Microprocessor Fast Load Current Changes

The microprocessors and SoC Integrated Circuits impose very demanding large and fast load current changes of 10A/usec to 50A/usec yet requiring ultra small transient voltage overshoot to settle very quickly. This is currently handled by up to 12 synchronous buck converter operating in parallel at 5MHz from 12V input resulting in poor efficiency, large size and high cost! This demand is currently met by third party vendors using the same multistage buck approach consisting of over 40 power management companies slicing the 10 billion dollar market among them by providing 12V to 2V, 100A drives. This legacy solution requires an Intermediate Bus Converter (IBC) to converter 48V bus voltage to 12V. Many companies are now attempting to eliminate such intermediate converter and provide direct 48V to 1V, 200A converter. Besides using inefficient two-stage conversion solution, all these also fail to address fast load transient requirements other than using the aforementioned multi-stage synchronous buck solution.

Direct 48V to 1V, 100A Converter with Single-stage Transient Solution

This article describes in a detail the direct 48V to 1V, 100A solution based on the new Hybrid Switching Method and its implementation in novel converter topology implementing a new magnetics structure designated Hybrid transformer. The present uP drives all use the 12V to 1V legacy solution since all other proposed 48V to 1V solutions can not handle fast transients.

Dual use for 12V to 1V, 200A uP drive solution

One additional advantage of the described solution for 48V to 1V conversion is that it could be directly utulized for the present 12V, 1V uP prices by simply adjusting the transformer turns ratio and using appropriate lower voltage rated switching devices. This will also allow a smooth transition from the present 12V to 1V to new advanced 48V to 1V uP drives needed in near future.

Introduction

Basic Operation of the Ćuk-buck 2 Converter

From the converter circuit drawing below one could make the wrong conclusion that it is just another ordinary converter operating with Pulse Width Modulation with regulation using duty ratio control at constant switching frequency. Moreover, the converter topology will be easily wrongly classified as a Tapped Inductor Buck Converter as technical articles from Google, Texas Instruments and number of technical publications published by them and a number of university research teams have copied it and incorrectly analyzed due to lack of recognition of the inherent resonance implicit in this converter. (1,2,3,4)

Hidden Resonant Circuit

Even the voltage conversion ratio can not be correctly derived without recognizing the fact that there is an invisible resonant circuit operating during OFF time period as illustrated in the converter drawing below incorporating the resonant inductor in series with primary winding. Note tat this inductor is NOT a physical inductor built externally, but is actualy representing the leakage inductance of the transformer. While in all other transformers of conventional PWM converters the leakage inductance is actually very undesirable as it stores the energy which must be dissipated each cycle resulting in large losses proportional to switching frequency and square of the peak resonant current. In this converter, the leakage indcutance is absorbed in the operation of the converter due to its resonance with the resonant capacitor Cr, Resonant current is continuous at all times and has no large current jump at switching instants as it does in Tapped Inductor buck converter. Fig b shows the equivalent circuit during OFF time interval when S2 switch and diode are conducting.

Resonant inductance Lr must satisfy flux balance during the OFF time so that flux balance on vr : vr = Vcr -Vc = 0 (1)

Vcr = Vc = V (2)

Using this result and provided that

C>> Cr (3)

the simplified equivalent resonant circut is obtained as in Fig.c the solution of which is:

vcr = V sin (wrt) (4)

icr= ip= -0.5 I cos (wrt) (5)

is = I – 0.5 I cos(wrt) (6)

Note that the diode current is the sum of the primary and secondary currents hence
id= ip + is = I (1-coswrt) (7)

Voltage Conversion Derivation

From equation 2, the flux balance on the transformer is shown in drawing below for 1:1 turns ratio of transformer. Hence

(Vg -2V) D = V 2 (1-D) (8)

and converion gain is

V=0.5 D Vg (9)

In general

V= D Vg N2 /(N1 +N2) (10)

Hence the voltage gain is the same as in PWM forward converter operating with duty ratio D and transform primary turns N1 +N2 and secondary turns N2.

Linear voltage gain with duty ratio D confirmed by measurement

Measurement of the voltage conversion ratio for n=2 is displayed below, confirming linear voltage gain characteristic with duty ratio change.

Hybrid Transformer Resonant Currents

Hybrid transformer, is a close approximation of the Faraday transformer and has inherent feature that its primary and secondary currents are sinusoidal at the resonant frequency and are scaled by the transformer turns ratio. Moreover, their peak magnitudes are directly proportional to the DC load currents and therefore automatically scale-up or down with the change of the DC load current. This directly leads to the load transient elimination in a fraction of a single switching period. Shown below are the three key sinusoidal-like waveforms for the special case of 1:1 transformer turns ratio, which were analytically derive earlier. This is made for easier understanding of the fundamental principles. The general case for N1 to N2 turns ratio can then be easily derived and equations generalized using the usual transformer current scaling law via its primary to secondary turns ratios.

Converter Simulation Model

Converter is modeled using .plecs simulation program from PLEXIM shown below:

Confirmation of Key Sinusoidal Current Waveforms by Simulation

Simulation for the 4 different DC load currents for 100A, 50A, 25 A and 10A shows extraordinary feature of the converter that all steady state current waveforms (primary, secondary and diode current) are directly scaling down proportionally to the DC load cuirrent I. This is then the reason why this converter makes possible large trasient elimination in a single switching period. Moreover, this is accomplished with a single module and does not require 12 modules of the synchronous buck converters to obtain acceptable transient response.

The diode current waveforms exhibit another extraordinary feature. The diode as a switch is turned ON at zero current and turned OFF at zero current for all load currents. Hence the diode switching losses are eliminated for any DC load currents from full load to no load due to Zero Current Switching (ZCS) . In fact, diode does not even know that it is switching, since at switching instances does not even conduct any current.

This also leads to another extraordinary feature not present in any switching converter. The instantaneous diode current is always positive. Hence the discontinuous conduction mode is eliminated for any load.

Obviously, as the delivered DC load current is reduced, so are proportionally, superimposed AC ripple current. Hence there will never be a large circulating when the diode is replaced with synchronous rectifer MOSFDET. Hence an ultra high 99% efficiency is maintained from full load to light load.

Voltage Regulation

The simulation below demonstrates that reduction of duty ratio from 0.1 to 0.05 leads to proportional reduction of output voltage from 2.4V to 1.2V.

Capacitor Power Transfer Paradox

This converter utilizes the resonant capacitor in series with the primary winding. Note that all power must be transferred from input to output by current passing through this capacitor Cr. Note that this capacitor Cr voltage rating is approximately 1V and yet it converts input voltage of 50V. Hence its voltage rating is 50 times lower than the input source voltage. In addition, the capacitor value needed is on the order of 100uF. Hence very small size capacitor is needed for this 100W, 1V 100A converter.

The key operation of the converter can be best understood when the models of the two circuit subintervals are analyzed with inclusion of the leakage inductance of the transformer Lr modeling the inevitable leakage inductance associated with any transformer as described in original drawing. Note also that this resonant inductance Lr resonates with the resonant capacitance Cr on the primary resulting in resonant period Tr given by

Tr=2PI Sqrt(LrCr) (11)

PI=3.14 (12)

Note that the large magnetizing inductance of the transformer is effectively shorted during OFF time interval leading to a purely sinusoidal AC current on the primary as illustrated in waveforms and resulting sinusoidal current on secondary. This has a crucial consequences for converter operation as described below:

Leakage losses eliminated!

When this resonant period Tr is equal to TOFF time of switch S2, avery unique optimum operating pont is achieved.

Tr =Toff (13)

Leakage inductance is essential part of the primary resonance. Since this current is continuous at both switching instances there are no losses normally associated with all other switching converters. For the first time these losses which are normally proportionally growing with the increased switching frequency in all other transformer coupled converters are completely eliminated! The bonus is that the primary current is purely sinusoidal with a zero average current. Hence there is no DC bias contribution from the transformer primary reducing energy storage in transformer dramatically. Obviously there is a strong incentive for keeping this ideal operating point for all input votage and load current conditions as described next.

Optimum Control Method

The ideal operating point given by (3) can be preserved by simply keeping OFF period constant that is

Toff = Constant (14)

Clearly, the variation of the ON-time period will then result in direct output DC voltage control. Obviously, this calls for variable switching frequency control. Nevertheless, for large step-down through duty ratio, such as D=0.1 this effectively results in near constant switching frequency with around 10% frequency variation.

Two Transformers in One

The DC voltage conversion is following the rectangular voltage excitation of the hybrid transformer as used for derivation of the c voltage conversion.

The hybrid transformer currents, however, are sinusoidal also satisfying the turns ratio of the hybrid transformer.

Zero Diode and MOSFET Switching Losses

The yellow waveform for the diode current clearly confirms that the diode is both turning ON and turning OFF at ZERO current hence no switching losses. In fact, as its voltage at both switching instances is also ZERO, the diode does not even know that it is switching! Practical consequence: no spikes and no EMI generated.

Obviously, the proper drive of active switches operating with Zero Voltage Switching ( ZVS) operation eliminate switching losses of active devices as well as spikes and EMI noise typically associated with them.

Bottom line, all switching losses and leakage losses are eliminated leaving only resisitve losses. Hence the ultra high efficiency over 99% as well as low EMI noise.

No Discontinuous Conduction Mode

Simulations demonstrated that diode current always operates in Continuous Conduction Mode (CCM) for all loads. As expected, there is also no change in the voltage conversion gain as seen in all simulation waveforms. Furthermore, the high efficiency at the full load is also maintained at very light load as there are no large circulating currents as those present in synchronous buck converter at light load.

Synchronous Rectifer Implementation with MOSFET and GaN Transistors

All MOSFET Implementation

Of the remaining conduction losses, by far the largest loss contribution for low output voltages like 1V is due to conduction losses of the rectifer CR diode. This can be mitigated by use of the MOSFET with ultra low ON resistance. It is important to note the distinction between operation of this MOSFET in Cuk-buck2 converter and the synchronous buck converter MOSFET. As described above, the converter never gopes into Discontinuous Conduction Mode (DCM) for any load. Hence the synchronous rectifier MOSFET never conducts the MOSFET transistor current as all current is always diverted through its body diode.

All GaN Transistors Implementation

Therefore, the ideal implementation of this switch would be to use Dual Gate GaN device such as recently introduce Panasonic device which emulates the diode conduction without having limitations of the body diode of MOSFETs. This in fact lands itself to a new Power Integrated Circuit (PIC) implementation.

Power Integrated Circuit with GaN devices

The GaN switching devices have a unique properety that all power devices can be built on a same die. This eliminates the very lossy and space wasting coonection for making connection between separately packaged discrete devices. Moreover, the high-side drive circuit and the direct drive for Synchronous GaN DEVICE can all be built on the same substrate making it a single Power and Drive Chip.

The bonus operating GaN devices at 100kHz switching frequency is not only that gate drive losses are practically eliminated, but also the fact that paralleling a number of devices DOES decrease conduction losses greatly without increasing gate drive losses as would be the case for MOSFET implementations.

Is feedback needed for regulation of this converter?

The short answer is no! The reason feedback was always used in conventional converters is twofold:

1. Low efficiency in the 80% and low 90% requires the change of duty ratio to compensate for the drop of the voltage due to large differences in converter losses between ful load and say 20% load.
2. All present converters with diode rectification result in Discontinuous Inductor Mode at no load to light load and need substantial duty ratio change to maintain voltage regulation.
3. Change of input voltage requires also change of duty ratio to keep output voltage constant despite input voltage changes.

The last problem is easily solved using the classical feedforwad control by controlling PWM sawtooth ramp magnitude with input voltage. Hence, the voltage changes are instantly compensated quickly within a single switching cycle! The first two problems are solved by having ultra high efficiency and no DCM operation of this converter.

With efficiencies in 99% and above range no regulation using feedback control is needed! Not only it is not needed but it is also not desired due to several drawbacks caused by feedback:

1. Oscillation and stability problems resulting in potential catastrophic failures!
2. Grossly limiting bandwidth to 10% or so of the fundamental open loop bandwidth resulting in much diminished transient response.

The remaining issue is how to change output voltage to desired discrete levels demanded by microprocessors. This can also be handled by controlling PWM sawtooth ramp, this time using a separate reference analog voltage to control even continuously the output DC voltage and not just in discrete steps!

Videos of Two Simulations

First video simulation made at the load current of 15A is shown below confirming continuity of the primary current exhibiting no current jumps as well as diode turning-on and turning-off at zero current.

Second simulation made at 50% load of 7.5A shown below confirms no change in fundamental operation and preservation of the same sinusoidal current waveforms at light load and all the way down to practically no load.

Obviously, the converter can be scaled up in current to 100A and 240W at 2.4V as illustrated in with the same salient results provided the resistive losses are likewise reduced by use of appropriately scaled resistive losses of the components!

Zero Output Ripple Voltage With Two Identical Modules

Only two identical converters operated in parallel but shifted in time for exactly half a switching period (see schematic drawing below) are sufficient to reduced output ripple voltage to ultra low value despite using 20 times less capacitance then needed in a single synchronous buck converter.

Note the simplicity of the practical implementation of two modules in parallel. Only once converter needs to be made, as the second module is IDENTICAL converter, whose operation is shifted exactly by a half switching period. Hence the control needs to include simply a sync pin to implement this parallel connection.

Note the complexity of a multistage synchronous buck with 12 modules, requiring complex phase shifting of 12 modules!

The current waveforms below show near complete cancellation of the ripple current and drastic reduction of the filtering capacitor needed on output to achieve ultra low ripple voltage.

Output Filtering Capacitor Paradox

The current waveforms below show near complete cancellation of the ripple current and drastic reduction of the filtering capacitor needed on output to achieve ultra low ripple voltage. In the previously simulated example for 48V to 2.4V, 100A converter output power of two modules would be 540W. Yet the filtering capacitors needed would have only 2.4V rating and would use capacitors which are 0 times or more reduced in value than a filtering capacitor need for a single buck converter.

Hence both resonant capacitor Cr and the output capacitor C are would be ultra compact even at high power of 500W.

Elimination of the Large and Fast Transient in a Single Cycle

The Ćuk-buck 2 converter is the very first converter which has a unique property: The peak resonant current drawn from the source is directly proportional to the DC load current as given by equation (6). Hence the sudden change of the DC load current as experienced in microprocessors, will eliminate voltage transient in less than half the switching period. This feature is inherent to both Hybrid Switching Method in general as well as Storage-less Switching Method in which both intervals have their own defined resonances. The experimental measurements recorded for converter using Storage-less Switching Method shown below is representative of the inherent Fast Transient performance with both methods.

Experimental Verification at 100kHz switching frequency

The early breadboard prototype is shown below demonstrating small size of the magnetics even at moderate switching frequencies of 100kHz. The experimental waveforms also confirm Zero Voltage Switching (ZVS) of the two MOSFET transistors.

Small magnetic size despite 100kHz switching!

The size of magnetics is directly proportional to flux per output number of turns so for 1 turn secondary N2=1 and 100kHz , we get for 0.5V output

Voltsec/N2 = 0.5V x 10usec= 5Vusec (17)

Hence for AC flux density of 0.25T this translates into core cross section of only 20mm2.

This is in fact identical what just a separate inductor in single buck converter would need. Here there is only one additional primary winding of 1 turn. For both windings being foil windings, there is actually minimal increase of the window area to implement 1:1 turn ratio transformer.

Switching frequency overkill beyond 150kHz

There is a reason for singling 150 kHz as an upper limit! Below that frequency, there is no requirements regarding radiated EMI noise, as this is deemed too low frequency band to interfere. ZVS of ideal switches and zero current switches of the diode, eliminate spike noise and conducted noise by definition anyway, so expensive filtering measures for conducted and radiated noise mitigation can be much reduced or even entirely eliminated.

Obviously, current GaN and SiC Devices proponents would continue their hype that MHz switching frequencies are needed to reduce size of magnetics and hence justify need for high speed of their devices. Nothing is further from the truth as they would work much better at below 150kHz. In addition they have to learn that conventional topologies not only abuse them in by ten times or higher overkill in frequency but also with simultaneous 10 times or more overkill in their device voltage requirements such as for 48V to 1V direct buck with GaN Devices. Brute-force with both is not solution but huge detriment for wider use of those otherwise excellent switching devices.

Applications

One might get the wrong impression that the application of the Cuk-buck 2 is limited to microprocessor drives due to its unique property to handle large and fast load transients.This is not the case, as its ultra high efficiency, small size and low cost make it ideal choice also for all automotive applications for direct 48V to 1V conversion, Point of Load applications, FPGA power supplies and many other applications.

Conclusion

The key results are summarized here. All other present approaches have three fundamental drawbacks for 48V to 1V conversion:

1. They use two-stage conversion approach with an Intermediate Bus Converter ( IBC) for 48V to 12V conversion and separate 12V to 1V multiphase synchronous buck converters with up to 12 stages operating at 5MHz switching frequency.
2. They are not able to provide fast transient response naturally in a single-stage but require a number of parallel stages (8 or more) at ultra high 5 MHz switching frequency.
3. Their size, cost and weight are order of magnitude larger than achievable by two identical Ćuk-buck 2 modules.

US patent 9,231,471 B2

First page of US patent is reproduced below confirming the following:

1. Application date: March 28, 2011
2. Prior publication Date: Oct. 4, 2012 document US 2012/0249102 A1
3. Publication Date: Oct. 4 2012

From the First Masterclass POWER ELECTRONICS: 50 YEARS IN 3 DAYS!

2nd Masterclass will include expanded second part on NEW POWER ELECTRONICS covering Power System on Chip (PSoC) which is analogous to System on Chip (SoC) of the signal processing in microprocessor designs and other innovations! Here is a preview.

Buck, boost, flyback and forward converters retiring!

The time has finally arrived to recognize that inductors with magnetic cores and “miraculous” air-gaps built-in to avoid core saturation with even minute DC currents are the key reason that has stalled Power Electronics for last 50 years! These inductors showed in the drawings below of four basic converters are NOT AC inductors Faraday invented in 1831!? They are “DC inductors” which saturate magnetics cores even with minimum DC currents making them into an effective short instead of a large inductance desired! The saturation is avoided with a concoction of inserting an air-gap into magnetic cores.

Air Gap Effect on Inductance

The inserted air-gap has a dramatic effect on huge reduction of inductance for even small air-gaps! See enclosed video demonstration of effect of air-gaps on DC inductors! Hence the need for 100-fold increase of switching frequency to 5MHz from 50kHz to make up for loss of that inductance with increased DC load current.

The air-gaps in magnetic cores is not the “magic” that magnetic “experts” have led you to believe for decades but simply very effective “killers” of inductance! No fancy physics of B vs H “miracles” and “basic Physics 101” as some “experts” claim!? My first masterclass graduates already learned that on Sept 25 to 27, 2018 Masterclass in great details!

The four minute video below is the single most important reason why all present non-isolated switching DC-DC converters such as buck, boost and buck-boost must be abandoned and replaced by Power Systems on Chip (PSoC) solutions. The secret is in eliminating magnetic cores and the associated air-gaps from inductors at 50kHz and NOT 50MHz as fully explained in powerelectronics.com article enclosed below!

Powerelectronics.com article on PWM Resonant converter

The article describing the first true Power Supply on a Chip converter, PWM resonant converter is published in Power Electronics.com on-line magazine in April 2017. Here is a link: https://www.powerelectronics.com/power-management/step-down-dc-dc-converter-eliminates-ferrite-cores-50khz-enabling-power-supply-chip

References TBD

Comments on Slobodan Cuk’s article

Sam Mallicoat – CTO at Information Display Company

“Pure genius SC!”

Slobodan Cuk – CEO at TESLAco

“@Sam Mallicoat Thanks a million! I feel I just started! More is to come! Photo of my Caltech days following in Nikola Tesla’s footsteps!”

Marino Dotta – Senior Power Electronics Designer

“È talmente semplice e funzionale che sembra impossibile il non averla inventata prima.”

Slobodan Cuk – CEO at TESLAco

“@Marino Dotta You say: “It is so simple and functional that it seems impossible not to have invented it before”. Historical point. I invented Ćuk converter on April 1, 1975. My employer’s California Institute of Technology (Caltech) Patent Office refused to even conduct basic search to find out if it is patentable, as it was considered so basic as you say, ” impossible not to have been invented before”. Only when the isolated extension, or Isolated Ćuk converter, was disclosed, still with only two switches, did Caltech considered doing patent search! Well, it took me another 30 years or so before I dared to add one more switch in 2008. This, however, has required first major rethinking of topologies and inventing new Switching Methods as being even more fundamental than converter topologies!

Boris Radonic – Software Engineer

“Very old, but still the best topology.”

Slobodan Cuk – CEO at TESLAco

“@Boris Radonic Thank you very much for your compliment. Much appreciated! I will, however, take exception to topology being old. My patent US 9,231,471 B2 was filed on March 28, 2011 and was officially published by Patent Office on Oct 4, 2012. Yet, it was issued on January 5, 2016 due to patent office backlogs! By international patent regulations that US is now following, patent expires 20 years from date of application, which makes it valid until March 28, 2031, still some 13 years left! Unfortunately, this does speak volumes of the present state of the Power Electronics hung-up on new switching devices driven to unsustainable MHz switching frequencies but ignoring fundamentals: new switching methods, corresponding novel converter topologies and magnetics circuits. Yet the future of Power Electronics could not be brighter with young generation continuing in our footsteps! I am still here to guide it.”

Walter Kawai Chu – Chief Innovator at Stapes Technology

“Dr. Cuk, I am in your side. Get back your license royalties deserved!”

Slobodan Cuk – CEO at TESLAco

“Walter Kawai Chu Thank you for your support and advice. My patents, license royalties and my masterclasses allowed me to keep my independence and are helping now to generate a grass-root support for moving Power Electronics out of the limbo it has been for last 50 years. I invite all Power Electronics lovers and enthusiasts to join this movement. I guarantee you it will be very rewarding! In 1988, 30 years ago, when the enclosed photo was made, I stood up at conferences and published papers against even then hyped-up 10MHz converters! Now I have many proofs that even 100kHz is more than sufficient for ultra small, efficient and low cost switching converters! There is also more to come. Stay tuned!”

Clifford Emeric – Independent Researcher

“Really? This is not a matter of recapturing an ancient patent so lost past royalties can be recaptured, and people are getting confused thinking a 2 switch buck tapped-inductor topology is somehow the same as Professor Cuk’s new 3 switch hybrid transformer Cuk-Buck2 topology. Maybe that extra switch is a clue to the difference! The buck tapped-inductor is not a transformer because when switch is off one half winding is DISCONNECTED, so ipso facto how is it a transformer when one leg is open circuit – to wit: a transformer requires both windings to be mutually active AT THE SAME TIME. The old tapped-inductor Buck just bounces energy through one winding into the magnetic core, and then bounces it back out with the other winding, and never are two windings mutually coupled during the same time interval operating as a transformer. How does a disconnected winding end stay in play? To the contrary Cuk-Buck2 during off period uses synchronous MOSFET to keep that winding end connected. Even if a sync MOSFET is added to the old Buck it still would be different as it would lack the series resonant capacitor that resonates with primary winding. Cuk-Buck2 in ON period charges both capacitor and inductor (both halves) linearly, and in OFF period primary leakage inductance and series capacitor resonant dissipating the core’s stored energy sinusoidally (unlike old Buck with square wave operation). This is just a small part of this topology’s true inner secrets with new features, like a one cycle transient response while achieving this without the need for a output feedback control circuit (only voltage feed forward is needed). Anybody want to try that with the old Buck in open loop operation? Cuk-Buck2 as you can see is RADICALLY DIFFERENT, and there is yet more to this topology.

Here is another secret! During a load transient the o/p load current loop path through o/p capacitor, o/p winding, and diode, is reflected backwards into the primary current loop path through sync MOSFET, primary winding and diode, through the transformer’s mutual coupling in true Faraday transformer fashion. In essence due to Lenz’s law input and output magnetomotive forces on the core cancel out, and thus transient impact on the core material is insignificant. Can you see one important big difference with the tapped-inductor Buck? How could one have a backwards reflected o/p load current transient appearing on the input side of the transformer in the OFF period when that winding-end is open circuit? This disconnect in part explains why transient response of the tapped-inductor Buck is poor. Now the single cycle transient response of the Cuk-Buck2 may be better appreciated, since the input winding IS NOT OPEN CIRCUIT as in the old Buck. So do you still think the Cuk-Buck2 is old topology? No sir it is brand new with fantastic transient response.”

Slobodan Cuk – CEO at TESLAco

“@Clifford Emeric Clifford, you are one of the very few who fully understand and appreciate unique features of this converter. I provided in this latest revision more simulation data to support your analysis. Thank you.”

Muhammad M. Roomi (Dr) – Postdoctoral Research Fellow

“I am recent PhD scholar who worked on single stage energy conversion and Sir, with much respect this is one fine piece of invention.”

Slobodan Cuk – CEO at TESLAco

“@Muhammad M. Roomi (Dr) Glad to hear that you recognize the importance of the single-stage conversion and are making your own contributions! I have completely revised this article to provide further detailed derivation and explanations. I would appreciated our comments after you have reviewed this latest revision.”