingly,

[0068] C_prc=C_sec·N²

[0069] N²=67²=4489

[0070] C_mcreflected to primary side=24.19pf·67²=0.10854μf

[0071] C_effectivereflected to primary side=43.72pf·67²=0.1962μf.

[0072] The additional capacitance to be added to the primary side is:

[0073] 0.1962μf−0.1085μf=0.0876μf.

[0074] As illustrated in FIG. 7b, the equivalent primary circuit has a resonant frequency= $\frac{1}{2\pi \sqrt{\mathrm{LC}}}=\frac{1}{2\pi \sqrt{43.2\mathrm{\mu H}·\mathrm{.1962}\mathrm{\mu F}}}=54.62\mathrm{Khz}.$

[0075] FIG. 8 illustrates an alternative embodiment of the present invention in which resonant circuit 100 of the power supply device incorporates two transformers (labeled as 20a) which are connected directly in parallel to obtain an increased maximum output power capability. As illustrated in FIG. 8, the center taps of the primary windings of transformers T1 and T2 are connected as a common node. Similarly, an end of sub-winding P1 of transformer T1 is connected to an end of sub-winding P1 of transformer T2 as a common node and an end of sub-winding P2 of transformer T1 is connected to an end of sub-winding P2 of transformer T2 as a common node. Similarly, the start and finishing ends of the secondary winding of transformer T1 and transformer T2 are respectively connected together.

[0076] FIGS. 9A-9B illustrate another alternative embodiment of the present invention in which the power supply device incorporates three transformers (labeled jointly as 20b) which are connected directly in parallel to obtain an increased maximum output power capability. The three or more transformers are connected directly in parallel in a manner similar to FIG. 8. For example, the center taps of each of the primary windings are connected together as a common node. The ends of each sub-winding P1 and each sub-winding P2 are connected as common nodes, respectively.

[0077] As discussed above with respect to FIGS. 1-4, the operation of the zero voltage level detector allows the transformer of the resonant oscillator 100 to maintain a cycle to cycle energy balance. Due to the maintenance of this energy balance, an additional transformer can be connected directly in parallel to the first transformer. The parallel connected transformer configuration can be viewed as a single element, comprised of the lumped inductive and capacitive elements of the individual transformers. As is the case with the single transformer, the parallel connected transformers of the oscillator circuit 100 operate in a resonant mode at the resonant frequency of the lumped elements. The transformer cores can thus be prevented from saturating and any detrimental interaction between the transformers can be eliminated. The parallel connected transformers can thus achieve a high conversion efficiency. Moreover, since any number of transformers can be connected directly in parallel, the power supply device of the present invention can be easily adapted to provide a higher output power with minimal engineering effort because the core circuitry (e.g., the transformer) remains the same.

[0078] FIGS. 10A-10B illustrate the circuit and operation of a power supply device in accordance with the present invention which includes two parallel connected transformers T1, T2 as part of the resonant oscillator 100. The resonant frequency of the resonant oscillator 100 illustrated in FIG. 10A is 61 kHz as calculated by

[0079] $\frac{1}{2\pi \sqrt{L_{\mathrm{PRI}}\left(\mathrm{equiv}\right)C_{\mathrm{PRI}}\left(\mathrm{equiv}\right)}}.$

[0080] Thus, the resonant period of oscillation is {fraction (1/61)}KHZ=16.393μS. The turn ratio of transformers T1 and T2 is 67.

[0081] FIG. 10B illustrates the operation of the circuit having two transformers as part of the resonant oscillator 100, which is similar to that described in FIG. 4B above for the circuit having a single transformer.

[0082] At time t₀, switch commutator driver 60 simultaneously closes SW1 and opens switch SW2. Thus, voltage V3 is clamped to zero, and voltage V1 and voltage V2 rise sinusoidally at the resonant frequency of 61 KHZ. The peak voltage V2 is two times the level of voltage V1. In the exemplary embodiment, voltage V1 is 9.03 volts and thus the peak voltage V2 is 18.06 volts (2×9.03 volts).

[0083] After 8.19 μS (half of the resonant period T of 16.393 μS), the voltage V1 reaches a zero level. This is detected by the zero voltage level detector 60 which in turn closes switch SW2 and opens switch SW1. Voltage V2 is now clamped to zero while voltage V1 and voltage V3 rise sinusoidally at the resonant frequency of 61 KHZ. Like V2, the peak voltage of voltage V3 is 18.06 volts (2×9.03 volts).

[0084] After the resonant period has elapsed (16.393 μS after time t₀), the voltage V1 again returns to a zero level. This is detected by the zero level detector 60 which outputs a signal to enable the switch driver 60 to reclose switch SW1 and reopen switch SW2. The resonant cycle will then repeat as discussed above.

[0085] The voltage V4 is equal to V2 (or V3) times the turn ratio N. In this exemplary embodiment, the voltage V4 will thus be equal to 1210 (18.06v×67). The voltage V4 is then multiplied by the multiplier 30 to provide a high voltage output HV OUT to be applied to the load. In the exemplary embodiment illustrated in FIG. 10A, the multiplier will convert the voltage V4 having a peak voltage of 1210 to a peak voltage HV OUT of 6.5 KV.

[0086] FIGS. 11A-12 illustrate the development and implementation of the circuit of FIG. 10. As illustrated in FIG. 11A, due to sinewave of the resonant circuit, voltage V_DCis: $V_{\mathrm{DC}}=\frac{2V_1}{\pi }=\frac{V_2}{\pi }=\frac{V_3}{\pi }=\frac{18.06}{\pi }=5.748v.$

[0087] As illustrated in FIG. 11b,voltage V₁is a half sinewave at 2× the resonant frequency: $t_R=\frac{1}{F_R}=t_1=\frac{\pi }{t_1}.$

[0088] V₁as a function of time is:

[0089] V₁(t)=V_pSIN(wt).

[0090] The calculation for the angular frequency is: $W=2\pi f=\frac{2\pi }{2t_1}=\frac{\pi }{t_1}.V_p=\frac{\pi }{2}{V}_{\mathrm{DC}}\mathrm{or}{V}_{\mathrm{DC}}=\frac{2V_p}{\pi }.$

[0091] The calculation for V_pkrelated to output voltage is: $V_{\mathrm{pk}}=\frac{\left(\frac{\left(I_o·Z_{\mathrm{MULT}}+V_o\right)}{M}\right)}{2N}.$

[0092] The primary winding output voltage as a function of input voltage is: $V_{\mathrm{pk}}=\frac{\pi }{2}{V}_{\mathrm{IN}}=\frac{\left(\frac{\left(I_o·Z_{\mathrm{MULT}}+V_o\right)}{M}\right)}{2N}.$

[0093] The primary output voltage as a function of input voltage is: $V_p=\frac{\pi }{2}{V}_{\mathrm{IN}}=\frac{\left(\frac{\left(I_o·Z_{\mathrm{MULT}}+V_o\right)}{M}\right)}{2N}$ $V_p=\frac{\pi }{2}{V}_{\mathrm{IN}}=\frac{\left(\frac{\left(I_o·Z_{\mathrm{MULT}}=V_o\right)}{M}\right)}{2N}$ $\frac{\pi }{2}{V}_{\mathrm{IN}}\frac{\left(\frac{\left(I_o·Z_{\mathrm{MULT}}+V_o\right)}{M}\right)}{2N}$ $V_{\mathrm{IN}}·\frac{\pi }{2}2N=\frac{I_o·Z_{\mathrm{MULT}}+V_o}{M}$

[0094] OR

[0095] V_O=(V_IN·π·N·M−I_O·ZMULT).

[0096] FIG. 12 illustrates an electrical equivalent model of multiple connected transformers. As will appreciated by those skilled in the art, the equivalent inductance is L_PRI(Equivalent)=L_PRI1+L_PRI2+L_PRIN. Similarly, the equivalent inductance for the secondary winding of the transformers is L_SEC(Equivalent)=L_SEC1+L_SEC2+L_SECN and the equivalent capacitance of the secondary winding of the transformers is $C_{\mathrm{SEC}}\left(\mathrm{Equivalent}\right)=\frac{1}{\frac{1}{C_{\mathrm{SEC}}1}+\frac{1}{C_{\mathrm{SEC}}2}+\frac{1}{C_{\mathrm{SEC}}N}}.$

[0097] The effective resonant frequency of the secondary winding of the transformers is Resonant Frequency: $F_{\mathrm{RSEC}}\left(\mathrm{Equivalent}\right)=\frac{1}{2\pi \sqrt{L_{\mathrm{SEC}}\left(\mathrm{equiv}\right)C_{\sec }\left(\mathrm{equiv}\right)}}.$

[0098] The effective resonant frequency of the primary winding of the transformer is: $F_{\mathrm{RPRI}}\left(\mathrm{Equivalent}\right)=\frac{1}{2\pi \sqrt{L_{\mathrm{PRI}}\left(\mathrm{equiv}\right)C_{\mathrm{PRI}}\left(\mathrm{equiv}\right)}}.$

[0099] While the invention has been described in its preferred embodiments, it is to be understood that the words which have been used are words of description rather than limitation, and that changes within the view of the appending claims may be made without departing from the true scope and spirit of the invention.