Mutual capacitor and its applications

Transcription

1 Mutual capacitor and its applications Chun Li, Jason Li, Jieing Li CALSON Technologies, Toronto, Canada E-ail: Published in The Journal of Engineering; Received on 27th October 2013; Accepted on 29th April 2014 Abstract: This study presents a new ac circuit eleent the utual capacitor, being a dual of the utual inductor, which is also a new ac transforer. This eleent is characteristic of the utual-capacitance coupling of a ulti-capacitance syste. A unity-coupled utual capacitor works as an ideal current or voltage transforer, and incidentally acts as wavefor separating when inductor eployed or wavefor converting fro square-wave to quasi-sine or wavefor filtering, between ports. As a transforer, the utual capacitor is easy to design, easy for heat cooling, ore accurate for current or voltage transforation, dissipating less energy as well as saving aterials, suitable for high-power and high-voltage applications. Experients to deonstrate perforances of unity-coupled utual capacitors are also given. 1 Multi-capacitance systes and utual capacitor Of an ac network, a linear capacitor C, when supplied with an ac voltage source across its terinals, is described as i = C(dv/dt) [1], which characterises the i v relationship between its two terinals or between a single conductor and soewhere at infinity in syste; where C is its sybol and also denotes its capacitance value, i is the current flowing through it, v is the voltage across its terinals and we assue that it is non-dissipative (Fig. 1a). As well, there exist ulti-capacitance ac systes in engineering, such as transission lines in electric power transission, ulticonductor systes in coplicated circuital environent and issues on distributions of stray capacitances over adjacent circuits investigated in solid-state electronics etc. A ulti-capacitance ac syste consists of K + 1 conductors, where assuing the potential of the K + 1st conductor is referenced [Note: a ulti-capacitance ac syste in an infinite space ay consist of K conductors, where the potential at infinity is referenced.], potentials of the other conductors are in turn v 1, v 2,, v n,, v K. For an independent, linear and lossless ulti-capacitance ac syste, such as the ulticonductor syste in Fig. 1b, the current flowing into conductor accordingly is dv i = C 1 1 dt + C dv 2 2 dt + +C dv n n dt ( ) dv + +C K = 1, 2,..., K K, dt n = 1, 2,..., K (1A) Or, if only the perforances in sinusoidal steady-state operation are concerned, the current has its phasor-doain notation as I = jvc 1 V 1 + jvc 2 V 2 + +jvc n V n + +jvc K V K, ( ) = 1, 2,..., K n = 1, 2,..., K (1B) where C n, when = n, is tered self-capacitance, the relationship between conductor and the reference potential; C n when n, is tered utual-capacitance [2], the relationship between conductors and n; and they both contribute to producing the current i or I. It ust be noted that the C n in ulti-capacitance syste (1) (A or B) (Fig. 1b) differs obviously fro the C discussed in Fig. 1a, for in (1) there exists not only the self-capacitance C n ( = n) in selfexcitation a perforance on its current by potential of its own conductor, but also the utual-capacitance C n ( n) in utual excitation a perforance on the current by potentials of the other conductors, whereas the C eans only the perforance on its current by potential of its own conductor. It is easy to find or iagine that behaviours of a ulticapacitance syste described in (1) are quite siilar to those of a well-known ulti-inductance syste. A physical phenoenon of the utual inductance is a behaviour of agnetic-field coupling; and a physical phenoenon of the utual-capacitance falls in the category of electric-field coupling. That the coupling of utual inductance of ulti-inductance systes was tiely discovered and fully exploited benefited fro the effect of ferroagnetic edia, which allows ankind to have had the utual inductor widely used for long [Note: a utual inductor, or coupling inductors as popularly called, norally eans a device of two or ore coils, but in this paper we ephatically ean those of two coils, being agnetically coupled between each other, including the conventional inductance transforer.]. Although the coupling of utualcapacitance of ulti-capacitance systes scatters uch sparser in nature and both agnitudes and coupling levels of its behaviours are far less than those of the coupling of utual inductance of any iron-cored utual inductor. It is why for so long no such practical device a capacitance transforer, which ay or should be tered utual capacitor, exists appearing and perforing siilar to the utual inductor. Even so, the author thinks, it will not prevent us fro getting to know of and exploit the principle of ulticapacitance systes. Anyone that expects a utual capacitor ay ideally have thought that a utual capacitor should be the sae as or siilar thoroughly to a utual inductor both in appearance and in perforance; it should work ideally dual to a utual inductor or inductance transforer. However, it is not true and there is no such device! Maxwell s electroagnetic-field equations ( D = ρ and B = 0) reveal that electric field acts ostly analogously to agnetic field, but not all. Therefore an ideal utual capacitor built up in an integrated structure and working ideally dual to a utual inductor under any condition does not exist! However if we exaine and analyse this issue in a point for practical use, we ay set this device as such a odel: (i) it can be a dual of the utual inductor acting well enough in ost behaviours, especially for practical use although not ideally in every aspect and (ii) it can succeed in perforance as a transforer in engineering. In this paper, we also call such a device the utual capacitor.

2 Fig. 1 Capacitance relationship of ac networks a Between two conductors b For conductor in a ulti-conductor syste For better understanding of ulti-capacitance systes and for a purpose of practical use, we designate K = 2 for (1), the utual capacitor will be put forth and discussed in this paper. Accordingly, its theory and feasibility will be discussed as well which include circuit configurations of the utual capacitor and its definition and scheatic sybols; concepts of its unity coupling and principle of its transforation of voltage or current its prerequisite conditions for unity coupling, ethods and forulae; its characteristics of wavefor separation; and the experiental deonstrations to show its perforances in practical uses. Owing to liited space, ore and further investigations will be presented following this publication. 2 Basics of the utual capacitor 2.1 Mutual capacitor A utual capacitor is a two-port network eleent with no power loss and coupled by electric field between ports of input and output. To investigate the new network coponent ore clearly, we use Figs. 2 and 3 to be its scheatic sybols or circuit odels for two types of the utual capacitor, in which the position of circles or dots represents the polarity notation of the port voltages, and circles or dots ean different types of the utual capacitor as well. As seen in both figures, v 1 and v 2 are port voltages, respectively, for input and output and i 1 and i 2 each are port currents for the. The utual capacitor shown in Fig. 2 ay be configured in its siplest scheatic diagras as in Fig. 4. Its equation is expressed in phasor doain as (2), wherein it is presented by port currents, tered the current type of utual capacitor ; also referred to as the Δ (or π) utual capacitor. [Note: (2) is obtained fro (1B) by designating K = 2 and aking notations in accordance with those illustrated in Fig. 4; or it also can be a dual brought out directly, by the principle of duality, fro the characteristic equation of the utual inductor or lossless voltage inductance transforer.] { I 1 = jvc I V 1 jvc M V 2 = jv(c A + C M )V 1 jvc M V 2 (2) I 2 = jvc M V 1 jvc II V 2 = jvc M V 1 jv(c B + C M )V 2 where the second step of both forulae is in accordance with Fig. 4a, described with the structural paraeters C A, C B and C M which can exist all over the three-diensional (3D) space, that is, { C A <, C B <, C M < }; C I and C II are tered the Fig. 3 Circuit sybols for voltage type of utual capacitor self-capacitance coefficients and C M tered the utualcapacitance coefficient of the current type utual capacitor; and they are, respectively, defined in the following I 1 = jvc I V 1 (3) V 2 = 0 I 2 = jvc II V 2 (4) V 1 = 0 I 1 = jvc M V 2 V 1 = 0 I 2 = jvc M V 1 V 2 = 0 Obviously there are { C I = C A + C M = C A //C M C II = C B + C M = C B //C M The coupling coefficient of the current type utual capacitor is defined as C M (5) (6) k c = (7) C I C II and when k c = 1, the current type utual capacitor is known as unity-coupled or in unity coupling, that is, fully coupled or in full coupling. The utual capacitor shown in Fig. 3 ay be configured in its siplest scheatic diagras as in Fig. 5. Its equation is expressed Fig. 2 Circuit sybols for current type of utual capacitor Fig. 4 Circuit ipleentation for current type of utual capacitor

3 coupling between ports.] Besides, for electronic circuits, a negative capacitance value can also be achieved through a negative ipedance converter (NIC) [3] of integrated circuits (ICs); in ters of which a utual capacitor constituted operates theoretically at any frequency. Fig. 5 Circuit ipleentation for voltage type of utual capacitor in phasor doain as (8), wherein it is presented by port voltages, tered the voltage type of utual capacitor ; also referred to as the Y(or T) utual capacitor. Equation (8) is also a dual fro (2) (in a siilarity of operational rules in atheatics) [Note: (8) can also be obtained fro the characteristic equation of the utual inductor by substitutions of a relationship L = 1/(ω 2 C).] V 1 = 1 jvc 1 I 1 1 jvc I 2 = 1 jv ( ) C a C I 1 1 jvc I 2 V 2 = 1 I jvc 1 1 I jvc 2 = 1 I 2 jvc 1 1 ( ) jv C b C where the second step of both forulae in (8) is in accordance with Fig. 5a, described with the structural paraeters C a, C b and C which exist in 3D space but not all, that is, {C a 0, C b 0, C 0}; C 1 and C 2 are tered the self-capacitance coefficients and C tered the utual-capacitance coefficient of the voltage type utual capacitor; and they are, respectively, defined in the following Obviously there are I 2 (8) V 1 = 1 I j v C 1 (9) 1 I2 = 0 V 2 = 1 I jvc 2 (10) 2 I 1 = 0 V 1 = 1 I jvc 2 I 1 = 0 V 2 = 1 (11) I jvc 1 I 2 = 0 C 1 = C ac = C C a + C a C C 2 = C bc = C C b + C b C (12) The coupling coefficient of the voltage type utual capacitor is defined as C k v = (13) C 1 C 2 and when k v = 1, also the voltage type utual capacitor is known as unity-coupled or in unity coupling, that is, fully coupled or in full coupling. Although both the denoination and the definition of the utual capacitor are described with capacitances, they are ostly ipleented with capacitors as well as inductors, for, at a fixed frequency ω, a positive inductance functions exactly as a negative capacitance does, naely C = 1/(ω 2 L). [Note: even an arrangeent of three inductors in a Δ (or Y ) configuration is a utual capacitor, other than a utual inductor, unless there exists agnetic 2.2 Unity-coupled utual capacitor Unity coupling of the current type utual capacitor and its current transforation property: For the current type utual capacitor shown in Fig. 4 to be unity-coupled, anipulating (6) and (7) by letting k c = 1, establishes its prerequisite condition for unity coupling as C I C II C 2 M = 0or 1 C A + 1 C B + 1 C M = 0 (14) In a sense of being unity-coupled, its definition (2) will becoe I 1 = C ( ) A jvc C A + C A V 1 + jvc B V 2 B I 2 = C ( ) (15) B jvc C A + C A V 1 + jvc B V 2 B Note that the algebraic su of both current ters in above parentheses is not zero, for jωc A V 1 +jωc B V 2 = I CA + I CB = 0, eaning physically that the net current flowing into reference potential keeps not being zero [Note: according to geoetrical syetry of its structure, it eans that the current type utual capacitor cannot be designed as a current transforer of a ratio equal to one.]. When this reains true, a unity-coupled current type utual capacitor will have its current transforing ratio between ports as n c = I 1 = C A = C I I 2 C B C M C = sgn(c M C I ) I = sgn(c A C B ) C II C I C II (16) This is also called the ratio of the unity-coupled current type utual capacitor. It indicates that this ratio is set up only when it is unitycoupled, and deterined only by its structural paraeters C A and C B,orC I and C II, independent of its operating frequency or/and its load across output port. The unity-coupled current type utual capacitor is an ideal current transforer Unity coupling of the voltage type utual capacitor and its voltage transforation property: For the voltage type utual capacitor shown in Fig. 5 to be unity-coupled, anipulating (12) and (13) by letting k v = 1, establishes its prerequisite condition for unity coupling as C 1 C 2 C 2 = 0orC a + C b + C = 0 (17) In a sense of being unity-coupled, its definition (8) will becoe ( C V 1 = b I 1 + I ) 2 (C a + C b ) jvc a jvc ( b C V 2 = a I 1 + I ) 2 (C a + C b ) jvc a jvc b (18) Let the algebraic su of both voltage ters in above parentheses

4 keep not being zero, that is, (I 1 /jωc a ) = (I 2 /jωc b ), eaning physically to keep it true as V Ca = V Cb [Note: according to geoetrical syetry of its structure, it eans that the voltage type utual capacitor cannot be designed as a voltage transforer of a ratio equal to one.]. When this keeps true, a unity-coupled voltage type utual capacitor will have its voltage transforing ratio between ports as n v = V 1 V 2 = C b C a = C = sgn(c C 1 ) = sgn(c a C b ) C 1 C 2 C 1 C 2 C 1 (19) This is also called the ratio of the unity-coupled voltage type utual capacitor. It indicates that this ratio is set up only when it is unitycoupled, and deterined only by its structural paraeters C a and C b,orc 1 and C 2, independent of its operating frequency or/and its load across output port. The unity-coupled voltage type utual capacitor is an ideal voltage transforer. 2.3 Characteristics of wavefor conversion of the unity-coupled utual capacitor The utual capacitor, particularly a unity-coupled utual capacitor, is constructed by cobining capacitors of positive and negative capacitances. In a case of power application, based on today s technology, a negative capacitance can be ipleented only by an inductor, for C = 1/(ω 2 L), leading to an LC network (where L is the inductance and C is capacitance) fored as a utual capacitor. It is a known fact that an LC network is a haronic filter; but what if it does a great attenuation of high haronics while norally transforing the fundaental current and voltage between ports? Or, what if a current or voltage source of ac square-wave is applied onto its input and its output supplies a current or voltage whose agnitude has been transfored as needed but with its wavefor shaped of a quasi-sine? The unity-coupled utual capacitor or capacitance transforer perfors a transforation of current or voltage while converting the wavefors fro square-wave to quasi-sine, which has never been in such a case for the inductance transforer. We take the voltage type utual-capacitance transforer for an exaple to investigate the perforance of wavefor conversion fro square-wave to quasi-sine. In Fig. 5a, we choose C a = 1/ (ω 2 L a ), C b > 0 and C > 0 to for a unity-coupled utual capacitor, where ω is the electric angular frequency of the fundaental of the square-wave at a fixed frequency applied to the input. Thus, the phasor expression for (17) is ω 2 L a (C b + C ) = 1, which is the base to design a unity-coupled voltage type utual capacitor. This transforer is a step-up (0 < n v < 1); and the frequency spectru analysis of its responses to a square-wave voltage should be derived out fro Fig. 5a and (19) in the phasor doain. Assuing its input being applied a voltage of syetrical square-wave and its haronic has an order nuber as k( = 1, 2, 3, ), we have a voltage aplitude relationship between the kth haronic and its fundaental at the output as (20), where R is the load at the output. It is provable that a unity-coupled voltage type utual capacitor has a good perforance of reoving the forward voltage haronics, either by chart, list or even by experients as long as suitable paraeters L a, C b and C of the utual capacitor are given. Meanwhile, we can also prove that it has an excellent capacity of reoving the backward current haronics. Assuing its output is supplied a current of syetrical square-wave, we derive out the current aplitude relationship of the kth haronic reflected to the input side over its fundaental as I 1k I 11 [ ( )] (k/n v )1 k 2 1 n, k = 1, 2, 3,... (21) v If setting the circuit paraeters in Fig. 5a as C a = 1/(ω 2 L a ), C b = 1/(ω 2 L b ) and C > 0, we can design a voltage transforer either step-up or step-down, but it ust be noted that the voltages on both ports in this situation are anti-phased fro each other (n v <0). Based on alost the sae as above phasor-doain analysis, a voltage aplitude relationship between the kth haronic and its fundaental at the output and a current aplitude relationship of the kth haronic reflected to the input side over its fundaental are, respectively, derived out as (22), where R is the load at the output. It is provable that it has a better perforance of reoving/attenuating the forward voltage haronics than that of the transforer of (20), either by chart, list or even by experients, whereas they both have the sae capacity of reoving/attenuating the backward current haronics. 2.4 Unity-coupled utual capacitor against the circuits in resonance Generally speaking, by taking a tap fro the joint of two capacitors in series, we can also obtain a voltage transforation, which is the concept of a voltage divider. However the voltage divider can only step-down a voltage, together with its voltage ratio changing dependently on both the power angular frequency ω and the load. A utual capacitor, however, when unity-coupled, transfors a current or voltage independently, either step-up or step-down, which copletely is its own internal echanis. This echanis is essentially analogous to the circuit resonance, but it is controllable and it is safe at any situation, or we ay say, it is a special resonance set with three independent eleents under a condition of (15) or (18), naely (2) and (14) or (8) and (17). As seen in (2) and (8), there exist soe zeros and poles, which are actually those values satisfying equations C I C II =(C A + C M )(C B + C M )=0 or C 1 C 2 =(C a + C )(C b + C ) = 0. Aong these zeros or poles, anywhere a su of two structural paraeters equals zero is where a circuit resonance occurs, which interprets that the circuit resonance is not defined in the unity-coupled utual capacitor, as well as not a safe circuital phenoenon. V 2k V 21 ( )[ ] ( k/n v 1 k2 (1 n v ) + jvla /n 2 v R ) (k2 1), k = 1, 2, 3,... C b n v = C b = v 2 L C a C b = a C b + C (20) V 2k V 21 I 1k I 11 [ ( )] (( ( )) ) (k/n v )1 k 2 1 n v + jk2 v(l a + L b )1 k 2 /nv (1/R), k = 1, 2, 3,... [ ( )] (k/n v )1 k 2 1 n, n v = L a v L b (22)

5 Fig. 6 Capacitor voltage transforer 2.5 Mutual capacitor against the capacitor voltage transforer The idea of coupled capacitor including its two-port networks was presented by Maxwell in his Treatise of the id 1800s. The theory for the had also been uch developed by followers once for soe tie, especially in circuit synthesis counity. In 1936, being an expectation or proposition, the CVT being only a ter was presented [4], with no clai ade for presenting any new principles of design. One of its earliest practical devices was given and published in 1949 [5]. Today the CVT has been widely used as an instruent transforer in power systes (Fig. 6). In operation, the CVT works in alost the sae way as a unity-coupled voltage type utual capacitor does; however, it is uch different fro the utual capacitor in any aspects: (i) In foring principle, the CVT eploys an inductor or copensating reactor L connected to the joint of a capacitor voltage divider to copensate the circuit into resonance, whereas the utual capacitor is built of independent capacitors of positive and negative capacitances under unity-coupled conditions. (ii) In practice, the CVT has a daper winding added to its Tr plus a daper which are a ust to suppress haronics (see Fig. 6), being configured ore coplicated than the utual capacitor of just three coponents working in linear status. (iii) The CVT has only one arrangeent, whereas the utual capacitor in Fig. 5 has ore any one or two of its three capacitors being given negative capacitance will ake an arrangeent. (iv) The CVT cae out of a technique being a device only while the utual capacitor originates fro a new theory being a circuit eleent dual to the utual inductor as well as a new transforer copleentarily with the conventional transforer for engineering uses; which is why, although another half century has already passed, the idea of coupled capacitor had still to be stopped at the CVT which so far can serve only as instruent voltage transforers (of output quite sall to lessen interference of haronics) while the utual capacitor can and will be used, with its various configurations, either for instruents or for power applications, not only in voltage but also in current transforation, plus functioning wavefor conversion. [Note: owing to space liitation, only the basics of the principle of the utual capacitor are presented in this paper.] 3 Experients The above presentation of the utual capacitor is proposed under a condition of linearity. As a atter of fact, a practical coponent or device in engineering is norally non-linear, and as its operating state as well as surroundings especially the teperature change, its non-linearity keeps unstable or its perforing property goes nonlinear, leading errors to be in existence. The detailed analyses on errors of the utual capacitor, approaches to iniise the, forulae for ore accurate calculations etc. will be given by author in another publication. Fortunately, practical non-polar capacitors, particularly polypropylene-dielectric, in engineering have high linearity within their operating regions, norally being treated as linear coponents which can eet ost engineering s requireents. A negative capacitance can only be obtained by eploying an inductor for power application; however, iron-cored inductors norally are non-linear, and will bring in a great deal of haronics when directly used, as well as spoil the prerequisite condition of the utual capacitor. The author copleted the following experients by eploying an approach the linearisation processing of inductors, that is, by eploying a structure of agnetic circuit of airgapped cores or agnetic powder cores, so as to have the inductors operate within the required regions of linearity, resulting in that the utual capacitor works in linearity. The inductors for negative capacitance, after cores being air-gapped, will have uch lower inductances although, which really is not a bad thing since utual capacitors for power applications, especially in low frequency, are ade up of large value capacitors, accordingly by (17), the values of negative capacitance are large as well by C = 1/(ω 2 L); thus extreely, in the case of huge power applications, air-cored coils ay be used to lower the cost, raise the efficiency (because of lower loss of iron and copper) and also iprove the heat-cooling condition of the utual capacitor. The following three experients, although not of big power output (which ay be calculated fro the data given), were copleted at high efficiency (P out /P in > 90%) under very liited laboratory conditions of the author. 3.1 Unity-coupled current type utual capacitor An experient for the unity-coupled current type utual capacitor in Fig. 4a was copleted, at a frequency f = 50 Hz, with paraeters of C A = 1/(ω 2 L A ), C B > 0 and C M = 1/(ω 2 L M ). Its specific circuit paraeter values, experiental conditions, easureent results and one of its wavefor records are given in Fig. 7, where the records of (V 11, V 21 ), (V 12, V 22 ) and (V 13, V 23 ), respectively, represent three pairs of the wavefor data (V 1 :V 2 ) easured at low, iddle and high sections of operating range of the utual capacitor (the photo displays the second pair). The current ratios calculated fro the wavefors easured are lined in turn as I 1 /I 2 = , , , whereas the ratio calculated by applying (16) yields n c = I 1 /I 2 = This unity-coupled current utual capacitor has a relative error of δ c 0.33%, which is a reasonable result. 3.2 Unity-coupled voltage type utual capacitor An experient for the unity-coupled voltage type utual capacitor in Fig. 5a was done, at a frequency f = 50 Hz, with paraeters of C a = 1/(ω 2 L a ), C b > 0 and C > 0. And its specific circuit Fig. 7 Experiental circuit diagra, data and a wave for of a unity-coupled current type utual capacitor

6 Fig. 8 Experiental circuit diagra, data and a wave for of a unitycoupled voltage type utual capacitor paraeter values, experiental conditions, easureent results and one of its wavefor records are shown in Fig. 8, where the records of (V 11, V 21 ), (V 12, V 22 ) and (V 13, V 23 ), respectively, represent three pairs of the wavefor data (V 1 :V 2 ) easured at low, iddle and high sections of operating range of the utual capacitor (the photo displays the first pair). The voltage ratios calculated fro the wavefors are lined in turn as V 1 /V 2 = , , , whereas the ratio calculated by applying (19) yields n v = V 1 /V 2 = This unity-coupled utual capacitor has a relative error of δ v 1.73%, higher than expected although also reasonable, ostly because of the reason n v being close to one [Note: it is sensitive for errors when n v is in the vicinity of 1, see the note following (18).]. 3.3 Characteristics of wavefor conversion of the unity-coupled utual capacitor The voltage type unity-coupled utual capacitor in Fig. 8 has its voltages and currents on both ports, all shaped like sine wave when it has a linear load R across its output. However in Fig. 9a, V 1 is a square-wave voltage source fro an inverter and V 2 has a load as actually is an inductive rectifier. Thus, the output has the current, i II (see V II in Fig. 9b), easured as an approxiate squarewave, whereas its input has a current, i I (see V I in Fig. 9b), displayed as an approxiate sine-wave (although both are not good enough). This illustrates that the utual capacitor actually functions well of a wavefor separation or wavefor conversion between the output square-wave current and the input quasi-sinusoidal current or being tered the backward haronic current filtering. The wavefors in Fig. 9c were easured for a contrast between its input voltage V 1 and its output voltage V 2 and V 1 being shaped approxiately like a square-wave, whereas V 2 approxiately like a sine wave, which proves that this utual capacitor has a good ability of forward haronic voltage filtering [Note: wavefors, both squared and sinusoidal, in photos showing distorted or non-ideal originate ostly fro the weak output capability of the square-wave voltage source (see V 1 in Fig. 9c) or the source V 1 has too large an internal ipedance of soe non-linearity. The author had another experient siilar to this but in high frequency resulting with satisfactory wavefors.]. 4 Conclusions The utual capacitor, or the utual-capacitance transforer, being a new circuit eleent or coponent, will enrich the applications of the network coponents and also cause the principle of duality into practical use since it akes a dual of the utual inductor. Being a new ac transforer it can do ost of the perforances which the utual inductor, or the conventional inductance transforer, does in diverse aspects; and it can also function soething that the inductance transforer cannot. Featuring distinctly, the utual capacitor and the utual inductor copleent each other. A utual inductor or an inductance transforer is safer for use in a case of relatively sall power and low voltage, and it has a saller size, costs less and operates reliably, whereas at high-voltage circustances, particularly for huge power use, it has a series of probles such as in heat cooling, and it costs too uch for aintenances. The utual capacitor or the capacitance transforer, copared with its inductive partner however, has higher transforation accuracy (for its ease to be unity-coupled), higher efficiency (for its less iron-and-copper power loss), uch better cooling perforances (thanks to its separated and assebled structure), uch lower use of aterials (for its less uses of copper, iron, cheicals Fig. 9 Experient on wavefor conversions with a voltage type utual capacitor a Circuit diagra and data b Current i I against i II (V I = 16.0 V, V II = 1.13 V) c Voltage V 1 against V 2 (V 1ax = 76.0 V, V 2ax = 68.8 V)

7 and equipent for cooling) and uch less aintenance jobs needed; it also can raise power factor of the grid, an advantage in huge power applications and concurrently iproves the wavefor quality of electricity, although it is not suited to be ade as a sall transforer, especially for use at low frequency, because it ust be set a constant LC value satisfying (14) or (17). A power utual capacitor ust operate at a fixed frequency of high stability; and the anufacture of big power utual capacitors ust be based on the developent of the anufacturing level of non-polar capacitors and so forth. The utual capacitor will be good for use in the following aspects: (i) current/voltage transforations for easureent or test, especially of high voltage or high current; (ii) highpower and high-voltage applications of voltage/current transforers or constant-current generators for power transission, concurrently with wavefor separation; (iii) applications of power deliveries and wavefor conversions/separations between switching devices in power electronics (uch lower switching-transition power-losses especially at high frequencies); and (iv) IC transforers constructed of IC capacitors and IC NICs [3, 6]. The author believes it ay have a long way to go for the utual-capacitance transforer to function as ain transforers at power substations, but we can expect alost any day when it plays a role as rectifier transforer such as for electroplating, selting, locootive powering or laboratory use etc., as well as acts as the gate-guard transforers of power consuers to keep haronics generated by all the users doestic apparatuses off the grid. 5 References [1] Boylestad R.L.: Boylestad s circuit analysis (Prentice-Hall, Inc., USA, 2003, 3rd Canadian edn.) [2] Bakshi M.V., Bakshi U.A.: Eleents of power systes (Technical Publications Pune, India, 2009, 1st edn.) [3] Kawai K.: Negative ipedance converter, US Patent Nuber: B2, USA, 2006 [4] Wellings J.G., Mortlock J.R., Mathews P.: Capacitor voltage transforers, J. Inst. Electr. Eng., 1936, 79, (479), pp [5] Billig E.: The design of a capacitor voltage transforer. Proc. IEE Part II: Power Engineering, Deceber 1949, vol. 96, (54), pp [6] Wofford B.A., Nguyen R.: High precision integrated circuit capacitor, US Patent No B1, USA, 2004