[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-656892-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-656892-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","es1803-complex-impedance-and-phasors","ES.1803 - Complex Impedance and Phasors","","Exponential response formulas and complex arithmetic are used to define and compute complex gain and phase lag in linear time-invariant systems with sinusoidal inputs. The material explains how phasor diagrams encode phase angles and phase differences and how impedance generalizes resistance. It derives the complex gain for a second-order example system, rewrites the result in polar form, and introduces general impedance as a system function. It then connects impedance back to Ohm’s law and develops series/parallel combination principles.",{"@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/es1803-complex-impedance-and-phasors/656892/",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/es1803-complex-impedance-and-phasors/656892.png","ImageObject",300,407,{"name":42,"@type":43},"nayy☆","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-25",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",6,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How is complex gain defined for a linear time-invariant system with sinusoidal input?","Question",{"text":63,"@type":64},"The output of a complexified input is written using an exponential response formula, and the complex-valued factor multiplying the input is called the complex gain.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What does a phasor diagram show about circuit components?",{"text":68,"@type":64},"It graphically represents the phase angle and phase differences between the voltages across components and the output current.",{"name":70,"@type":61,"acceptedAnswer":71},"How does impedance relate to resistance and Ohm’s law?",{"text":72,"@type":64},"Impedance is defined via the system’s complex gain for a voltage-to-current relationship; for a simple resistor circuit, the impedance equals the resistance and reproduces Ohm’s law.","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},656892,1793104043,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,104,109,114,119,122,126],{"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":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":102,"slug":103},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":97,"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":55,"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":115,"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":143,"read_time":144},962090893153,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","ES.1803: Complex Impedance and Phasors  \nJeremy Orloff  \n1 Introduction  \nWe are going to use the exponential response formula and complex arithmetic to understand the notions of impedance and phasor diagrams for electrical circuits.  \nWe’ll see that capacitors and inductors in a circuit have an impedance, which generalizes the notion of resistance across a resistor. (In fact, we will consider resistance to be a typeof impedance.) We will see that the rules for combining impedances in series or parallel are exactly the same as those for resistance.  \nThe phasor diagram tells us about the output current and the voltages across the various components in a circuit. It captures graphically the phase angle and phase differences between them.  \n2 Complex gain  \nIf we have a linear, time invariant system with input 􀓷 cos(􁆂􀔣) then we know the output is 􀔖(􁆂)􀓷 cos (􁆂􀔣 − 􁅿(􁆂)) . Here, 􀔖(􁆂) is the real gain and 􁅿(􁆂) is the phase lag.  \nIf we use complex replacement to change the input 􀓷 cos(􁆂􀔣) to 􀓷􀔔􀖄􁇖􀖏 , then the output is 􀓼(􀔘􁆂)􀓷􀔔􀖄􁇖􀖏 . We call 􀓼(􀔘􁆂) the complex gain.  \nExample 1. Consider the system 􀔍″ + 8􀔍′ + 7􀔍 = 􀓻′(􀔣) , with 􀓻 (􀔣) the input. Find the real gain, phase lag and complex gain.  \n(Note: For circuits, we will use capital letters for variables and parameters. We will also add tildes to indicate a complexified sinusoidal function.)  \nSolution: To find gain and phase lag, we let the input be 􀓻 (􀔣) = 􀓷 cos(􁆂􀔣) . We start by complexifying the equation:  \ñ ̃ ̃ ̃  \n􀔍″ + 8􀔍′ + 7􀔍 = (􀓷􀔔 􀖄􁇖􀖏 )′ = 􀔘􁆂􀓷􀔔 􀖄􁇖􀖏 , where 􀔍 = Re(􀔍) .  \nWe then use the exponential response formula to solve this. The characteristic polynomial is 􀔅 (􀔡) = 􀔡 2 + 8􀔡 + 7 . So the ERF implies  \ñ  \n􀔍􀖋 (􀔣) =  \n􀔘􁆂􀓷􀔔 􀖄􁇖􀖏  \n􀔅 (􀔘􁆂)  \n􀔘􁆂  \n= ⋅ 􀓷􀔔 􀖄􁇖􀖏 .  \n7 − 􁆂 2 + 8􀔘􁆂  \n􀔘􁆂  \nThis gives us the complex gain, 􀓼(􀔘􁆂) =  \n7 − 􁆂 2 + 8􀔘􁆂 .  \nNow, we write 􀓼(􀔘􁆂) in polar form, 􀓼(􀔘􁆂) = |􀓼(􀔘􁆂)|􀔔 −􀖄􁇓(􁇖) .  \n|􀓼(􀔘􁆂)| = √(7 − 􁆂 􁆂2)2 + 64􁆂2 . Since Arg(7 − 􁆂2 + 8􀔘􁆂) = tan−1 ( 7 􁆂􁆂 2 ) in Q1 or Q2, we have  \n􁅿(􀔘􁆂) = − Arg(􀓼(􀔘􁆂)) = −(Arg(􀔘􁆂)−Arg(7−􁆂2 +8􀔘􁆂)) = − 􁅺2 +tan−1 ( ~~ ~~7~~ ~~􁆂􁆂~~ ~~2 ) in Q1 or Q4 .  \n3 GENERAL DEFINITION OF IMPEDANCE 2  \nSo,  \n􀖋 (􀔣) = √~~ ~~(7~~ ~~−~~ ~~􁆂~~ ~~􀔦2)2~~ ~~+~~ ~~64􁆂~~ ~~2 ⋅􀓷􀔔 􀖄(􁇖􀖏−􁇓(􁇖)) ⇒ 􀔍􀖋 (􀔣) = Re (􀖋 ) = √~~ ~~(7~~ ~~−~~ ~~􁆂~~ ~~􀔦2)2~~ ~~+~~ ~~64􁆂~~ ~~2 􀓷 cos(􁆂􀔣−􁅿(􁆂)) .  \nThus, the gain is 􀔖(􁆂) =  􀔦  and the phase lag is 􁅿(􁆂) .  \n√(7 − 􁆂2 )2 + 64􁆂 2  \nSide notes: 1. 􀓼(􀔘􁆂) = 􀔖(􁆂)􀔔 −􁇓(􀖒) .  \n2. If we let the input be 􀔔 􀖎􀖏 , where 􀔢 is any complex number, then the output is 􀓼(􀔢)􀔔 􀖎􀖏 .􀓼(􀔢) is known as the transfer or system function of the system.  \n3 General definition of impedance  \nLater we will discuss the meaning of impedance in circuits. For now, we can give a definition in terms of systems.  \nFor a linear, time invariant system, we assume the input is 􀓷 cos(􁆂􀔣) . For this input, the complex impedance is 􀔏(􁆂) = 􀓼(1􀔘􁆂)~~ ~~ , where 􀓼(􀔘􁆂) is the complex gain.  \nIn Example 1, the complex impedance is 􀔏(􁆂) = 􀓼(1􀔘􁆂) = 7 − 􁆂 2 + 8􀔘􁆂 .  \nThe real impedance is |􀔏(􁆂)| = |􀓼(1􀔘􁆂)| = 􀔖(1􁆂) .  \n3.1 Resistance as impedance  \nConsider a circuit that consists of a, possibly time varying, input voltage 􀔋􀖄􀖉 (􀔣) in volts anda resistor with resistance 􀔇 in Ohms. The voltage drives a current 􀓾(􀔣) around the circuit.  \nI (t)  \nOhm’s law says 􀔋􀖄􀖉 (􀔣) = 􀔇􀓾(􀔣) or 􀓾(􀔣) = 1􀔇 ⋅ 􀔋􀖄􀖉 (􀔣) .  \nWe can consider this to be a system with input 􀔋􀖄􀖉 (􀔣) and output 􀓾(􀔣) . (It is a very simple linear, time invariant system.)  \ñ ̃ ̃  \nIf we have complexified input 􀔋􀖄􀖉 (􀔣) = 􀔔􀖄􁇖􀖏 , then the complexified system is 􀔇􀓾 = 􀔋􀖄􀖉 (􀔣) .̃ 1 ̃ 1  \nSo, 􀓾(􀔣) = 􀕳 􀔋􀖄􀖉 (􀔣) . This implies the complex gain is 􀓼(􀔘􁆂) = 􀔇  \nThus, the impedance 􀔏(􁆂) = 􀓼(1􀔘􁆂) = 􀔇 . For this simple circuit, the impedance is the same as the resistance.  \n4 SIMPLE CIRCUIT PHYSICS 3  \n3.2 Resistance in series and parallel  \nFor future reference, let’s recall that two circuits with two resistors 􀔇 1 and 􀔇2 can be reduced to an equivalent circuit with a single resistor 􀔇 .  \nIf 􀔇 1 and 􀔇2 are in series, then 􀔇 = 􀔇 1 + 􀔇2 .  \nVin (t)","cbCaikVZnyXwcXhd","https://ap.wps.com/l/cbCaikVZnyXwcXhd","pdf",262621,"English","# 1 Introduction\n# 2 Complex gain\n# 3 General definition of impedance\n## 3.1 Resistance as impedance\n## 3.2 Resistance in series and parallel\n# 4 Simple circuit physics\n## 4.1 The basic differential equations\n# 5 Complex impedance and Ohm’s law for circuits","[{\"question\":\"How is complex gain defined for a linear time-invariant system with sinusoidal input?\",\"answer\":\"The output of a complexified input is written using an exponential response formula, and the complex-valued factor multiplying the input is called the complex gain.\"},{\"question\":\"What does a phasor diagram show about circuit components?\",\"answer\":\"It graphically represents the phase angle and phase differences between the voltages across components and the output current.\"},{\"question\":\"How does impedance relate to resistance and Ohm’s law?\",\"answer\":\"Impedance is defined via the system’s complex gain for a voltage-to-current relationship; for a simple resistor circuit, the impedance equals the resistance and reproduces Ohm’s law.\"}]","ES.1803 - Complex Impedance and Phasors | PDF",1792971343,23]