CA2619872C - Methods and configurations of lc combined transformers and effective utilizations of cores therein - Google Patents

Abstract

The LC combined transformer is a combination of capacitors, inductors and an electrically-isolated mutual inductor, i.e. conventional transformer, which in principle is a unity-coupled mutual capacitor or a cascade connection of an ideal transformer and unity-coupled mutual capacitor(s). To improve the imperfections of the widely-used transformers, by employing the simplest passive-circuit design to attain a perfectly-functional match between mutual capacitors and the mutual inductor, this invention achieves the optimal features of current or/and voltage transformation, and introduces a new function of waveform conversion from square-wave to quasi-sinusoid. The ideal current transformer herein suits sinusoidal current measurements, the ideal voltage transformer herein suits sinusoidal voltage measurements, and they also could be upgraded to ideal transformers for both current and voltage transformation. They can be designed as both energy transferable as well as waveform convertible, being applied in power systems and/or power electronics. Herein also states the design approach of integrated inductor and mutual inductor, and the use of push-pull inductor, materials being fully utilized and sizes decreased.

Description

METHODS AND CONFIGURATIONS OF LC COMBINED TRANSFORMERS
AND EFFECTIVE UTILIZATIONS OF CORES THEREIN
SUMMARY OF THE INVENTION
This invention relates to a transformer which is a combination of capacitors, inductors and an electrically-isolated mutual inductor (namely, conventional transformer), and called LC combined transformer which, in principle, is a unity-coupled mutual capacitor or a cascade connection of an ideal transformer and unity-coupled mutual capacitor(s).
Realizations of the present LC combined transformer of this invention can be divided into three fundamental categories or types according to their functional focuses: current transformation category/type (ideal current transformer), voltage transformation category/type (ideal voltage transformer) and, voltage and current transformation category/type (ideal transformer); besides, though to some extent, they all have the function of waveform conversion from square wave to quasi-sinusoid. Aiming at the imperfections of the widely-used transformer in practical engineering, the invention presents some improvements in principle employing the easiest passive-circuit design approaches to realize the optimum characteristics of current or/and voltage transformation that eliminate the reactive error in principle, optimize structural parameters so as to reduce real-power loss error to minimum, as well as limit non-linear errors of both the inductors and mutual inductor. To ensure the realizations of their best features, this invention also details the needed specific device selections, linearization processing of inductors, and the integration design approach for the coils and magnetic cores of the inductor and the mutual inductor, not only to achieve in compensation of the errors comprehensively, but also in cost savings with the goal of small devices. The ideal current transformer designed by this invention is suited for sinusoidal current measurement; the ideal voltage transformer suited for voltage measurement; and further upgraded can evolve them into both voltage and current transformation, to realize power transferred with voltage and current in-phased, decreasing the ac line reactive current. The invention also introduces into the designs the new characteristic of waveform conversion from square-wave to quasi-sinusoid, by which the transformers for both waveform conversion (or waveform isolation) and power delivery can be designed, suitable for applications in power systems and/or power electronics, such as in dc transmission, the passive filtering of ac voltage or current, etc. Meanwhile, the use of push-pull inductor is brought out, which is a solution to the problem of the core's unsymmetrical magnetization in double-ended converter under the alternately driving and also an improvement on the issue of cross-conductance of the driving switches.
BRIEF DESCRIPTION OF THE DRAWINGS
The following drawings, which form an important part of this specification, aid to elaborate the presented invention in details:
Fig. 1 (prior art) is a schematic circuit symbol of a mutual inductor (or conventional transformer) and its equivalent circuit diagram expressed by using an ideal transformer.
Fig. 2 is the diagram of general circuitry arrangement of the LC combined transformer and those of its equivalent circuits for lossless analysis and for loss analysis.
Figs. 3 (a) and (b) are diagrams of the equivalent circuits for lossless analysis and for loss analysis of current transformation-A type of the LC combined transformer (Ideal Current Transformer A); (c) and (d) are diagrams of the configurations employing the design approach of integrated inductor and mutual inductor.
Figs. 4 (a) and (b) are diagrams of the equivalent circuits for lossless analysis and for loss analysis of current transformation-B type of the LC combined transformer (Ideal Current Transformer B).
Fig. 5 gives diagrams of the equivalent circuits for lossless analysis and for loss analysis of the in-phase mode of voltage transformation type of the LC combined transformer.
Figs. 6(a), (b) and (c) are diagrams of the equivalent circuits for lossless analysis and for loss analysis of the anti-phase mode of voltage transformation type of the LC
combined transformer; (d) is that of its configuration employing the design approach of integrated inductor and mutual inductor; (e) is the simplest diagram when coLb ¨ licoCb = O ; (f) is the diagram when coLb, = co-Lb¨licoCb> O; (g) is a diagram for (f) when the integration design approach of inductor and mutual inductor employed.

2 Figs. 7 (a) and (b) are duplicates of Figs. 5 (a) and (b); (c) is a diagram of their equivalent circuit expressed by employing an ideal transformer; (d) is for (c), when coCp2 = , namely, Eq.(60) satisfied, evolved into the equivalent circuit diagram of in-phase mode of voltage and current transformation type of the LC combined transformer.
Figs. 8 (a) and (b) are duplicates of Figs. 6 (a) and (b); (c) is a diagram of their equivalent circuit expressed by employing an ideal transformer and also of the trends or methods evolving to be an ideal transformer; (d) is in (c) with a compensation capacitor, like Cp , Cpa or Cpb inserted in parallel connection to satisfy corresponding one of Eqs. (66), (67) and (68), the evolved equivalent circuit diagram of anti-phase mode of voltage and current transformation type of the LC combined transformer (ideal transformer); (e) and (f), respectively corresponding to Figs. 6 (f) and (g), are diagrams of the ideal transformer configuration.
Fig. 9 (a) is a duplicate of Fig. 3 (a); (b) is a diagram of its equivalent circuit expressed by employing an ideal transformer; (c) is in (b) with a compensation capacitor, Csa or Csb , inserted in series connection to satisfy either Eq. (72) or (73), the evolved equivalent circuit diagram of voltage and current transformation-A type of the LC combined transformer (Ideal Transformer A).
Fig. 10 (a) is a duplicate of Fig. 4 (a); (b) is a diagram of its equivalent circuit expressed by employing an ideal transformer; (c) is in (b) when n, = k, namely Eq. (78) satisfied, the evolved equivalent circuit diagram of voltage and current transformation-B type of the LC combined transformer (Ideal Transformer B).
Fig. 11 (a) is the diagram of a principle and experimental circuit using either Fig. 5 or Fig. 7 to implement the waveform conversion from square-wave to quasi-sinusoid; (b) is an improved circuit upgraded from (a) by employing the push-pull inductor; (c) are the control and drive signals used for transistor switches in (a) and (b); (d) is the hysteresis loop of the core of inductor L a in (a) in steady-state operation; (e) is the hysteresis loop of the core of inductor 28a in (b) in steady-state operation.
Figs. 12-1 - 12-9 are illustrated drawings for "6-4. Principle of the Mutual Capacitor':
Fig. 12-1 The schematic symbols for current type of mutual capacitor: (a) referenced in common; (b) referenced separately.

3 Fig. 12-2 The schematic symbols for voltage type of mutual capacitor: (a) referenced in common; (b) referenced separately.
Fig. 12-3 Circuit realizations for current type of mutual capacitor: (a) in delta (A) or pi (u) configuration and referenced in common; (b) equivalent from (a) but referenced separately.
Fig. 12-4 Circuit realizations for voltage type of mutual capacitor: (a) in tee (T) or wye (Y) configuration and referenced in common; (b) equivalent from (a) but referenced separately.
Fig. 12-5 Equivalent circuit diagrams of the unity-coupled mutual capacitors by employing controlled sources: (a) current type; (b) voltage type.
Fig. 12-6 Circuit symbols for ideal mutual capacitor: (a) current type; (b) voltage type.
Fig. 12-7 Equivalent circuit diagrams of the unity-coupled mutual capacitors by employing ideal mutual capacitor, to clarify those in Fig. 12-5: (a) current type; (b) voltage type.
Fig. 12-8 Example for making the dual of a circuit with a coupling component: (a) the primary;
(b) how to make the dual; (c) the dual.
Fig. 12-9 (Prior art) Two realizations of Negative Impedance Converter (NIC) for negative capacitance, and the equivalent circuit.
BACKGROUND OF THE INVENTION
It is well known that the electric transformer, i.e. the conventional voltage/current transformer, widely-used in electrical engineering is actually a mutual inductor, i.e. Tr in Fig. 1(a), with its coupling coefficient k less than but close to 1 [which is also the prior art of this invention]. In order to address this issue more clearly, for the time being, let's review its electric characteristic equations when neglecting power loss. As the port variables of a mutual inductor supposed as corresponding to those illustrated in Fig. 1(a), in electrical theory, its electrical characteristic equations in a sinusoidal steady-state circuit are presented as {VI =- jcoLl I 1 ¨ j coM /2 (1) V2 = j COM /1 - j (OL 2 '2 (2) where LI and L2 respectively represent self-inductances of the primary winding and the secondary winding of the mutual inductor, M is the mutual inductance between them both;
CO = 27cf . And attention must be paid to its coupling coefficient k and turns ratio n, which are defined as

4 kM
= __________________________________________________________________ (3) VL,L2 N, \IL, n = = ¨ (4) Obviously, the mutual inductor in Fig. 1(a) can be electrically equalized as in Fig. 1(b) [Note: Fig. 1(c) is also an equivalent circuit], with its equations accordingly equivalently transformed as follows:
Vi, = V, - jco(1-4,/, = jc.okL, I, - jcok\IL,L2 /2 = AjC0k j:i /1 - jC0kX/2) { (5) Vb = V2 jC0(1 -142/2 = jC0k-ILIL2 /1 - jOkL2 /2 = 1.T.2.(jC0kITL7/1 - fa/kV-LT/2) (6) , 1 1 = v. + L2 r2 + = v a .
¨I
1 /2 = /0 + -1 /2 (7) jokL, jL1 jeokL, n n From Eqs. (5), (6) and (7), for a practical voltage/current transformer or mutual inductor, its voltage Va V
ratio is n = N1 = # -1, and its current ratio is /, =10 + -1 /2 # -1 /2 , (/0 #O); which means N2 Vb V2 n n that it is actually not precise either using a voltage transformer for voltage transformation or using a current transformer for current transformation, and that errors exist in it substantially, which is determined by the deficiency in its structural principle. The error caused from its leakage inductances (1-k )L1 and (1-k )L2 and magnetization inductance kLI is called reactive error [Note:
Reactive error not only worsens the transforming precision but also increases reactive current of the supply so as to cause power loss and wastes for transmission line materials]. In addition, there exist the power-dissipation error, or resistive error, from its copper loss and iron loss as well as its non-linearity error from its non-linear cores. Therefore, to meet its required precision, the conventional transformer had to resort to lots of methods for improvements while being designed.
Furthermore, due to complexity of the network loads, there disperse great numbers of high-order harmonics in the supply network. The high-order harmonics not only contribute to energy wastes but also endanger the safety of facilities and loads, causing misoperations and mishaps, seriously interfering with signal transmissions. The conventional transformer is powerless against high-order harmonics except for its insulations threatened and cores overheated. It would have been a dream that, provided that only a few of passive components are added, it could come out that the conventional transformer will become one both transferring power from input to output and also functioning as harmonic isolation from between, i.e. a function of waveform conversion from square-wave to sinusoid being added. It was just a matter of regret, being long expected but not realized, in the past.
DESCRIPTION OF PRIOR ART
ln engineering, The simplest design approach to achieve the transformer's precision is to alter the turns ratio of primary winding N1 to secondary winding N2 , or ratio n of the transformer. Seen from Va above equations, even though n=¨= N1 VI
#¨ , if the value of (NI/N2) or n is a little bit altered in Vb N2 V2 design, to make it true as V= n to a certain extent, it is said as the needed ratio n. This is a simple and effective method, being employed by almost all the manufacturers and designers, but with a limited applicability. Practical experiences and further analysis discover that the three errors stated above, especially the reactive error, cause the transformation precision of the conventional transformer seriously depending upon variations of its burden and also upon the working frequency co of sinusoidal current [Note: It is an easier discussion assuming that the burden is a purely resistive load].
It is why this method is mainly used to design only transformers for power delivery or those for monitoring measurements of low level.
Also, particular designs of transformers aiming at particular applied situations are employed, such as changes in structures, styles, shapes, positions or materials of the cores, windings, and even enclosures; either utilizations or improvements of the coupling between electric fields or magnetic fields, of the shielding, or even of the insulating measures, and the like; so as to obtain some particular properties, including transformation precision, of the transformer functionally improved.
Lots of patent literatures lay in this category, with those Canadian and US, such as CA1278350, US6879235, US6903642, US6611408 etc.
With the rapid development of electronic technology in the recent three decades, attempts of improving the transformer's precision assisted by electronic designs have succeeded greatly resulting in patents everywhere in the world, of which some exploiting piezoquartz or Hall sensors, some employing ultrared or laser transmission, or being controlled by micro-computers; whose advantages includes high transformation and measuring precisions obtained for transformers but with inconveniences that electronic circuits need independent power supplies, that they are complicatedly designed, and that they have a relative low reliability while working in inferior circumstances. Patents fallen in this category includes CA2021712, US6989623, US6965225, US6348768, etc. They all are different in principle from this invention, no need to be further discussed.
In current engineering, applications similar to this invention herein are the capacitor divider and the capacitor voltage transformer; of which it is the latter in particular that operates at a principle basically the same as one of the applications of this invention, i.e. the in-phased ideal voltage transformer, does, with a difference that the capacitor voltage transformer was designed from realization of the compensation to the capacitor divider to a final object of the voltage transformation while this invention is focused on realization of some function of a subunit network, or mutual capacitor Note: refer to 6-4. Principle of the Mutual Capacitor], and finally to achieve an LC combined transformer functioning as an ideal transformer of voltage and current transformation. Examples related to the capacitor voltage transformer in the Canadian and US patent literatures are instanced as: CA1104650, CA838786, CA1137556, US6919717.
Moreover, only one Canadian and US patent addressing waveform conversion from square-wave to quasi-sinusoid is found as CA1104661 (US4250455), which is actually on active filtering technology. Of this invention, some other applicable techniques and approaches are accessorily introduced, especially the design approach of integrated inductor and mutual inductor and the usage of push-pull inductor, on which nothing was found in the preliminary search.
DETAILED DESCRIPTION OF THE INVENTION
The general circuit arrangement of the LC combined transformer is illustrated as in Fig. 2(a) or (b), with the load not included. Circuit components 1 and 3 are inductors La and Lb, with inductance value > 0 meaning positive, and the value = 0 meaning short-circuited. Circuit components 2, 4 and are capacitors Cm, Cb and C, , with capacitance value > 0 meaning positive (including C
short-circuited), and the value = 0 meaning open-circuited. 6 is the core of the mutual inductor, 7 its primary winding N1 (with inductance LI> 0), and 8 its secondary winding N2 (with inductance L2 > 0) and, 6,7 and 8 constitute a mutual inductor Tr or 26 (shown in the dotted-line box) or conventional transformer whose cores must be linearized. All the circuit components and the mutual inductor herein can be real devices, although their magnitudes or values may be worked out respectively by one or more components based on the principles of series-parallel connections, with their application equivalent for the definition herein, and with the corresponding power loss. Their electrically-interconnections are: One end of inductor 1 and one end of capacitor 2 are together connected to one end of inductor 3; the other end of 3, one end of capacitor

5, and one end of capacitor 4 connected to each other; and the other end of 4 connected to one terminal of the winding 7; and the other end of 7 connected to the other end of 5 and to the other end of 2, taken for the common terminal; designating the other end of 1 and the common terminal as the input port of the LC combined transformer, two terminals of the winding 8 being designated as its output port; and with the stipulation that input and output ports herein can be designated at will when needed. Where capacitor 5 should be maybe as it is seen herein, or equivalently moved if needed to parallel with the input or output port. And when capacitor 5 moved away or open-circuited, the position of capacitor 4 may be interchanged with that of inductor 3, or equivalently moved to series with the input or output port owing to doing so with the circuitry function unchanged except for a different magnitude. The mutual inductor Tr or 26 (or transformer) is a double-winding, and it can also be a multi-winding, as long as it can be theoretically converted to a double-winding mutual inductor and utilized of this invention. Any circuit designed out of the configurations of this invention must be working under the circumstance with a constant frequency co (or f) of periodical sinusoidal wave or of a periodical wave at least unless in peculiar applications.
The technology scheme of this invention lies in that by utilization of the mutual inductor 26's leakage inductances 9 of (1¨k )L, and 11 of (1¨k )L2 and the magnetization inductance 10 of kLi, mated with externally connected capacitors or/and inductors, in accordance with the principle of the mutual capacitor [Note: As a lumped-constant circuit element, a mutual capacitor is a brand-new ac two-port network component whose performance is completely dual to the known mutual inductor. See "6-4. Principle of the Mutual Capacitor"], one or two cascaded mutual capacitors can be constructed with the function of ideal current or voltage transformation; and also being cascaded with the ideal transformer 27 which is peeled off the leakage and magnetization inductances 9, 10 and 11 from 26 and enclosed in the broken-line box, an ideal current transformer, an ideal voltage transformer or an ideal transformer can eventually be achieved.
Fig.2(b) is the diagram of equivalent circuit for lossless analysis of Fig.
2(a), and Fig. 2(c) is that for loss analysis. In order to make easier analysis and designs hereafter, let's assume that the LC
combined transformer always has a resistive load, R. The arrangement of the specific circuit or variant of every type and mode of the LC combined transformer must be designed in accordance with its featured focuses or its main functions, while the main functions are to be determined by the employed LC unit system or module/block/subunit, named mutual capacitor.
The LC combined transformer, according to its functional focus, can be divided into three fundamental categories or types: current transformation category/type (ideal current transformer), voltage transformation category/type (ideal voltage transformer), and voltage and current transformation category/type (ideal transformer); The first type has two circuit arrangements of transformation-A type and transformation-B type, the latter two types include in-phase mode and anti-phase mode respectively, and the third type also includes transformation-A type and transformation-B type arrangements.
1. Current Transformation Type LC Combined Transformer (Ideal Current Transformer) The Current transformation type of the LC combined transformer, or the ideal current transformer, has its main duties as performing sinusoidal current transformation, current monitoring and measuring or test for instruments, and it also can be designed for ac power delivery, as an ac constant-current generator, or apparatus for current waveform conversion or isolation from square wave to quasi-sinusoid as well.
1-1. Current Transformation-A Type LC Combined Transformer Here details the design of the current transformation-A type LC combined transformer with V2 side in Fig.2 as input port and VI side as output. Therefore, in Fig. 2, take inductor 1 and capacitor 4 short-circuited (namely, L a = ra = 0, C b¨+ r b = ), capacitor 5 open-circuited (C p = 0, r p-"-C ), to obtain the analysis circuit diagram as in Fig. 3.
In Fig. 3(a), the transformer 26's secondary magnetization inductance 10, leakage inductance 9, inductor 3, and capacitor 2 constitute an LC subunit/subsystem, called delt (A) or pi ( n ) mutual capacitor. And the current ratio of this mutual capacitor can be calculated as n, = ¨1 = ¨1 (1+ +1¨(o2C.(L2+ Lb) (8) 12 k L21 iCOL2 If component parameters are set to meet the condition co2Cm (L2 + L ,,) = 1 (9) 1 ( L
the ratio will be n, =¨ I =¨ 1+
(10) 12 k L2 And including the ideal transformer 27, the current ratio of the entire circuit in Fig. 3(a) will be ( 1 + 01) ¨I = ¨I ,= ¨I = ¨1 n, =
/2 / /2 n nk L2 This result indicates that the circuit in Fig.3, when the condition Eq. (9) met, is an ideal transformer of current transformation, called transformation-A ideal current transformer or ideal current transformer A, independent of both the working frequency w and the load R. And the ratio is determined only by the selected values of the mutual inductor's turns ratio (n= N1 = ), the coupling coefficient (k = ), the self-inductance L2, and the series inductor L b . [Note: Once in a while, when Lb=0(rb=0 accordingly, See Fig. 2(a)), the circuit built will be as in Fig. 3(c).]
But, all the above conclusions are obtained in the ideal situation. As a matter of fact, the frequency of steady-state sinusoidal current is slightly undulate (for 60Hz or 50Hz line frequency has a relative error Af = ¨Au) 1%); capacitors have their capacitance values changeable with the waving ambient temperature; iron-cored inductors are of small non-linearity that their inductance values are changeable with magnitudes of the current flowing through the coil windings therein (i.e.
with the changes of operating points); in addition, wires, cores as well as capacitors in reality are power-dissipated (see Fig.3(b)); which all would deviate the current ratio from Eq. (11). Here come the errors theoretically derived as follows:

An, The relative error of the current ratio on frequency change is 2coC __ Aco R= (12) c The relative error of the current ratio on capacitance change is An co C
,,R = AC (13) ng The relative error of the current ratio on relative permeability change of the core material is An a coCmR Apr (14) x+ ,L1,. Pr where, a =1,11g is the ratio of the core magnetic circuit length to the air-gap length; and 11 r is the relative permeability of the inductors' core material. Moreover, the prerequisite for obtaining this equation is that inductors of L2 and L b are made of the same core material and of the same a value.
The relative error of the current ratio on the devices' power-loss from Fig.
3(b) is Ang (r2 + r, + rk + r,õ)(coC,õ)2 R (15) tic The prerequisite for obtaining this equation is that quality factors of the inductors of L2 and Lb are coL2 cnLõ
equal and far greater than 1, i.e. = >> 1; and also that the loss tangent of capacitor Cm rk rb should be very small, or (0 Cm rm = tg 6 ¨ 0 .
Design Key Points [Note: refer to 6-1. Design Instructions of the LC Combined Transformer and General Rules for Its Device Selections]: Attentions should be paid to error equations (12) - (15) on that ( Cm R) is a key parameter expression for designing errors of the mutual capacitor, called error-designed parameter expression of the mutual capacitor; when it is small the error will be small;
meanwhile, Eq. (9) shows that the inductance value of (L2 + L b) will be large so as to waste materials and increase sizes. Therefore, proper compromise will be needed in practical designing.
Device Selections: The criterion of device selections for transformation-A
ideal current transformer is to meet the requirements of above theoretical designing as far as possible, raising the inherent features that properties of devices vary along with ambient or/and working conditions in materials, physical structures, as well as manufacture methods etc, namely improving the linearity, and decreasing devices' power dissipation or reducing influence of devices' power-loss over operation.
Device selection of capacitor of Cm includes that a proper capacitance value should be determined according to the measuring accuracy or error requirement designed from (12)¨(15), and the right product be chosen according to the requests of, the range of ambient temperature change, working frequency, voltage grade, value precision grade and dielectric loss angle etc. In this case, due to Cm in parallel with the low-valued resistive load R (ammeter A) (see Fig.3(c)), the objective of voltage grade is apt to be met, and the dielectric loss angle tangent, tg 6 <
10-3, of non-polar capacitors of most modern manufacturers is good enough for this application;
then by Eq.(13), according to the determined value and the range of ambient temperature change, select the capacitor with appropriate dielectric material.
The values of parameters L b, L2 n and k of the serried inductor and the mutual inductor are to be determined from Eqs. (9) ¨ (11), where the value k must be pre-determined accurately through experiment so as to reduce blindness in the follow-up designing.
Device selection of the mutual inductor and the serried inductor is a key step for designing in this case, including determination of the coil copper wires, core materials, physical structures and their production methods. The L1 and L2 of the mutual inductor must be of a core material with low-loss and high saturation magnetic flux density, identical to that of the inductor L
b , with precise calculation of the amount of copper and core to be used, managing to ensure the quality factors of L2 and L b to be equal and far greater than one, or coL2= wLb 1. Both the series inductor 3 r2+rk rfi and the mutual inductor 26 must be of a structure of cores plus air-gaps, which is referred to as linerization processing of inductors/mutual-inductors [Note: refer to 6-2.
Formulas for Lazerization Processing of InductorsIMutual-Inductors], for air-gapped inductor is calculated as NT 2 c NT 2 c NT 2 c NT 2 c NT 2 c /1-101v 2 " 2 = POIY 2 '12 /ivy 2 2 poiv 2 2 P01/ 2 2 L, =
-'g2 4- 1/%2 /Pr 1g2 (lr, 2 /1g2 )/Pr /g2 az /Pr 11,2[(1g 2 /11 2 ) I/Pr 1F2 Wa2 '/pr]
PON b2 S h PONb2Sb POlvNT b2 c h poN b2S b POlvNT h2 c b L b =
1gb +11,b /Pr lgb[1+ (1m, figb)/ Air] 1 gb [1+ abl Pr]
11,b[(1gb /11,b) +1/ Pr] 112 Wab 1/Pr where, 11; and 1g represent the core length and air-gap length respectively, and ai=lpillgi ( i=2, b ); N , is coil winding turns number; S , is core cross-sectional area. Assuming a=a2=11,211g2=141gb=ab, and substitute above two formulas of L2 and Lb into Eq. (11) as / 1 ri Lb 1 1 S ( N 1 ______________________________________ 1 + lh 2 Sb Nh 2 = ________________ = ________________ 1+ g2 h ________________________________________ (1

6) /2 nk L2 nk 1gb S2 \,N2 nk 1bb S2 \N2 Eq. (16) indicates that the current ratio of this LC combined transformer illustrated in Fig. 3 is absolutely determined by the structural parameters of Li and L2 of the mutual inductor 26, and of L b of the series inductor 3, theoretically independent of the value iir of the core material; which is because the introduction of the air-gap, i.e. the linerization processing of inductors, causes the inductances much more stable, and also because of a principle of cancellation of similarity employed during the design and coil winding of inductors. The relative error of the final current ratio of the entire current transformer influenced by the change of relative permeability of core is obtained from Eq. (14).
1-2. Design Approach of Integrated Inductor and Mutual Inductor Figs. 3(c) and (d) are diagrams of the current transformation-A type LC
combined transformer employing the design approach of integrated inductor and mutual inductor.
The integrated inductor and mutual-inductor includes the mutual inductor's core magnetic circuit 6, the series inductor's core magnetic circuit 12, the mutual inductor's primary winding 7, the two-in-one common coil winding 8 which serves as both the mutual inductor's secondary winding and also the series inductor's winding, as well as the auxiliary winding 13.
The magnetic circuits of the integrated inductor and mutual inductor may be made from any core material, with any possible shape and any cross-sectional areas, and also may be unequal in length to each other; but the ratios of both, of the core magnetic circuit length to the air-gap length respectively, should be equal or approximately equal. The mutual inductor's turns ratio, coupling coefficient, primary self-inductance, secondary self-inductance, and all the current and power relationships are still determined as those of the conventional mutual inductor, but its output total inductance be determined, under the condition of the magnetic circuits with sound linearity, by the sum of the mutual inductor's secondary self-inductance determined as a conventional mutual inductor plus the inductance determined by windings 8 and 13, and core 12 all together.
The so-called integration design of the inductor and mutual inductor is actually having the cores of the series inductor and the mutual inductor integrated together, and also having their coil windings integrated together, as a result that they look like only one mutual inductor with a function of the mutual inductor plus the series inductor. Assuming N2 = N b , that is /, 1 (1 + Lb = 1 /g2 Sb 1= 1[S61 0 6a ) /2 nk \ L2 nk 1gh S2 nk[ 11h S2]
which is the equation of the current ratio of the current transformation-A
type LC combined transformer employing the design approach of integrated inductor and mutual inductor. From this equation, only k could be adjusted when n ( = N 1 /N2 ), /F , /g and S are made fixed. However, the variation of k means changing the air-gap length, also meaning the condition of Eq. (9) spoiled.
Now, assuming Nb = N2+ AN again and substituting it into Eq.(16), we have 1 __ [, 1g2 Sh( , Ale ______________ 1 1+1,,2 Si, (1 + AN
¨=¨ -= i+ = 1+ (160 12 nk L2 ) nk 1gh S2 N2 nk 11th S 2 \, As seen in this equation, the variation of A N, i.e. changing turn number of the auxiliary winding, changes only the inductance of L b , by which it is succeeded to have the coil micro-adjusted, with the layout of the coil windings as in Fig. 3(d).
Like the design of every other product, the design of this product has to be improved through repeated experiments so finally to be as expected. Moreover, a suggestion is made, if possible, that the same kind of magnetic powder core material should be employed for the two pairs of cores of F1 and F2 illustrated as in Fig. 3(c) or (d); whose advantage is easy to have an equal a value for both.
It saves materials to design an LC combined transformer by employing the integration design of inductor and mutual inductor (a coil winding of L b saved) so that the total size decreases because the air-gapped cores set the current transformer free from heavy burden of the balance of the magnetic potentials or ampere-turns, and meanwhile the requirements of the window areas of the cores and of the insulation grades decrease accordingly. However, these advantages can be brought into play only at high-current applications because an LC fixed value must be set, by Eq. (9), for the current transformation-A type LC combined transformer. It is also easy to notice from Eqs.(10) and (11) that the current transformation-A type LC combined transformer, as a matter of fact, performs two current transformations that 1/n is the first current transformation, namely the current ratio of the conventional current transformer, and the second is that of the mutual capacitor which is determined by Eq. (10), so that a very high rating of current transformation ratio could be achieved.
In the integrated inductor and mutual inductor (Fig.3(c) or (d)), the function of a mutual inductor occurs between coil windings N1 and N2 while N2 by its own functions as two inductances in series P
as LToial =L11 +L1.2 = oN22SFN2S 0 b 2 (17) /gi +1,11,u, 1g2 +1, 2 /p, where, meanings of the symbols are the same as previous, and the subscripts in accordance with the core number F1 and F2 [Note: this equation is obtained under the condition of a good linearity]. And proof of this equation omitted.
1-3. Current Transformation-B Type LC Combined Transformer The circuitry design of the current transformation-B type LC combined transformer is also presented as the form with V2 side in Fig.2 as input port and V1 side as output. In Fig. 2, make inductors 1 and 3 short-circuited (namely, La = ra = O, Lb = rbi = 0), capacitor 5 open-circuited (Cp =
0, rp¨ +co), to obtain the analysis circuit diagram as in Fig. 4.
In Fig. 4(a), the mutual inductor 26's secondary magnetization inductance 10, leakage inductance 9, capacitors 2 and 4 constitute an LC subunit/subsystem, called delta (A) or pi ( n ) mutual capacitor.
And the current ratio of this mutual capacitor can be calculated as n, =L =-1 I 2 1 j COC. __ I m R} (18 co ) 12 k L2Ch wL2 \ Ch If component parameters are set to meet the condition (CC
co 2L2 (CL, _L C,,, )= co2L2b m (1 9) \ Cb + Cm then n, =¨= 1 (20) 12 k co 2L2C CO 2 kL,Cm k(Cb+C,n) And notice that n < 1 in most cases. Thus the current ratio of the entire circuit in Fig. 4(a) will be ¨ ¨ = ¨ ¨ = n--= ____________________________________________________ (21) , /2 / /2 , n nk(Ci, +Cm) And this result denotes that the circuit in Fig.4, when the condition Eq. (19) is met, is also an ideal transformer of current transformation, called the transformation-B ideal current transformer or ideal current transformer B, independent of both the working frequency 6) and the load R. And the ratio is determined only by the selected values of the mutual inductor's turns ratio (n = N I = ), the coupling coefficient (k = ) , the series capacitance Cb, and the parallel capacitance Cm.

Here give the errors theoretically derived as follows: The relative error of the current ratio on __________________________________________ C Aco frequency change is An Ji +b Cm )R]2 = 2 = ______________ (22) flcCb CO
(,) The relative error of the current ratio on capacitance change is \ __ C AC
lc + [co (C b + C õ, )R.1-12 (23) Cb C
c c The relative error of the current ratio on relative permeability change of the core material is Anc + [co ________ b + C .)R]2 C Ap (24) p C a + pr ,u, where, a =1, fig is the ratio of the core magnetic circuit length to the air-gap magnetic circuit length; it rthe relative permeability of the inductors' core material.
Moreover, the prerequisite for obtaining this equation is that inductors of (1-k)L2and kL2 are of the same a value.
The relative error of the current ratio on the devices' power-loss obtained from Fig. 4(b) is An r (r2 + rb + rk + rõ,)(coC õ,)2 R (25) nc r The prerequisite for obtaining this equation is that quality factor of the inductor L2 is far greater than coL 2 1, i.e. 1; and also that the loss tangent of capacitors Cb and C, should be very small, that + rk is coCbrb= ()Cmr,n=tg 8 O.
Design Key Points [Note: see 6-1. Design Instructions of the LC Combined Transformer and General Rules for Its Device Selections]: Attentions should be paid to error equations (22) ¨ (25) on that (111+ [a)(C, +C. )R]- _____________________________________________________ =C
m ) is the error-designed parameter expression of the mutual capacitor;
C b when the values of (C, + ) and m set small the error will be very small; meanwhile, Eq. (19) Cb shows that the inductance of L2 will be large so as to waste materials and increase the sizes.
Therefore, proper compromise will be needed in practical designing.
Device Selections: Device selections of capacitors 4 or Cb and 2 or Cm include proper determination of their values on designed measuring accuracy or error requirements, choosing the right products according to the requests of, the range of ambient temperature change, working frequency, voltage grade, value precision grade and dielectric loss angle etc, and characteristics of both capacitors changing with the environment expected as keeping in accordance. Requirements for the mutual inductor 26 is of a precise k value, L2 with a good linearity, and low power loss.
2. Voltage Transformation Type LC Combined Transformer (Ideal Voltage Transformer) The voltage transformation type of the LC combined transformer, or the ideal voltage transformer, has its main uses of performing sinusoidal voltage transformation, voltage monitoring and measuring/test for instruments; and it also can be designed for ac power delivery, or as an apparatus for voltage waveform conversion or isolation from square-wave to quasi-sinusoid as well. The voltage transformation type of the LC combined transformer includes two realizations of circuit arrangements of in-phase mode and anti-phase mode.
2-1. In-Phase Mode of the Voltage Transformation Type LC Combined Transformer In the circuit diagram of Fig. 2, let inductor 3 short-circuited (i.e. L b = r bi = 0), capacitor 5 open-circuited (i.e. C p = 0, r +C
) to obtain the in-phase mode of the voltage transformation type LC combined transformer illustrated in Fig. 5(a). In order to analyze it, let's split capacitor 4 into capacitances 4a and 4b (namely, Cb splited into Cbi and Cb2 and C = C hl L C
61 = C h1C h2), and Cbl +Ch2 equivalently reflect the leakage inductance 11 on the right side of the mutual inductor 26 onto the left side as inductance 14, shown as in Fig. 5(b); where inductor 1, capacitors 2 and 4a constitute the first LC subunit/subsystem, called tee (T) or wye (Y) mutual capacitor;
capacitance 4b, two leakage inductances 9 and 14 of the mutual inductor 26, and its magnetization inductance 10 constitute the second wye (Y) mutual capacitor; and the third part is the ideal transformer 27.
For the first T mutual capacitor, assuming that it has an equivalent load of resistance 121, its voltage ratio will be __________________ nõ,=¨v=(1¨co2LaCb,)+ [1 c 2LaKbi +CA-1 (26) x .1 )C bl R, If choosing the component parameters to meet the condition co2La (Cb, +Cm )=1 (27) Chl we have no =1¨ co2LaC n, = (28) C bl +C m Then, the relative error of the voltage ratio on frequency change is ( V ( \ ct) Any, 1 I
ill + ________________ 2 __ 1A (29) nvi \co nvICm RI , \nvl ) co The relative error of the voltage ratio on capacitance change is i / \ ( \
An V ,' 1 ' 1 AC AC b, AC õ, 1+ 1 __________________________ ; __ when = ____________ (30) no c \ c.o noC,,, R, ) il,,, j C.
\ Chl Cm ) The relative error of the voltage ratio on relative permeability change of the core material is ( 2 ( \
Any, 1 a 1 A,u,.
1+ 1 ______________________________________________________________ (31) nvl I, \ CI) nv,C m R11 (a + fir) ny, ) Pr where, a =11õ11g is the ratio of the core magnetic circuit length to the air-gap magnetic circuit length; 11 , is the relative permeability of the inductors' core material.
The relative error of the current ratio on the devices' power-loss obtained from Fig. 5(c) is Any, ra (32) /1,1 , no 2R1 The prerequisite for meeting Eq. (32) is that the loss angle tangents of capacitors Cbj and Cm are equal or approximately equal, that is tg 6 bj = CO C birbi-----G)Cmrm=tg 6m, as well as tg ¨> 0.
Also, it is noted that, when output power of this mutual capacitor is P, R1 =Vx2 = V 2 1 (33) P n2 o P
Design Key Points [Note: see 6-1. Design Instructions of the LC Combined Transformer and General i V i \

Rules for Its Device Selections]: From the error equations,, 1+ 1 will be c'D floc R11 no i found out as the error-designed parameter expression of this mutual capacitor;
if the value of ( Cm R1) set large the error will be small, but its capacity of load carrying will be limited; to improve which there exist some ways, increasing the value of Cm or/and co .
Device Selections: Device selections of capacitors 2 and 4 require the value precision grade, temperature coefficient taken as high as possible based on the requests of design. The temperature coefficients of Cbl and Cm are needed to be in accordance, and the loss angle tangents should be equal or approximately equal, that is tg 6 bi =C bl rb1 Cm rm= tg 6 , as well as tg 6 O.
Meanwhile, the maximum voltages on the capacitors CIA and Cm are calculated as the following equations (assuming the mutual capacitor's maximum load as Rim).
Ubl 2Vx 2V, 2V, 0 ) (34) > ____________ ¨
max (-"L' bl Rim CDC bln vl Rim NC mnvi Rim .\ 2 2V '\ 2 2V 1 ¨ n 1 = , iii1+ vl U max 2Vx11+ __________ = , 1+ _____________________________________ (35) CM R1 m n 1, cocb,Rim n \ coCmn R1 ni The core of inductor 1 or La should be selected a low-loss core material, with its magnetic circuit length ratio a of the iron core to the air gap chosen by Eq. (31) to meet the design requirements and also to meet the material specifications.
Assume R2 as the equivalent load of resistance for the second mutual capacitor; its voltage ratio is 1 1 1¨co2(1¨k2).L1C132 n v2 , (36) V k co2L1Cb2 j CDC b2 = R2 _\
If setting the component parameters to meet the condition W2 ¨ k 2 =1 (37) Vx 1 we have n v2 = ¨ =I 1 2 __ =k (38) co y k \ L1Cb2 The relative error of the voltage ratio on frequency change is \ 2 /
I Aco Anv2 c LI = ¨ 1 (39) nv2 \ 2 ) \ 2 k ) co The relative error of the voltage ratio on capacitance change is \ 2 , Anv2 1 += ACb2 (40) nv2 c R2 j C b2 The relative error of the voltage ratio on relative permeability change of the core material is em1 +(COL 2 12 a Apr An,2 1=
(41) nv2 1 \ R2 Jc- ) (a + Pr) Pr where, a = 1, 11 g is the ratio of the core magnetic circuit length to the air-gap magnetic circuit length; 1-1 r is the relative permeability of the inductors' core material.
The relative error of the current ratio on the devices' power-loss obtained from Fig. 5(c) is Anv2 rh2 + 1) (42) nv2 k 2R2 The prerequisite for Eq. (42) is that the quality factors of inductances (1-k)Li and kLi are equal.
Design Key Points [Note: refer to 6-1. Design Instructions of the LC Combined Transformer and General Rules for Its Device Selections]: The error-designed parameter expression of this mutual capacitor is , ,N2 1 , ( coL, , 1+ ' cuL ' 1 lwhich denotes that, to minimize the error, 2 the value of ' should be as R k2 \ 2 ) small as possible, and the k value as large as possible.
Device Selections: Device selection for capacitance Cb2 is the same as that for Cbi, because they will be merged together as one in the end, and the maximum voltage on Cb2 is calculated as follows U 62 max = 04_42 v. = k L = ___ V (43) V k n 2 The core material for L1 or the mutual inductor 26 should be selected, from Eqs. (41) and (42), with a high permeability and low core loss material. The prerequisite for Eq. (42) is that the quality factors of inductances (1 -k)LI and kLi are equal, or koo ¨ Pr, = co kLi Irk , which is not easy to get into practice because r1 is mainly the copper loss while rk is mainly iron loss.
Try to decrease the difference between both as far as possible so as to be more accurate to estimate error by Eq. (42).
Now from Eqs. (28) and (38) as well as the ideal transformer's ratio n, the voltage ratio of entire in-phase mode of the voltage transformation type LC combined transformer will have the equation as V, V, Vx 1/v IcnCh, n = =
no nv2 n= knn = _____________________________________________________ ( _________________________________________________________________________ ¨44) v V2 Vy V2 Chl+Cm Eq. (44) indicates that the circuit illustrated in Fig. 5, under the conditions of above discussion, is an ideal voltage transformer independent of the working frequency co and the load R. It also shows that polarities of voltage transformation of V1 and V2 are in-phased, therefore, called the in-phase mode of the voltage transformation type LC combined transformer or in-phased ideal voltage transformer.
2-2. Anti-Phase Mode of the Voltage Transformation Type LC Combined Transformer In Fig. 2, let capacitor 5 open-circuited (i.e. C p= 0, r-- + ) , though not excluding a round-off design of having capacitor 4 shot-circuited (i.e. C b y r b2 = 0), to obtain the anti-phase mode of the voltage transformation type LC combined transformer illustrated in Fig.
6(a). Imitating what's done for the in-phase mode, equivalently reflect the leakage inductance 11 on the right side of the mutual inductor 26 onto the left side as inductance 14, shown as in Fig. 6(b);
where inductors 1 and 3, plus capacitor 2 constitute the first LC subunit/subsystem, called tee (T) or wye (Y) mutual capacitor; capacitor 4, the leakage inductances 9 and 14 of the mutual inductor 26, and its magnetization inductance 10 constitute the second LC subunit/subsystem, called tee (T) or wye (Y) mutual capacitor; and the third part is the ideal transformer 27.
Still, assume that the first tee (T) mutual capacitor has an equivalent load of resistance RI, then the voltage ratio will be no = ¨ =(i¨ co2LbC.)+ jco(Lt, +Lb¨ co2LbLb \ (45) ( L b L
If choosing component parameters to meet condition co2C. a = co2Cm (La //4)=1 (46) \La + Lb we have no =1¨ co2 L aC = (47) Lb Thus, the relative error of the voltage ratio on frequency change is ( An 1 vl 1 nvi ¨1 = 2 1 AN
(48) tivl \co nvlCmR1 n vl The relative error of the voltage ratio on capacitance change is \ 2 r 1 AC .
An vi no ¨I
+ 1 (49) no c \,,a) noC mRi \, no Cm The relative error of the voltage ratio on relative permeability change of the core material is Anvi nvl ¨ 2 1 1 Apr 1+ ________________________ , 1 ________________________________ (50) , nvi ct) noC.Ri ( a + Pr ) no ) where, a = /dig is the ratio of the core magnetic circuit length to the air-gap magnetic circuit length; ii r is the relative permeability of the inductors' core material. And the prerequisite for obtaining Eq. (50) is that La and Lb have cores of the same material and also of the same a value.
The relative error of the current ratio on the devices' power-loss obtained from Fig. 6(c) is Ano 2(1¨ no )1-1, (51) r R, The prerequisite for Eq. (32) is that the quality factors or Q-values of inductors 1 or La and 3 or Lb should be equal, that is 6) La/ra= coL Orb= Q, as well as r m= rallrb be managed to achieve.
Besides, the value of R1 still could be worked out by Eq. (33).
Design Key Points [Note: refer to 6-1. Design Instructions of the LC Combined Transformer and General Rules for Its Device Selections]: This mutual capacitor has an error-designed parameter expression ( =\ 2 nvl _________ ¨1 1 as 111+ , which shows that, to have a small error, the values of Cm and n vi con C R , 170 m have to be large. In addition, if the positions of Lb and Cb switch to each other in the circuit, circuit function stays unchanged so that inductor 3 of Lb and the mutual inductor 26 could be constructed by integration design of the inductor and mutual inductor as schematically illustrated in Fig. 6(d).
Device Selections: Device selection of capacitor Cm requires the value precision grade, temperature coefficient taken as high as possible based on the requests of design. The maximum ' \2 I ( \ 2 COLh voltage on Cm will be determined as Um. 21ix 1+ __ = __ 1+ __________ (52) no \
Moreover, Eq. (51) requires that Cm's equivalent series resistance, r m= ra fir b , to which a solution is to insert a proper resistor connected in series with it, with the only concerning that you should weigh and balance the necessity of paying a price of power dissipation. Inductors of La. and Lb are selected as stated before, with the requests of the same a value and of the same Q-value.
The second subunit is the same as that in the in-phase mode [Note: but now in Fig. 6, Cb must take the place of Cb2 in Fig. 5]. Thus, borrow the result from that as is in the in-phase mode and obtain the voltage ratio of the anti-phase mode of the voltage transformation type LC
combined transformer as VI VI Vy La = n =
knnyi = ¨kii¨ ( n ==53) n Vi '11,2 V2 Vy Vy V2 Lb This equation indicates that the circuit illustrated in Fig. 6, when satisfying the conditions of above assumed, is also an ideal voltage transformer, with the polarities of voltages of input and output anti-phased, which is why, called the anti-phase mode of the voltage transformation type LC
combined transformer or anti-phased ideal voltage transformer.

If going one more step further, make coL, = 0 in Figs. 6(a) and (b); from Eqs. (37), (46) coCb and (47), to get the following Lõ = ¨ k2)L, (54) and co2C. (1¨ k2)Li = 1 1-1/1/0 (55) Hence the circuit has its simplified arrangement (see Fig. 6(e)). Similarly, once more assume 01,1. = col,b __ 1 >
0 , i.e. when L bx = L b co21 r = Lb ¨(1¨ k2)L1 > (56) ct)C b b the circuit could leave out Cb as in Fig. 6(f) as well as in Fig. 6(g) by the integration design of inductor and mutual inductor.
3. Voltage and Current Transformation Type LC Combined Transformer (Ideal Transformer) The voltage and current transformation type of the LC combined transformer, or the ideal transformer, is actually the technological extension expanded either from the voltage transformation type LC combined transformer to the current transformation type, or from the current transformation type LC combined transformer to the voltage transformation type. Accordingly, for the former there exist two configurations of circuitry designs of in-phase mode and anti-phase mode; while for the latter there also exist two circuitry realizations of transformation-A type and transformation-B type.
3-1. In-Phase Mode of the Voltage and Current Transformation Type LC Combined Transformer Firstly review the in-phase mode of the voltage transformation type LC
combined transformer and redraw the circuit diagrams in Figs. 5(a) and (b) as in Figs. 7(a) and (b). In Fig. 7(b), of the first tee (T) mutual capacitor consisting of inductor 1, capacitors 2 and 4a, currents i r =VI - Vm _VI -Vx (Vm -V ji CDCbl = 1 \
1 v1 _______________________________________ __ ix 11 i col, a =
jcoLa jcoLa ' i (1)C- - - hi ny i j col, a co2Lac, ( C m 1 1 ( C \
= j co -- V, + ______________ IX = jcoC ' 1,11/1+ I x , c, = m (57) \,nvl ) nvl no no i I x = n I , ¨ jcoC mV, = no' 1 ¨ ..10(noC .)172 = noi 1 ¨ jcoC p2V2 , (C
1,2 = n,,,C m) (58) From Eqs. (28) and (58), an equivalent circuit, between Vi and V, in Fig.

7(c), of the ideal transformer 15 and its secondary-side paralleled capacitor 16 or Co is evolved. In the same way, of the second tee (T) mutual capacitor consisting of capacitance 4b, the mutual inductor's two leakage inductances 9 and 14, and also the magnetization inductance 10, there is a current as V, ¨ vk Vx ¨V, (vk ¨ Vy ) /.õ = _______________ = ____________ ______________________________________ ja)(1-4, 1õ
j WC b2 jc0Cõ W`Cb211-4,1 C \
, ny2 j , __ 1 '1v2' ' jc00¨ 41 1 CO2 C õ (1 ¨ 1c)Li _ Vx 1 Vx 1 ' = , ______ ,+ I = __ + __ I (L ¨ n v2 . kL1 = k 2L1 ) (59) Y
jcpknv24Li) ny2 Y iC Lpl n v2 pl That is to say, from Eqs. (38) and (59), achieve the equivalent circuit of inductor 17 in parallel with the primary of the ideal transformer 18, evolved from that between V, and Vy in Fig. 7(b). Then, assume that the component parameters satisfying the condition coCp2 =1/coLpi , i.e.
co2Cp2Lp1 =co2nock2L1 = c 20 ¨ no )Cbik2Li ¨ co2 Cb1Cm k 24 a__ (02 -k2 LI(C al 1 C m) =1 (60) Cm +C.
and notice Eq. (27) and Ch = - C
61 1 Cb2 , we achieve that, when ( CbCm I
CC
\
W 2Lt = w2LIK b I_Cm)=1 , Cb = b1 m (61) \Ch +Cm ) (Cbi+C.)Ik 2 ¨Cbl ) Fig. 7(c) is in circuitry equalized as Fig. 7(d) with its voltage and current equations as VI Vx V ynkC bi = n C, ¨ = ¨ = ¨ = ¨ = n a, = ny2 = = ny = ________________________________ (62) V2 Vx Vy V2 Cbl+Cm k + Cm I I I 1 1 1 1 Chl __________________ +Cm k C
= _______ .I x V = = = = ( (63) Ix. Iy 12 11y112,2 n n kC
v n b They appear completely as the forms of ideal transformer's equations, termed the in-phase mode of the voltage and current transformation type LC combine transformer or in-phased ideal transformer.
And from Eqs. (27) and (61) we have 11¨ nvl 1 Cb L a = 2 = C 2Cm 1 (64) CD2Cm _ k2(Cb+Cm)_ Design Key Points: The in-phase mode of the voltage and current transformation type LC
combine transformer (see Fig. (7)) is just the improvement or upgraded from the in-phase mode of the voltage transformation type LC combine transformer. Hence, its error analysis, design key points, and device selections all are the same as the according contents respectively of the latter stated above, with a difference that the former has functioned as the input and output current in-phased just one-step further beyond the latter.
However, the two mutual capacitors of the in-phased ideal transformer in Fig.
7 are implicated with each other during the specific designing, especially on the adjustment. In practical engineering, especially on spot test or adjustment, deviations of parameter values, influenced by lots of factors, are fated, although parameter value precision grades are ensured as high as possible in the course of designing and manufacturing; and micro-adjustments are unavoidable. Here present two methods shown in the following that can be used for on-site micro-adjustments.
Method I: Take Lp as a micro-adjusted inductor with its value far below LI, and connect Lp in series with the primary winding N1 of the mutual inductor. Then Eq. (36) will become Vx 1 1 \ +1¨(02Ch2K1¨k2)LI p 12v2 = = 1 (36a) V k CO2 -1-L]-L1Cb2 / jalCh2 = R2 Accordingly, Eq. (37) could be as co2c2[(1¨k2Y1 +Lp]=1 (37a) L
Eq. (38) as nv2 = ¨ = 1 2 _________________________________________ = k 1 (38a) or, V k L1C132 \ k2 L1,, Method II: Put a micro-adjusted inductor Ls (<< L2) in series with the secondary side of the mutual inductor. Then Eq. (36) will be turned as ( 1 0 +L, co2C1,24(1+Ls 14¨k2 ) nv2 -= = ¨1 1 ______________________________________________________ (36b) V k co2L1Cb2 jCpCb2 .R2 k2 Eq. (37) as co2L,C,2 1 __ = co2L,Cb2(1 ¨knx2)=1 (370 1+LsIL2) Vx 1 1 \
and Eq. (38) as 11,2 = ¨ = I ______________________________________ 2 (380 V k co L,C,2) 1 + LsIL2 Moreover, the two methods stated above are suited only when the k value of the mutual inductor is slightly greater than originally tested or L1 a bit less than designed. To match their uses, the coil winding of L1 should be pre-set a tap at the position of just a little bit fewer turns next to an end to make it have an inductance slightly less than originally designed. In this way, once either of the two cases above-mentioned happens, the pre-set tap in series with the Lp, take Method I as an example, could be connected to where N1 ought to so that flexible micro-adjustments could be realized.
Obviously, such a way has also slightly changed the ratio of the entire transformer; when necessary, revision should be made.
3-2. Anti-Phase Mode of the Voltage and Current Transformation Type LC
Combined Transformer In the same way, redraw the circuit diagrams of the anti-phase mode of the voltage transformation type LC combined transformer in Figs. 6(a) and (b) as in Figs. 8(a) and (b).
In Fig. 8(b), of the first tee (T) mutual capacitor consisting of inductors 1 and 3, capacitor 2, current v ( La !ILI) Ix nvl 11 ____________ n L p2 = (65) iC,01(La //Lb VInvi yl /1 2 jco Lp2 no By Eqs. (47) and (65), electrically equalize the first mutual capacitor in Fig. 8(b) as an arrangement of ideal transformer 19 and its secondary in parallel with inductance 20 illustrated in Fig. 8(c). Of the second tee (T) mutual capacitor in Fig. 8(b) consisting of capacitor 4, both of the mutual inductor's leakage inductances 9 and 14, and the magnetization inductance 10, the expressions of Ix and Lpi are identical to Eq. (59) so that its equivalent circuit could be the same as in fig. 7(c) of inductance 17 or Lpi in parallel with the primary of ideal transformer 18, and the circuit in Fig. 8(b) will be in circuitry equalized as in Fig. 8(c). Furthermore, if a reactive compensation capacitance 5 or Cp inserted in parallel connection at the position of Võ in Fig. 8(c), or according to practical necessity, either capacitance 5a or Cpa at VI, or capacitance 5b or Cpb at V2 , with their values as 1 1 ( 1 1+L, C p = __ 2 (66) (Lpi ilLp2) co2 ______ L Lb ) ( =\ 2 / I
Cpa =Cpinvl2 =
(67) ji b A, a J
n2 [1 k2O+LalL,) C b = k2 n2 C = (68) p p 02 L Lb After compensated, functions of the circuit in Fig. 8 can be specifically and equivalently described as the form of ideal transformers illustrated in Fig. 8(d), with its voltage and current relationships as V, V, Vx V y nkL a = no = 11,2 11 = 11, = _______________________________________________ (69) V2 Vx Vy V2 Lb I I I I 1 1 1 1 Lb x Y = __________________________________________________ (70) = = =
J2 I: 1y 2 n vl n v2 n nv nkL
These equations show the relationships of out-of-phased voltages and currents, termed the anti-phase mode of the voltage and current transformation type LC combined transformer or anti-phased ideal transformer. As well, here present the circuit arrangements of the ideal transformers upgraded from Figs. 6(f) and (g) respectively as in Figs. 8(e) and (f).
Design Key Points: In the same way as in the in-phase mode, the anti-phase mode of the voltage and current transformation type LC combine transformer (see Fig. (8)) is also just the improvement or upgraded from the anti-phase mode of the voltage transformation type LC
combine transformer.
Hence, its error analysis, design key points, and device selections all are the same as the according contents respectively of the latter stated above, with a difference that the former has functioned as the input and output current anti-phased just one-step further beyond the latter.
3-3. Voltage and Current Transformation-A Type LC Combined Transformer Firstly review the current transformation-A type of the LC combined transformer and redraw the circuit diagram in Fig. 3(a) as in Fig. 9(a). In Fig. 9(a), of the delta (A) or pi ( ) mutual capacitor consisting of inductances 3, 9, 10, and capacitor 2, voltage V = jC0(i ¨ iijkL2 = /CO(I ¨ /2)1CL2 ¨ iCO(1õ ¨12)kL2 7 \ 7 = jC0 / --k kL2 ¨ jco(fcoCõy2)kL2= j co 1¨ ¨1 kL2I + co2 kL2Cõy2 (71) nc ) c)n - -= jo L.01 + ¨1 V2; L,1= (1 ¨ ¨1)1cL2 n, n, -From Eqs. (10) and (71), obtain the equivalent circuit, between V and V2 in fig. 9(b), of ideal transformer 22 and in series with its primary winding the equivalent input inductance 21 or Ls1 of the mutual capacitor. Next, let's insert a compensation capacitance 23a or Csa in series connection at point a of input port, or when necessary, insert a compensation capacitance 23b or Csb in series connection at point b of output port, with their values as C' sa = 1 = (72) CO2 {(1 ¨ k)Li + n2Lsi i CO 24 (1 ¨ kInc) C sb =(73) C 2 n c2 Ki ¨ 10-L '2 + I , sd w2n,L2(11c ¨1c) Functions of the circuit in Fig. 9(b) after compensation can be equivalently expressed as the form of ideal transformers in cascaded connection, with the voltage and current relationships as VI = VI . V = n 1 = nkL2 {
/1 /1 / 1 = "cLLb d+- LL2 T2 = ¨/ = 7-2 = ¨n'11 c nkL2 b 2 (74) (75) They completely appear as the forms of an ideal transformer's equations, referred to as the voltage and current transformation-A type of the LC combined transformer, or transformation-A ideal transformer or ideal transformer A, when the circuit in Fig. 9 satisfying the condition either of Eqs.
(72) and (73).
Design Key Point: The voltage and current transformation-A type LC combined transformer (Fig.
(9)) is just the improvement or upgraded from the current transformation-A
type of the LC combined transformer. Hence, its error analysis, design key points, and device selections all are the same as the according contents respectively of the latter stated above, with a difference that the former has functioned as the input and output voltage in-phased just one-step further beyond the latter.
3-4. Voltage and Current Transformation-B Type LC Combined Transformer In the same way, redraw the circuit diagram of the current transformation-B
type LC combined transformer in Fig. 4(a) as in Fig. 10(a). In Fig. 10(a), of the delta (A) or pi ( n ) mutual capacitor consisting of inductances 9 and 10, and capacitors2 and 4, voltage V = jco(/ ¨/h )kL2 = jco(/ ¨/2)kL2 ¨ jco(iõ ¨/2)kL2 ( = ic) I kL2¨ ico(joclly2)kL2 = ico 1-- ki,21 +co' kL2c õy, ,zcJ
= jcoLoI +-1V2 ;[Lsl = (1 ¨ )kL 2 , when n, >11 (76) n, ( = j 1 _________ .I +V2 n, con,Cõ, n, _____________________________ I +,V2 ; Csi = ncCõ n ______________________________ , when , <1 (77) jcoC, n, 1 ¨ tl, In most cases, there exists n, < 1 ; thus the equation above should be expressed as taking on the series equivalent capacitance Csi as in Eq. (77) so that in Fig. 10, the delta (A) mutual capacitor between V and V2 can be replaced by an equivalent circuit of ideal transformer 25 and in series with its primary the equivalent input capacitance 24 or Csi, with the mutual inductor's primary leakage inductance (1-k)Li in Fig. 10(a) being equalized as its secondary leakage inductance (1-k)L2 in Fig.
1n 10(b). Next, assume jcp(1-42 + _______________ = 0 , i.e. 02(1 k)L2Csi = w2(1 k)L2 '2C =1 , or icoCsi 1¨n, co2(1-42n,2Cm =1¨n, ; and notice Eq. (20), namely co2L2Cm = _______________ , being substituted in as kn, 1¨k1 k __ = n =1 n, , or say when n, = k, or in= ¨1 (78) Fig. 10(b) could be equivalently replaced as Fig. 10(c), with the network port voltage and current equations as ( VI VI V 1 , ¨ ¨ = ¨ n = ¨ -= nk 1+ _____________________________________ (79) V2 V V2 nc Ch /
I I I 1 Cb (80) /2 / /2 n nk(Cõ +Cm) These are also equations of an ideal transformer, which is why the circuit in Fig. 10, when satisfying condition Eq. (78), is referred to as the voltage and current transformation-B
type of the LC

combined transformer, or transformation-B ideal transformer or ideal transformer B.
Design Key Point: The voltage and current transformation-B type LC combined transformer (Fig.
(10)) is also just the improvement or upgraded from the current transformation-B type of the LC
combined transformer. Hence, its error analysis, design key points, and device selections all are the same as the according contents respectively of the latter stated above, with a difference that the former has functioned as the input and output voltage in-phased just one-step further beyond the latter.
4. Function of Waveform Conversion from Square-Wave to Quasi-Sinusoid All the three categories or types of the LC combined transformers presented by this invention possess the function of waveform conversion or waveform isolation from square-wave to quasi-sinusoid [Note: take fundamental filter of square-wave as a typical example of waveform conversion, and rectifier transformer as atypical application of waveform isolation]. The following details analysis and explains of only one example for its operating principle and effect [Note: see 6-3. Functions of Waveform Conversion from Square-Wave to Quasi-Sinusoid of the Mutual Capacitor (Continue)].
Let's investigate the working status of the in-phase mode voltage transformation type LC
combined transformer in Fig. 5 under the driving of a cycling or periodic square-wave source v, ).
Assuming that v1(t) is a voltage of symmetrical cycling square-wave implemented on the input port of the mutual capacitor, with a cyclic frequency co = 2 Tr f= 2 n /T and its Fourier's series as (t) = '14,1 sin co t + Vi3 sin3cot +1415 sin 5cot +===+ sin kcot +=== , (m=1,3,5,...) (81) where, V11, v13, v15 '"' mean the magnitudes of the fundamental, third harmonic, fifth harmonic --etc. In addition, the magnitude ratio of m-th harmonic to fundamental for a symmetrical cycling square-wave is Vim /V11 = 1/m.
From Eqs. (26) to (28), magnitude of the m-th harmonic of the output voltage V, of the first mutual capacitor in Fig. 5 under the implement of v1(t) will be worked out as I
IVin V. = ________________________________________________________ (82) ( - 2 - no )12 1 co C mR, 11õ, )on j or expressed as Vxõ, =
Vim nvl V x, ( =\ 2 [1 - M2 - no )12 + _____________ I ____ 1 I
co C jm 1 no = = ______________________________________________________________ (83) 1 ( 1 1( (1 - m\- 2 - 2 0 - no )12 +
CO C R n m \, )\..n1 By this equation, calculate when n vi = 0.75, 0.5, 0.25, a)c,R, = 0.1, 1, 2,10, 100 , the values of xm for the mutual capacitor as recorded in the following form:
Vx, vi co Cn,R, IVxmA/xil, when m =

0.1 1.0000 .0278 .0089 .0042 .0024 .0015 1 1.0000 .1630 .0273 .0093 .0043 .0023 0.75 2 1.0000 .1884 .0282 .0095 .0043 .0023 1.0000 .1995 .0286 .0095 .0043 .0023 100 1.0000 .2000 .0286 .0095 .0043 .0023 0.1 1.0000 .0062 .0020 .0010 .0006 .0004 1 1.0000 .0379 .0080 .0029 .0014 .0008 0.50 2 1.0000 .0445 .0085 .0030 .0014 .0008 10 1.0000 .0475 .0087 .0030 .0014 .0008 100 1.0000 .0476 .0087 .0030 .0014 .0008 0.1 1.0000 .0010 .0003 .0002 .0001 .0001 1 1.0000 .0085 .0022 .0009 .0004 .0002 0.25 2 1.0000 .0119 .0026 .0010 .0005 .0002 10 1.0000 .0144 .0028 .0010 .0005 .0003 100 1.0000 .0145 .0028 .0010 .0005 .0003 Form 1 List for calculations of 11.7,./V,(11 by Eq. (83) when nvi and co Cõ,Ri have different values Design Considerations: From the results of the listed data, the influence on the output voltage by the harmonics of fifth and over is almost negligible; the influence of the third harmonic increasing accompanied with increase of riv1( generally, negligible when nv1 0.5); the change of (ú C,,R1) shows the load carrying capacity of the mutual capacitor not bad, with the load heavier the better fundamental filtering characteristic of the mutual capacitor. However, the heavier load for the mutual capacitor, the worse errors for it will occur determined by Eqs. (29) through (32). Therefore, during designing in practice, balances need to be made on or between the filtering characteristic, the load capacity, and the ratio errors.
5. Utilization of Push-Pull on Inductor The utilization of push-pull on inductor is also termed use of the push-pull inductor. Fig. 11(a) is a principle schematic and also an experimental circuit of the waveform conversion from square-wave to quasi-sinusoid using the circuit either in Fig. 5 or in Fig.7. Fig. 11(b) is an experi- mental circuit, to perform the square-wave to quasi-sinusoid conversion, improved from Fig.
11(a) by employing the push-pull inductor.
In Fig. 11(a), when the control-input terminal P of switch 29 or TR is input the signal with a waveform like P, the waveform of input voltage VD of the LC combined transformer is an asymmetrical pulsed square-wave sequence, while the input current I , is a single-polar periodic saw-tooth waveform, by which the core of inductor 28 or L is magnetized with a locus curve or hysteresis loop as shown in Fig. 11(d) . Within a cycle in steady-state operation of the circuit in Fig.
11(a), commencing at point Br in Fig. 11(d) with switch 29 or TR closed and switch 30 or D open while I , increasing, the magnetic flux density, accompanied with the change of the magnetic field strength, moves up the curve V to point a; and then switch 29 or TR open and switch 30 closed as well as I , decreasing, the flux density moves down the curve II back to point Br. This illustrates that the core's magnetization phenomenon occurs only in the first quadrant, which means that the core is not effectively utilized.
To overcome this drawback and make full use of the core, it will result in a good effect to use a full-bridge or half-bridge circuit to feed the dc/ac inversion. However, a bridge circuit has a shortage that it needs a complicated switch-control-and-driving circuit, for the reason that the reference voltages of its two sets of alternately working switches are not at a common potential.
To achieve this same goal, a use of the push-pull inductor is another choice (see Fig. 11(b)), which includes: 0 one center-tapped inductor 28a or La ; two sets of electrically-symmetric driving switches or switching devices such as transistors 31 and 33 [Note 1: Examples for "electrically-symmetric" are as those of driving switches, passive switches and their driving signals etc in double-ended circuits such as half-bridge, full-bridge and push-pull arrangements. Note 2: Suppose that the circuit herein belongs to positive logic and employs npn bipolar junction transistors (BJTs) though this application is not limited on positive logic nor to bipolar transistors employed only]; two sets of electrically-symmetric passive switches or rectifiers such as diodes 32 and 34; with the value and current rating of inductance La or 28a, and electrical specifications of the switches all determined by the requirements of design. 0 one end of inductor 28a electrically connected to the collector of transistor 31 and also to the anode of diode 32, the other end of 28a to the collector of transistor 33 and also to the anode of diode 34, the emitters of transistors 31 and 33 electrically connected together to the reference level, the cathodes of diodes 32 and 34 electrically connected together to a high potential, the center-tap of inductor 28a connected to another appropriate level [Note: In this example, to the junction between capacitances Cb and Cm], the bases or control-input terminals of transistors 31 and 33 separately connected to corresponding control-and-drive signals with two periods as a cycle, electrically-symmetrical to each other and alternately working. 0 the push-pull inductor employing a technique of the bi-periodically time-shared driving as described as: the PWM
control-and-drive signals for switches 31 and 33 in Fig. 11(b) separately be chosen as those like PI
and P2; although the total current, IA in Fig. 11(b), of the push-pull inductor remains the same as I , in Fig. 11(a), the magnetization mode of the core of inductor 28a or La is changed (see Fig. 11(d)) as:
during the steady-state operation of the circuit in Fig. 11(b), when only switch 31 or TRi turned on, the core's magnetization locus goes up curve I from point -Br to point a; then switch 31 or TRI off and diode 32 or DI turned on, while magnetizing down curve II from point a back to point Br at the moment that the first period of the circuit operation ends; symmetrically, the second period starts when only switch 33 or TR2 turned on, the core's magnetizing continuously moving down curve III
from point Br to point b; thereafter, switch 33 or TR2 off and diode 34 or D2 turned on, while the locus going up curve IV from point b back to point -Br at the end of the second period of the circuit operation and also of one cycle of the bi-periodically time-shared driving.
[Note: Herein the working sequence of switches is described by investigating the core's magnetization loci; it also can be described simply by stating the switch operations as: switch 33 being off for the first period while switch 31 on not longer than (T - t off) before turning off; for the second period switch 31 off while switch 33 on not longer than (T - t off) before turning off, with the end of second period as the end of a cycle of the bi-periodically time-shared driving;
where T is the time of switch operating period of the circuit and t off is the minimum time for the driven inductance to release all its stored energy].
In this example, the inductance value of inductor 28a in Fig. 11(b) is equal to that of inductor 28 in Fig. 11(a). In most cases, inductor 28a may use same cores and share the same coil turns number as those for inductor 28 , with the differences that, two coils of N turns, if N is the coil turns number for inductor 28 , wound bifilarly in parallel or separately in sections; and the wire cross-sectional area of the 28a coils equal to half that of 28's; and the wound twin coils connected series-aiding, with the connected point as the center-tap.
The technique of bi-periodically time-shared driving, in the utilization of push-pull on inductor, extends the core's magnetization as widely as to all four quadrants, or full range of its magnetization characteristic, greatly upgrading its effectiveness, and with its size relatively decreased as well as the loss and cost accordingly declined. In addition, it eliminates problem of the core's unsymmetrical magnetization phenomenon in conventional push-pull driving mode and greatly alleviates the cross-conductance of driving switches. Therefore, this technique is also suited for driving any other double-ended circuits, including bridge, half-bridge, and conventional push-pull, etc. As well, the usage of push-pull inductor, besides for the mutual capacitor or the LC
combined transformer, could be exploited in other circuits, such as in active power factor correction (APFC) circuit, and the like.
6. Explanations 6-1. Design Instructions of the LC Combined Transformer and General Rules for Its Device Selections 1). The design of the LC combined transformer is substantially that of mutual capacitors, in which the first step is to study and digest the requirements of the design specifications and target, in particular of the errors, and then, in accordance with them to determine all the parameters of the mutual capacitors.
2). Every specialized error of voltage/current ratio of the LC combined transformer is the sum of those respectively accorded of all the contained subunits, mutual capacitors and mutual inductors; and the total error is the sum of all the specialized errors or of all the errors of all the subunits. It has been verified, in theory and by experiences, that the ratio errors of the LC combined transformer originate significantly from frequency swing and power dissipation, which particularly appears apparent while heavily loaded with a low equivalent load resistance for power transferring. Principal measures to decrease its errors include: stabilizing the frequency, operating at a higher frequency, modifying parameters of mutual capacitors to optimize error designed parameter expressions of all the mutual capacitor subunits, as well as using low loss materials and devices, etc.
3). Capacitors to be used should be with capacitance values as accurate as possible, with minimum temperature coefficients, minimum tg 6 values or satisfying specified design requirements, and with suitable working voltages.
4). Cores of inductors and mutual inductors of an LC combined transformer should be employed the same soft magnetic material with high magnetic permeability that exhibits evenly around the operating range, with low loss, and with high saturation flux density. The relative permeability ti , of the cores should be taken by determining the mean value of the maximum relative permeability and the minimum relative permeability of inductors working between 2% or 5% (in accordance with the needs or precision requirements) and 100% of their rated currents, i.e. ,u, =(urn., +
)/2.rmin Linearization processing of the cores must meet the error requirements, with strictly control of the amounts used for copper and iron so as to realize the requested Q values by design.
5). Manufacture of a mutual inductor must come through models and experiments to obtain accurate design data, with the values of k, and of Li and L2 as precise as possible.
6). Adjustments and tests of the LC combined transformer should be separately based on its mutual capacitor subunits. Due to the decentralization of component parameters, mutual capacitors must be adjusted and tested within rated load ranges, in terms of principle of input and output voltages/currents in-phased, and measure the errors.
7). The design instructions and key points herein or included in this description state only those particular related to this invention, nothing of conventional methods.
6-2. Formulas for Linerization Processing of Inductors/Mutual-Inductors 1). Determine the product, SS , , of the core's cross-sectional area and window area:
For an inductor, -µ12 LIh 2 SS ,> X10-6 (1"114) or ( X 102cm4) (84) Bh j where, S --- core's cross-sectional area (m2, in general, S = 0.95 X area measured in practice);
S --- core's window effective area (m 2, S = K x area measured in practice, window utilization coefficient K 0.7 ¨ 0.9);
L --- inductor's inductance value (H);
1h rms value of the current at highest operating point in the winding (A);
Bh --- flux density at highest operating point of the basic magnetization curve (T);
j --- current density for coil copper wire (A/mm 2 or X 10 6 A/M2 For an mutual inductor, ( SS, 2 1+ ¨=(X10- 6m4) or (X102cm4) (85) 2/P ) 71" fBh j where, Iz rms value of the magnetizing current of the winding (A);
Ip rms value of the real-power current of the winding (A);
P --- power transferred through mutual inductor (W);
f--- sinusoidal frequency in operation (Hz).
2). Determine the coil turns number N, and the copper wire's diameter d:
LIh N > (86) BhS
where N --- coil turns number;
for an mutual inductor, Ih =Iz ; or by V
N= ____________________________________________________________ (87) A/2 it- f BõS
where V --- rms voltage of the winding (V). And d -=21/Ih Z,1.13 (mm) (88) j where Ih VI2 ____ +12 when for an mutual inductor;
with an assumption of a copper wire di d, check effectiveness of the window area.

3). Determine the core's air-gap length 1,, equivalent relative permeability 11 r; and check the inductance L:
du, ON lh ¨Hõ1,) 1 = _______________________________________________________________ (89) Bh where, 1, --- air-gap length (m);
1 F --- mean length of the core's magnetic circuit (m);
H h --- magnetic field strength at highest operating point of the basic magnetization curve (A/m);
and Pr. = (90) prIg +I, ,u,,,uõN2 S
L= ________________________________________________________________ (91) 11, 6-3. Functions of Waveform Conversion from Square-Wave to Quasi-Sinusoid of the Mutual Capacitor (Continue) Functions of waveform conversion from square-wave to quasi-sinusoid of the LC
combined transformer are actually those of the mutual capacitors. Following the detailed discussion of the in-phased mode voltage transformation type LC combined transformer's waveform conversion function given in the description, as a supplement, here presents the corresponding discussion of the anti-phased mode voltage transformation type LC combined transformer's waveform conversion functioned from square-wave to quasi-sinusoid, as well as a related discussion of the current transformation-A type LC combined transformer's current conversion functioned from square-wave to quasi-sinusoid.
The first mutual capacitor of the anti-phased mode voltage transformation type LC combined transformer in Fig. 6 also possess the characteristic of waveform conversion from square-wave to quasi-sinusoid, explained with an expression resulted from Eqs. (45) to (47) as 1/1.1 V ¨ _____________________________________________________________ (92) - 2 - no)12 + m(1¨ m 2 ) 1 1 vl )ct)C R
m or as V

_______ = Vint n vl V xl v11 ( \ 2 ( 1-12 _m2 (1 ¨ nvl )12 + m(1 ¨ m ) ¨1 1 n v, ct) C ,õR, j (93) 1 n vi = =

( , \ 2 r ¨ 2 ¨ nvl )12 in ¨ 1 ) __________________________________________ 1 vi \ci) C R1 )12 Design Considerations:
By listing the data of IVõm/Või calculated from this equation with different values of n vi and ( co CmRi ) of the mutual capacitor, conclusions could be drawn as: This mutual capacitor owns a much better characteristic of waveform conversion than that of in-phase mode except at a point no = 1;
the farther away from the point of n vi = 1, the better the characteristic is; and with a heavier load, and a more optimum characteristic will achieve; in addition, its voltage ratio ranges either greater than 1 or less than 1. However, it is also found from Eqs. (48) to (51) that the ratio error of the mutual capacitor turns out worse as its load goes heavier. Therefore, to promote the load capacity, it is necessary to make every effort either to increase the capacitance Cm, or to make the frequency 6-) higher, or to enhance the rivi , or to have all the three synthesized together so as to achieve the goal of both realizing waveform conversion and having the minimum errors.
The current transformation-A type LC combined transformer in Fig. 3 has a characteristic of current conversion from square-wave to quasi-sinusoid which, from Eqs. (8) to (10), is mathematically deduced as /2m = ___________________________________________________________ (94) ( nc = +[kwCR
_ /2. /m ____________ 1 -I- [kCOCinR _______ (95) ____________________________ cc __ r-2 in 2 m=i 1+ kwC,,,R _____ j_ where, r is a definite limited value. The final expression cc of this equation means that, when m larger /
than some definite number, the value of 2m is roughly inversely proportional to the square of m, which obviously shows this mutual capacitor having a function of current conversion from square-wave to quasi-sinusoid. As a matter of fact, supposed that the mutual capacitor is applied in inverse direction, i.e. the input and output ports switched to each other, its current conversion function will be much better; which could be soundly explained through an observation that the delta ( A ) mutual capacitor of the inversely-directional application is actually the dual mate of the tee (T) mutual capacitor in Fig. 11(a) [Note:
refer to Fig. 12-8 in "6-4. Principle of the Mutual Capacitor "].
6-4. Principle of the Mutual Capacitor 1). Definition of the Mutual Capacitor Definition: A mutual capacitor is a two-port network component with no power loss and coupled by electric field between ports of input and output.
Figs. 12-1 and 12-2 are schematic symbols or circuit models of the two types of mutual capacitors, in which the polarity notation, dots or circles, represents the same port voltage polarities, and dots or circles mean different type of them.
The first pair of the definition equations of the mutual capacitor is differentially expressed as Eq. (96), wherein they are described with currents as the characteristic of them, so as to be termed the current type of the mutual capacitor. Fig. 12-3 illustrates the simplest circuit configurations of the current type of the mutual capacitors, referred to as the delta ( d ) or pi ( Jr) mutual capacitor.
{. = c dv, c dv2 1 dt m dt (96) dv i2 = -C,,, jdv2 C - /
- dt dt The second pair of the definition equations of the mutual capacitor is integrally expressed as Eq. (97), wherein they are described with voltages as the characteristic of them, so as to be termed the voltage type of the mutual capacitor. Fig. 12-4 illustrates the simplest circuit configurations of the voltage type of the mutual capacitors, referred to as the tee (T) or wye (Y) mutual capacitor.

, vi =¨ fildt + __ fi2dt CI Cm . (97) V2 =¨ fi1 dt + ¨ fi2dt Cm C2 It must be pointed out that, though both the denomination and the definition of the mutual capacitor are described with capacitances, they are actually realized with capacitors as well as inductors, for, at a constant frequency w , an inductor functions exactly as a capacitor of negative capacitance, namely c _ 1 .
(2 L
[Note: even though the arrangement of three inductances in a delta ( A ) or tee (T) configuration is that of a mutual capacitor, not of a mutual inductor unless there exists magnetic coupling between ports.] Besides, of electronic circuits, a negative capacitance can be realized with a negative impedance converter (NIC, See Fig. 12-9) of integrated circuits; in terms of which a mutual capacitor constituted works at any frequency. Sometimes, a mutual capacitor may be called C-transformer as well, and a mutual inductor called L-transformer.
The self-capacitance coefficients, CI and C11, the mutual capacitance coefficient Cm of the current type mutual capacitor are respectively defined as dvt 1= CI (98) dt v2 =0 i2 = Cõ dv2 (99) " dt v, =0 i, 1 =-C dv2 i2 = - CA: dt v, =O
dvi dt v2 =0 (100) For the current type mutual capacitor in Fig. 12-3, obviously there are C/ = CA + Cm (101) CH - CB +CM (102) The coupling coefficient of the current type mutual capacitor is defined as Cm (103) k, = VC, Cõ
and, with having 1 k cl = 1 known as the current type mutual capacitor unity-coupled or in unity coupling, i.e. fully-coupled or in full coupling.

The ratio of the current type mutual capacitor is defined as II
n, = ¨ (104) i2 and pointed out that the ratio of the current type mutual capacitor has a practical significance.
The self-capacitance coefficients, Ci and C2, the mutual capacitance coefficient Cm of the voltage type mutual capacitor are respectively defined as 1 . , v1 =- kat (105) C 1 i2 = 0 .
V2 = 1 ¨ ft2dt (106) 1 VI = c!÷, fi2dt il =0 (107) v, =-- fiidt C . i2 = 0 For a voltage type mutual capacitor in Fig. 12-4, obviously there are CaCai C1 = ________ =C 1 Cm (108) Ca +Ca, c2= ___ C bCm= Cb 1 C. (109) CI, + Cai The coupling coefficient of the voltage type mutual capacitor is defined as C
k, ¨ In (110) VC, C2 and, with having i kyl = 1 known as the voltage type mutual capacitor unity-coupled or in unity coupling, i.e. fully-coupled or in full coupling.
The ratio of the voltage type mutual capacitor is defined as nv = ¨v1 (111) v2 and pointed out that the ratio of the voltage type mutual capacitor has a practical significance.
2). The Unity-Coupled Mutual Capacitors (1). Prerequisite of Unity Coupling of the Current Type Mutual Capacitor and lts Current Transformation Characteristic For the current type mutual capacitor illustrated in Figs. 12-1 and 12-3, suppose l k el = 1, or have it unity-coupled. Then, from Eqs. (101) ¨ (104) and (96), obtain il CA
nc= ¨=¨= sgn(CA CB) \Ic i (112) i 2 CH CH
This denotes that the ratio of the current type mutual capacitor IS its current ratio than nothing else provided that it is unity-coupled. And also noticed that this ratio is determined completely by its structural parameters C1 and CH, independently of its operating frequency and the load across its output port.
To make sure the current type mutual capacitor being unity-coupled, its structural parameters must meet the condition obtained from Eqs. (100), (101) and (103) as _____ + __ + __ = 0 (113) CA Cõ Cm (2). Prerequisite of Unity Coupling of the Voltage Type Mutual Capacitor and Its Voltage Transformation Characteristic For the voltage type mutual capacitor illustrated in Figs. 12-2 and 12-4, suppose l k vl = 1, or have it unity-coupled. Then, from Eqs. (108) ¨ (111) and (97), obtain CI, C
n, = 1---i = __ = sgn(C õ C b) ¨2- (114) v2 Ca Cl This denotes that the ratio of the voltage type mutual capacitor IS its voltage ratio than nothing else provided that it is unity-coupled. And also noticed that this ratio is determined completely by its structural parameters C1 and C2, independently of its operating frequency and the load across its output port.
To make sure the voltage type mutual capacitor being unity-coupled, its structural parameters must meet the condition obtained from Eqs. (108), (109) and (110) as Ca + C, + C. = 0 (115) (3). Equivalent Circuits of Unity-Coupled Mutual Capacitors Expressed with Controlled Sources First, let's look at the current type mutual capacitor. Assuming the current flowing through the coupling capacitance Cm in Fig. 12-3 as i (in a direction to V2), we have v, = __ f(i, - i)dt C
=c1 (i, + i2) dt CB = ¨1 (i + i2) dt A CA CB
= (1+ ¨1)1 dt B v2 CA n, CA

j i, dt- ¨1 v2 (116) ( n, A
\ 1 nc ) ( n If letting Cs = , CA (117) 1+ tt from Eqs. (112), (116) and (117), a controlled-source equivalent circuit for unity-coupled current type mutual capacitor is set as in Fig. 12-5(a).
Next, we discuss the voltage type mutual capacitor. Assuming the voltage across the coupling capacitance C1, as v, shown as in Fig. 12-4, we have i = Ca ¨ (v, -v) dt = C0 --(v1-v2) + = C, --(v2 -v) dt dt (118) =C0 ¨d[v, (1--1)]+ ¨i2 dt nVCb 1 d 1 = (1--)Ca ¨v1 - ¨i2 11, dt n v Ifletting C,. = 1-1 Cc, (119) itv ) from Eqs. (114), (118) and (119), a controlled-source equivalent circuit for unity-coupled voltage type mutual capacitor is set as in Fig. 12-5(b).
(4). Ideal Mutual Capacitors If we are drawing out in further abstract from the unity-coupled current type mutual capacitor shown in Fig. 12-3, letting CA +CO , CB +CO
, and meanwhile Cm - co ( restricted by Eq. (113)), and assuming the limit of ratio of (CA/CB ) existing, we have the ratio of Eq.
(96), or ( ii/ i 2), approaching its limit in a result as Eq. (112), that is ( \
. ii 11111 = 11111 Crn = it, (120) ct _____ )1'4 2 CA ->+. \:--"B ) CB __ ) oo while Eq. (116) being evolved as 11M vf = ¨ -- v2 (121) nc CB -)4,0 meaning the port relationships shaped as the simplest descriptions, shortly as 'ii = nei2 1 1 (122) v1 = ¨ ¨ V2 n, _ Also, we do the same thing for the unity-coupled voltage type mutual capacitor shown in Fig. 12-4, letting C. --> +0 , Cb ¨> +0 , and meanwhile Cm ¨* ¨ 0 ( restricted by Eq.
(115)), and assuming the limit of ratio of( Ca/Cb ) existing, we have the ratio of Eq. (97), or (v 1 / v 2), approaching its limit in a result as Eq. (114), that is ( vi .
11M -= 11M ---?(-.6' =nv (123) >03 \,. ¨a , C1,¨>+0 C,, HO
while Eq. (118) being evolved as 11Mi 1 = ¨ ¨i2 (124) C0 -*-FO n, C,, -*+O
meaning the port relationships also shaped as simplest descriptions, shortly as Iv1= nv v2 i I= ¨ -- i2 n, (125) Making a comparison between above two pairs of equations concluded in short, of Eqs. (122) and (125) indicates that they both have the same mathematical equations, just with a negative reciprocal ratio to each other. Thus, provided that we ignore some difference between properties (of types, current and voltage), and stress their sameness in mathematics and commonness in physics (having the same mathematical equations and belonging to common capacitive two-port network components coupled by electric field), present one in between and representing them both of two-port network components, i.e. the ideal mutual capacitor, as well as its mathematical equations and the model of network component.
Port equations of the ideal mutual capacitor are given as {ii = it i2 1 (126) v1 = --v2 n and its schematic symbols or circuit models given as in Fig. 12-6. Attention should be paid to Fig. 12-6, in which what the polarity notation is like denotes its type, with the symbol in Fig. 12-6(a) corresponding to Eqs. (122) when n = n, , and with the symbol in Fig. 12-6(b) corresponding to Eqs. (125) when n = -1/nv. .
Fig. 12-7 illustrates the equivalent circuits, represented with ideal mutual capacitors, of unity-coupled mutual capacitors, replacing those in Fig. 12-5.
3). The Mutual Capacitors and the Duality Principle for Electric Networks Introduction of the mutual capacitor complements and perfects the duality principle of electric networks, which assists the principle of duality to get more effectively applied into practice. An example for a circuit including a coupling component is shown as in Fig. 12-8.
Fig. 12-8(b) illustrates a dualizing approach being employed, termed the branch-dualizing rule, which is briefly stated as follows:
Dual branch is given by 90 counterclockwise turning primary branch if the primary is in accordance with the passive sign convention [Note: It's a convention that a branch's current just enters the "+" end of its voltage], such as [Note: "<=>"means "dual to each other"]:
resistance <=> conductance; primary of mutual inductor <--> primary of mutual capacitor;
inductance <=> capacitance; driving switch off <=> driving switch on.
CD Dual branch is given by 90 clockwise turning primary branch if the primary is not in accordance with the passive sign convention, such as:
current source <=> voltage source; passive switch on <=> passive switch off;
secondary of mutual inductor <=> secondary of mutual capacitor.
[Note: Examples of switch categories: driving switch ¨ transistor; passive switch diode].

Claims (4)

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE PROPERTY OR
PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:

1. A species of electric transformer, termed LC combined transformer, used for transferring electric signal or energy of periodic sine wave, and proportionally altering the amplitude(s)/magnitude(s) of current or/and voltage of the periodic sine wave between input and output ports when neglecting power loss, being a current type unity-coupled mutual capacitor characterized as current transformation, or a voltage type unity-coupled mutual capacitor characterized as voltage transformation, or a circuit of LC combined transformer in cascade connection of an ideal transformer plus one current type unity-coupled mutual capacitor or two voltage type unity-coupled mutual capacitors, comprising said current type unity-coupled mutual capacitor consisting of three linear capacitances in delta (A) configuration with a sum of the reciprocals of said three capacitances being zero;
said voltage type unity-coupled mutual capacitor consisting of three linear capacitances in wye (Y) configuration with a sum of said three capacitances being zero;
either of said unity-coupled mutual capacitors including negative capacitance(s) which may be realized through negative impedance converter(s) (NIC) or by employing linear inductor(s) when operating at a constant frequency;
said circuit of LC combined transformer comprising current transformation-A
type LC
combined transformer consisting of a mutual inductor Tr , an inductor L b and a capacitor C m , if designating the primary winding terminals of said Tr as the input port of said current transformation-A type LC combined transformer, said Tr's secondary winding and said L b and said C m being connected in series to form a closed loop before designating the two terminals of said C m as the output port; said current transformation-A type LC combined transformer presenting a ratio of current transformation between ports as under the prerequisite of the parameters satisfying .omega.2C m(L2 + L b)= 1 ;

said circuit of LC combined transformer comprising current transformation-B
type LC
combined transformer consisting of a mutual inductor Tr and two capacitors C b and C m , if designating the primary winding terminals of said Tr as the input port of said current transformation-B type LC combined transformer, said Tr's secondary winding and said C b and said C m being connected in series to form a closed loop before designating the two terminals of said C m as the output port; said current transformation-B type LC combined transformer presenting a ratio of current transformation between ports as under the prerequisite of the parameters satisfying said circuit of LC combined transformer comprising in-phase mode voltage transformation type LC combined transformer consisting of a mutual inductor Tr , an inductor L a , and two capacitors C b and C m , if taking one end of said L a as the input terminal, the other end is connected with one end of said C b and also one end of said C m , the other end of said C b connected to one terminal of the primary winding of said Tr , the other end of said C m connected to the other terminal of said primary winding with this joint taken as the common terminal, then designating said input terminal and said common terminal as the input port of said in-phase mode voltage transformation type LC combined transformer, as well as designating the two terminals of the secondary winding of said Tr as the output port; said in-phase mode voltage transformation type LC combined transformer presenting a ratio of voltage transformation between ports as under the prerequisite of its parameters satisfying said circuit of LC combined transformer comprising anti-phase mode voltage transformation type LC combined transformer consisting of a mutual inductor Tr , two inductors L a and L b , and two capacitors C b and C m , if taking one end of said L a as the input terminal, the other end is connected with one end of said L b and also one end of said C m , the other end of said L b connected to one end of said C b , the other end of said C b connected to one terminal of the primary winding of said Tr , the other end of said C m connected to the other terminal of said primary winding with this joint taken as the common terminal, then designating said input terminal and said common terminal as the input port of said anti-phase mode voltage transformation type LC combined transformer, as well as designating the two terminals of the secondary winding of said Tr as the output port; said anti-phase mode voltage transformation type LC combined transformer presenting a ratio of voltage transformation between ports as under the prerequisite of its parameters satisfying and .omega.2(1 - k2)L1C b = 1 ;
for all above formulas where I1 and I2 respectively represent the sinusoidal currents of entering said input port and leaving said output port , V1 and V2 respectively represent the sinusoidal voltages across said input and output ports, L1 and L2 respectively are the self-inductances of primary and secondary windings of said mutual inductor Tr , k and n respectively are the coupling coefficient and turns ratio of said Tr , L a and L b respectively represent corresponding inductances of said inductors L a and L b , C b and C
m respectively represent corresponding capacitances of said capacitors C b and C m , C b1 and C b2 respectively represent the first and the second of two series-equivalent components of said capacitance C b or C b = and .omega. is the electric angular frequency of the periodic sine wave applied to this transformer.

2. The electric transformer according to claim 1, wherein the inductor L b and the mutual inductor Tr may be linearly integrated into an integrated inductor and mutual inductor, comprising:
the core magnetic circuit F1 of said Tr, the core magnetic circuit F2 of said L b, the first winding N1 of said Tr, the two-in-one common coil N2 serving both as the second winding of said Tr and also as the winding of said L b , and the auxiliary winding .DELTA.N (set when needed) of said L b , being structurally built as that, said N1 is just wound around said F1 with its terminals designated as one port of said integrated inductor and mutual inductor, said N2 wound around the paralleled and adjacent-to-each-other portions of both said F1 and F2, said .DELTA.N just wound around said F2, plus said N2 and .DELTA.N connected in series-aiding with their terminals after series designated as the other port, with a result that the turns ratio n, the coupling coefficient k, the primary self-inductances L1 and L2 respectively of said N1 and N2 , and relationships of currents and powers of said mutual inductor Tr will all remain the same as those of its original mutual inductor without being integrated, the inductance L b of said inductor L b will be determined just by said F2 and (N2 +.DELTA.N) as that of a normal inductor, except that the total inductance of said other port of this integrated inductor and mutual inductor will be the sum of said inductances L2 and L b when both said core magnetic circuits are linear.

3. The electric transformer according to claim 1, wherein the inductor L a may be a center-tapped inductor, thus being termed a use of push-pull inductor, including:
1 the center-tapped inductor, two electrically-symmetric switching devices such as power bipolar junction transistors (BJTs) --- each with a diode connected in series-aiding or in reverse-parallel for a purpose of protection, and two electrically-symmetric auxiliary switching devices such as diodes ;
2 being constructed as that the center-tap of said inductor is electrically connected to a high potential, one end of said inductor connected to collector of the first BJT and also to anode of the first diode, the other end of said inductor connected to collector of the second BJT as well as to anode of the second diode, emitters of both said BJTs connected together to the reference potential, cathodes of both said diodes connected together to another appropriate high potential, and bases of both said BJTs respectively connected to corresponding control-and-drive signals;
3 and employing a technique of bi-periodically time-shared driving to drive the push-pull inductor.

4. The use of push-pull inductor according to claim 3 , wherein the technique of bi-periodically time-shared driving is described as:
a pulse-width modulation (PWM) control and drive with two switching periods being a cycle of a sequence, stated as follows:
for the first period the second switching device keeping OFF while the first switching device being ON no longer than (T - t off) before being turned off; for the second period the first switching device keeping OFF while the second switching device being ON no longer than (T -t off) before being turned off, with the end of the second period as the end of a cycle of the bi-periodically time-shared driving; where T is the time interval of switching period of the circuit, and t off is the minimum time for the driven inductance to release all its stored energy.