Perfectly Coupled-Inductor SEPIC converter
Reduces to WHICH Converter?
Analysis of Equivalent Circuit Transformations
by Professor Slobodan Ćuk
Perfectly Coupled-Inductor SEPIC Converter
Abstract
For quite a while now, the various “shortcuts” were circulated on LinkedIn media, such as IEEE Power Electronics Society and various websites extolling the virtues of the PWM switch small-signal model! In particular, the claim is made that the “magic shortcuts” could reduce frequency response of a Perfectly Coupled-Inductor Nonisolated SEPIC converter (CI Converter from now on!) to that of a Boost converter as included in a prior discussion to this group!?
However, the detailed analysis bellow reveals:
- The PWM small-signal switch model is not even needed! The sequence of correct equivalent circuit transformations would, even without any small-signal model, reduce CI Converter to ordinary nonisolated polarity inverting buck-boost converter! The addition of small-signal model only contributes to hide transformation mistakes.
- It then follows, even without small-signal model that CI Converter frequency response is that of buck-boost converter and not boost with its right-half-plane zero! No magic there either!
- This analysis points out to the critical incorrect circuit transformations leading to this erroneous result.
- It also points out at other incorrect simplifying assumptions, which led to boost converter circuit!
- As invalid transformations and simplifications are correctly identified here the correct final result of the nonisolated flyback converter is obtained!
Checking for contradictions of final result
This analysis also points out that when one obtains “surprising” result like boost frequency response, one should thoroughly scrutinize for obvious contradictions! In this case, it is simply impossible that the final result could have small signal frequency response of the boost converter and DC conversion gain of an entirely different buck-boost converter as claimed by PWM switch model!?
Detailed Analysis of CI Sepic converter
This analysis will also introduce truly “magical shortcut”, which can be used to determine what switching converters qualify for implementation of the Coupled-Inductor and Integrated Magnetics Methods (see 1979 patent posted to this group)!
Fig. 1 Basic SEPIC converter and its AC circuit
Test to verify Coupled-Inductor applicability?
First, we establish the fact that the qualifying converters must have at least two inductors before coupling. Coupling of Inductors does not change DC voltage gain which existed before the coupling. Instead, coupling only affects the distribution of the AC ripple currents between two winding as I discussed in numerous publications and in my patent on Ćuk converter and Coupled-Inductor SEPIC converter (see thread posted before!)
What makes given switching converter converter eligible for implementation of Coupled-Inductor Method I introduced over 44 years ago!
Fig.1 has original SEPIC converter! By shorting capacitors and DC voltage source, we obtain bottom AC circuit model. This is a true magic! Note that the two inductors are connected in parallel in AC model! Note also that the two switches, ideal switch S and diode CR, will generate identical common rectangular-wave voltage drive for any duty ratio D. This then qualifies both inductors to be placed on a common core in a transformer-like configuration. Note also that there could be only one turns ratio that of 1:1. Arbitrary turns ratio is not allowed, unlike in real AC transformer. Another difference is that the DC currents of each separate inductor are added together as dictated by the dot connections!
The two parallel inductors could obviously be replaced with a single common inductor L in case of a perfectly coupled winding with no leakage, when L1=L2 = L. The trick is now to determine the correct way of implementing such coupling in original DC-DC SEPIC converter. However, we first introduce the two incorrect transformations!
Two incorrect transformations
Fig. 2 further explains that point. Common inductance can be shown in circuit model as either on primary side (top circuit) or on secondary side (bottom side) of the ideal 1:1 isolation transformer! Note that marked termination points 1 & 2 for L1 and 3 & 4 for L2 determine also correct dot connection of the isolation transformer. Hence two possible circuits could be contemplated as in Fig 2 below.
Fig. 2 Two incorrect transformations with floating transformer (no common ground)
Model in which common inductance was on a secondary side (bottom drawing) is obviously wrong, since the input voltage source is NOT connected to converter at all!) On would then think that model in Fig. 2 (top) in which common inductance is retained on primary side would be OK, since now input source is connected and one would have big sigh a relief. This is, however, also wrong! Both transformations have the same problem! Both inductors in original topology are floating inductors and cannot be directly replaced with either of two inductor circuit positions.
It is interesting to note that the converter of Fig. 2 (top) is does not have steady-state and hence it does not even exist! The capacitor Cc is charging during OFF- time of switch but lacks discharge path!?
Wrong boost equivalent circuit model?
It is interesting how a desire to further simplify the wrong model of Fig.2 (top) leads to wrong boost converter model. The dubious argument is made that capacitor Cc for some reason (!?) does not participate in power conversion (although it did in SEPIC converter!) and can be therefore eliminated by shorting it to result in boost converter or Fig. 3!?
Fig. 3 Wrong boost converter model using false assumption that capacitor Cc can be SHORTED?!
This is PWM switch has led to incorrect conclusion that the Perfectly Coupled SEPIC converter using PWM switch model will have-boost frequency response!?
Correct Transformation
However, the input inductor can be relocated to the bottom leg so that terminal 1 and terminal 4 are common forming a 3-port network with a common terminal 1 & 4. Now, both transformations discussed above have the same identical correct solution obtained by shorting terminals 2 and 4 as in Fig. 4.
Fig. 4 Relocation of the input inductor to bottom leg and shorting the other two terminals 2 and 3.
This ultimately placies voltage source across coupling capacitor Cc and results in buck-boost final equivalent circuit. Note that capacitor Cc is now placed directly across ideal voltage source. This is a real reason why coupling capacitor Cc disappears in the correct buck equivalent buck-boost converter. The fact is, that in the case of the nonideal voltage source this capacitor would remain in the model and would result in buck-boost converter with inclusion of capacitor Cc and third-order dynamic model!
Final correct buck-boost equivalent circuit model
Fig. 5 Final correct buck-boost equivalent circuit.
Conclusion
Number of incorrect transformations and incorrect simplifications like shorting capacitor Cc led to wrong boost small-signal model!? Moreover, after deriving incorrectly boost small-signal frequency response, Vorperian continues to prove correctly that the DC voltage conversion is that of buck-boost!?
This would be the very first converter ever, which has frequency response of boost converter and DC voltage gain of buck-boost converter! That alone should have given the red flag to Vorperian and Ridley that something went amiss in their “magic” shortcuts!
Note regarding PWM switch model!
PWM switch model starts, as in this example, with small-signal model first, while steady-state DC model is always an after-thought!?
Yet, it is steady-state model which defines equilibrium around which small signal perturbations are made and should be the starting point!
State-Space Averaging Model
This is the case with State-Space Averaging Method, which determines first correct general steady-state (DC) model for any PWM switching converter, for those known 50 years ago as well as those invented ever since! It does not depend on “shortcut tricks” to massage the switch into an artificial 3-terminal double pole single-throw switch!
What about isolated converters or even 3-switch converters with isolation like forward and flyback converter with voltage clamp? Note that 3-switches cannot be manipulated into a single-pole double-throw ideal switch. The isolation in flyback and forward simply makes any such PWM switch transformation impossible and/or incorrect!?
Think how you would find correct frequency response of the Coupled-Inductor Isolated Sepic converter or even Integrated Magnetic SEPIC converter. With PWM switch model it is impossible. With State-Space Averaging Method this becomes a trivial extension by describing correctly mutual coupling inductances in matrix formulation.
Myth of the Moving Time Average!
Ever since my introduction of State-Space Averaging method in 1976 PhD thesis and 1976 IEEE conference paper, many revision attempts were made to explain averaging circuit method. They all followed original justification for boost small signal model in Gene Wester PhD thesis invoking “moving time average! It was claimed ever since that this somehow eliminates the ripple currents and voltages and results in smooth ripple-free continuous waveforms.
This continues even to the present day even though it has no scientific basis whatsoever! Another related argument is that just replacing PWM switch with dynamic model circuit gives you better “circuit insight” equal lacks any scientific basis!
It appears that my advisor and mentor late Prof. David Middlebrook was the only one who correctly understood my State-Space Averaging Method! In the preface of our 1981 first 2 volume paperback edition of Power Electronics books Prof. Middlebrook lamented:
“… If the models for all such converters are the same, it should be possible to derive this unique model without having to specify in advance any particular converter. This problem was solved in a very elegant manner by Slobodan Ćuk. In his 1976 PhD thesis he introduced the analysis method of State-Space Averaging which in a single stroke eliminates the switching process from consideration and exposes the desired dynamic response. From this model came the same unique small signal equivalent circuit model, which is now called the canonical model. Again, with the clarity of insight, the form of the model becomes “obvious”. It contains the three essential properties of any DC to DC converters, namely DC conversion, low pass filter and conversion ratio adjustment by a control signal.”
Extension of SSA to Hybrid Switching Method
State-Space Averaging is naturally extended now to special kind of resonant switching I introduced and called Hybrid Switching Method! This method has all the benefits of resonance but using duty ratio PWM control. It does also have 3 switches (one transistor on primary and two diodes on secondary) which automatically disqualifies PWM switch model! In addition, the resonance is essential as it fundamentally changes DC conversion ratio! PWM switch model as its names says is limited to PWM switching converters with no resonance and with special limiting single-pole double throw switch with a common terminal!
This SSA extension is not suitable for classical true resonant and quasi-resonant switching converters and latest LLC converter topologies, in which resonance extends across both switching subintervals. This is just fine, as these resonance methods are in many ways deficient to Hybrid Switching Method which contains the resonance in only one subinterval!
Conclusion
The PWM switch model and other preferable” shortcuts” are just a myth propagated by engineers who do not want to invest time to learn and expect that SPICE model of such simplified circuit model (albeit wrong) will give them the fast and correct results. Recent 2018 article on modelling says: “Compared with State-space averaging model, the PWM switch model is simpler and circuit oriented with physical insight.” No one ever explained what those better circuit and physical insights are?!
Another myth is that State-Space Averaging Method is very complicated and should not be used. The truth is that this method will give you correct analytical results for both steady-state and dynamic model for all converters of practical interests!
Would you ever want to have perfect coupling?
The answer is no, never! The question remains to be explained why this is the case if the above analysis makes the case for simpler 2nd order dynamic model of Perfectly Coupled SEPIC converter!?
