PHY305F Electronics Laboratory I. Section 2. AC Circuit Basics: Passive and Linear Components and Circuits. Basic Concepts

Transcription

1 PHY305F Elecrnics abrary I Secin ircui Basics: Passie and inear mpnens and ircuis Basic nceps lernaing curren () circui analysis deals wih (sinusidally) ime-arying curren and lage signals whse ime aerage alues are zer paricularly impran because much f ur elecrical pwer is in he frm f lages and currens Energy-srage cmpnens capacirs inducrs

2 Ideal apacirs - capacir is a deice ha can sre energy in he frm f a charge separain when apprpriaely plarized by an elecric field (i.e., a lage). Simples capacir = w parallel cnducing plaes f crsssecinal area separaed by air r anher dielecric maerial. Ne: dielecric maerial is n an elecrical cnducr, bu cnains a large number f elecric diples which becme plarized in he presence f an elecric field. Parallel-plae capacir wih air gap d (air is he dielecric) (frm Rizzni Figure 4.) _ d _ ircui symbl = ε d ε = permiiiy f air = x 0 - F/m Ideal apacirs - apacirs ac as pen circuis fr D currens because he insulaing dielecric will n allw he curren flw. If he lage a he capacir erminals changes wih ime, s will he charge accumulaed a he w capacir plaes. harge separain caused by he plarizain f he dielecric is prprinal he applied elecric field and hence he lage Q = r q() = () is he capaciance f he elemen and is a measure f he abiliy f he capacir sre charge. The SI uni f capaciance is he farad (F): F = /. Mre cmmn: micrfarads ( µf = 0-6 F) r picfarads ( pf = 0 - F) T increase capaciance, real capacirs are fen made f ighly rlled shees f meal film wih a dielecric sandwiched beween.

3 Ideal apacirs - 3 Fr applied lage (), he change wih ime f he sred charge is analgus a curren. Differeniae equain fr q() ge he defining circui law fr a capacir (defines he i- relainship): dq() d() i () = = d d Inegrae ge he lage acrss a capacir: () = where i (' = ) d' = ( ) = i i ) d' Iniial lage jus indicaes ha sme charge is sred in he capacir a ime. (' (' ) d' fr mbining apacirs apacirs cnneced in series cmbine in he same way as resisrs cnneced in parallel. 3 EQ = 3 apacirs cnneced in parallel add. 3 (frm Rizzni Figure 4.) EQ = 3 3

4 Energy Srage in apacirs The energy sred in a capacir can be readily deried: W () = = = W () = P d' (') i (') d' d (' ) (') d' d' [ ()] Ideal Inducrs - n inducr is a deice ha can sre energy in a magneic field. Usually made by winding a cil f wire arund an insulaing cre. s curren flws hrugh he cil, a magneic field is se up. In an ideal inducr, he resisance f he wire is zer, s a cnsan curren hrugh he inducr flws wihu causing a lage drp. Thus, an ideal inducr acs as a shr circui fr D currens. Magneic flux lines Irn cre inducr i ( ) (frm Rizzni Figure 4.8) ircui symbl ( ) = _ di d 4

5 Ideal Inducrs - pplicain f a ime-dependen lage acrss he inducr will generae a curren, defined by: di() () = d is he inducance f he cil. The SI uni f inducance is he henry (H): H = -s/. Ne he dualiy wih he i- relainship fr capacirs: he rles f i and are reersed beween he w Inegrae ge he curren acrss an inducr: i() = where I (' ) d' = = i( ) = (' ) d' (') d' I fr i () = d() d mbining Inducrs Inducrs cnneced in parallel cmbine in he same way as resisrs cnneced in parallel. Inducrs cnneced in series add. EQ = 3 3 EQ= 3 3 (frm Rizzni Figure 4.9) 5

6 Energy Srage in Inducrs The energy sred in an inducr can be deried by firs deermining he insananeus pwer in he inducr: di() P () = i() () = i() = d Hence he al sred energy is: W () = = W () = P (') d' d d' i [ (')] [ i ()] d' d d i [ (' )] Time-Dependen Signal Surces nenin fr indicaing ime-dependen signal surces: () _ i () ( ), i( ) _ Generalized ime-dependen surces Sinusidal surce (frm Rizzni Figure 4.8) 6

7 Peridic Signal Waefrms Peridic ime-dependen signals appear frequenly in pracical applicains and are useful apprximains f many phenmena. peridic signal saisfies he equain: x () = x( nt) n =,, 3 where T = perid f x(). Signal r waefrm generars can pride peridic lages (r currens) f ariable perid and ampliude T T 3 T 4 T Time Sawh wae 0 T Square wae T Triangle wae T T T T 3 T Pulse rain 0 _ T Sine wae (frm Rizzni Figure 4.9) T Time Time Time Time Sinusidal Waefrms Sinusidal waefrms are he ms impran ype f imedependen signals. General equain: x() = cs( ω where = ampliude, ω = radian frequency (=πf), φ = phase. () x = cs( ω) _ T Reference csine x () = cs( ω φ = π (radians) T _ rbirary sinusid T (frm Rizzni Figure 4.0) 7

8 erage and RMS alues Tw ways quanify srengh f a ime-arying elecrical signal: () Time-aeraged (r D) alue Measuremen f mean lage r curren er sme ime perid. T Defined as x() = x(' ) d' where T=perid f inegrain T 0 erage alue f a sinusidal signal is always zer, bu aerage pwer is n! Need anher way quanify signal srengh. () R-mean-square (r RMS) alue ccuns fr flucuains f he signal abu is aerage alue. Defined as T xrms = [ x(' )] d' T 0 square r f he aerage (mean) f he square f he signal RMS f a sinusidal signal is always imes is peak alue. ircuis wih Dynamic Elemens nsider his circui cnaining a capacir. R pply K: S () = R() () i R s() i Hence: S () = Ri() i (' )d' Ne: i Inegrae: R = i d di Ne ha d S i (')d' i = i () = d d R R d S d We culd als apply K: ir = = i = R d d = S d R R Sluin f eiher f hese differenial equains will deermine all lages and currens in he circui. R 8

9 Sinusidal Frcing f ircuis - nsider he same circui wih a sinusidal R lage surce: S () = cs( ω) i R Subsiue his in he preius diff l equain: s() i d Ne: i R = i = csω d R R The frcing funcin is sinusidal, s assume he sluin is : () = sinω Bcsω = cs( ω Subsiuing his in he equain fr c (), gruping cefficiens f like erms, and ning ha he cefficiens f cs & sin mus = 0: ωr s () = () () ω (R) c().67 5 B = ω (R) 3.33 ime (ms) R Sinusidal Frcing f ircuis - This sluin mehd wrks fr simpler circuis bu can require sluin f higher-rder differenial equains if many circui elemens are presen, paricularly capacirs and inducrs. Hweer, i leads he fllwing summary f he key pins in circui analysis. In a sinusidally frced linear circui, all branch lages and currens are sinusids a he same frequency as he frcing signal. The ampliudes f hese lages and currens are a scaled ersin f he exciain ampliude The lages and currens may be shifed in phase wih respec he frcing signal. 9

10 I is pssible represen sinusidal funcins using nly he frequency, ampliude, and phase, wih help f cmplex numbers. llws cmplex algebra replace sluin f differenial equains. Euler s Ideniy is he basis fr his nain: I defines he cmplex expnenial e jθ as a pin in he cmplex plane ha can be represened by real and imaginary cmpnens: θ e j = csθ jsinθ his ideniy is jus a rig. relain in he cmplex plane (equaes plar and recangular frms f a cmplex number) jθ ecr f magniude e = cs θ sin θ = Euler s Ideniy Im j sin _ cs Re _ j e j = cs j sin Phasrs pply Euler s Ideniy a generalized sinusid: j( ωφ) jω jφ cs( ω = Re e = Re e e This fllws frm: j( ωφ) Re e = Re cs( ω j sin( ω S we can express a generalized sinusid as he real par f a cmplex number whse argumen r angle is gien by (ωφ) and whse lengh r magniude is he peak ampliude f he sinusid. The cmplex phasr ha crrespnds sinusidal signal cs( ω is defined as he cmplex number e jφ : φ e j = cmplex phasr nain fr cs( ω = φ This definiin is a simplificain, as i remes he perar Re and he erm e jω frm he full expressin abe. [ ] [ ] [ ] [ ] = cs( ω 0

11 Phasr Mehdlgy () ny sinusidal signal may be mahemaically represened in ne f w ways: a ime-dmain frm: () = cs( ω jφ a frequency-dmain r phasr frm: (jω) = e = φ Ne he jω in he nain (jω), which indicaes he e jω dependence f he phasr. Bld uppercase indicaes phasr lages r currens. () phasr is a cmplex number, expressed in plar frm, cnsising f a magniude equal he peak ampliude f he sinusidal signal and a phase angle equal he phase shif f he sinusidal signal referenced a csine signal. (3) When using phasr nain, i is impran ne he frequency ω f he sinusidal signal, as his is n explici in he phasr eqn. Superpsiin f Signals - Example: superpsiin f w sinusidal surces f differen phase and ampliude bu f he same frequency. π Gien: () = cs( ω φ ) = 5cs(377 ) () = Wrie he lages in phasr frm: jφ (jω) = e = φ = 5 (jω) = cs( ω φ = φ ner recangular frm: dd: (jω) = 5.0 j4.49 = 8.98e S e jφ () = 8.98 cs(377 S π 6 ) ) = 5 cs(377 π 4 = 5 π 4 π (jω) = 0.6 j0.6 (jω) = 4.49 j3.88 jπ / 6 = 8.98 ) π 6 _ () () s ()

12 Superpsiin f Signals - The preius apprach cann be used in he case f w sinusidal signals ha are n a he same frequency. nsider a lad frced by w curren surces in parallel, wih i() = cs( ω) i() = cs( ω) ad curren: i () i () lad i() = i() i() r I = I I Here we cann use addiin f phasrs because he erm e jω is implicily presen. This is clear frm he full j0 j0 expresssins fr he phasr currens I = e,i = e j0 jω [ ] [ ] j0 j ω I = Re e e, I = Re e e The equain fr i () is he nly unambiguus eqn. fr lad curren. T analyse a circui wih muliple sinusidal surces a differen frequencies using phasrs, he circui mus be sled separaely fr each signal. Then he resuls fr each surce are added. Impedance Impedance is a parameer defined fr resisrs, capacirs, and inducrs ha can be regarded as a cmplex resisance. I allws herems fr D circuis be exended circuis. The cncep f impedance is equialen saing ha capacirs and inducrs ac as frequency-dependen resisrs, i.e., as resisrs whse resisance is a funcin f he frequency f he sinusidal exciain. Generalizing Ohm s aw circuis gies: ( ω,) = Z( ω)i( ω,) where Z is he impedance (uni = Ohms). S ( ) S ( ) S ( ) S ( j ) ~ ~ ~ ~ i ( ) i ( ) i ( ) circuis ( j ) circuis in phasr/impedance frm R Z is he impedance f each circui elemen (frm Rizzni Figure 4.33)