Basic Circuit Elements - Prof J R Lucas

Transcription

1 Basic Circui Elemens - Prof J ucas An elecrical circui is an inerconnecion of elecrical circui elemens. These circui elemens can be caegorized ino wo ypes, namely acive elemens and passive elemens. Some Definiions/explanaions of elecrical erms Charge (uni: coulomb, C; leer symbol: q or Q ) The elecric charge is he mos basic quaniy in elecrical engineering, and arises from he aomic paricles of which maer is made. Poenial Difference (uni: vol, ; leer symbol: v or ) The poenial difference, also known as volage, is he work done (or energy required) o move a uni posiive charge from one poin o anoher (across a circui elemen). Thus he change in work done dw when a charge dq moves hrough a poenial difference of v dw v.dq Curren (uni: ampere, A; leer symbol: i or I ) The elecric curren is he rae of charge flow in a circui. dq i, q i. Energy (uni: joule, J; leer symbol: w or W ) The Energy is he capaciy o do work. Thus in elecrical quaniies his may be expressed as dw v.dq v.i. Power (uni: wa, W; leer symbol: p or P ) The elecric power is he rae of change of energy. dw p v. i Common usage of leer symbols I is common pracice o use he simple leers (such as v, i, p, w) o represen quaniies which are varying wih ime, and capial leers (such as, I, P, W) o represen quaniies which are consans. Bu his need no always be done and is a useful pracice raher han a rule. Wih represenaion of elemens, obviously his pracice does no exis as hey are no ime variables. d The leer p is also commonly used o represen he differenial operaor. The leer j is normally used for he imaginary operaor - as i is almos invariably used o denoe curren. Basic Circui Elemens Professor J ucas November 007

2 Passive Circui Elemens The mos basic of he passive circui elemens are he resisance, inducance and capaciance. Passive elemens do no generae (conver from non-elecrical energy) any elecriciy. They may eiher consume energy (i.e. conver from elecrical form o a non-elecrical form such as hea or ligh), or sore energy (in elecrosaic and elecromagneic fields). esisance (uni: ohm, Ω; leer symbol:, r ) (a) (b) (c) Figure Circui symbols for esisance The common circui symbols for he esisor are shown in figure. Figure (a) is he common symbol used for he general resisor, especially when hand-wrien. Figure (b) is he mos general symbol for he resisor, especially when in prined form. Figure (c) is he symbol used for a non-inducive resisor, when i is necessary o clearly indicae ha i has been specially made o have no or negligible inducance. A resisor made in coil form, mus obviously have a leas a small amoun of inducance. The basic equaion governing he resisor is Ohm s aw (see also page 5).. This may also be wrien as G., G where G is he conducance (uni: siemen, S ) p().. i () G. v () w()... i (). G. v (). I is o be noed ha p() is always posiive indicaing ha power is always consumed and energy always increases wih ime. Inducance (uni: henry, H; leer symbol:, l ) (a) (b) Figure Circui symbols for Inducance (c) The common circui symbols for he Inducor are shown in figure. Figure (a) shows a coil which is he simples symbol (and mos common when hand-wrien) for he inducor. A simpler represenaion of his is shown in figure (b) and is used o simplify he drawing of circuis. The symbol shown in figure (c) is someimes used in prined form, especially on ransformer nameplaes, bu is no a recommended form as i could lead o confusion wih he common resisor. The basic equaion governing he behaviour of an inducor is Faraday s law of elecromagneism. dφ e () Basic Circui Elemens Professor J ucas November 007

3 When here are N urns in a coil, e.m.f. will be induced in each urn, so ha he volage across he coil would be N imes larger. Also, if he volage is measured as a drop, he negaive sign vanishes, so ha dφ v N The flux produced in he magneic circui, is proporional o he curren flowing in he coil, so ha we may express he rae of change of flux in erms of a rae of change of curren. d i φ i, v This is wrien as a basic elecrical circui equaion as d i v p i or i i i p Since he volage across an inducor is proporional o he rae of change of curren, a sep curren change is no possible hrough an inducor as his would correspond o an infinie volage. i.e. he curren passing hrough an inducor can never change suddenly. You migh ask, wheher his would no occur if we swiched off he curren in an inducor. Wha would really happen is ha he ensuing high rae of change would cause a very large volage o develop across he swich, which in urn would cause a spark over across he gap of he swich coninuing he curren for some more ime. p(). d i w().. i. i.di ½. i d I can be seen ha w() is dependan only on i and no on ime. Thus when he curren i increases, he energy consumed increases and when i decreases, he energy consumed decreases. This acually means ha here is no real consumpion of energy bu sorage of energy. [If we compare wih waer ap, opening i and leing he waer run ino he ground would correspond o waer consumpion, where as filling a bucke wih he waer, and perhaps puing i back ino he waer ank, would correspond o waer sorage]. Thus an inducor does no consume elecrical energy, bu only sores i in he elecromagneic field. Sored energy w() ½. i Capaciance (uni: farad, F; leer symbol: C, c ) C (a) (b) Figure 3 Circui symbols for Capaciance The common circui symbols for he Capacior are shown in figure 3. When a volage is applied across a capacior, a posiive charge is deposied on one plae and a negaive charge on he oher and he capacior is said o sore a charge. C Basic Circui Elemens Professor J ucas 3 November 007

4 The charge sored is direcly proporional o he applied volage. q C. v Since q i. he basic equaion for he capacior may be re-wrien in circui erms as d v v i or i C C d Since he curren hrough a capacior is proporional o he rae of change of volage, a sep volage change is no possible hrough a capacior as his would correspond o an infinie curren. i.e. he volage across a capacior can never change suddenly. p(). d v w().. v. C C. v.dv ½C. v d I can be seen ha w() is dependan only on v and no on ime. Thus when he volage v increases, he energy consumed increases and when v decreases, he energy consumed decreases. This acually means ha here is no real consumpion of energy bu sorage of energy. Thus a capacior does no consume elecrical energy, bu only sores i in he elecromagneic field. Sored energy w() ½C. v Summary For a resisor, v i, i G v For an inducor, v p i, i v p For a capacior, Curren hrough an inducor will never change suddenly. v i, i Cp v Cp olage across a capacior will never change suddenly. Impedance and Admiance These may all be wrien in he form v Z(p) i, i Y(p) v where Z(p) is he impedance operaor, and Y(p) is he admiance operaor. Impedances and Admiances may be eiher linear or non-linear. This is defined based on wheher he values of, and C (slope of characerisic) are consans or no. Figure 4(a) inear Sysem Figure 4(b) Non-inear Sysem Basic Circui Elemens Professor J ucas 4 November 007

5 Ohm s aw This law was firs described by Professor Georg Simon Ohm (79-867) in a pamphle Die galvanische Kee mahemaisch bearbeie (ranslaion The Galvanic Circui invesigaed mahemaically) in 87. In any conducing medium, he curren densiy J r is relaed o he elecric field E r by he conduciviy σ of he medium (which can be a ensor in anisoropic maerials) as r r J σ E (vecor form of Ohm s aw) Noe: This form of he equaion is only valid in he reference frame of he conducing maerial. If he maerial is moving a velociy v relaive o a magneic field B, J σ.( E v B) The oal curren I flowing across a surface S, perpendicular o i, is given by I J. ds J. S for a conducing rod of uniform cross-secion S In an elecric field E, he difference of poenial beween he ends of a conducor is given as E. dl E. l for a uniform field I σ. S l Thus σ., so ha I. and. I S l l σ. S where he consans σ.s G conducance, l resisance, and ρ l σ. S σ conduciviy Thus. I giving he now familiar form of Ohm s aw. Kirchoff s aws Gusav ober Kirchoff (84-887) was he firs o publish a sysemaic formulaion of he principles governing he behaviour of elecric circuis, based on already available experience. His work embodied wo laws, namely a curren law and a volage law. In a circui supplied from a baery, exernal o he baery only an elecrosaic field exiss. The field ξ may be derived from he gradien of a scalar poenial φ. i.e. ξ φ The line inegral of he field hrough he baery from he negaive erminal o he posiive erminal, or he volage rise, is equal o he emf of he baery. Thus he poenial change over a closed loop becomes zero, giving us Kirchoff s volage law emf Σ volage drop exernal o circui or Σ v 0 where v is he volage change across any circui elemen. The ne flow of curren ino a volume is accompanied by an increase of charge wihin he same volume. J. ds ρ d S Applying Gauss s heorem ρ J d ρ d or 0 J d Since i is rue for all, i follows ha ρ J 0 ρ For seady currens, 0 so ha. J 0 Which resuls in he Kirchoff s curren law Σ i 0 where i is he curren of any circui elemen conneced o he node. Basic Circui Elemens Professor J ucas 5 November 007

6 Acive Circui Elemens An Acive Circui Elemen is a componen in a circui which is capable of producing or generaing energy. [Producing energy acually means convering non-elecrical form of energy o an elecrical form]. Acive circui elemens are hus sources of energy (or simply sources) and can be caegorised ino volage sources and he curren sources. olage sources are hose ha keep heir erminal volage very nearly he same as heir inernal volages ( E), while curren sources keep heir erminal currens very nearly he same as heir inernal currens (I I S ). Thus volage sources would have series impedances which are relaively small, while curren sources would have shun admiances which are relaively small. Ideal volage sources would have zero inernal impedance while ideal curren sources would have zero inernal admiance. Sources can also be caegorised as being independen sources, where he generaed volage (or curren) does no depend on any oher circui volage or curren; and dependen sources, where he generaed volage (or curren) depends on anoher circui volage or curren. While he circle is used as he circui symbol for independen sources, he diamond is used as he circui symbol for dependen sources. Independen source For an independen volage source (or curren source), he erminal volage (or curren) would depend only on he loading and he inernal source quaniy, bu no on any oher circui variable. (Independen) olage Sources An ideal volage source (figure 5(a)) keeps he volage across i unchanged independen of load. e() e() Z(p) Figure 5(a) Ideal volage source e() for all e() Z(p). However, pracical volage sources (figure 5(b)) have a drop in volage across heir inernal impedances. The volage drop is generally small compared o he inernal emf. (Independen) Curren Sources An ideal curren source (figure 6(a)) keeps he curren produced unchanged independen of load. I() I() Figure 5(b) Pracical volage source Figure 6(a) Ideal curren source I() for all Y(p) Figure 6(b) Pracical curren source I() Y(p). Basic Circui Elemens Professor J ucas 6 November 007

7 Pracical curren sources (figure 6(b)) have a drop in curren across heir inernal admiances. The curren drop is generally small compared o he inernal source curren. Dependen Source A dependen volage source (or curren source) would have is erminal volage (or curren) depend on anoher circui quaniy such as a volage or curren. Thus four possibiliies exis. These are (figure 7) o olage dependen (conrolled) volage source o Curren dependen (conrolled) volage source o olage dependen (conrolled) curren source o Curren dependen (conrolled) curren source Example Figure 7 Dependen sources For he circui shown in figure 8, deermine he curren I and o. Answer Applying Kirchoff s volage law, gives I I 4 o 5 o 4Ω 4 o Also, from Ohm s law 5.I o 7.5 Ω Thus by subsiuion, we have I.5 A and o.5 o Figure 8 Circui for example α o γ o βi o αi o Example For he circui shown in figure 9, deermine he curren I and o. Answer Applying Kirchoff s volage law, gives 5Ω 6 5 (I 0.8 o ) 4 o Also, from Ohm s law.i o Thus by subsiuion, we have I.0 A and o Example 3 For he circui shown in figure 0, deermine he curren I. Answer Applying Kirchoff s volage law, gives E I (α) I Also, from Ohm s law 6 E I 0.8 o Ω 4 o Figure 9 Circui for example I α I o o Figure 0 Circui for example 3 o αi o Basic Circui Elemens Professor J ucas 7 November 007

8 By soluion of he equaions And I o E ( α) α o E ( α) Operaional Amplifier (Op Amp) An operaional amplifier is an acive Invering Inpu circui elemen ha behaves as a volageconrolled volage source. An operaional Oupu Non-invering Inpu amplifier can be used o add, subrac, muliply, divide, amplify, inegrae and Null differeniae signals and are hus very Figure Circui connecions of Op Amp versaile. A pracical Op Amp, commercially available in Inegraed circui (IC) packages would have he inpus and pus as shown in figure. The Null deermines he offse. The pu volage of he Op Amp is linearly proporional o he volage difference beween he inpu erminals and by he gain A. However, he pu volage is limied o he range o of he supply volage. This range is ofen called he linear region of he amplifier, and when he pu reaches o hese limis, he op amp is said o be sauraed. The equivalen circui of he Op Amp is shown in figure. I has a dependen volage source A d. An ideal Op Amp has infinie gain (A ), infinie inpu resisance ( in ), and zero pu resisance ( 0). Parameer Symbol Ideal Typical Open-loop gain A 0 5 o 0 8 Inpu esisance in 0 6 o 0 3 Ω Oupu esisance 0 0 o 00 Ω Supply olage, 5 o 4 d in A d Figure Equivalen Circui of Op Amp Invering Amplifier An invering amplifier circui is shown in figure 3. In an invering amplifier he pu volage decreases when he inpu volage increases and vice versa. The equivalen circui of he invering amplifier is as shown in he figure 4. in Figure 3 Circui of Invering Amplifier Basic Circui Elemens Professor J ucas 8 November 007

9 in d d d i in d Ad If 0, and in, A d A giving A A in d d Thus A in A d A in A ( A) If A as for an ideal Op Amp, Gain in d 0 A Thus in ideal Op Amps, d is usually aken as a virual earh, even when analyzing. in i d in A d Figure4 Equivalen Circui Non-invering Amplifier In he non-invering amplifier (figure 5), he inpu volage is applied direcly o he non-invering () inpu and a small par of he pu volage is applied o he invering () inpu from he poenial divider. For an ideal Op Amp, wih 0, in and A i can be shown ha Gain Summing Amplifier If anoher inpu resisor, equal o he value of he original inpu resisor, is added o he inpu of he Op Amp, a Summing amplifier is obained (figure 6). ( ina inb ) If he wo inpu resisors are no equal ina ( A inb B ) in Figure 5 Non-invering Amplifier A ina B inb Figure 6 Summing Amplifier Basic Circui Elemens Professor J ucas 9 November 007

10 Differenial Amplifier So far, only one inpu erminal has been considered, eiher invering or non-invering. I is also possible o connec inpu signals o boh erminals a he same ime. The resulan pu volage is proporional o he difference beween he wo inpu signals A and B when he resisance values are appropriaely chosen. This ype of Op-Amp circui is commonly known as he Differenial Amplifier (figure 7). For his configuraion, for an ideal Op Amp, in, 0, and A. A, giving d 0 bu wih no curren d going ino he Op Amp as in Thus if he volage a each inpu erminal is, I A ina, I This gives A inb 3 B 3 inb B B 3 ina I A A I B B inb 3 d Figure 7 Differenial Amplifier Subsiuion gives A ina A A A B 3 3 inb A ina When A B and 3, he ransfer funcion becomes ( ) inb ina Naural Behaviour of --C Circuis The naural behaviour of a circui does no depend on he exernal forcing funcions, bu on he sysem iself. [I is like, if we ake a pendulum and give i an iniial swing, and hen le go, he behaviour of he pendulum depends only on is naural frequency. However, if we keep on pushing i a some oher frequency, hen he behaviour would also depend on he frequency of he forcing funcion]. Thus in order o deermine he naural behaviour, we mus use a forcing funcion which does no have is own frequency. The wo forcing funcions ha lend hemselves o purpose are he sep funcion and he impulse funcion. Uni Sep Funcion The uni sep H() (similar in appearance o a sep in a H() saircase) has an ampliude zero before ime zero, and an ampliude uniy afer ime zero. H() 0, < 0 H(), > 0 Figure 8 Uni Sep Basic Circui Elemens Professor J ucas 0 November 007

11 Uni Impulse Funcion The uni impulse δ() has an ampliude zero before ime zero, and an ampliude of infiniy a ime zero, and zero again afer ime zero. I also has he propery ha he area under he curve is uniy. δ() 0, < 0 δ() δ(), 0 δ() 0, > 0 0 also, δ ()., which gives δ (). 0 The uni impulse funcion also has he following properies. f ( ) δ ( ). f(0), and f ( τ ) δ ( ). f(τ) Series - circui Consider he exciaion of a series - circui by a sep exciaion e() E.H(). We can wrie he differenial equaion governing he behaviour as di. i es ( ) E.H() This equaion has a paricular inegral of E/ corresponding o seady sae, and a complemenary funcion corresponding o p i i 0 i.e. he soluion o he equaion is of he form E i( ) A. e The consan A can be deermined from he iniial condiions. E E i.e. a 0, i 0 A () e Figure 9 Uni Impulse Consider now he exciaion of he series - circui by an impulse exciaion e() E.δ() The complemenary funcion is he same as before, bu he paricular inegral is now differen and equal o 0. The new coefficien A can be obained from he iniial condiions which are now differen. The response o a uni impulse is also he same as he derivaive of he response o he uni sep. Thus he uni impulse response is d E E e e Oher circuis can also be similarly solved by wriing he differenial equaions governing he behaviour. Furher informaion is available on he secion on problems on circui ransiens. e s () Figure 0 Series - circui e s () E Figure Sep esponse Basic Circui Elemens Professor J ucas November 007