Lecture #4 Capacitors and Inductors Energy Stored in C and L Equivalent Circuits Thevenin Norton

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1 EES ntro. electroncs for S Sprng 003 Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 EES ntroducton to Electroncs for omputer Scence Andrew R. Neureuther Lecture # apactors and nductors Energy Stored n and L Equvalent rcuts Thevenn Norton opyrght 00, Regents of Unversty of alforna EES ntro. electroncs for S Sprng 003 Lecture : 0/03/03 A.R. Neureuther Game Plan 0/03/03 opyrght 00, Regents of Unversty of alforna Verson Date 0/0/03 Monday 0/03/03 apactors and nductors; Equvalent Sources Schwarz and Oldham:.., 3. Wednesday 0/0/03 NL Elements; Graphcal Solutons; Power Schwarz and Oldham: 3.3. Next (th) Week R Transent Schwarz and Oldham: 8. plus Handouts Problem Set # Out /7/03 Due //03 :30 n box near 7 ory. Flow;. KL;.3 KVL;. resstor crcut;. Power Problem Set #3 Out //03 Due //03 :30 n box near 7 ory 3. and 3. chargng capactors; ; Equvalent rcuts;

2 EES ntro. electroncs for S Sprng 003 Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 What Goes n the rcut Element Boxes? va v a v v b v 3 v c v b v c ref. node opyrght 00, Regents of Unversty of alforna EES ntro. electroncs for S Sprng 003 Lecture : 0/03/03 A.R. Neureuther BAS RUT ELEMENTS Verson Date 0/0/03 Voltage Source urrent Source Resstor Wre apactor nductor (always supples some constant gven voltage lke deal battery) (always supples some constant gven current) (Ohm s law) ( short no voltage drop ) (capactor law based on energy storage n electrc feld of a delectrc S&O.) (nductor law based on energy storage n magnetc feld n space S&O.) opyrght 00, Regents of Unversty of alforna

3 EES ntro. electroncs for S Sprng 003 APATOR opyrght 00, Regents of Unversty of alforna Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 Any two conductors a and b separated by an nsulator wth a dfference n voltage V ab wll have an ual and opposte charge on ther surfaces whose value s gven by Q = V ab, where s the capactance of the structure, and the charge s on the more postve electrode. A smple parallelplate capactor s shown. f the area of the plate s A, the separaton d, and the delectrc constant of the nsulator s ε, the capactance uals = A ε/d. Symbol or A onsttutve relatonshp: Q = (V a V b ). (Q s postve on plate a f V a >V b ) dq But a dv = so = uvalent to Q = v where we use the assocated reference drectons. d a b V ab EES ntro. electroncs for S Sprng 003 Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 ENERGY STORED N A APATOR You mght thnk the energy (n Joules) s QV, whch has the dmenson of joules. But durng chargng the average voltage was only half the fnal value of V. QV = V Thus, energy s. opyrght 00, Regents of Unversty of alforna 3

4 EES ntro. electroncs for S Sprng 003 Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 ENERGY STORED N A APATOR (cont.) More rgorous dervaton: Durng chargng, the power flow s v nto the capactor, where s nto termnal. We ntegrate the power from t = 0 (v = 0) to t = end (). The ntegrated power s the energy t = t E = t = t Fnal ntal v = v Fnal ntal dq Fnal = v dq ntal but dq = dv. (We are usng small q nstead of Q to remnd us that t s tme varyng. Most texts use Q.) v Fnal E = v dv = ntal V Fnal V ntal opyrght 00, Regents of Unversty of alforna EES ntro. electroncs for S Sprng 003 Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 APATORS N PARALLEL (t) V Add urrents (t) = Equvalent capactance defned by = (t) V(t) learly, APATORS N PARALLEL opyrght 00, Regents of Unversty of alforna

5 (t) EES ntro. electroncs for S Sprng 003 V V APATORS N SERES Equvalent to Add Voltages Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 (t) V = = So =, = learly, = Equvalent capactance defned by = V V and = =, d(v V ) V = so = ( APATORS N SERES ) opyrght 00, Regents of Unversty of alforna EES ntro. electroncs for S Sprng 003 Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 apactance and nductance apactors: two plate example; Store energy n the electrc feld Q = V, = / and V = (/) ntegral of voltage omputer example ma current chargng pf V(t) = (/)t = (03 A/0 F) t = 0 9 V/s t At D.. tme dervatves are zero => s open crcut n parallel add; seres / = sum (/ ); short together (nfnte current but conserve charge) nductors: col example; Store energy n the magnetc feld; Flux = L, V = L d/ and = (/L) (ntegral of voltage) At D.. tme dervatves are zero => L s short crcut L n parallel /L = sum (/L ); seres add; connect n seres when have dfferent currents => L L = (L L ) NEW opyrght 00, Regents of Unversty of alforna

6 EES ntro. electroncs for S Sprng 003 Resstors n Seres Example of V Graphs Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 f two resstors are n seres the current s the same; clearly the total voltage wll be the sum of the two R values.e. (R R ). Thus the uvalent resstance s R R and the V graph of the seres par s the same as that of the uvalent resstance. Of course we can also fnd the V graph of the combnaton by addng the voltages drectly on the V axes. Lets do an example for K K resstors K K V V V (ma) K ombned K K K V (Volt) opyrght 00, Regents of Unversty of alforna EES ntro. electroncs for S Sprng 003 Example of V Graphs Smple rcut, e.g. voltage source resstor. Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 f two crcut elements are n seres the current s the same; clearly the total voltage wll be the sum of the voltages.e. V S R. We can graph ths on the V plane. We fnd the V graph of the combnaton by addng the voltages V S and R at each current. Lets do an example for =V, R=K VV S K V (ma) V K ombned K V V (Volt) opyrght 00, Regents of Unversty of alforna 6

7 EES ntro. electroncs for S Sprng 003 SS R R A B R 3 Smplest Equvalent rcuts Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 Thevenn Norton An aduately uvalent crcut s one that has an vs. V graph that s dentcal to that of the orgnal crcut. N V OUT (ma) R Th V Th V OUT N RN ombned rcut V OUT V TH = V O N = S R TH = R N = ( S /V O ) S VO V (Volt) opyrght 00, Regents of Unversty of alforna EES ntro. electroncs for S Sprng 003 opyrght 00, Regents of Unversty of alforna Lecture : 0/03/03 A.R. Neureuther Verson Date 0/0/03 vs. V and Equvalent rcuts vs. V for deal voltage source s a vertcal lne at V = V SV vs. V for deal current source s a horzontal lne at = S vs. V for a crcut made up of deal ndependent sources and resstors s a straght lne. The smplest crcut for a straght lne s an deal voltage source and a resstor (Thevenn) or a current source and a parallel resstor (Norton) The easest way to fnd the vs. V lne s to fnd the ntercepts where = 0 (open crcut voltage V T ) and where V = 0 (Short crcut current N ) The shortcut for fndng the (slope) = R T = R N s to turn off all of the dependent sources to zero and fnd the remanng uvalent resstance between the termnals of the elements. 7