dv 7. Voltage-current relationship can be obtained by integrating both sides of i = C :

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1 EECE202 NETWORK ANALYSIS I Dr. Charles J. Kim Class Noe 22: Capaciors, Inducors, and Op Amp Circuis A. Capaciors. A capacior is a passive elemen designed o sored energy in is elecric field. 2. A capacior consiss of wo conducing plaes separaed by an insulaor (or dielecric). In many applicaions, he plaes may be aluminum foil while he dielecric may be air, ceramic, paper, or mica. 3. Commercially available capaciors are, by he dielecric maerials hey are used of, polyeser capaciors (ligh and sable), film capaciors, and elecrolyic capaciors (high capaciance).. When a volage source (v) is conneced he a capacior, he amoun of charge sored (q) is direcly proporional o he applied volage: q = Cv, where C, he consan, is he capaciance of he capacior. In oher words, capaciance is he raio of he charge on one plae of a capacior o he volage difference beween he wo plaes. 5. The uni of he capaciance is he farad(f), in honor of he English physicis Michael Faraday (79-867). F = Coulomb/Vol. dq dq d( Cv) 6. The equaion q = Cv can now changed, since i =, o i = = = C. 7. Volage-curren relaionship can be obained by inegraing boh sides of i = C : v( ) = i( x) dx + v( 0 ) C 0 8. The energy sored in a capacior, since he insananeous power delivered o he capacior is p = vi = vc, can be: w pd C v d L v Cv ( ) Cv ( ) Cv = τ = τ = = = Imporan properies of a capacior (a) Noe ha, from i = C, when he volage across a capacior is no changing wih ime, he curren hrough he capacior is zero. ----> A capacior is an open circui o DC. (b) Noe ha, from v( ) = i( x) dx + v( 0 ), he volage on he capacior canno change C 0 abruply; insead, he volage mus be coninuous. (c) However, he curren hrough a capacior can change insananeously. (d) An ideal capacior does no dissipae energy. I akes power from he circui when soring (or charging) energy and reurns previously sored energy when delivering (or discharging) power he circui.

2 0. The equivalen capaciance of series-conneced capaciors is he reciprocal of he sum of he n reciprocals of he individual capaciances: = Ceq k= Ck. The equivalen capaciance of parallel-conneced capaciors is he sum of he individual capaciors: C eq = C k n k= 2. Example Problems: The wo series-conneced capaciors are conneced o he erminals of a 2500 black box a =0. The resuling curren i() for >0 is known o be i( ) = 900e [ua] (a) How much energy was iniially sored in he series capaciors? (b) Find v () for >0 (c) Find v 2 () for >0 (d) find v() for >0 (e) How much energy is delivered o he black box in he ime inerval 0<<? 2

3 B. Inducor. An inducor is a passive elemen designed o sore energy in is magneic field. 2. A pracical inducor is usually formed ino a cylindrical coil wih many urns of conducing wires. 3. The volage across an inducor is direcly proporional o he ime rae of change of he curren ) hrough he inducor: v( ) = L, where L is he consan of proporionaliy called he inducance of he inducor, which is he propery whereby an inducor exhibis opposiion o he changes of curren flowing hrough i.. The uni of inducance is he henry (H), named in honor of he American invenor Joseph Henry ( ). henry equals vol-second per ampere. 5. The curren-volage relaionship is: i( ) = v( y) dy + i( 0 ) L 0 di 6. The energy sored in an inducor, since he power delivered o an inducor is p = vi = L i, di can be: w = pdτ = ( L ) idτ = L idi = Li ( ) Li ( ) = Li Imporan properies of an inducor. ) (a) Noe ha, from v( ) = L, he volage across an inducor is zero when he curren is consan. ----> An inducor acs like a shor circui o DC. (b) Noe ha, from i( ) = v( y) dy + i( 0 ), he curren hrough an inducor canno change L 0 insananeously. ) (c) Noe ha, however, from v( ) = L, he volage across an inducor can change abruply. (d) An ideal inducor does no dissipae: he energy sored in i can be rerieved a a laer ime. The inducor akes power from he circui when soring energy and delivers power o he circui when reurning previously sored energy. 8. The equivalen inducance of series-conneced inducors is he sum of he individual inducances: n L eq = L k k= 3

4 9. The equivalen inducance of parallel inducors is he reciprocal of he sum of he reciprocals n of he individual inducances: = L Leq k= k 0. Pracical Problem a. Background (i) ICs are recangular pieces of silicon. (ii) Elecrical conac beween he silicon and meals pins are made wih fine gold wire, called wire bond (Fig. A). (iii) The chip is hen coaed in plasic o proec from physical damage. (Fig. B) (iv) Since wires are no perfec conducors, hey have resisance and inducance. (v) In mos cases, wire resisance and inducance are negligibly small. (vi) However, he curren (being used by he chip (or processor)) changes quickly, wire inducance can play a significan role. b. Analysis (i)we now examine he influence of he small wire inducance o he volage across a high speed microprocessor. (See Fig. C for an equivalen circui wih he wire bond modeled by he 0 nh inducor) (ii) The chip supply volage is represened by 5V volage source. (iii) The curren i() represens he curren being used by he microprocessor. And his curren demand changes, as he microprocessor execues various funcions. An example of curren change is shown in Fig. D.

5 (iv) Then, volage across he chip can be expressed by: ) ) Vchip ( ) = 5 vl = 5 L = di ( ) (v) calculaion and chip volage able for ime periods: di ) i = 0 - ns - 2 ns 2-6 ns 6-20 ns ( 8 = = 2.5 ) L V chip () (vi) The resuling chip volage is illusraed in Fig.E. As we see, he volage swings much and he chip volage is no sable a all. (vii) Then, how can we have a more sable chip volage? (viii) The answer comes from he chip volage equaion. If we reduce he inducance L, hen he sudden volage sho or drop would be reduced. (ix) Le s add one more wire bond beween he chip and he meal pin. (See Fig. F) (x) See Fig. G. for a new equivalen circui. (xi) Then, he curren will be equally divided in o wo inducors. di ( ) (xii) calculaion and chip volage able for ime periods: di ) i = 0 - ns - 2 ns 2-6 ns 6-20 ns ( 8 ) L () =.25 0 = V chip (xiii) The resuling chip volage is illusraed in Fig.H. As we see, he volage swings less and he chip volage is more sable. By adding more wire bonds, we could furher sabilize he chip volage. 8 5

6 . Example Problem: The volage a he erminals of he 300 uh inducor of he circui (a) is shown in (b). The inducor curren i is known o be zero before ime =0. Derive he expression for i (for >0) and skech i. SOLUTION: 6

7 2. Anoher Example Problem: Iniially here was no energy sored in he 25 H inducor when i was placed across he erminals of he volmeer (wih full-scale of 50 V). A =0, he inducor was swiched insananeously o posiion b where i remained for second before reurning insananeously o posiion a. Wha will be he reading of he volmeer be a he insan he swich reurns o posiion a? The d Arsonval movemen has he raing of 50mV@ ma. Noe ha V s =20 [mv]. SOLUTION: 3. Example Problem: Find i() wih volage v()=0 for <0, and v()=20e -0 for >0. Assume ha i(0)=0 SOLUTION: 7

8 C. The Res of he Operaional Amplifier. In he previous chaper, we discussed abou he following op amp circuis: summer and subracor. 2. We will discuss wo more op amp circuis ha had been widely used in analog compuers: inegraor and differeniaor. 3. An inegraor is an op amp circui whose oupu is proporional o he inegral of he inpu signal. (a) Consider a circui below. This is he familiar invering amplifier circui, replacing he feedback resisor by a capacior. vs (b) A node-volage equaion a node : i R + i C = 0, where ir = 0 o and ic = C. Rs vs o (c) Therefore, he curren equaion becomes: = C ----> o = vs R RC (d) Inegraing boh sides gives vo ( ) vo (0) = vs ( ) RC 0 (e) Assuming v o (0)=0 (discharging he capacior prior o he applicaion of he inpu signal), we have vo ( ) = vs ( ). RC 0. A differeniaor is an op amp circui whose oupu is proporional o he rae of change of he inpu signal. (a) Consider anoher circui shown below. s vo (b) Applying KCL a node : i R + i C = 0 ir = 0 s and ic = C. R vo s s ( ) (c) Therefore, we have: = C ----> vo ( ) = RC R (d) Cavea: Differeniaor circuis are elecronically unsable because any elecrical noise wihin he circui is exaggeraed by he differeniaor. Hence, he differeniaor circui is no as useful and popular as he inegraor. I is seldom used in pracice. 8