開關(guān)電容變換器的建模與控制研究

1、本科畢業(yè)設(shè)計(jì)（論文）開關(guān)電容變換器的建模與控制研究research on modeling and control of switched capacitorsconverters學(xué) 院：專 業(yè)：學(xué)生姓名：學(xué) 號(hào)：指導(dǎo)教師：北京交通大學(xué)2017年4月學(xué)士論文版權(quán)使用授權(quán)書本學(xué)士論文作者完全了解北京交通大學(xué)有關(guān)保留、使用學(xué)士論文的規(guī)定。特授權(quán)北 京交通大學(xué)可以將學(xué)士論文的全部或部分內(nèi)容編入有關(guān)數(shù)據(jù)庫(kù)進(jìn)行檢索，提供閱覽服務(wù), 并采用影印、縮印或掃描等復(fù)制手段保存、匯編以供查閱和借閱。（保密的學(xué)位論文在解密后適用本授權(quán)說明）學(xué)位論文作者簽名：簽字日期： 年 月 日指導(dǎo)教師簽名：簽字日期： 年 月 日

2、獲得全文及仿真模型請(qǐng)聯(lián)系作者：q:374381054 v:lidedong 一學(xué)院：專業(yè):指導(dǎo)教師（簽名）：學(xué)生姓名：學(xué)號(hào)：_ 提交日期：畢業(yè)設(shè)計(jì)（論文）基本內(nèi)容和要求：開關(guān)電容變換器可以獲得很高的電壓傳輸比,在光伏發(fā)電和燃料電池并網(wǎng)領(lǐng)域有 著廣泛的應(yīng)用前景和良好的市場(chǎng)應(yīng)用前景。本設(shè)計(jì)以典型的開關(guān)電容變換器為研究對(duì)象，在分析其解構(gòu)特點(diǎn)和工作原理基礎(chǔ) 上，開展考慮實(shí)際物理模型的建模研究，并據(jù)此研究變換器的控制策略，掌握關(guān)鍵參 數(shù)和元器件的設(shè)計(jì)原則，開展仿真實(shí)驗(yàn)研究，并對(duì)該方案的經(jīng)濟(jì)性進(jìn)行比較分析。要求：熟練使用matlab/simulink或者其他電力電子仿真工具，具有一定的動(dòng)手 能力，結(jié)合本設(shè)

3、計(jì)的理論分析，開展相應(yīng)的仿真研究。畢業(yè)設(shè)計(jì)（論文）重點(diǎn)研究的問題：分析掌握開關(guān)電容變換器的工作原理掌握開關(guān)電容變換器的關(guān)鍵參數(shù)設(shè)計(jì)方法就開關(guān)電容變換器的控制方案開展仿真實(shí)驗(yàn)研宂獲得全文及仿真模型請(qǐng)聯(lián)系作者：q:374381054 v:lidedong中文摘要摘要：在能源危機(jī)的大背景k,尋找清潔的可再生的新能源已成為全世界的迫切要 求。新能源依賴于現(xiàn)代電能變換技術(shù)的發(fā)展與應(yīng)用，而開關(guān)電容變換器(switched capacitors converters, scc)是其中的核心部分。開關(guān)電容變換器可以獲得很高的電壓俾 輸比，在光伏發(fā)電和燃料電池并網(wǎng)領(lǐng)域有著廣泛的應(yīng)用前景和良好的市場(chǎng)前景。開關(guān)電

4、容變換器可以獲得很高的電壓傳輸比，在光伏發(fā)電和燃料電池并網(wǎng)領(lǐng)域有著廣泛的應(yīng)用 前景和良好的市場(chǎng)前景。本文研究目的是提出基于平均電流概念的開關(guān)電容變換器的建模方法，用來幫助準(zhǔn) 確理解開關(guān)電容變換器的相關(guān)輸出特性，同時(shí)分析等效串聯(lián)電阻(equivalent series resistance, esr)、系統(tǒng)參數(shù)(電容參數(shù)、開關(guān)頻率和占空比)等對(duì)scc輸出特性影響。 通過數(shù)學(xué)建模，來指導(dǎo)scc控制算法的設(shè)計(jì)。利用psim建立全電路仿真模型，結(jié)合理 論計(jì)算與仿真結(jié)果討論建模方法的可行性。本文的研究方法是，首先通過舉例說明開關(guān)電容變換器的工作原理，然后依據(jù)電路 理論和電力電子理論對(duì)模型進(jìn)行必要的簡(jiǎn)化，

5、最后討論各個(gè)參數(shù)對(duì)電容開關(guān)變換工作狀 態(tài)的影響，進(jìn)而得出開關(guān)電容變換器關(guān)鍵參數(shù)的設(shè)計(jì)原則，使得參數(shù)的設(shè)計(jì)可以滿足實(shí) 際應(yīng)用的需要。關(guān)鍵詞：幵關(guān)電容變換器；建模；參數(shù)設(shè)計(jì)；仿真；控制獲得全文及仿真模型請(qǐng)聯(lián)系作者：q:374381054 v:lidedong 一abstractabstract: in the context of the energy crisis，finding new and clean renewable energy has become an urgent requirement around the world. development of renewable en

6、ergy depends on the development and application of modern electrical energy conversion technology and the switched capacitor converter (scc) is one of the key devices，which can reach very high voltage transfer ratio. thus it has a wide application prospect and good market prospects in photovoltaic p

7、ower generation and fuel cell grid integrations.in this paper, a modeling method of switched capacitor converter based on the average current concepts, is proposed to help better understand the correlation output characteristics of scc exactly, to help understand the correlation output characteristi

8、cs of switched capacitor converters exactly, moreover，it also helpful for analyzing the output characteristics related with the equivalent series resistance, capacitance parameters，the switching frequency and duty cycle, etc. then the mathematical modeling is also provided for guiding the control al

9、gorithm design of the switched capacitor converters. a full circuit simulation model based on psim is established, which verified the feasibility of modeling methods by comparison discussing the theoretical calculation and simulation results.the research methods of this paper are illustrated as foll

10、ows. firstly，analyzing the working principle of switched capacitor converters by examples, and then it further simplifies the circuit according to the circuit theory and power electronics theory. moreover，it discusses the impact of various parameters on the capacitor switching converter operating st

11、ates. finally， the design principles of key parameters are deduced for making the parameters design to meet the needs of practical application.keywords： switched capacitor converters, modeling，parameter design, simulation，control目 錄中文摘要錯(cuò)誤!未定義書簽。abstract5 61 引言錯(cuò)誤!未定義書簽。1.1課題研究背景與意義錯(cuò)誤!未定義書簽。1.2開關(guān)電容變換器

12、建模研宄現(xiàn)狀錯(cuò)誤!未定義書簽。1.3 主要研究?jī)?nèi)容錯(cuò)誤!未定義書簽。1.4經(jīng)濟(jì)性分析錯(cuò)誤!未定義書簽。2 基本理論錯(cuò)誤!未定義書簽。工作原理錯(cuò)誤!未定義書簽。電路初步簡(jiǎn)化錯(cuò)誤!未定義書簽。10 分支電路分析錯(cuò)誤!未定義書簽。11 電阻簡(jiǎn)化錯(cuò)誤!未定義書簽。12 電容簡(jiǎn)化錯(cuò)誤!未定義書簽。13 簡(jiǎn)化后的工作電路與分支電路錯(cuò)誤!未定義書簽。2.3等效電阻分析錯(cuò)誤!未定義書簽。2.1 理論基礎(chǔ)錯(cuò)誤!未定義書簽。2.2 公式推導(dǎo)錯(cuò)誤!未定義書簽。2.3 對(duì)等效電阻的討論錯(cuò)誤!未定義書簽。2.4建模分析錯(cuò)誤!未定義書簽。3.3 提出模型錯(cuò)誤!未定義書簽。3.4 對(duì)電流的分析錯(cuò)誤!未定義書簽。3.5 對(duì)電

13、壓的分析錯(cuò)誤!未定義書簽。3.6 輸出特性總結(jié)錯(cuò)誤!未定義書簽。2.5 小結(jié)錯(cuò)誤!未定義書簽。3 仿真分析錯(cuò)誤!未定義書簽。3.1全電路仿真錯(cuò)誤!未定義書簽。cc模式下的全電路仿真錯(cuò)誤!未定義書簽。pc模式下的全電路仿真錯(cuò)誤!未定義書簽。nc模式下的全電路仿真錯(cuò)誤!未定義書簽。3.2仿真數(shù)據(jù)分析錯(cuò)誤!未定義書簽。4.4 電流波形分析錯(cuò)誤!未定義書簽。4.5 電容電壓分析錯(cuò)誤!未定義書簽。4.6 輸出電壓分析錯(cuò)誤!未定義書簽。4.7 仿真數(shù)據(jù)分析總結(jié)錯(cuò)誤!未定義書簽。小結(jié)錯(cuò)誤!未定義書簽。4 誤差分析錯(cuò)誤!未定義書簽。4.1模型誤差錯(cuò)誤!未定義書簽。4.2開關(guān)管esr對(duì)模型誤差的影響錯(cuò)誤!未定義

14、書簽。1) 開關(guān)管頻率力=5khz時(shí)的誤差分析錯(cuò)誤!未定義書簽。2) 開關(guān)管頻率/；.=25khz時(shí)的誤差分析錯(cuò)誤!未定義書簽。3) 開關(guān)管頻率.=250khz時(shí)的誤差分析錯(cuò)誤!未定義書簽。4.3各個(gè)電容器esr對(duì)模型誤差的影響錯(cuò)誤!未定義書簽。小結(jié)錯(cuò)誤!未定義書簽。5 開關(guān)電容變換器的控制錯(cuò)誤!未定義書簽。3.1.1 控制電路部分錯(cuò)誤!未定義書簽。5.2啟動(dòng)邏輯部分錯(cuò)誤!未定義書簽。6 結(jié)論錯(cuò)誤!未定義書簽。轉(zhuǎn)嫌8致 謝錯(cuò)誤!未定義書簽。pf寸 10獲得全文及仿真模型請(qǐng)聯(lián)系作者：q:374381054 v:lidedong 一參考文獻(xiàn)1j丘濤文.開關(guān)電源的發(fā)展及技術(shù)趨勢(shì).電力標(biāo)準(zhǔn)化與技術(shù)經(jīng)濟(jì)

15、，2008(6).2j丘東元，張波.開關(guān)電容變換器組成原理及發(fā)展趨勢(shì)/中國(guó)電源學(xué)會(huì)全國(guó)電源技術(shù)年會(huì).2007. 13梅純.開關(guān)電容變換器的研究dj.華巾科技大學(xué)，2007.4曾毅.車載輔助控制電源建模分析與研宄d.屮南大學(xué)，2010.15j曹劍平.dc-dc開關(guān)變換器建模與數(shù)字仿真分析研宄dj.中南大學(xué)，2008.16j劉術(shù)平.開關(guān)電容dc-dc變換器的研究ldj.華南理工大學(xué)，2003.17周嘉農(nóng)，曾小平.dc-dc開關(guān)變換器的建模與分析的動(dòng)態(tài)評(píng)述llj.華南理工大學(xué)學(xué)報(bào):自然科學(xué) 版，2000, 28(8):111-116.l8jmiddlebrook r d, cruk s m. a g

16、eneral unified approach to modeling switching convertor power stagesljj. international journal of electronics, 1976,42(6):521-550.l9guangyong zhu, adrian ioinovici. steady-state characteristics of switched-capacitor, electronic converters j. journal of circuits system & computers, 2011,07(2):69-

17、91.evzelman m, ben-yaakov s. average-current-based conduction losses model of switched capacitor convertersljj. ieee transactions on power electronics, 2013, 28(7):3341-3352.arntzen b, maksimovic d. switched-capacitor dc/dc converters with resonant gate drivejj. ieee transactions on power electronic

18、s, 1998, 13(5):892-902.12jmakowski m s, maksimovic d. performance limits of switched-capacitor dc-dc convertersc/ power electronics specialists conference, 1995. pesc '95 record. ieee. 1995:1215-1221 vol.2.13ben-yaakov s. behavioral average modeling and equivalent circuit simulation of switched

19、capacitors converters. ieee transactions on power electronics, 2012, 27(2):632-636.l14h吳剛，阮新波，葉志紅.采川開關(guān)電容的非隔離型高升壓比直流變換器ln.巾國(guó)電機(jī)工程學(xué)報(bào)， 2015, 35(2):442-450.15jm全書海，黃亮，等.燃料電池用新型高升壓比dc/dc變換器jj.武漢理工大學(xué)學(xué)報(bào):信息與 管理工程版，2012, 34(4):464-466.116ben-yaakov s. behavioral average modeling and equivalent circuit simulat

20、ion of switched capacitors convertersj. ieee transactions on power electronics, 2012, 27(2):632-636.17劉健，陳治明，鐘彥儒.開關(guān)電容dc-dc變換器的分析j.電子學(xué)報(bào)，1997(2):83-85.l18j evzelman m, ben-yaakov s. simulation of hybrid converters by average modelsjj. ieeetransactions on industry applications, 2014, 50(50):1106-1113.1

21、. 劉健，陳治明.開關(guān)電容dcdc變換器的穩(wěn)態(tài)分析j.電工電能新技術(shù)，1997(1):11-15.2. ngo k d t，webster r. steady-state analysis and design of a switched-capacitor dc-dc converterjj. ieee transactions on aerospace & electronic systems, 1994, 30(1):378-385 vol.l.3. wu w c，bass r m. analysis of charge pumps using charge balancej.

22、2000, 3:1491-1496 vol.3.4. ioinovici a. switched-capacitor power electronics circuitsjj. circuits & systems magazine ieee， 2001，1(3):37-42.231 ben-yaakov s，evzelman m. generic and unified model of switched capacitor convertersfc/ energy conversion congress and exposition, 2009. ecce. 2009:3501 -

23、 3508.241丘東元，張波，鄭春芳，等.諧振開關(guān)電容變換器新型pwm控制策略j.中國(guó)電機(jī)工程學(xué)報(bào)， 2006, 26(2):116-120.251王磊，張波，丘東元.dc/dc開關(guān)電容變換器的控制技術(shù)丨幾電力電子技術(shù)，2004, 38(6):103-105. 261王磊，丘東元，張波.dc/dc開關(guān)電容變換器拓?fù)浼翱刂萍夹g(shù)j.電氣應(yīng)用，2004(5):68-71.271耿莉，陳治明，劉健.開關(guān)電容dc-dc變換器的輸出特性及控制模式j(luò)】.西安理工大學(xué)學(xué)報(bào)，2000, 16(4):330-334.281劉健，陳治明，嚴(yán)百平.集成開關(guān)電容dc-dc變換器jl微處理機(jī)，1996(1):47-50

24、.291張杰.高性能開關(guān)電容dc-dc變換器的研究與設(shè)計(jì)d.浙江大學(xué)，2010.301邵濱，楊雨佳，王穎，等.一種開關(guān)電容型直流變換器的系統(tǒng)研宂與設(shè)計(jì)j1.電路與系統(tǒng)學(xué)報(bào)， 2008, 13(4):124-129.311劉術(shù)平，梁冠安，彭軍.開關(guān)電容dc/dc變換器的理論研究j1.電源技術(shù)應(yīng)用，2003(3):36-40. 321王忠閣.開關(guān)電容ac-ac變換器研究d1.中國(guó)計(jì)量學(xué)院，2014.附錄a外文文獻(xiàn)i. introductionswitched capacitor converters (dcno.ed as scc singular and plural instancesk al

25、so referred io a> *chargc pumps." have inherent losses that make ihcir use pmhibiiivc in many appliculions. even so. scc arc preferred in a number of cam?s cue io their ic compaiibiliiy. rclalivelv small size, and the absence of magnetic clcmcni.s. scc can be modeled a、a network that is conf

26、igured by the switches loasci of subcircuils lhai charge and discharge flying capacitors. the average behavior of scc systems was analyzed in numerous earlier siudio (see. c.g. 11 卜111 |k in which (he expression、of the voltage transfer ratios and the expected losses were derived, l ull circuit simul

27、ation can be carried out by various software packages lo verify he theoretical calculations and to lest alternative designs. however. full simulaiion lhai includes the switching iransienis suffers from at least iwo deliciencies: the need for long simulation lime and convergence problems due to the f

28、ast transitions caused by (he switches. as experienced in the ease of su itched inductor converters 112卜115|. average circuit simulation could help io alleviate ihec pwblems and. in fad. cun provide additional information and a belter insight into the simulated convener.manuurripl received .november

29、 is. 2010; revised april 5. 2011 and july 15. :“iisc|、k"iha i. 2o| | d.iu«u tunvnkcrmim bn(ur> 2<m2 ibiswork w；x %uppohcd by the hracl science bounibimn under grant 47m)8 and grant 5.7/11 keaxnmcfhled for publkolkwi by as«iale editor k (letlet* aelzarc.the author 1、with the powe

30、r electronic、ijihoratory. dep;inmen( of electried and cofpulcr l:npnccfing. ocn-gurion unacrily of the n<g<*. dccr-shco, wi05 hmcl (e-mail: sbycc.bguc.il).cot« xersions of one or more of (he figure), in thi> piper are axiljble online al hi(p:/4cccxplorc.iccc .orj.ilijikl object identif

31、ier 10.110vtpel.2011.2171996里 he objective ot this study wa，to develop an average model of scc sysicnu (hat would be compatible with any circuit simulator and be capahle of pa)viding penineni daia such as the output voltage, the uverage current in the subcircuils. the average voltage across the capa

32、ciiws. and lhe efficiency. funhemuxe. average circuit models can help to gain a belter under>ianding of (he role of switch resistance, capacitance value, clock duration. and duty cycle. the proposed simulation model is in fact a translation of the analytical results of (101 into average equivalen

33、t circuits. as such, the accuracy of the model is identical lo the results cf 11()| and it is limited by the assumptions of its denvaiion (such that (he subcircuils can he approximated by a firsi-onlcr/fc circuit).ii. basic tiilorlticai- considijiationsfor ihc sake of clarity and brevity, we conside

34、r first a 1:1 scc 巧、iem depicted in fig. 1. the convener includes a flying capacitor c with a lossy componeni /?>、< luo swikhessj and with aon" rvsisianccs of /?, | a ulspcc*ivcly. an outputlitter capacitor co with a?ksro and load resistance ro as the switches run al a frequency ofthe cir

35、cuit toggles between two wbcircuits;one during fj. when si is *on. see fig. 2(a)|.and (he other during tt. when s/ is |scc fig. 2(b)|. and r： in fig. 2 represent the uxal loop rcsisianecs/?i =札| +/?2 = .2 + "ksi< + "ksr.(2)each of the subcircuils can be represented by a generic charging

36、 circuit (see fig. 3) that includes a voltage source av, (where/is 里 for duration and 2 for 7*2 >, a resistor rt. and a capacitor c,1111 trassactx>ss os kmfr i u(tronics vcx 27. no. 11 brvary 2012lettersbehavioral average modeling and equivalent circuit simulationof switched capacitors convert

37、ersshmucl (sam) ben-yaakov. fellow.ieeeab!ract gcnerlv bvhu%loral ttenige drcull model 扇 smitched cu|uivltor confertvr (sd*) h propomxl und drmon4nitr<i ii unity conversion s(c. the model h lui%vd <>n (hr average rur- rent、c>nvepl unci vun l>v used io culculalc or umulalc the tt%eruge

38、 %nlur* >f |hr sc c* %iirlnhlc mich u* oulpul volliigr, cupadlor %«ll> hhd subcircuit current*. the model h %nlid for all operational ninuvs vf an scc (cuinplvlc. puitiul, and iiu thui gel and s coin, patihlcullh uny drcull mmulutor ihut includes dependent m>urcc<u excellent uxreemv

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40、ion 7*.，fig. 3. gcncnc cqucilor.charging equivalent virvuil.fig. 5 average etuivnkm tirvuil model vf ihc 1:1 scc (al linuiblicin oflirm mw of nuinw <s). (b> kmuljiicci of second row of (m> (c) output %ecli(m third mw of (x|. (d) cap alloc charge kiiaskc. rurth nm of (s).pig. 4. (>s>ib

41、le charging current、hapc、. (a) complettf charge (cc> <im partial charge (pc), (c) no charge (nc).and since the loial average curreni through c is zero,c曹 j = ,cvi = ,<»(6)cofxxntraiing lirm on the nc ease (|mx l ig. 4(c). iik curreni、 /m4 within durations t and arc rclaied to ihc avera

42、ge curreni lc by a factor am:(7)based on the abene. one can summarize the voltage and current rclaiionships for the nc ease by the following set ol equations:with an initial condition of zero voltage. for the 1:1 convenerrvini10lil0 01一 1(100i"it(i人摘2.0、001人、鬌2 ci =cccoctc(3)(4)av. is (he volta

43、ge across the relevant switch jum before it i、 turned on.once a switch is turned on. (he equivalent capacitor will stan charging (by a positive or negalive ) and a cunrem f,(/) will build up. impending on the relationship between (he duration /； and the time consiani ktc. ihc curreni can lake one of

44、 three possible shapes. for ti >> the charging uill he completed within 7 i see fig. 4(a) this case is denoted as cc. hbr 7; 2 z?rct the charging will be partial |?c, fig. 4<b)|. for ti « ffici, there will be no effective charging (nc) and the current will be practically consiani |sce

45、fig. 4(c)|. in this latter ease, ihe capacitor voltage will slay about constant within t,.assuming a sieady-siaic condition, (he awragc value of ihc currcm during 12. /c«v2 (averaged over a switching cycle 廠 > will be equal (o (he output current zo. namelythe firm two rows of aforemeniioned

46、matrix arc kvl for t and t： durations (see hg. 2). while the (bird and ibunh rows reslaic (5) and (6).the set of (8» can be emulated by an equivalent cirvuil ihul includes dependent sources and passive elements as follows. the hrsi row is emulated by the circuit of fig. 5(ahhat includes a depen

47、dent source ex which replicates vin. a resistor /?*. and a dcperxkni vokagc mhiox e2 which replicates i he voliagc on ca(xicilor c. the insianianaxis current /to| i scaled by g lo represent the average curreni (over t j lhai loads ihc source vhl. similarly, (he second row is emula.cd by the circuit

48、of fig. 5(b) and the third row by fig. 5(c). the lounh row which is related to the capaciiofs charge balance is emulated by a capacitor c lhai is fed by iwo opposing current sources (c|. c；) that represeni the average currcnis (over t.) of the two subdreuils.the dcliniiions of ihc dependem sources a

49、rc thusr.ex = bj = vc; s vc; e, = kg篇 s aa|zm|； g2 =cl = u，l;gi =(10)= le.(5)il should he noticed that the cqui、aleni circuit of hig. 5 includes insianiancoux currcnis in the kvl circuits |z, in fig. 5(a) andku i trassacrionson k)u| k 11ix ironic s. voi 27. no 2. u hki aky2ci26miih trassmnk>ns on

50、 kmi r ilmironics. vol 27. m>. 2. fibkuaky 2012l;if. 6. bclm% h>r of equivalentkri of a hakikuii smooth vur%c(>0. *a” point: cc operation. "b” poiiu nc operation ctnubling %uk*irvuil(b)j and average currcnis al the input |scc fig. 5(a>|. output |scc fig. 5(c). and charge balance |se

51、e hg. 5(d)| circuits. this is necessary to keep correct kvl relationships on ihe one hand and. on the oilier, to nuiiniain correcl charge-balance and correct power relationships. il should also be noticed lhai (he model or fig. 5 assumes a consiani voltage on (he capacitors which is the coitcd situa

52、tion in the nc case and compal hie uiih the notion of "average model."the equivalem circuit of fig. 5 includes quite a number of dependent sources that might cause simulation conversion problems. these can be eased by eliminaiing sone of the dependem sources by. c.g. combining the i wo sec

53、tions of fig. 5(a) and replacing e： by (he capcilor.iii. exti nsion t() till： gl ni rai switching caskthe model developed in section ii was based on ihc assumption that ihe rntxleled scc operates in the nc region (see fig. 4(c)|. tlie extension of ihe incxlel to :he general case lo include the cc an

54、d modes |scc l ig. 4(a) and (b,| can be carried oul hy applying ihe relationships bcluecn the operational conditions and power dissipation as follows.as was proven in an earlier publication |101. the average power pt dissipated by a given subcircuil / during a su itching phase duration z, can be exp

55、ressed aspl = (/r-,)，ajfe'' = ic： (l"the term in bracket can be defined as (he equivalem rcsisiancc of switching phase i： = <lathe behavior of ihe equivalent resisiancc function (12) as a function of j shows an incrcusc in value when this parameter becomes larger (see l ig. 6). apply

56、ing taylor senes expansion, the ass mpit >ik value of this ainctioni? forte nc ease (7 « ric.) is found as the limit of (12) when j«1 (see fig. 6. horizonial line at tlx: ixnioin of ihe curve);t n札，nc =凡¥ = .(13)turning now io (he cc ease (7, >> /f,c i » 1). the behavior

57、 of the equivalent resistance (zf, ,(< ) follou s the function:thi、i、the n、ing pan of the h" plol shown in hg. 6.assuhk ncm (hat a subcircuil of the given scc operates under some arbitrary j conditions such that its equivalent ivsislancc is and that its position on the plot of fig. 6 is at p

58、oint a. that is, ouimde the nc region. considering the behavior of (12). it folio認(rèn)、that (he same power dissipation will be observed in an emulating subcircuil that operates ；u poini b (see fig. 6) undernc condition, if its loop rvmslarxe obeys the relationship /r(15)this implies that (he cc and pc eases can be emulated by an equivalent circuit that is assumed io operate under nc condi- lionx. if the loop resistance is adjusted for equal