快速交—直流PWM整流感應(yīng)發(fā)電機(jī)系統(tǒng)的高性能控制中英文翻譯

1、第32頁中文譯文快速交直流pwm整流感應(yīng)發(fā)電機(jī)系統(tǒng)的高性能控制摘要：本篇主要描述了一種快速的交直流感應(yīng)發(fā)電機(jī)整流系統(tǒng)的直流電壓調(diào)節(jié)控制方法，該整流系統(tǒng)中發(fā)電機(jī)的銅耗可減到最小。通過輸入輸出線性化的補(bǔ)償，使得同步參考結(jié)構(gòu)中的等價模型體線性化、解耦，達(dá)到高性能的轉(zhuǎn)速矢量控制。穩(wěn)態(tài)分解提供了發(fā)電機(jī)的操作規(guī)則，計(jì)算機(jī)仿真已證明了不同負(fù)載條件以及轉(zhuǎn)速變化下的控制策略的有效性。文章還展示了一些實(shí)驗(yàn)結(jié)果。索引詞：感應(yīng)發(fā)電機(jī)，交直流pwm整流器，勵磁，輸入輸出線性化，巴特沃思多項(xiàng)式，磁飽和、緒論越來越多的研究和實(shí)際應(yīng)用已轉(zhuǎn)移到可再生能源系統(tǒng)，例如風(fēng)能。又由于發(fā)電機(jī)具有低成本，維護(hù)費(fèi)用少，結(jié)構(gòu)簡單牢固，無刷（鼠

2、籠式）等固有的優(yōu)點(diǎn)，增加了感應(yīng)電機(jī)特別是發(fā)電機(jī)的使用。感應(yīng)電機(jī)的轉(zhuǎn)差率決定了它是以發(fā)電機(jī)還是電動機(jī)運(yùn)轉(zhuǎn)，正的轉(zhuǎn)差率說明電機(jī)以電動機(jī)形式運(yùn)轉(zhuǎn)，而負(fù)的轉(zhuǎn)差率標(biāo)志著電機(jī)工作在發(fā)電狀態(tài)。眾所周知，感應(yīng)電機(jī)可以用作自激發(fā)電機(jī)，也就是說（a）通過定子接線端三個連接電容（b）通過使用反用換流器/整流器系統(tǒng)，發(fā)電機(jī)能夠被勵磁1。在使用反用換流器/整流器系統(tǒng)的情況下，直流側(cè)的電容是否作為三相電容取決于反用換流器的開關(guān)信號，整流器的單相直流電容給感應(yīng)電機(jī)提供必需的勵磁。大量廣泛的觀察說明，過去超過25年里在自激，電壓組合建模，感應(yīng)電機(jī)的穩(wěn)態(tài)分析等不同領(lǐng)域所做的大量工作已經(jīng)被展示在2。而且，在以前的出版著作中，描述

3、過感應(yīng)發(fā)電機(jī)整流系統(tǒng)的矢量控制。該系統(tǒng)用于產(chǎn)生直流而且它的整流器也提供勵磁3。該系統(tǒng)特別適用于風(fēng)能方面的研究應(yīng)用，因此來研究變化的轉(zhuǎn)子轉(zhuǎn)速的反應(yīng)控制器。4中也做了感應(yīng)發(fā)電機(jī)整流器在方向場控制下的穩(wěn)定性研究，強(qiáng)調(diào)了感應(yīng)發(fā)電機(jī)在高速應(yīng)用時存在可能的不穩(wěn)定性。本文中采用的控制方法已經(jīng)詳細(xì)展示，闡明了在系統(tǒng)的等效模型中，用輸入輸出線性化的方法將非線性部分從線性部分中分離出來。經(jīng)測試，在電機(jī)負(fù)載變化以及轉(zhuǎn)速變化的情況下，提出的控制策略依然可行。電機(jī)運(yùn)轉(zhuǎn)在最低銅耗下5。通過調(diào)節(jié)轉(zhuǎn)子使用最小損耗功能，可達(dá)到損耗最小。穩(wěn)態(tài)分析處理自激發(fā)電機(jī)在飽和條件下的運(yùn)行狀況6。研究了感應(yīng)電機(jī)輸出功率的容量以及在不同負(fù)載條

4、件下電機(jī)操作時各個參數(shù)的作用。本篇分析主要突出了勵磁量對系統(tǒng)的勵磁需求的作用，該系統(tǒng)以調(diào)制信號的大小衡量勵磁要求。通過固定磁鏈，電機(jī)勵磁需要的調(diào)制量能被確定下來。連同飽和作用，研究了系統(tǒng)在最低銅耗下的情況。本篇按以下方式組織：第二和第三部分分別詳細(xì)地描述了三相快速整流器模塊和感應(yīng)發(fā)電機(jī)。在第三部分中也提到了組合系統(tǒng)模塊。第四部分做了電機(jī)的穩(wěn)態(tài)分析。第五部分明確地表述了控制策略，第六部分通過仿真和實(shí)驗(yàn)結(jié)果證明了提出的控制策略。、三相快速整流器模型在系統(tǒng)中三相快速整流器用于交直流轉(zhuǎn)換。每個整流器由六個活動開關(guān)（反并聯(lián)續(xù)流二極管）組成，開關(guān)控制采用基于載波控制的三角正弦脈沖寬度調(diào)制方式（即pwm方式

5、）。上、下橋臂的三個開關(guān)器件的作用分別被定義為和，開關(guān)只有在開通時才起作用，當(dāng)它關(guān)斷時不起作用。而且，同一橋臂上下兩個開關(guān)器件不能同時導(dǎo)通。在三相參考坐標(biāo)系中，根據(jù)開關(guān)所起的作用，整流器的電壓等式可表述為：開關(guān)作用經(jīng)連續(xù)傅立葉變換后可近似為一個平均值和一個時變值。在三角正弦pwm方式中，隨時間變化的調(diào)制信號和三角波做比較生成的信號控制開關(guān)通斷。根據(jù)直流電壓和相應(yīng)的調(diào)制指標(biāo)(mqs, mds)，將（13）式轉(zhuǎn)化到同步參考坐標(biāo)系中，q軸和d軸電壓可被表述為：轉(zhuǎn)化到同步參考坐標(biāo)系下，整流器直流側(cè)的電壓等式可以描述為：是整流器負(fù)載。 、三相感應(yīng)發(fā)電機(jī)模型除電流方向不同外，三相感應(yīng)發(fā)電機(jī)的模型方程式和感

6、應(yīng)電動機(jī)一樣。系統(tǒng)模型如圖1展示。圖1 交直流感應(yīng)發(fā)電機(jī)整流系統(tǒng)同步參考坐標(biāo)系中發(fā)電機(jī)的方程式可以表述為： 是電機(jī)轉(zhuǎn)速，是qd坐標(biāo)中電機(jī)電壓的角頻率， 分別是q，d軸上的轉(zhuǎn)子磁鏈，分別是q，d軸上的定子電流。就像早期所提出的，這種控制方法考慮將轉(zhuǎn)子磁通和控制（定子）電流設(shè)置為控制變量。根據(jù)所要求的狀態(tài)變量感應(yīng)發(fā)電機(jī)的模型方程式可以表述為：其中在方程式（8）（12）中的參數(shù)按下式定義：、穩(wěn)態(tài)勵磁穩(wěn)態(tài)分析是采用感應(yīng)發(fā)電機(jī)按（14）（16）所示的復(fù)雜模型方程式進(jìn)行的，在（16）式中考慮了氣隙磁通飽和的作用?？紤]到電機(jī)的飽和效應(yīng)及最小銅耗的條件，分析主要在于決定發(fā)電機(jī)勵磁所要求的調(diào)制指標(biāo)的大小值。圖

7、2 是一個2馬力的感應(yīng)電機(jī)在飽和條件下磁感隨磁通量的變化曲線。電壓方程式在參考坐標(biāo)系下的變化角度體現(xiàn)了q軸隨磁通鏈變化的校準(zhǔn)。因此，假定d軸的磁通量是零，則d軸的磁感恒定。而q軸上磁感是磁通的函數(shù)，可近似由多項(xiàng)式（13）表示。 是轉(zhuǎn)差頻率，可定義為。圖2 磁感的倒數(shù)隨磁通量的變化曲線穩(wěn)態(tài)下可導(dǎo)出，（14）（16）中p為零?？紤]到電機(jī)氣隙磁鏈的磁飽和，分解的目的在于使電機(jī)運(yùn)轉(zhuǎn)時的總銅耗最小。當(dāng)轉(zhuǎn)子轉(zhuǎn)差的導(dǎo)數(shù)為零時，電機(jī)的總銅耗（如式17所示）最小。選擇合適的電機(jī)轉(zhuǎn)差率可以實(shí)現(xiàn)這種條件，如式18所示。如圖三所示，轉(zhuǎn)差率是磁感的函數(shù)。圖3 轉(zhuǎn)差率隨磁通量的變化曲線為了獲得調(diào)制指標(biāo)（m）的大小和磁通量

8、的關(guān)系，（16）式中的定子電流根據(jù)定子磁通量表達(dá)。重新整理（15）式得：將（19）式帶入（14）式中得：定子電流用定子磁通表達(dá)為：將（20）式帶到（21）式，并將（16）式中的定子電流消去，可得等式： 用方程式（22），可以達(dá)到在最小銅耗的條件下，通過改變調(diào)制指標(biāo)量的大小產(chǎn)生磁通量變化的效果。當(dāng)磁鏈改變時磁感的對應(yīng)數(shù)值能夠被計(jì)算出來。用磁感的值，可以計(jì)算在最低銅損時的轉(zhuǎn)差和對應(yīng)的調(diào)制指標(biāo)。假定恒定的轉(zhuǎn)子轉(zhuǎn)速，圖4展示了變化的負(fù)載電阻和變化的磁通量對調(diào)制指標(biāo)m的作用。圖4所示，隨著負(fù)載電阻的增加，調(diào)制指標(biāo)的計(jì)算值減小。從圖4也可看出，在恒定負(fù)載下，兩個不同的磁感值可以獲得相同的調(diào)制指標(biāo)。圖7說明

9、了不同磁感的作用。同樣地，圖5展示了某個恒定的磁通下轉(zhuǎn)速的變化效果。此時，取磁通量為一個恒定的值0.25wb。根據(jù)等式（22）中的電感，負(fù)載電阻和調(diào)制指標(biāo)的平方的乘積組成的表達(dá)式 ，給出了在分解中用到的另外一個參數(shù)。這個參數(shù)可以用來衡量系統(tǒng)以固定的轉(zhuǎn)速和轉(zhuǎn)差率運(yùn)轉(zhuǎn)時的負(fù)載電阻。圖6顯示了不同的轉(zhuǎn)差率對產(chǎn)生的影響。圖4 轉(zhuǎn)差率折算到最小銅耗下，負(fù)載電阻變化時，調(diào)制指標(biāo)m的值隨磁通鏈的變化曲線圖5 在最小銅耗下，電機(jī)負(fù)載電阻變化，磁通量為恒值 0.25wb時，調(diào)制指標(biāo)m的大小隨轉(zhuǎn)速的變化曲線。圖6 電機(jī)轉(zhuǎn)速恒定，轉(zhuǎn)差率變化時，隨磁通鏈的變化曲線在磁通量固定，負(fù)載恒定的情況下，轉(zhuǎn)速只與曲線上的一個點(diǎn)

10、對應(yīng)，每個點(diǎn)都對應(yīng)確定的電流和磁通量。因此d軸轉(zhuǎn)子的參考磁通能用（18）和（19）式計(jì)算。根據(jù)最小銅耗下的轉(zhuǎn)差率，算出對應(yīng)的磁通量如下式所示。對于曲線上的每個點(diǎn)，都有一個直流電壓的工作范圍。用磁通量表達(dá)定子電流如（25）式所示，將（25）式帶入（16）式得出一個用磁感和調(diào)制指標(biāo)量表達(dá)直流電壓的等式（26）。在式（26）中，通過改變調(diào)制指標(biāo)的q軸分量并計(jì)算相應(yīng)的d軸分量，可以得到直流電壓的范圍。也就是說當(dāng)時同一個調(diào)制指標(biāo)對應(yīng)兩個不同的磁感磁通鏈的值。磁通鏈越高，電壓越高，在大多數(shù)情況下，這是不可取的也是不可行的。從圖7可以看到磁通鏈對直流電壓范圍的影響。圖7顯示了不同的負(fù)載電阻下，磁通量較小時直

11、流電壓變化的等高線。圖7 轉(zhuǎn)速恒定，負(fù)載電阻在范圍變化時，直流電壓隨調(diào)制指標(biāo)m的q軸分量的變化曲線、控制方案的公式表述用提出的控制方案，如圖8所示，可以達(dá)到通過控制調(diào)制信號的q軸和d軸分量控制直流電壓的目的?？刂品桨覆捎棉D(zhuǎn)子磁通鏈和定子電流作為控制變量。感應(yīng)發(fā)電機(jī)的模型方程式（8）（12）是非線性的，用輸入輸出的線性去耦技術(shù)可以獲得輸入控制變量和輸出控制變量間的一個控制關(guān)系8。用這種方法，古典的線性控制系統(tǒng)理論可以用在決定每個控制器的結(jié)構(gòu)和pi控制器的固定增益參數(shù)方面。當(dāng)被控參數(shù)是直流電壓和轉(zhuǎn)子磁通量和時，這兒的控制變量是 。從方程式（8）（12）中，控制器的輸出可以定義為：將（45）式代入（

12、6）式中得：直流電壓控制器的輸出可用于計(jì)算q軸參考電流。從（1011）式，推導(dǎo)出轉(zhuǎn)差頻率和d軸參考定子電流為：通過（8）和（9）式，q軸和d軸的電壓分量的控制式可以表達(dá)為：定義控制器增益如下，推導(dǎo)出狀態(tài)變量的傳遞函數(shù)如（33）所示：將pi控制器轉(zhuǎn)移函數(shù)的系數(shù)與butterworth多項(xiàng)式9的系數(shù)作比較，可以決定pi控制器的持續(xù)增益。在這個控制方案中控制器的傳遞函數(shù)是二次的，因此將控制器傳遞函數(shù)的分母系數(shù)與二次butterworth多項(xiàng)式（34）作比較。在butterworth方法中，傳遞函數(shù)的特征值一律位于s平面的左半部分，并且在一個以原點(diǎn)為中心，為半徑的圓周上。圖8 感應(yīng)發(fā)電機(jī)系統(tǒng)的控制方案

13、、仿真和實(shí)驗(yàn)結(jié)果我們已經(jīng)用matlab/simulink，對于針對感應(yīng)發(fā)電機(jī)交直流整流系統(tǒng)提出的控制方案實(shí)施了仿真。圖9中的仿真結(jié)果展示了感應(yīng)發(fā)電機(jī)的起動過程。從0到0.45秒轉(zhuǎn)速以傾斜的直線上升到240rad/s，一直到2s轉(zhuǎn)速保持在240rad/s。圖9的也展示了在系統(tǒng)起動響應(yīng)過程中q軸和d軸電壓分量的變化，電磁轉(zhuǎn)矩的變化，并且直流電壓在0.3秒時達(dá)到穩(wěn)態(tài)值。圖10展示了在負(fù)載和轉(zhuǎn)速變化時控制方案的動態(tài)響應(yīng)。當(dāng)轉(zhuǎn)速恒定系統(tǒng)運(yùn)行在穩(wěn)態(tài)時，在1s將負(fù)載電阻從100變化到25，則轉(zhuǎn)速在2s時從240rad/s變化到200rad/s，又在4s時從200rad/s增加到240rad/s。為了適應(yīng)負(fù)載

14、和轉(zhuǎn)子速度的改變，受控直流電壓有效地追蹤變化指令，圖示為200v。圖8所示的控制方案已經(jīng)用一個40mhz的dsp芯片tms320 lf2407aevm實(shí)現(xiàn)。這個2馬力的感應(yīng)發(fā)電機(jī)被用于控制整流器輸出的220v的直流電壓。實(shí)驗(yàn)波形如圖11所示。圖9 起動過程的仿真結(jié)果 從頭開始：（a）q軸電壓，（b）d軸電壓，（c）轉(zhuǎn)速，（d）電磁轉(zhuǎn)矩，（e）直流電壓，（f）相電壓。圖10 負(fù)載和轉(zhuǎn)速變化時系統(tǒng)的動態(tài)響應(yīng) （a）q軸電壓，（b）d軸電壓，（c）轉(zhuǎn)速，（d）電磁轉(zhuǎn)矩，（e）直流電壓，（f）相電壓。對應(yīng)于發(fā)電機(jī)的初始勵磁，電容器的耐壓值指示為20v，發(fā)電機(jī)以1319rmin的速度運(yùn)轉(zhuǎn)。圖11展示了感

15、應(yīng)發(fā)電機(jī)以1319 rmin的轉(zhuǎn)速運(yùn)轉(zhuǎn)時的穩(wěn)態(tài)波形。直流電壓穩(wěn)定在 220 v并跟蹤參考電壓的變化。線電壓恒定在110 v。a相電流是一個3安培的穩(wěn)態(tài)值。（d）顯示了對應(yīng)于頂端設(shè)備的調(diào)制信號。圖11中，線電壓有一些高頻分量，這是由快速整流器的開關(guān)暫態(tài)造成的。圖11 2馬力感應(yīng)發(fā)電機(jī)穩(wěn)態(tài)響應(yīng)波形參考直流電壓 = 220，轉(zhuǎn)速 = 1319 rpm，負(fù)載電阻 = 80（a）直流電壓（137.5 伏/格）；（b）線電壓（350伏/格）；（c）a相電流（7.5安/格）；（d）a相調(diào)制信號（1.36/格）、結(jié)束語本文提供了在電機(jī)最小總銅耗的條件下，系統(tǒng)的一種詳細(xì)分解方法，并經(jīng)過一步步的推算得到系統(tǒng)的控制

16、方案。此發(fā)電機(jī)整流器系統(tǒng)已經(jīng)經(jīng)過不同負(fù)載條件和不同轉(zhuǎn)速的測試。轉(zhuǎn)速改變而直流電壓穩(wěn)定證明了控制器的魯棒性很好。在仿真和實(shí)驗(yàn)結(jié)果中也展示了負(fù)載變化對系統(tǒng)的影響。用最小銅耗條件下轉(zhuǎn)子磁通的d軸參考分量的計(jì)算值，已經(jīng)說明實(shí)現(xiàn)了銅耗最小。分解中也考慮到飽和效應(yīng)，通過磁感的q軸分量隨磁通鏈變化而改變解決。本文也展示了磁通鏈變化和不同的負(fù)載條件下發(fā)電機(jī)系統(tǒng)的自然響應(yīng)。整流器調(diào)制指標(biāo)的大小可以衡量發(fā)電機(jī)需要的勵磁，所以發(fā)電機(jī)的勵磁情況能用調(diào)制指標(biāo)值詳細(xì)地表示。附 錄實(shí)驗(yàn)中用的是230v，4極，2馬力的三相感應(yīng)電機(jī)，具體參數(shù)為：定子電阻 轉(zhuǎn)子電阻 不飽和磁感 定子每相漏電感 轉(zhuǎn)子每相漏電感 直流電容值 c =

17、 7800f英文原文high performance control of a boost ac-dcpwm rectifier-induction generator systemjyoti sastry, olorunfemi ojo, zhiqiao wudepartment of electrical and computer engineering/center for energy systems researchlaboratory for electric machines and power electronicstennessee technological univers

18、itycookeville, tn 38505, u.s.aphone: (931)-372-3869, fax: (931)-372-3436, e-mail: abstractthis paper presents a control methodology for the dc voltage regulation of an induction generator-ac-dc- boost rectifier system in which the copper loss of the generator is minimized. with the aid

19、 of an input-output linearization technique, which linearizes and decouples the model equations in the synchronous reference frame, a rotor flux vector control type high performance is achieved. steady-state analysis provides some insights into the operability regime of the generator. the effectiven

20、ess of the control scheme under different load conditions as well as varying rotor speeds has been demonstrated by computer simulations. some experimental results are included. index words: induction generator, ac-dc pwm rectifier, excitation, input-output linearization, butterworth polynomials, mag

21、netizing flux saturation.i. introductiongrowing research and real application interests in alternative energy systems such as wind energy has increased the use of the induction machine as a generator because of inherent advantages of such as low cost, reduced maintenance, rugged and simple construct

22、ion, brush-less rotor (squirrel-cage) and so on.the operation of an induction machine as a motor or generator is determined by the operating slip of the machine.a positive operational slip would indicate the operation of the machine as a motor, and a negative slip would indicate the generating mode

23、of the machine.it is well known that an induction machine can be made to work as a self-excited generator, i.e. the generator can be excited by the (a) connection of three capacitors at the stator terminals of the machine (b) by using an inverter/rectifier system 1. in the case of the inverter/recti

24、fier system, the dc side capacitor appears like three phase capacitors due to the switching signals of the inverter, and the single dc capacitor of the rectifier provides the required excitation for the induction generator. an extensive overview illustrating the vast amount of work done in different

25、 areas over the last 25 years such as self-excitation, voltage buildup modeling, steady state analysis of an induction machine has been presented in 2.also, in previously published work, the vector control of the induction generator rectifier system to produce dc power in which the rectifier also pr

26、ovides the excitation has been reported 3. the system has been studied specifically for applications related to wind energy, thereby studying the controller response for varying rotor speeds. also the stability of an induction generator-rectifier under field orientation control has been studied in 4

27、, highlighting the possible instability of an ind uction generator used in high-speed applications. the control method adopted in this paper has been laid out in detail, illustrating the inp ut-output linearization method used in separating the linear from the non-linear terms in the system model eq

28、uations. the proposed control scheme has been tested for its effectiveness by varying load conditions as well as varying the rotor speed of the machine. the mach ine has been operated at a condition of minimum copper loss 5. the condition of minimum loss is achieved by regulating the command rotor f

29、lux using a loss minimization function. the steady state analysis deals with the operation of the self-excited generator under conditions of saturation 6. the induction machine has been studied for its output power capability and the effect of the parameters of the machine on the operation of the ma

30、chine under different load conditions. the analysis in this paper aims at highlighting the effect of the magnetizing flux on the excitation requirements of the system with the magnitude of the modulation signal as a measure of the required excitation. by fixing the magnetizing flux linkage the requi

31、red modulation index for excitation of the machine can be determined. along with the effect of saturation, the system has been studied under a condition of minimum copper loss. the organization of this paper is as follows; sectio ns ii and iii detail the models of the three-phase boost rectifier and

32、 induction generator respectively. the model of the combined system has also been included in section iii. section iv deals with the steady state analysis of the machine. section v gives the formulation of the control scheme, and section vi validates the proposed control scheme using simulatio n and

33、 experimental results. ii. model of three-phase boost rectifierthe three-phase boost rectifier is used as the ac-dc converter in the system. each rectifier comprises of six active switches (with their anti-parallel diodes) that are switched using carrier-based sine-triangle pulse width modulation (p

34、wm). the switching functions of the three top and three bottom devices are defined as and and respectively. the switching function has a value of one when the switch is turned on and it is zero when it is turned off. also, the switching function of the bottom device is complimentary to that of the t

35、op device. the voltage equations for the rectifier in the abc reference frame can be expressed in terms of the switching functions as:the switching function can be fourier- series approximated as in (3b) comprising of an average value and a time-varying component. the time varying component is the m

36、odulation signal, which compared with the triangle in the sine-triangle pwm scheme generates the switching function. transforming (1-3) to the synchronous reference frame, the q and d-axis voltages can be expressed in terms of the dc voltage and the corresponding component of the modulation index (m

37、qs, mds ) as,the voltage equation on the dc side of the rectifier after synchronous reference frame transformation is given bythe load of the rectifier is .、model of three-phase induction genetratorthe model equations for a three-phase induction generator are the same as for an induction motor, exce

38、pt for the direction of the current flow. the system model is shown in figure 1.fig 1. induction generator-ac-dc rectifier systemthe equations for the generator expressed in the synchronous reference frame are 7:the rotor speed is , the angular frequency of the qd0 motor voltages is and the q-d roto

39、r flux linkages are and . the stator q and d axis currents are and ,respectively.as mentioned earlier, the control scheme under consideration deals with the rotor fluxes and stator currents as control variables. the model equations for the induction generator can be expressed in terms of the desired

40、 state variables as:wherethe parameters and used in equations (8)-(12) are defined as:、steady state excitationthe steady state analysis is carried out using the complex form model equations of an induction generator (14)-(16). the effect of magnetic air-gap flux linkage saturation is taken into acco

41、unt 6. the analysis aims at determining the value of the magnitude of the modulation index required for the excitation of the generator, taking into account the effect of saturation and a condition of minimum total copper loss in the machine. under saturated conditions, the magnetizing inductance va

42、ries with the magnetizing flux as shown in figure 2 for a 2 hp induction machine. the reference frame transformation angle of the voltage equations assumes the alignment of the q-axis with the magnetizing flux linkage. hence, the d-axis magnetizing flux is assumed to be zero, and the d-axis magnetiz

43、ing inductance is constant. however the q-axis magnetizing inductance is a function of the magnetizing flux, which is approximated by a polynomial given in (13). where is the slip frequency defined as .during steady-state, the derivatives, p in (14-16) are zero. alo ng with accounting for magnetic s

44、aturation of the air-gapflux linkage in the machine, the analysis aims at the operation of the machine at minimum total copper loss. the total copper loss (17) in the machine is minimum, when its derivative with the rotor slip is zero. this condition is achieved by appropriate selection of the opera

45、ting slip of the machine, given by equation (18). the slip is plotted as a function of the magnetizing inductance in figure 3.fig 2. variation of the reciprocal of the magnetizing inductance with the magnetizing fluxfig 3. variation of the slip with the magnetizing flux.to obtain a relationship betw

46、een the magnitude of the modulation index (m) and the magnetizing flux, the stator currents in equation (16) are expressed in terms of the stator fluxes. rearranging equation (15)substituting equation (19) in (14)the stator currents can be expressed in terms of the stator fluxes as:sub stituting (20

47、 ) in (21) and eliminating the stator currents in (16), using equation (22) the effect of the changing magnetizing flux on the magnitude of the modulation index is obtained under a condition of minimum copper loss. the magnetizing flux linkage is varied and the corresponding values of magnetizing in

48、ductances are calculated. using the values of the magnetizing inductances, the slip and the corresponding modulation index is calculated for minimum total copper loss condition. figure 4 shows the effect of changing load resistance and changing magnetizing flux linkage on m assuming a constant rotor

49、 speed s can be seen from figure 4, an increase . as can be seen from figure 4, an increase in the load resistance decreases the values of the modulation index calculated. also from figure 4, at a constant load, the same value of modulation index is obtained for two different values of magnetizing i

50、nductance. the effect of the different magnetizing inductances is illustrated in figure 7.similarly, the effect of change in rotor speed for a constant magnetizing flux is illustrated in figure 5. in this case, the magnetizing flux is chosen to have a constant value of 0. 25 wb.expressing the produc

51、t of the load resistance and the square of the modulation index in terms of the magnetizing inductance using equation (22), gives another parameter that can be used in the analysis. the parameter can be used as a measure of the load resistance that the system can be operated with for a fixed value o

52、f rotor speed and slip.the effect of the different operating slips on r0 is shown in figure 6.fig 4. variation of the magnitude of the modulation index m withthe magnetizing flute linkage for varying load (rl=35-65)resistances, and slip calculated under minimum loss.fig 5. variation of the magnitude

53、 of the modulation index m with rotor speed for a constant value of magnetizing flute 0.25 wb, for varying load resistances (rl =35-65) under minimum loss.fig 6. variation of ro with the magnetizing flux linkage for varyingslips at a constant rotor speed =200 rad/sec. fixing the magnetizing flux, at

54、 a constant load and rotor speed identifies with a single point on the m vs curve. each point corresponds to a certain current and flux. therefore the d-axis rotor reference flux can be calculated using equations (18) and (19). the flux is calculated corresponding to the operating slip at minimum lo

55、ss.also for every point on the m vs curve, there is a range of dc voltages for that particular operating point. expressing the stator currents in terms of the magnetizing flux (25) and substituting in equation (16) results in an equation for the dc voltage in terms of the magnetizing inductance and

56、modulation index (26). the range of do voltages is obtained by varying the q-axis component of the modulation index and calculating the corresponding d-axis component, in equation (26).for the same value of the modulation index there are two values of magnetizing inductance/magnetizing flux linkage.

57、the higher magnetizing flux linkage yields a higher voltage,which in most applications is not acceptable and infeasible.the effect of the magnetizing flux linkage on the range of dc voltages can be seen in figure 7. the dc voltage contours are plotted for the lower value of the magnetizing flux, for plotted for the lower value of the magnetizing flux, for different load resistances. fig 7. var