Basic Circui Elemens - J ucas An elecrical circui is an inerconnecion of circui elemens. These circui elemens can be caegorised ino wo ypes, namely acive 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, V; leer symbol: v or V ) The poenial difference, also known as volage, is he work done in moving a uni posiive charge from one poin o anoher. 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 doing work. 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 V, 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 reserved for he imaginary operaor - as i is almos invariably used o denoe curren. Passive Circui Elemens We will sar wih he passive circui elemens. The mos basic of he passive circui elemens are he resisance, inducance and capaciance. These are passive elemens and do no generae 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). Basic Circui Elemens Professor J ucas November 200
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. =. This may also be wrien as = G., G = where G is he conducance (uni: siemen, S ) p() =. =. i 2 () = G. v 2 () w() =.. =. i 2 (). = G. v 2 (). 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 2 Circui symbols for Inducance (c) The common circui symbols for he Inducor are shown in figure 2. Figure 2(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 2(b) and is used o simplify he drawing of circuis. The symbol shown in figure 2(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 = ( ) 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 Basic Circui Elemens Professor J ucas 2 November 200
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. = ½. i 2 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 2 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. 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 C Basic Circui Elemens Professor J ucas 3 November 200
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. = ½C. v 2 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 2 Summary For a resisor, v = i, i = G v For an inducor, v = p i, i = v p Curren hrough an inducor will never change suddenly. For a capacior, v = i, i = Cp v Cp Volage 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 Acive Circui Elemens An acive circui elemen is one which produces elecrical energy in a circui. [Producing energy acually means convering non-elecrical form of energy o an elecrical form]. Acive elemens can be furher caegorised ino volage sources and he curren sources. Basic Circui Elemens Professor J ucas 4 November 200
Volage 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 Figure 5(b) Pracical 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. Curren Sources An ideal curren source (figure 6(a)) keeps he curren produced unchanged independen of load. I() I() Figure 6(a) Ideal curren source Y(p) Figure 6(b) Pracical curren source = I() for all = I() Y(p). However, 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. 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 saircase) has an ampliude zero before ime zero, and an ampliude uniy afer ime zero. H() = 0, < 0 H() =, > 0 H() Figure 7 Uni Sep Basic Circui Elemens Professor J ucas 5 November 200
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 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) + 0 Figure 8 Uni Impulse Series - circui Consider he exciaion of a series - circui by a sep exciaion e() = E.H(). e s () 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 Figure 9 Series - circui p i + i = 0 e s () i.e. he soluion o he equaion is of he form E E i( ) = A. e + The consan A can be deermined from he iniial condiions. i.e. a = 0, i = 0 E E A = ( ) = e 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 deails areavailable on he secion on circui ransiens. Basic Circui Elemens Professor J ucas 6 November 200