[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-672284-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-672284-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","transformer-equation-notes","Transformer Equation Notes","","Transformer Equation Notes gives a detailed, step-by-step derivation of the ideal transformer equations using source and load inductors with series resistances. It defines the inductors, loop voltages and currents, introduces mutual inductance through the coupling coefficient k, and derives the input impedance Zin by combining loop equations and simplifying assumptions (perfect coupling, and conditions under which RL becomes negligible or high frequency behavior). It then establishes current and voltage relationships and discusses how voltage mapping depends on whether the source resistance RS is small.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/technology/","Technology",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/transformer-equation-notes/672284/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/transformer-equation-notes/672284.png","ImageObject",300,407,{"name":42,"@type":43},"Rhys","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-03","2026-11-02",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",5,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What circuit elements and parameters are defined for the transformer derivation?","Question",{"text":63,"@type":64},"The notes define two series inductors with resistances: the source (primary) inductor LS with resistance RS and the load (secondary) inductor LL with resistance RL. They also define source and load voltages (VS and the voltage across RL) and currents IS and IL, plus the turns counts NS and NL and the input impedance Zin.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How is mutual inductance M introduced and related to the coupling coefficient k?",{"text":68,"@type":64},"Mutual inductance M is assumed to represent how the source inductor influences the load inductor and vice versa. It is then expressed using a coupling coefficient k (from 0 to 1) through M^2 = k^2 LS LL, with k=1 treated as perfect (ideal) coupling.",{"name":70,"@type":61,"acceptedAnswer":71},"What assumptions are used to simplify the input impedance Zin?",{"text":72,"@type":64},"The derivation makes Assumption 1 that the transformer is perfectly coupled (k=1). It also uses Assumption 2a that RL is small compared with LS and LL or, equivalently, Assumption 2b that operation is at high frequencies so the RL term can be eliminated, leading to a simplified expression for Zin.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},672284,1793588744,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,100,104,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":102,"slug":103},6,50,"technology",{"id":105,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},7,"Healthcare",40,"healthcare",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},8,"Research & Report",30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":55,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":101,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":105,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":80,"read_time":143},687207024643,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Transformer Equation Notes  \nThis file contains a more detailed derivation of the transformer equations than the notes or the experiment 3 write-up. It will help you to understand what assumptions were needed while deriving the ideal transformer equations we use. To do the derivation, we  \nwill use the figure pictured below:  \nfigure 1  \nAs seen in figure 1, the transformer has two inductors: a source (or primary) inductor (LS) and a load (or secondary) inductor (LL) . Each inductor loop is in series and has some resistance. The resistance of the source loop is represented by RS and the resistance of the load loop is represented by RL. The source has an input voltage of VS. The voltage in the load loop is measured across RL and is induced in the load inductor by the source inductor. The current through the source loop is given by IS. The current through the load loop is IL.  \nWhen you do any derivation, it is best to know what result you are trying to achieve. The ideal equations for a transformer in terms of the figure above are:  \na =  \nNL VL LL  \n= = =  \nNS VS LS  \nIS  \nIL  \nZ =  \nin  \nRL  \n2 a  \nwhere a is a constant and N is the number of turns on each inductor in the transformer. Therefore, NS is the turns on the source inductor and NL is the number of turns on the load inductor. Zin is the input impedance. It is the steady state impedance or “resistance”of the source inductor coil when it is coupled with the load coil in the transformer.  \nInductor Impedance  \nWe are going to use the loop equations for the circuit to derive our final result. If the two circuits are separated, as depicted in figure 2, the steady state loop equations are simply: VS = VRS + VLS = IS RS + IS j ωLS and 0 = VLS + VRS 0 = IL j ωLL + IL RL . (Note: V=IZ.) What effect does putting them together in a transformer have? We will assume that the effect of the transformer is to create some mutual inductance, M. The mutual inductance is the net amount that the source inductor pushes on the load inductor and visa versa. We do not know what M is, but we do know it is an inductance and that it works against the primary and load inductances. We can see that because the currents are going  \nin opposite directions. This idea is pictured in figure 3 below:  \nfigure 3  \nNow we can write our loop equations again:  \n1]VS = IS RS + IS j ωLS − IL j ωM and 2] IS j ωM = IL j ωLL + IL RL  \nWe will use these equations to derive all the relationships about transformers, but first, we need to further define M. We know M depends on both LS and LL. We also know that in a real transformer, there will be some influence of one inductor on the other, but there will also be some losses due to the fact that they are connected through a medium (such as air or iron) . Therefore, we will define M in terms of a constant, k, the coupling  \ncoefficient. We will let k take on a number from 0 to 1. When k is 1, the influence of one inductor in the transformer on the other is (unrealistically) ideal. This means that all of the inductance from one pulls down on the other. We call this a perfect transformer. Mathematically, we will define M by the following expression: M 2 = k 2 LS LL . The reason for this exact choice will become evident later when we do the derivation.  \nFinding the Input Impedance  \nFirst we want to find an expression for Zin, the net impedance of the source inductor in the transformer. This impedance is the combined influence of M and LS. We know that whatever Zin is, it must be the “resistance” of the source inductor in the circuit. Therefore, we know the total impedance of the circuit must be.  \nZT = RS + Zin Zin = ZT − RS ZT = ~~ ~~VISS~~ ~~ Zin = ~~ ~~VISS~~ ~~ − RS  \nIf we can find an expression for VIS~~ ~~ − RS , then we will have the input impedance.  \nS  \nWe can use the source loop equation to find this:  \nNow we have an expression for Zin in terms of the source inductance and the mutual inductance, but it is still in terms of complex impedances. We can use t","cbCaiuuFK67nxjEA","https://ap.wps.com/l/cbCaiuuFK67nxjEA","pdf",59942,"English","# Inductor Impedance\n## Ideal transformer equations and notation\n## Loop equations with mutual inductance M\n## Defining mutual inductance via coupling coefficient k\n# Finding the Input Impedance\n## Deriving Zin from circuit equations\n## Assumptions used to simplify\n# Current and Voltage Relations\n## Current relation between primary and secondary\n## Voltage relation and the role of RS","[{\"question\":\"What circuit elements and parameters are defined for the transformer derivation?\",\"answer\":\"The notes define two series inductors with resistances: the source (primary) inductor LS with resistance RS and the load (secondary) inductor LL with resistance RL. They also define source and load voltages (VS and the voltage across RL) and currents IS and IL, plus the turns counts NS and NL and the input impedance Zin.\"},{\"question\":\"How is mutual inductance M introduced and related to the coupling coefficient k?\",\"answer\":\"Mutual inductance M is assumed to represent how the source inductor influences the load inductor and vice versa. It is then expressed using a coupling coefficient k (from 0 to 1) through M^2 = k^2 LS LL, with k=1 treated as perfect (ideal) coupling.\"},{\"question\":\"What assumptions are used to simplify the input impedance Zin?\",\"answer\":\"The derivation makes Assumption 1 that the transformer is perfectly coupled (k=1). It also uses Assumption 2a that RL is small compared with LS and LL or, equivalently, Assumption 2b that operation is at high frequencies so the RL term can be eliminated, leading to a simplified expression for Zin.\"}]","Transformer Equation Notes | PDF",18]