-to-converters to which this principle applies are described. New set of coupled
inductor converters is presented.ABSTRACT. The new simple physical
description of zero-ripple technique in Switching Mode Power Supplies is proposed. Basic
types of-to-converters to which this principle applies are described. New set of coupled
inductor converters is presented.
There are three well known basic types of-to-converters. Usually their
output contains a capacitor, to which a load is connected. At switching frequencies the
output capacitor has insignificantly low impedance in comparison to load resistance, so, a
single equivalent element, having voltage Vo and low resistance, can be assumed as voltage
source as shown on Fig.1.
The basic topologies are far from ideal due to both input
and output non zero current pulsations. The most common way to decrease high frequency
pulsations is to apply additional LC-filters to the output . Unfortunately this technique
not only makes
heavier, but also slows down their dynamic response.
A much better way way to minimize pulsations is to include coupled filters, which can easily be integrated into converter topology. The best known device, implementing such feature, is Cuk-converter (Fig. 3a) [IEEE Transactions on Magnetics, Vol. MAG-19, No. 2, March 1983, pp. 57-75]. Depending on the relationship between the coupling coefficient (k) and the effective turns ratio (n), the converter provides either an input or output with zero-ripple. Isolated modifications of converters with EI-core inductance from the same article are shown on Fig. 2 with input (2b) and output (2c) zero-ripple current.
Although Cuk-converters were widely covered in various technical articles, my own experience showed me that some blank spots in the field still existed. Although two-windings Cuk-topology behaved in strict accordance to the theory, I couldn't manage zero-ripple currents in any of multiple windings converter structure with a common toroid core. So I started my own investigations, the main results of which are presented below. Note, that I observe only common core inductance converters.
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It is known, that zero-ripple feature in two windings coupled filter can be achieved on condition of equivalent values of effective turns ratio (n) and coupling coefficient (k). Unfortunately this purely mathematical identity hides the physical nature of zero-ripple effect. So I present my own explanation of the phenomenon. Assume Lm - the value of Buck-converter filter inductance (Fig. 1a) with wo turns.To implement filter zero ripple output current, let us make an arbitrary tap in the winding of inductance and then connect an external inductance Lext between it and the load tap (Fig. 3a).
This combination of two inductances is equal to single inductance with equivalent value: Lo=LextЧLm/[Lext+(1-Np)ІЧLm], where Np=wp/wo. As if Lext is connected to small part of turns wo, the voltage applied to Lext is ws/wo times of Lo. So for any fixed tap, there probably exists such Lext, whose ripple current (DiLext) is equal to input ripple current (DiLo) of equivalent inductance Lo. This would mean, that ripple current in winding ws is equal to zero. So, solving the simple equation: DiLo=DiLextЧ(1-Np)ЧLo/Lext, we finally get the condition for zero ripple current in winding ws: Lext=NpЧ(1-Np)ЧLm, where Lm=Lo/Np. Our next move is to make the ripple current of Lext bypass the load. For this case we add capacitor C in series with Lext (Fig.3b). If C is an ideal one with infinite capacitance, its voltage is always equal to output voltage Vo, so nothing changes in the filter behavior after modification, except ripple current no more flows through the load. Fig. 3c-d shows other equivalent modifications of the same coupled filter. The main advantage of the above simple analysis is that it can easily be corrected for real elements, containing parasitic parameters. Contrary, attempts to derive the zero ripple condition from known k=n identity for non ideal elements is not so obvious in the case. More general analysis including some parasitic parameters is to be obserwed below.
Further I'll show, that in order to implement zero-ripple current in multiple winding single core inductance, the condition k=n is necessary but not enough.
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While learning more about coupled filters, I tried to apply this principle to any basic topology from Fig. 1. As a result, it appeared, that every basic topology has at least two modifications with either input or output zero-ripple property. Below I present the summary table of all possible modifications of zero ripple converters and their equivalent schemes (Fig. 4). Although I derived that schemes myself, most of them were proposed earlier by Anatoly Polikarpov (Moscow Power Energy Institute). The only exclusion is Buck I, which I could not find in available literature. So I claim no authorship for these topologies except collecting them together and proposing equivalent schemes, which can be assumed as simplified versions of the originals.
As can be seen from the above pictures, Flyback converter has four zero-ripple modifications, while Buck and Boost have only two each. It is also easy to notice that some converters are topologically dual on condition of interchanging diode and transistor and corresponding input and output voltage sources. For example Buck I is dual to Boost II, Boost I is dual to Buck II, Flyback Ia dual to Flyback IIa, Flyback Ib dual to Flyback IIb. This observation was very helpful in generating the complete set of zero-ripple converters. Note that Flyback Ib and IIb represent classical Cuk converter. Earlier equations for Lo and Lext are still valid for any of the above topology. On assumption C as ideal capacitor, the analysis of the above schemes is similar to that of Fig. 1. Parasitic parameters of real elements can easily be added.
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As if in a converter with two windings inductance, one of them always has to pass ripple current, the main idea of adding a third winding is to obtain both input and output zero-ripple. Let us see what happens after adding third winding to an inductance. The only topology applicable to this procedure is Cuk converter from Fig. 2a with equal turns in both windings. Splitting capacitor in two series ones and connecting their common point to the ground through the newly added third winding we do not infringe volt-second balance on inductance [see the above reference]. The additional winding is ideal for passing ripple current. According to above analysis the new winding has to have less turns and contain corresponding Lext in series, as shown on Fig. 5a. From the equivalent scheme Fig. 5b it can be seen, that again we have the only winding ws with zero-ripple current. So, can the three winding Cuk converter provide the desired zero-ripple function in reality? The answer will be "no", because zero-ripple current of ws is sum of both input and output counter-phased ripple currents, due to C1 and C2 finite capacitances. Note that even ideal C1 and C2 with infinite capacitances do not guarantee ripple extermination, because windings are another cause of non zero input and output ripples.
So, are there any ways to obtain both input and output zero-ripple feature? The answer is "yes". As if the ripple current circulates in Vo-Vi contour, inserting additional choke in series with Vo or (and) Vi can dramatically decrease the ripples. We can see now that single core multiwinding coupled inductance always provides the only equivalent cirquit branch with zero ripple current. So it can be stated that converter with N zero-ripple outputs needs to have N+1 winding inductance and at least N-2 separate chokes connected in series with any output.
It is clear, that additional choke have large dc current bias, and on condition of large values can severely increase size and weight of resulted converter. However there is a simple method of reducing dc current component in the above scheme. Implementing both additional chokesas single core coupled inductance Lf solves the problem, because direct currents in each winding have opposite directions. The final set of non isolated converters with both input and output zero ripples is shown on Fig. 6.
On condition Vi/Vo=w1f/w2f the bias current completely disappears, and Lf can be made insignificantly small.
It may be asked now, why nobody uses similar technique in
basic converters from Fig. 1. Although adding coupled inductance filter Lf to any of basic converters decreasesbias, both
input and output ripples on the contrary will be greatly increased. This is due to major
difference in topology structure, when there is no common trip of ripple current through
input and output of a converter.
Now let us turn to isolated converters.
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It may seem that adding a transformer to Fig. 2a converter automatically provides zero ripple feature in the resulted schemes Fig. 3b-c. Assuming for simplicity that transformer has equal turns in primary and secondary windings and incorporating Lext in Fig. 3b we get an isolated topology Fig. 7a. After obvious equivalent transformations (Fig.7a®b®c) we finally come to equivalent scheme Fig. 7c.
As can be seen from Fig 7c, zero-ripple current of ws flows through converter output only when transistor is off. When it is on the whole input ripple current flows through the output. It can also be shown that converter on Fig. 3c also lacks the zero-ripple feature. Although the condition k=n is valid in both cases, we can see now that it is not enough for providing zero ripple current in some two windings converter topologies. So the second necessary condition for such topologies is quite obvious and can be formulated as follows: the zero-ripple lead of converter inductance must have no nodes on its way to input or output in equivalent scheme.
To overcome the problem in the above topologies we have to incorporate Lext both in primary and secondary part of isolated converter. Fig. 8 shows fixed versions of isolated zero-ripple converters and their equivalent schemes.
As was shown beforehand, the best way to choose turns ratio
of transformer is from the equation: w2../w1..=Vo/Vi, where dots represent identical indexes. As far
as Lext and Lf in Fig. 8c have nobias, converter weight and size is compared to any
of previous ones. For better tuning the sensitive zero-ripple feature, it is recommended
to implement Lext on pot core with regulated
bar.
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It seemed interesting to me to compare the effectiveness of classical T-type LC-filter and coupled LC-filter containing identical elements (Fig. 9) and having equal input alternating voltage. Additional resistor on Fig. 9 represents parasitic resistance of Lext. Let us estimate each filter efficiency with a quality coefficient which shows the degree of reduction input current ripple compared to filter output. This coefficient is counter proportional to filter transfer function. Numerically it is equal to resulted module of output current divided by input current. Although input voltage inPWMconverters is practically square, sinusoidal vaweform make analysis much easier.
On assumption of linear elements, the quality coefficients
for each filter are frequency dependable:
.
As can be seen from the last expression, true zero-ripple feature can never be achieved in
real coupled filter due to non zero value of r. More over, even on condition of r=0, zero
ripple filtration is available only for a single frequency, corresponding to a single pole
of the above function. So, we must provide Kcoup=Ґ at nominal
converter frequency wnom=2pfnom, to which corresponds more precise expression for
external inductance value :
.
After substituting this result in the above expressions we get final formulas. To estimate
filtering efficiency of coupled filter over classical one we shall use relative
coefficient Keff=Kcoup/Kclas. Fig. 10 presents calculated
and experimental results for both filters.
It is seen from the graphics, that at resonance frequency
the coupled filter is nearly 20 times more effective than classical one (and rises further
with decreasing Np), but beyond the frequency
value fcrit its efficiency dramatically
decreases. This critical frequency can be found by solving equation Kcoup=Kclas numerically. However for
further generalizations we better need symbolic solution. So, assuming Lm=Ґ we can
greatly simplify the resulted expression and provide a sufficient preciseness (few
percent):
.
The above assumption also allows to obtain very simple expression
for relative filter efficiency:
.
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Due to strong resonance character of coupled filter
transfer function, coupled inductor converters are best for use as constant frequency
power supplies. Though it is possible to decrease frequency susceptibility by
increasing value of filtering capacitor C (minimize the second addendum in Lext expression) this will cause decrease of overall
efficiency (Keff) of coupled filter (see the
last expression). So, classical filters are more effective for
power supplies with wide frequency range.
Unfortunately such results are far from optimistic to me,
because I develop off-line power supplies based on zero-voltage switching technique only,
all of which are oftype. So I stopped
my researches in the field and till now produced no practical versions of my zero-ripple
converters. However I want to believe that my Flybacks from Fig. 8 are highly perspective,
especially both sides zero-ripple "c"-one...
© Igor Gorianski
September 1997
Revised
26.11.97
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