Michał Tadeusiewicz. Electric Circuits. Technical University of Łódź International Faculty of Engineering. Łódź 2009

Transcription

1 Mchał Tadeusewcz Electrc Crcuts Techncal Unersty of Łódź Internatonal Faculty of Engneerng Łódź 9

2 Contents Preface.. 5. Fundamental laws of electrcal crcuts 7.. Introducton Krchhoff s oltage Law (KL) Krchhoff s Current Law (KCL) Independence of KCL equatons...5. Independence of KL equatons Tellegen s theorem Crcut elements esstors Independent sources Power and energy Smple lnear resste crcuts esste crcuts. DC analyss Superposton theorem and ts applcaton 3 6. Three termnal resste crcuts The Theenn-Norton theorem Node method Smple nonlnear crcuts Controlled sources 59. Capactor Introducton Contnuty property Energy stored n a capactor Seres connecton of capactors Parallel connecton of capactors Inductor 69.. Introducton Contnuty property Hysteress Energy stored n an nductor Seres connecton of nductors Parallel connecton of nductors Operatonal-amplfer crcuts Descrpton of the operatonal amplfer Examples Fnte gan model of the operatonal amplfer Frst order crcuts dren by DC sources esstor-nductor crcuts esstor-capactor crcuts Snusodal steady-state analyss Prelmnary dscusson Phasor concept Phasor formulaton of crcut equatons 5.4. Impedance and admttance Phasor dagrams 5.6. Effecte alue Power n snusodal steady state Instantaneous and aerage power

3 6.. Complex power Measurement of the aerage power Theorem on the maxmum power transfer esonant crcuts Seres resonant crcut Parallel resonant crcut Coupled nductors Basc propertes Connectons of coupled nductors Ideal transformer Three-phase systems Introducton Y-connected systems Three-phase systems calculatons Power n three-phase crcuts 68 eference books

4 Preface Ths book presents an ntroductory treatment of electrc crcuts and s ntended to be used as a textbook for students, durng the junor years, at the Internatonal Faculty of Engneerng of the Techncal Unersty of Łódź. The book coers most of the materal taught n conentonal crcut courses and ges the fundamental concepts requred to understand and tackle the electrcal engneerng problems. Its prerequstes are the basc calculus, complex numbers, and some famlarty wth ntegral calculus and lnear dfferental equatons, whch are desrable but not essental. The objecte of the book s to feature theores and concepts of fundamental mportance n electrcal engneerng that are amenable to a broad range of applcatons. The book ncludes a large number of examples. They are proded to llustrate the concepts and to make the theory more clearer. On each page there s a blank area where a student can note down comments, explanatons and addtonal examples dscussed durng the lectures. The book can be thought of as consstng of three parts. Part (Chapters -, 3) ntroduces many basc concepts, laws, and prncples related to electrc crcuts. In addton dfferent methods of the DC analyss of resste crcuts are studed n detal. Part (Chapters -, 4) deals wth smple lnear dynamc crcuts and ther components. The transent analyss of the frst order crcuts s consdered. Part 3 (Chapters 5 to 9) focuses on snusodal crcuts n the steady-state and dscusses many dfferent aspects of AC analyss. At the end of ths part, three-phase systems are ntroduced and analysed. I gratefully acknowledge the support and encouragement of Dr. Tomasz Saryusz-Wolsk, Head of the Internatonal Faculty of Engneerng of the Techncal Unersty of Łódź. Łódź, 9 Mchał Tadeusewcz 5

5 . Fundamental laws of electrcal crcuts. Introducton An electrc crcut s an nterconnecton of electrc deces (elements) by conductng wres. Fgure. shows a crcut consstng of a oltage source, two resstors, a transstor, a capactor, and a transformer. Any juncton n the crcut where termnals of the elements are joned together s called a node. On the crcut dagrams they are marked wth dots. Fg... An example of a crcut Fg... eference drectons of current and oltage In the crcuts we consder currents flowng through the elements (branches) and oltages between any two nodes. The unt for oltage s the olt (), whereas the unt for current s the ampere (A). Fgure. shows the reference drecton of current and oltage represented by arrows. 7

6 If at some tme current s poste, then t flows nto the element by node. If the current s negate t flows out of the element by node. The reference drecton of the oltage across the element s represented by an arrow. If at some tme oltage s poste, t means that the electrc potental of node s larger than the electrc potental of node. If t s negate then the electrc potental of node s smaller than the electrc potental of node. The reference drecton of each current and each oltage can be assgned arbtrarly. When they are chosen as shown n Fg.., we say that we hae chosen assocated reference drectons. Ths s the conenton we wll follow throughout the whole course.. Krchhoff s oltage Law (KL) The fundamental laws goernng electrc crcuts are Krchhoff s oltage and current laws and Tellegen s theorem. In an electrc crcut we consder a path traersng some branches n successon. If the startng node of a path s the same as the endng node, the path s called a loop. Krchhoff s oltage Law For any electrc crcut, for any of ts loop, and at any tme, the algebrac sum of the branch oltages around the loop s equal to zero. To wrte KL equaton we select a loop and assume arbtrarly ts reference drecton, clock-wse or counter clock-wse. Next we assgn the plus sgn to the branch oltages whose reference drectons agree wth that of the loop and the mnus sgn to the others. Example. KL equaton for the loop,, 3 n the crcut shown n Fg..3: KL equaton for loop, 4, 5, 7: ( t) ( t) ( t). 3 () t ( t) ( t) ( t)

7 7 (t) (t) 5 (t) 6 (t) 3 (t) (t) 4 (t) Fg..3. An example crcut for llustratng KL KL can also be expressed n terms of oltages between nodes creatng a closed node sequence. A node sequence s called a closed node sequence f t starts and ends at the some node. Krchhoff s oltage Law (general erson) For any electrc crcut, for any closed node sequence, and for any tme, the algebrac sum of all node-to-node oltages around the chosen closed node sequence s equal to zero. Example. 3 4, Fg..4. An example crcut for llustratng KL 9

8 Let us consder closed node sequences:,, 5, 4, and,, 3, 6, 5, 4,. KL equatons:,, 5, 4, :,. 5 3,, 3, 6, 5, 4, : Krchhoff s Current Law (KCL) Another fundamental law goernng electrc crcuts s Krchhoff s Current Law, as follows. For any electrc crcut, for any of ts nodes, and at any tme the algebrac sum of all the branch currents meetng at the node s zero. In the algebrac sum we assgn the plus sgn to the currents leang the node and the mnus sgn to the currents enterng the node. Example.3 (t) (t) 3 (t) 3 4(t) : ( t) ( t) 3( t), or smply. 3 3:. 3 4 Fg..5. An example crcut for llustratng KCL

9 To formulate KCL n a more general form we consder gaussan surface defned as a balloon-lke closed surface, as llustrated n Fg..6. the gaussan surface 3 Fg..6. The gaussan surface KCL (general erson) For all crcuts, for all gaussan surfaces, for all tmes t, the algebrac sum of all currents crossng the gaussan surface at tme t s equal to zero. In the algebrac sum we assgn the plus sgn to the currents leang the gaussan surface and the mnus sgn to the currents enterng the surface. Example.4 In the crcut shown n Fg..6 we wrte KCL equaton. 3 The topologcal propertes of a crcut can be exhbted usng a graph obtaned by replacng each branch by a lne. Each branch of the graph has orentaton ndcated by an arrow on the branch. Ths arrow s the same as the reference drecton of the current flowng through the correspondng branch of the crcut. Thus, the graph can be used to wrte KL and KCL equatons.

10 .4 Independence of KCL equatons For a gen crcut we can wrte many KCL equatons. Hence, the queston arses how many of them are lnearly ndependent Fg..7. An example graph To answer ths queston we consder the graph shown n Fg..7 and wrte KCL equatons at each node, 3, If we add the frst two equatons together, we obtan. 3 4 Multplyng both sdes of ths equaton by ( ) yelds whch s exactly the thrd equaton., 3 4

11 It means that the thrd equaton s a lnear combnaton of the frst two equatons. Thus, not each equaton brngs new nformaton not contaned n the others and at least one equaton repeats the nformaton contaned n the others. Howeer, f we reject the thrd equaton, then the remanng ones are lnearly ndependent. Thus, the thrd equaton s redundant, t s useless and can be dscarded. Generally, the followng ndependence property of KCL equatons holds. For any graph wth n nodes KCL equatons for any n lnearly ndependent equatons. ( ) of these nodes form a set of n ( ).5 Independence of KL equatons Smlarly as n the case of KCL equatons the queston arses how to wrte a set of lnearly ndependent KL equatons. The smplest answer s as follows. We wrte KL equatons selectng the loops so that any equaton contans at least one oltage that has not been ncluded n any of the preous equatons. It can be shown that for a crcut hang b branches and n nodes b n lnearly ndependent equatons can be formulated. 3 I II 4 III I 8 Fg..8. A graph for llustratng ndependence of KL equatons Example.5 Let us consder the graph shown n Fg..8. In ths graph we wrte lnearly ndependent KL equatons usng the proded rule. As a result we obtan the followng set of equatons 3

12 , 7, 3 4, Tellegen s theorem Let us consder a graph hang b branches and n nodes. Let us use the assocated reference drectons. Tellegen s theorem Let {,,, b } be any set of branch currents satsfyng KCL at any node and let,,, be any set of branch oltages satsfyng KL at any loop. Then t holds { } b b k. k k Note that the set of branch currents and the set of branch oltages are assocated wth the gen graph but not necessarly wth the same crcut. For example, let us consder the graph shown n Fg..9 and two dfferent crcuts depcted n Fgs. and. hang ths graph. Tellegen s theorem enables us to wrte the followng equatons:, ~ ~ ~ ~ ~ ~ ~ ~, ~ ~ ~ ~, ~ ~ ~ ~

13 3 4 3 Fg..9. A graph hang three nodes and four branches ~ ~ 3 4 ~ 3 ~ ~ ~ ~ 3 ~ 4 Fg... A crcut hang the graph of Fg..9 Fg... A crcut hang the graph of Fg..9 5

14 . Crcut elements The components used to buld electrc crcuts are called crcut elements. In ths secton we defne smple two-termnal crcut elements: a resstor, ndependent oltage and current sources.. esstors An element s sad to be a resstor f ts oltage-current relaton s of algebrac type. Ths relaton s represented graphcally by a cure n plane, called the characterstc of the resstor. Any resstor can be classfed as lnear or nonlnear. A resstor s called lnear f ts characterstc s a straght lne through the orgn (see Fg..). Fg... Characterstc of a lnear resstor It s descrbed by the equaton or (.) G, (.) 6

15 where G. Equaton (.) s known as Ohm s law, s called the resstance and G s called the conductance. The unt for resstance s the ohm ( Ω) and for conductance the semens S. The symbol of a lnear resstor s shown n Fg... ( ) or G Fg... The symbol of a lnear resstor Any resstor whose characterstc s not a straght lne through the orgn s classfed as a nonlnear resstor. A typcal example of a nonlnear resstor s a dode, descrbed by the equaton K λ ( e ), (.3) where K and λ are poste constants. Fg..3 shows the symbol and the characterstc of a semconductor dode. Fg..3. Symbol and characterstc of a dode 7

16 A general symbol of any nonlnear resstor s depcted n Fg..4.. Independent sources oltage source Fg..4. General symbol of a nonlnear resstor An element s called a oltage source f t mantans a prescrbed oltage ( t) between ts termnals for any current flowng through the source. Consequently, a oltage source mantans a prescrbed oltage S ( t) between ts termnals n an arbtrary crcut to whch t s connected. The symbol of the oltage source s shown n Fg..5. S (t) S Fg..5. The symbol of a oltage source Generally, the prescrbed oltage s a tme aryng sgnal S ( t). In a specal case, t s constant S, called a DC oltage source. In such a case, the characterstc expressng the oltage between the termnals of the source n terms of the current flowng through the source s a horzontal lne, as shown n Fg..6. S Fg..6. Characterstc of a DC oltage source 8

17 The defned oltage source s an deal element not encountered n the physcal world. A real oltage source can be represented by an equalent crcut shown n Fg..7. S S Fg..7. Model of a real oltage source Certan deces hae S ery small and can qute effectely be approxmated by the deal oltage source. Let us consder a real oltage source termnated by a load, as shown n Fg..8. S S S Load Fg..8. A real oltage source termnated by a load Usng KL and Ohm s law we wrte the equaton, (.4) S that descrbes, on the plane, the straght lne, shown n Fg..9. Ths lne s a characterstc of the real oltage source, and s called a load lne. S 9

18 S slope - S S Fg..9. Load lne of a real oltage source Current source A current source s an element whch mantans a prescrbed current S ( t) for any ( t) between ts termnals. Consequently, a current source mantans a oltage prescrbed current S () t n an arbtrary crcut to whch t s connected. The symbol of a current source s shown n Fg... S (t) Fg... Symbol of a current source If the prescrbed current s constant, S ( t) I S the current source s called a DC current source. Its characterstc s shown n Fg...

19 I S Fg... Characterstc of a DC current source The defned current source s an deal element. A real current source can be represented by the crcut shown n Fg... I S S Fg... A real DC current source Let us consder a real DC current source termnated by a load, as shown n Fg..3.

20 I S Load S Applyng KCL at the top node we wrte Fg..3. A real DC current source termnated by a load Snce then I. (.5) S, (.6) S I S. (.7) S Equaton (.7) descrbes characterstc of a real DC source, as shown n Fg..4. Ths characterstc s a straght lne, called a load lne. I S I S Fg..4. Characterstc - of real DC current source

21 Let us multply both sdes of equaton (.7) by S and rearrange ths equaton as follows By denotng S S S I. (.8) S S S I we obtan equaton (.4) that descrbes a real oltage source. Thus, f S S I S, then the real current source and the real oltage source are equalent. 3

22 3. Power and energy Let us consder a crcut and draw two wres from ths crcut. As a result we obtan a two-termnal crcut called a one-port. If the one-port s suppled wth a source, the t t flows out of current ( ) flows nto the one-port by termnal A and the same current ( ) the one-port by termnal B. Therefore, we ndcate only one current ( t), as shown n Fg. 3.. The oltage () t between the termnals s also ndcated. A (t) B (t) Fg. 3.. A one-port wth ndcated the port oltage and port current are called a port current and a port oltage, respectely. The nstantaneous power enterng the one-port s equal to the product of the port oltage and port current The current ( t) and the oltage ( t) where ( t) ( t) ( t) ( t) () ( ) p, (3.) s n olts, t n amperes and p t n watts (abbreated to W). The energy delered to the one-port from tme to t s gen by the equaton (, t) p( τ ) dτ ( τ ) ( τ ) dτ w t t t t t t, 4

23 where a arable τ means tme. Hence, t holds dw p() t. dt If the one-port s a lnear resstor, specfed by or G, then p p ( t) ( t) ( t) ( t), (3.) ( t) ( t) ( t) G ( t). (3.3) Thus, the nstantaneous power s, n ths case, nonnegate for all t. If the one-port s a nonlnear resstor, represented by a characterstc located n the frst and thrd quadrants only, then ( t) ( t) and the power enterng the resstor s nonnegate, ts energy s a nondecreasng functon of tme and the resstor consumes the energy. Such a resstor s called passe. If some parts of characterstc le n the second or thrd quadrant, then for some t, ( t) ( t) <, the power enterng the resstor s negate and the resstor delers energy to the outsde world. Such a resstor s called acte. 5

24 4. Smple lnear resste crcuts Crcuts consstng of resstors and sources are classfed as resste crcuts. In partcular, they can be suppled wth DC sources only. The analyss of such a class of crcuts s called DC analyss. Let us consder a crcut consstng of two lnear resstors connected n seres, as shown n Fg. 4.. Fg. 4.. Two resstors connected n seres To analyse ths crcut we apply the KL and Ohm s law. Snce the same current traerses both resstors we wrte ( ). (4.) Hence, we hae. (4.) Equaton (4.) states that the seres connecton of two lnear resstors s equalent to resstor, oltages across the resstors are specfed by the equatons:. (4.3),. 6

25 Hence, t follows the relaton, (4.4) whch states that the seres connecton of resstors and can be consdered as a oltage dder. The oltage s dded n proporton to and. Formula (4.3) can be drectly generalzed to the seres connecton of n lnear resstors,,, n. (4.5) Fgure 4. shows the crcut consstng of two lnear resstors connected n parallel. n Fg. 4.. Two resstors connected n parallel oltages across resstors and are dentcal and equal to. The current accordng to KCL satsfes the equaton: Usng Ohm s law. (4.6), (4.7) 7

26 we obtan or where, (4.8) (4.9). (4.) Thus, two resstors connected n parallel are equalent to the resstor specfed by (4.). Usng (4.7), (4.9), and (4.) we wrte Hence, t holds,.. (4.) Thus, the parallel connecton of resstors and can be consdered as a current dder, where the currents are dded accordng to equaton (4.). Equaton (4.) can be drectly generalsed to the crcut consstng of n resstors,, connected n parallel n n. (4.) Fgure 4.3 shows a three-termnal resstor called a potentometer. Termnal 3 called a wper, can be shfted along the resstor, ddng t nto and. p x y 8

27 p x y 3 p x y y Fg A potentometer Fg Crcut contanng a potentometer Let us consder the crcut shown n Fg. 4.4, contanng a potentometer. The crcut can be consdered as a seres connecton of resstor and parallel connected resstors y and. Hence, the resstance faced by the oltage source s Formula (4.) enables us to fnd the current and then oltage y x y x. (4.) y (4.3) y x y y y y. (4.4) y x ( ) y y 9

28 5. esste crcuts. DC analyss In ths secton we study crcuts consstng of lnear resstors and ndependent DC sources. We formulate some theorems goernng these crcuts, whch enables us to analyse them effcently. 5. Superposton theorem and ts applcaton Let us consder a lnear crcut dren by n oltage sources and m current sources I S S,, I S S,, The superposton theorem states that any branch current and any branch oltage n ths crcut s gen by the expresson of the form, S S n Sn S S m S m, I S n S m h h h k I k I k I, (5.) ( ) ( j,, m) where coeffcents j,, n and h j k j,. are constants and depend only on crcut parameters. In other words, any branch current and any branch oltage s a lnear combnaton of the oltage and current sources. Example 5. Let us consder a crcut consstng of lnear resstors, a sngle oltage source and a sngle current source. We extract from ths crcut the sources and an arbtrary resstor, as shown n Fg. 5.. We wsh to fnd the current flowng through resstor usng the superposton theorem. 3

29 S I S Accordng to the superposton theorem Let ~ h S and ~ ki, then S Fg. 5.. A crcut wth extracted sources and a resstor ~ Note that f I and ~ f. S h S ki S. (5.) ~ ~. (5.3) S If I S, then the branch contanng the current source can be replaced by an open ~ crcut (see Fg. 5.). Thus, s the current flowng through resstor n the crcut wth the current source set to zero (remoed). If, then the branch contanng the oltage source can be replaced by a short crcut (see Fg. 5.3). Thus, ~ s the current flowng through resstor n the crcut wth the oltage source set to zero (shortcrcuted). The current can be regarded as a response of the crcut due to the ~ oltage source actng alone. The current ~ can be consdered a response of the crcut due S to the current source actng alone. I S S 3

30 ~ S ~ S I S I S Fg. 5.. The crcut shown n Fg. 5. dren Fg The crcut shown n Fg. 5.. dren by the oltage source S by the current source I S Generally, the response of the crcut due to seeral oltage and current sources s equal to the sum of the responses due to each source actng alone, that s wth all other oltage sources replaced by short crcuts and all other current sources replaced by open crcuts. Example 5. Let us consder the crcut shown n Fg. 5.4, dren by a oltage source and a current source. We apply the superposton theorem to fnd oltage. I S S S I S 3 Fg A crcut dren by two sources 3

31 Frst we set the current source to zero. As a result, the crcut s dren only by the oltage source (see Fg. 5.5). S ~ S ~ 3 Fg Crcut of Fg. 5.4 dren only by the oltage source S ~ In ths crcut the same current traerses all the resstors. Hence, we fnd, ~ S whereas the oltage ~ s, accordng to Ohm s law, gen by the equaton 3 ~ ~. S 3 Now we set the oltage source S to zero, obtanng the crcut shown n Fg ~ ~ I S 3 Fg Crcut of Fg. 5.4 dren only by the current source I S 33

32 In ths crcut the resstance faced by the current source equals ( ) 3 3. The product of ths resstance and the current s oltage S I ~ ( ) I S ~ 3 3. The superposton theorem leads to the equaton ( ) S I S ~ ~

33 6. Three termnal resste crcuts The three-termnal crcuts shown n Fgs 6. and 6. consst of three lnear resstors. The crcut of Fg. 6. s called a Y crcut, whereas the one depcted n Fg. 6. s called a Δ crcut. crcut crcut Fg. 6.. Y crcut Fg. 6.. Δ crcut 35

34 In a Y crcut t holds. (6.) 3 Hence, we fnd ( ) ( 3 ) 3 3, (6.) ( ) 3 ( 3 ) 3 3. (6.3) To express and n terms of and n the Δ crcut, we apply current sources and to ths crcut, as shown n Fg Fg Δ crcut dren by two current sources To fnd and we use the superposton theorem. Frst, we set current source to zero (see Fg. 6.4) and compute oltages ~ and ~ 36

35 ( ) ~, (6.4) ~ ~. (6.5) ~ 3 Fg The crcut shown n Fg. 6.3 wth Next, we set current source to zero (see Fg. 6.5) and compute ~ ( ) ~ and ~, (6.6) ~. (6.7) ~ 37

36 3 3 ~ ~ 3 Fg The crcut shown n Fg 6.3 wth Accordng to the superposton theorem we obtan ( ) ~ ~, (6.8) ( ) ~ ~. (6.9) The Y and Δ crcuts are sad to be equalent f the sets of equatons (6.4)-(6.5) and (6.8)-(6.9) are dentcal. In such a case, the correspondng coeffcents of these equatons are equal,.e. 3 ( ) 3 3 3, (6.) , (6.) 3 ( ) (6.) 3 38

37 3 Solng ths set of equatons for,, we fnd 3 3, (6.3) , (6.4) (6.5) Formulas (6.3)-(6.5) ge the resstances of the Y crcut whch s equalent to the Δ crcut. Solng the set of equatons (6.)-(6.4) for,, we fnd 3 3 3, (6.6) , (6.7) (6.8) Formulas (6.6)-(6.8) ge the resstances of the Δ crcut whch s equalent to the Y crcut. The Y crcut s sad to be balanced f 3 Y. Usng n such a case, (6.6)-(6.8), we obtan 3 3 Δ, where Δ 3 Y. (6.9) 39

38 7. The Theenn-Norton theorem The Theenn-Norton theorem s a ery mportant law goernng lnear resste crcuts. It can be regarded as two equalent theorems. Let us consder an arbtrary one-port consstng of lnear resstors and ndependent sources, as shown n Fg. 7.. Fg. 7.. A lnear resste one-port The Theenn theorem Any lnear resste one-port can be replaced by a seres connecton of a resstor and a oltage source, where s an nput resstance across the one-port after all C eq sources nsde t are set to zero, C s a oltage across the termnals of the one-port when the port s left open-crcuted. The equalent Theenn crcut s shown n Fg. 7.. The elements of ths crcut C and can be determned as llustrated n Fgs 7.3 and 7.4. eq eq 4

39 eq C Fg. 7.. The Theenn equalent crcut eq C Fg Fndng eq Fg Fndng C The Norton theorem Any lnear resste one-port can be replaced by a parallel connecton of a lnear resstor and a current source I. eq eq SC s defned as n the Theenn theorem. s a current flowng through the shortcrcuted one-port. I SC 4

40 The equalent Norton crcut s shown n Fg. 7.5, whereas Fg. 7.6 shows the crcut enablng us to fnd. I SC I SC eq I SC Fg The equalent Norton crcut Fg Fndng I sc The crcut shown n Fg. 7.6 can be replaced, on the bass of Theenn s theorem, by an equalent crcut shown n Fg eq I SC C Fg The crcut equalent to the crcut of Fg

41 Usng KL and Ohm s law we wrte C eq I SC. (7.) Hence, t follows the equaton eq C, (7.) I SC showng the relaton between,, and I. eq C Proof of the Theenn theorem Let us consder a resste one-port, shown n Fg. 7.8, contanng n oltage sources,,, and m current sources I, I,, I. S S S n S S S m SC Fg A resste one-port Fg The one-port of Fg. 7.8 suppled wth a current source We connect an addtonal current source to the one-port, as shown n Fg

42 On the bass of the superposton theorem we obtan n m h sj k j I k j j sj j. (7.3) If, then the port termnals are open-crcuted and C. Hence, we hae C n j h j sj m j k j I sj. (7.4) If we set all the sources nsde the one-port to zero, that s, I I I, then equaton (7.3) reduces to S S hence, S m k, (7.5) k and equaton (7.3) can be rewrtten n the form eq (7.6), (7.7) C where and are defned as n the Theenn theorem. Equaton (7.7) descrbes C eq the crcut depcted n Fg. 7.8, beng Theenn s equalent crcut. eq S S S n C eq Fg The Theenn equalent crcut 44

43 Note that the Theenn crcut shown n Fg. 7.8 s equalent to the crcut depcted n Fg. 7.9, beng the Norton crcut. Thus, the Theenn and Norton crcuts are equalent. C eq eq C eq I SC Fg The crcut equalent to the crcut shown n Fg. 7.8 Example Let us consder the one-port shown n Fg. 7.. E E C eq Fg. 7.. An example one-port Fg. 7.. Theenn s crcut The equalent Theenn crcut s shown n Fg. 7.. We fnd the elements and eq of ths crcut. Accordng to Theenn s theorem s the nput resstance of the one-port shown n Fg. 7.. eq C 45

44 E E C Fg. 7.. A one-port enablng us Fg A crcut enablng us to fnd eq to fnd C Hence, we hae eq. (7.8) To fnd C we consder a crcut wth open-crcuted termnals shown n Fg In ths crcut the same current traerses all the elements. To fnd ths current we apply KL and Ohm s law Snce the current flows through resstor E E. (7.9), we hae E E E E C E E. (7.) 46

45 8. Node method Krchhoff s laws and Ohm s law enable us to analyse smple resste crcuts. Howeer, such an approach s neffcent n the case of more complex crcuts. In ths Secton we deelop a general, ery useful and commonly appled method, called the node method. To explan ths method we consder a crcut hang n nodes and ntroduce a concept of node-to-datum oltage. For ths purpose we choose arbtrarly one of these nodes as a datum node. The potental of ths node s set to zero, hence, t s grounded. For the remanng n nodes we ntroduce node-to-datum oltages (or smply node oltages) e,, e n, between these nodes and the datum. The reference drectons of the node oltages are shown n Fg. 8.. n- e e e n- n Fg. 8.. eference drectons of the node oltages It s easy to see that oltage between any two nodes k and j can be expressed n terms of node oltages e and e. k j kj e e. (8.) k j 47

46 k kj j e k e j Fg. 8.. Illustraton of equaton (8.) The dea of the node method wll be explaned usng the crcut shown n Fg. 8.3, where the current sources, and the conductances - G are gen. S n S G 5 S G 4 e G 5 3 e e 3 G G G 3 3 S Fg An example crcut 48

47 We choose the bottom node as a reference, ntroduce node oltages and wrte KCL equatons at nodes,, 3 4 S S, 5 4, (8.) 5 3 S. Next, we express the branch currents n terms of node oltages ( ) ( ) ( ), e e G G, e e G G, e G G, e G e G G, G e G (8.3) and substtute them nto (8.) ( ) ( ) ( ) ( ). e e G e G, e e G e G e e G, e e G G e S S S (8.4) Fnally, we rearrange equatons (8.4) as follows ( ) ( ) ( ). e G G e G, e G e G G G e G, e G e G G S S S (8.5) The set of node equatons (8.5) contans three unknowns,,. e e 3 e 3 49

48 Let us replace the resstor G by a oltage source S, as shown n Fg S G 4 e G 5 3 e e 3 G S G 3 3 S Fg Crcut dren by current and oltage sources Equatons wrtten at node and 3 are the same as n the preous case. Hence, we only need to wrte an equaton at node. Snce the current cannot be expressed n terms of the branch oltage, t s consdered an addtonal arable. Thus, we obtan the followng set of equatons 3 G e G G 3 e 4 3 G4 ( e e ) S, S ( e e ) G5 ( e3 e ) G ( e e ). 5 3 S, (8.6) 5

49 Ths s a set of three equatons wth four unknown arables e, e, e3,. Therefore, we add another equaton of the form e S, (8.7) where s the gen oltage source. Substtutng (8.7) nto (8.6) we elmnate the S e e e3 arable and obtan the set of three equatons n three arables,, 3 G e G G 3 e 4 3 G4 ( e S ) S, S ( e S ) G ( e S ) 5 3 G ( e ). 5 3 S S, (8.8) S G 4 e 3 e e 3 G G G 3 3 S Fg Nonlnear resstor crcut 5

50 The node method can be also appled to crcuts contanng nonlnear resstors. It wll be explaned a an example crcut shown n Fg. 8.5, ncludng a nonlnear resstor (semconductor dode) descrbed by the equaton ( ) 5 5 e K λ. (8.9) To wrte the node equatons we ntroduce temporarly the current as an addtonal arable 5 ( ) ( ). e G, e G e e G, e e G G e S S S (8.) Next we express n terms of the correspondng node oltages 5 ( ) ( ) ( ) e e K e e K λ λ (8.) and substtute t nto (8.) ( ) ( ) ( ) ( ) ( ) ( ). e K e G, e K e G e e G, e G G e S e e e e S S S λ λ (8.) In ths way, we obtan a set of three node equatons, n three unknown arables, descrbng the nonlnear crcut shown n Fg

51 9. Smple nonlnear crcuts In ths secton we analyse ery smple crcuts consstng of nonlnear resstors by means of a graphcal approach. Seres connecton of resstors Fgure 9. shows a crcut consstng of two nonlnear resstors connected n seres. Fg. 9.. Two nonlnear resstors connected n seres The resstors are specfed by ther characterstcs depcted n Fgs 9. and 9.3. Fg. 9.. The characterstc Fg The characterstc 53

52 On the bass of KL we wrte. (9.) Snce the resstors are connected n seres, the same current traerses each of them, hence, t holds. (9.) On the bass of these equatons we can fnd the characterstc usng a graphcal approach. To trace the characterstc we add the oltages and specfed by the characterstcs and, respectely, for each alue of the current. The graphcal constructon s llustrated n Fg. 9.4.,, Fg Graphcal constructon for fndng characterstc - Parallel connecton of resstors Fgure 9.5. shows a crcut consstng of two nonlnear resstors connected n parallel. 54

53 Fg Two nonlnear resstors connected n parallel The characterstcs and are depcted n Fgs 9.6 and 9.7, respectely. Fg The characterstc Fg The characterstc Snce the resstors are connected n parallel, the oltages and are dentcal. (9.3) 55

54 Usng KCL at the top node we hae. (9.4) Thus, the characterstc can be traced n a graphcal manner by addng for each alue of the correspondng currents and (see Fg. 9.8).,, Fg Graphcal constructon for fndng characterstc - Operatng ponts Nonlnear crcuts dren by DC sources hae constant solutons (branch oltages and current), called operatng ponts. The basc queston of the analyss of ths class of crcuts s fndng the operatng ponts. If a crcut s smple, the operatng pont can be found usng a graphcal approach, To llustrate ths approach we consder a typcal basng crcut depcted n Fg. 9.9, ncludng a nonlnear resstor specfed by the characterstc shown n Fg. 9.. The lnear part of ths crcut, consstng of the DC oltage source E and resstor, s descrbed by the equaton E. (9.5) b b 56

55 b A a f( a ) E a B b a Fg Typcal basng crcut Fg. 9.. Characterstc of the nonlnear resstor belongng to the crcut of Fg. 9.9 a We transcrbe ths characterstc n the b b plane to the a a plane. Snce we obtan b a and b a a a, E. (9.6) Equaton (9.6) descrbes a straght lne, shown n Fg. 9.. On the same plane we plot the characterstc The pont of ntersecton (, ) a ˆ a ( ) f. a a î s the soluton of the set of equatons ( ) f, a a E, a a hence, the ntersecton s the operatng pont of the crcut shown n Fg

56 a E ˆ a î a E a Fg. 9.. Graphcal constructon for fndng the operaton pont 58

57 . Controlled sources Controlled sources are crcut elements ery useful n the modelng of electronc deces. A controlled source s a two-port consstng of two branches. A prmary branch s ether a short crcut or an open crcut. A secondary branch s ether a oltage source or a current source. The source waeform depends on a oltage or a current of the prmary branch. Thus, there are four types of controlled sources. The controlled sources can be classfed as lnear or nonlnear. All the controlled sources are shown n Fgs..4. a) b) r f r ( ) Fg... Current-controlled oltage sources (CCS) a) lnear, b) nonlnear a) b) g ( ) f g Fg... oltage-controlled current sources a) lnear, b) nonlnear 59

58 a) b) α ( ) f α a) Fg..3. Current-controlled current sources a) lnear, b) nonlnear b) μ ( ) f μ Fg..4. oltage-controlled oltage sources a) lnear, b) nonlnear Controlled sources are ery useful n modelng electronc deces. For example the Ebers-Moll model of an npn bpolar transstor contans two lnear current controlled current sources, as llustrated n Fg..5. α α F F E C F B E BE B BC Fg..5. The Ebers-Moll model of an npn bpolar transstor C 6

59 . Capactor. Introducton A capactor s a two-termnal element whch stores an electrc charge. The smplest example of a capactor s shown n Fg... It s made of two flat parallel metal plates n free space. (t) q(t) (t) d -q(t) Fg... Parallel plate capactor When a current ( t) s appled, then a charge ( t) C( t) q s nduced on the upper plate and an equal but opposte charge s nduced on the lower plate. The constant of proportonalty, called capactance, s gen approxmately by ε A C, d 6

60 where 9 ε 36π s the delectrc constant called permttty, A s the plate area and d s the separaton of the plates. The unts of capactance are farads, abbreated to F. Equaton F m q C (.) defnes the q characterstc of the capactor. The characterstc s a straght lne through the orgn wth a slope C, as shown n Fg... q Fg... Characterstc -q of a lnear capactor A capactor whose characterstc s a straght lne through the orgn s called a lnear capactor. Otherwse, the capactor s sad to be nonlnear. An example q characterstc of a nonlnear capactor s shown n Fg.3. 6

61 q Fg..3. Characterstc -q of a nonlnear capactor The symbols of a lnear and a nonlnear capactor and the reference drectons of ( t) and () t are shown n Fgs.4 and.5. In these fgures q s the charge on ths plate whch s ponted by the reference arrow of the current. (t) q(t) (t) q(t) (t) (t) C Fg..4. Symbol of a lnear capactor Fg..5. Symbol of a nonlnear capactor 63

62 The current ( t) s gen by the equaton () t Usng equaton (.) and (.) we obtan ( t) dq. (.) dt ( C) dq d d C. (.3) dt dt dt Equaton (.3) expresses capactor current n terms of capactor oltage. To express capactor oltage n terms of capactor current we replace the arable t wth τ obtanng ( τ ) Next we ntegrate both sdes of (.4) between and t Hence, t holds. Contnuty property t d ( τ ) C. (.4) dτ t ( t ) d ( ) ( τ ) τ dτ C dτ C d C( ( t) () ) d. τ ( ) C. (.5) () t ( ) ( τ ) dτ t Let us replace t n equaton (.5) by t dt t dt ( t dt) ( ) ( τ ) dτ (.6) C and assume that ( t) s bounded for all t, that s there exsts a constant () t < L for all t. L, such that 64

63 Subtractng equaton (.5) from (.6) we hae t dt ( t dt) ( t) ( τ ) dτ. (.7) C t dt ( t t) ( t) As dt, then ( τ ) dτ, hence, d. Thus, oltage across any lnear capactor s a contnuous functon of tme. It means that ths oltage cannot jump nstantaneously from one alue to another..3 Energy stored n a capactor Consder a capactor suppled wth a generator as shown n Fg..6. (t) Generator C (t) Fg..6. A capactor suppled wth a generator The energy delered by the generator to the capactor from tme t to t s gen by the equaton t t t t (, t) p( τ ) dτ ( τ ) ( τ d )τ A t where p s the nstantaneous power enterng the capactor., (.8) 65

64 Let ( t ), hence, no charge s stored n the capactor. We choose ths state as the state correspondng to zero energy,.e. w ( t ), where at t t s the ntal energy of the capactor. Let us replace t by another arable τ n equaton (.3) and rewrte t n the form Substtutng (.9) nto (.8) yelds (, t) ( ) ( τ ) A t ( τ ) d ( τ ) C dτ ( ) d τ Cd τ. (.9) t ( t ) ( t ) τ dτ Cd C C( ( t ). (.) ) t A capactor s an element that stores energy, but does not dsspate t. Hence, the energy stored n the capactor at tme t s gen by the equaton Snce ( t ) () t w( t ) A( t,t) A( t,t) w. (.) A( ) w() t C () t w and t, t s specfed by (.), we hae.4 Seres connecton of capactors. (.) Consder two capactors connected n seres as shown n Fg..7. (t) C C (t) (t) (t) Fg..7. Two capactors connected n seres 66

65 Snce the same current traerses both capactors we wrte, on the bass of (.5) t () t ( ) ( τ d )τ, (.3) C Usng KL yelds At t equaton (.5) becomes () t ( ) ( τ ) dτ. (.4) C ( t) ( t) ( t) t. (.5) ( ) ( ) ( ). (.6) We add equatons (.3) and (.4) together and apply (.6) Let then t () t ( ) ( τ ) dτ. (.7) C C C, (.8) C C t () t ( ) ( τ ) dτ. (.9) C Equaton (.9) descrbes the equalent capactor of two capactors connected n seres. The ntal oltage of ths capactor s specfed by (.6), whereas the capactance s gen by (.8). 67

66 .5 Parallel connecton of capactors Fgure.8 shows two capactors connected n parallel. Snce oltages across the. capactors are dentcal, both capactors hae the same ntal oltage ( ) (t) (t) (t) (t) C C Fg..8. Two capactors connected n parallel Currents flowng through the capactors are gen by the equatons d () ( t) t C, () t dt ( t) d C. dt Applyng KCL at the top node and the aboe equatons yelds where () t () t () t ( C C ) ( t) d( t) d C, dt dt C C C. (.) Formula (.) ges the capactance of the equalent capactor. 68

67 . Inductor. Introducton Fgure. shows an nductor made of wre wound around a core, made of a nonmetallc materal. (t) (t) φ Fg... An example nductor When the dece s suppled wth a tme aryng current source, a magnetc flux s nduced and crculates nsde the core. The magnetc flux lnkage φ ( t), beng the total flux lnked by all turns of the col, s proportonal to the current ( t ) L( t) φ, (.) where the coeffcent L s called nductance. The unts of magnetc flux are webers (W), the unts of nductance are henrys (H). 69

68 If the core shown n Fg.. s a torod made of materal hang the magnetc constant (permeablty) μ, then the nductance s gen by the formula N A L μ, (.) l 4 H where μ 4π, N s the number of turns, A s the cross-sectonal area of the m core, l s the mdcrcumference along the core. The equaton φ L (.3) defnes φ characterstc of the nductor. The characterstc s a straght lne through the orgn wth a slope equal to L (see Fg..). In such a case the nductor s classfed as lnear. φ slope L Fg... Characterstc of a lnear nductor Otherwse, f the characterstc s not a straght lne through the orgn, the nductor s called nonlnear. The typcal characterstc of a nonlnear nductor s shown n Fg..3. 7

69 φ Fg..3. Characterstc of a nonlnear nductor The symbols of lnear and nonlnear nductors are shown n Fg..4. (t) (t) (t) L (t) Fg..4. Symbols of lnear and nonlnear nductors Accordng to Faraday s nducton law () t ( t) dφ. (.4) dt 7

70 Substtutng (.3) nto (.4) yelds d d ( L) L. (.5) dt dt To express the current flowng through a lnear nductor n terms of the oltage across the nductor we replace t wth τ and rewrte equaton (.5) n the form d ( τ ) L. (.6) dτ Next we ntegrate both sdes of equaton (.6) between and t t t ( t ) d ( ) ( τ ) τ dτ L dτ L d L( ( t) () ) d. τ ( ) Hence, we hae ( τ ). Contnuty property t () t ( ) ( τ ) dτ. (.7) L Let us replace t n equaton (.7) by t dt t dt ( t dt) ( ) ( τ ) dτ (.8) L and assume that ( t) s bounded for all t, that s, there exsts a constant () t < M for all t. Subtractng equaton (.7) from (.8) yelds t dt t t dt ( t dt) ( t) ( τ ) dτ ( τ ) dτ ( τ ) dτ. L L t M such that 7

71 t dt Snce ( τ ) dτ as dt, then ( t d t) ( t). Thus, current flowng through L t any lnear nductor s a contnuous functon of tme. Ths means that nductor current cannot jump nstantaneously from one alue to another..3 Hysteress Ferromagnetc core nductors exhbt the hysteress phenomenon as depcted n Fg..5. φ φ φ 3 Fg..5. Hysteress phenomenon The characterstc shown n Fg..5 s obtaned by ncreasng the current from to, next decreasng ths current from to 3 and after that ncreasng from 3 to. Thus, the flux decreases accordng to the upper branch and ncreases accordng to the lower branch 3. As a result, a close cure s traced. The magnetc flux becomes zero for the negate alue and the poste alue of the current. 4 73

72 .4 Energy stored n an nductor Consder an nductor suppled wth a generator as shown n Fg..6. (t) Generator L (t) Fg..6. An nductor suppled wth a generator The energy delered by the generator to the nductor from tme t to t s gen by the equaton t t t t (, t) p( τ ) dτ ( τ ) ( τ ) dτ A t, (.9) where p s the nstantaneous power enterng the nductor. Let ( t ), consequently also the magnetc flux equals zero and no magnetc feld exsts n the nductor. Such a state can be consdered a state correspondng to zero energy stored. Let us substtute (.6) nto (.9). Then, we hae t ( t ) ( t ) d ( ) ( ) ( τ ) A t, t L τ dτ L d L L ( t) ( t ). dτ t ( t ) ( t ) ( ) Snce ( t ) the energy ntally stored n the nductor s also equal to zero, w ( t ) and the energy delered to the nductor from t t to t s (, t) L () t A t. (.) 74

73 An nductor s an element that stores energy, but does not dsspate t. Hence, the energy stored n the nductor at tme t s gen by the equaton () w( t ) A( t,t) L () t w t. (.).5 Seres connecton of nductors Consder two nductors connected n seres as shown n Fg..7. L L (t) (t) (t) (t) Fg..7. Two nductors connected n seres Snce dentcal currents flow through the nductors, both nductors hae the same ntal current ( ). KL leads to the equaton where Hence, we hae where () t ( t) ( t) ( t), d () ( t) t L, () t dt () t d( t) ( t) d L. dt ( t) d( t) d d L L ( L L ) L, dt dt dt dt L L L. (.) Formula (.) ges the nductance of the equalent nductor. 75

74 .6 Parallel connecton of nductors Fgure.3 shows two nductors connected n parallel. (t) (t) (t) (t) L L Fg..3. Two nductors connected n parallel We wrte KCL equaton ( t) ( t) ( t) and substtute nto ths equaton t () t ( ) ( τ ) dτ L and t () t ( ) ( τ ) dτ. L As a result we obtan where t () t ( ) ( τ ) dτ, (.3) L ( ) ( ) ( ) (.4) 76

75 and or L L (.5) L L L L. (.6) L L Thus, the parallel connecton of two nductors can be replaced by a sngle nductor hang the nductance specfed by (.6), wth the ntal condton (.4). 77

76 3. Operatonal-amplfer crcuts 3. Descrpton of the operatonal amplfer An operatonal amplfer s a mult-termnal semconductor dece. Fgure 3. shows the symbol of ths dece, ncludng nsde trangle nterconnected transstors, resstors and a power supply oltage source. The termnals aalable for external connectons are called: an nertng nput, a nonnertng nput, an output, and external ground. Inertng nput Nonnertng nput Output External ground Fg. 3.. An operatonal amplfer In Fg. 3. oltages and currents are ntroduced, where and denote the currents enterng the operaton amplfer, s the nput oltage, denotes the output oltage, and the output current. d - d Fg. 3.. eference drectons of oltages and currents of the operatonal amplfer 78

77 The operatonal amplfer s descrbed by the followng set of equatons:,, where ( d ) [ ε,ε ] functon f ( ) s lnear ( ) A ( ) f d, f s a functon shown n Fg For belongng to a ery small nteral d d f d, where A equals at least. () E sat -ε ε d (m) -E sat Fg The characterstc f ( ) ( ) d of the operatonal amplfer At ±Esat the functon f d saturates. Snce the slope A s ery large we can assume that A. As a result we obtan the deal model of the operatonal amplfer, f hang the characterstc ( ) shown n Fg d 79

78 The characterstc s pecewse-lnear and conssts of three segments. They defne three operatng regons called: a lnear regon, a saturaton regon, and a saturaton regon. f ( d ) E sat Saturaton regon Lnear regon d - Saturaton regon -E sat Fg The characterstc of the deal model of the operatonal amplfer The lnear regon s descrbed by the followng equatons:, d. The output oltage n ths regon satsfes the nequalty called the aldatng nequalty., (3.) E < < (3.) sat E sat 8

79 In the saturaton regon the operatonal amplfer s descrbed by the equatons,, (3.3) E sat, and the output oltage satsfes the aldatng nequalty >. (3.4) d The aboe relatonshps lead to the model of the operatonal amplfer operatng n the saturaton regon, as shown n Fg d > E sat Fg The model of the operatonal amplfer operatng n the saturaton regon In the saturaton regon the operatonal amplfer s specfed by the followng set of equatons:,, (3.5) whereas the aldatng nequalty s E sat, <. (3.6) d 8

80 Hence, we obtan the model of the operatonal amplfer operatng n the saturaton regon, depcted n Fg d < E sat Fg The model of the operatonal amplfer operatng n the saturaton regon 3. Examples To explan the method for the analyss of the crcuts contanng operatonal amplfers we consder two examples. Example 3. Fgure 3.7 shows a crcut called an nerter. We assume that operatonal amplfer operates n the lnear regon and express the output oltage n terms of the nput oltage. P - d Fg The nerter contanng the operatonal amplfer 8

81 Snce the operaton amplfer operates n the lnear regon, t s descrbed by the set of equatons (3.). Especally, hence, d and KCL appled at node P leads to the equaton. Consequently, t holds Usng KL we wrte.. Thus, the output oltage equals.. (3.7) The obtaned result s ald f satsfes the aldatng nequalty E < <. sat E sat Substtutng (3.7) we obtan or E sat < < E sat, Esat < < Esat. (3.8) 83

82 Thus, the output oltage s specfed by equaton (3.7) f the nput oltage s selected so that the nequalty (3.8) s satsfed. Example 3. Fgure 3.8 shows a crcut called a oltage comparator. - d (t) constant Fg The oltage comparator contanng the operatonal amplfer In ths crcut the operatonal amplfer s allowed to operate n all the three regons. We wsh to fnd the transfer characterstc f ( ). Usng KL we wrte d. (3.9) Below we consder 3 cases.. The operatonal amplfer operates n the lnear regon. In ths case d, hence usng (3.9) we obtan. Ths result s correct f the aldatng nequalty E < < E s satsfed. sat sat As a result, we fnd the segment of the transfer characterstc shown n Fg

83 . The operatonal amplfer operates n the saturaton regon. Thus, >, E and equaton (3.9) ges d sat >. As a result, we obtan the segment of the transfer characterstc shown n Fg The operatonal amplfer operates n the saturaton regon. Smlarly, as n case, we wrte the relatons: d <, Esat, <, whch lead to segment 3 of the characterstc (see Fg. 3.9). E sat 3 -E sat Fg The transfer characterstc f ( ) of the oltage comparator shown n Fg Fnte gan model of the operatonal amplfer In ths secton we consder the operatonal amplfer model specfed by the characterstc f ( d ) shown n Fg. 3.. Unlke the deal model the slope of the segment passng through the orgn s ery large, but fnte. 85

84 f ( d ) E sat Slope A -ε A > ε d -E sat Fg. 3.. The characterstc f ( ) d of nondeal model of the operatonal amplfer In the lnear regon the model s descrbed by the equaton and the aldatng nequalty A d ε < < ε. d Hence, t follows the equalent crcut shown n Fg. 3.. d f ( ) d Fg. 3.. The equalent crcut of the operatonal amplfer operatng n the lnear regon 86

85 In the saturaton regon the mathematcal representaton of the model s E sat, > ε, whereas n the saturaton regon the model s descrbed by d E sat, < ε, The correspondng equalent crcuts are shown n Fgs 3. and 3.3. d d d > ε E sat Fg. 3.. The equalent crcut of the operatonal amplfer operatng n the saturaton regon d d < ε E sat Fg. 3.. The equalent crcut of the operatonal amplfer operatng n the -saturaton regon 87

86 4. Frst order crcuts dren by DC sources Crcuts made of one nductor or one capactor, resstors and sources are called frst order dynamc crcuts. In ths secton we study crcuts dren by DC sources. 4. esstor-nductor crcuts We consder a smple crcut consstng of a DC oltage source, a resstor and an I (see Fg. 4.). nductor, wth an ntal current ( ) L L Applyng KL we wrte Snce we obtan, after smple rearrangements Fg. 4.. A frst order resstor-nductor crcut. L d L dt L d L, dt, ( ) I. (4.) 88

87 Equaton (4.) can be presented n the form where d, (4.) dt τ L s called a tme constant. We rewrte equaton (4.) n the form d dt and separate the arables and Next, we ntegrate equaton (4.4) fndng L t L τ (4.3) τ L τ τ τ d dt. (4.4) τ d τ dt, t ln K, τ where the constant K may be wrtten as K ln A, where A s another constant. 89

88 Hence, we hae t ln, A τ or To determne A we wrte equaton (4.5) at t τ Ae. (4.5) t and set ( ) I Substtutng A, gen by (4.6) nto equaton (4.5) yelds () I A. (4.6) () t I e τ t. (4.7) When t, we hae I, whch s the correct ntal condtons. When t, the second term on the rght sde of equaton (4.7) anshes, hence, t holds ( ). Thus, we hae t () t ( ) ( ( ) ( ) ) e τ, (4.8) where ( ) s the ntal alue of ( t), whereas ( ) s the steady-state alue of ( t) Snce the steady-state current s constant, the oltage across the nductor equals zero. L d L dt. 9

89 Thus, the nductor behaes lke a short crcut and the crcut consdered n the steadystate (as t ) conssts of the oltage source and the resstor (see Fg. 4.). ( ) Fg. 4.. Model of the crcut shown n Fg. 4. n the steady-state The plot of ( t) s shown n Fg ( ) I (t) ( ) β α x t Fg Plot of (t) and graphcal nterpretaton of the tme constant The ntal rate of changng of the current s gen by d dt tan α ( ( ) ( ) ) e τ ( ( ) ( ) ). (4.9) τ t τ t t 9

90 On the other hand d dt t Equalzng (4.9) and (4.) we fnd ( ( ) ( ) ) tan β. (4.) x x τ. Thus, to fnd graphcally the tme constant, t s necessary to draw the tangent of the current at t and determne ts ntercept wth the horzontal lne passng through the pont (, ( ) ). In a ery specal case, when, ( ) and equaton (4.8) becomes The plot of ( t) t () t I e τ s shown n Fg. 4.4 and the tme constant τ x. (t). (4.) I.368 I x Fg Plot of (t) n the crcut wth t τ τ At t τ, ( τ ) I e I e. 368I. Thus, n one tme constant the current has declned to.368 of ts ntal alue (see Fg. 4.4). 9