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Alternating Current Voltage Source : v(t) = V cos t Current Source : i(t) = I cos t Phasors: a graphical method for (combinations) of trigonometric functions I t i(t)=I cos t p212c32: 1 Full Wave Rectifier G Irav = (2/ I i t p212c32: 2 RMS values i2(t) Root-Mean-Square: I rms ((i (t )) 2 ) average i (t ) I cos t i(t) i 2 (t ) I 2 cos 2 t 1 I 2 ( (1 cos2t )) 2 2 1 I average(i 2 (t )) I 2 ( (1 0)) 2 2 I I rms 2 V Vrms 2 Household Power 120V = Vrms V 2Vrms 2120V 170V p212c32: 3 i I cos t v iR IR cos t V cos t V IR v(t)= Vcos(t) I R Current is in phase with Voltage V=I t R i(t)=I cos t v(t)=V cos t p212c32: 4 v EMF 0 v ( L di ) dt I cos t di v L LI sin t dt LI cos( t 90) V cos( t 90) V LI IX L i(t)= Icos(t) L X L Inductive Reactance V=IXL I t Current lags Voltage Voltage leads Current v(t)=V cos (t+90°) i(t)=I cos t p212c32: 5 vq C i I cos dq i dt I q sin t i(t)= Icos(t) q q C I t I v sin t C I cos( t 90) C V cos( t 90) I V IX C C X C Capacitive Reactance t V=IXL i(t)=I cos t v(t)=V cos t Current leads Voltage Voltage lags Current p212c32: 6 Element Resistor Inductor Amplitude V = IR V IX L Capacitor V IX C XC Reactance Phase R in phase X L L v leads i 1 XC i leads v C XL R p212c32: 7 L-R-C Circuit i = I cos(t) i(t)=Icos(t) v(t)=Vcos(t+) = IXLcos(t+) + IRcos(t) + IXCcos(t) L vL(t)=IXLcos(t+) R vR(t)=IRcos(t) C vC(t)=IXCcos(t) p212c32: 8 v(t)=Vcos(t+) = IXLcos(t+) + IRcos(t) + IXCcos(t) Z2 = R2+ X2 = R2+ (XLXC)2 tan() = X/R VL=IXL VC=IXC VL- VC =IX I V=I Z VR=IR VL=IXL V=I Z VC=IXC VL- VC =IX VR=IR I p212c32: 9 V IZ Z R2 X 2 R 2 ( X L X C )2 R 2 (L 1 C ) 2 X tan R X L X C L 1 C R R v V cos(t ) voltage leads (lags) current if X L XC ( X L XC ) p212c32: 10 LRC series circuit example R 300 C .5F V 50V L 60mH f 1591 Hz XL XC X Z I VR VC VC p212c32: 11 Power Instantane ous Power : p iv I cos(t )V cos(t ) 1 1 IV cos( ) cos( 2t ) 2 2 Average Power : 1 P IV cos( ) I rmsVrms cos( ) 2 cos( ) " Power factor" =R Z p212c32: 12 Series Resonance Z XL XC X R log() Z phi I I = V/Z log() p212c32: 13 I o 1 o LC Width of resonance : I I o 2 o Quality factor Q o L R HW: add Q, calculations to all rlc series HW problems p212c32: 14 LRC series circuit example (more) R 300 C .5F L 60mH f 1591 Hz cos( 53) Z 500 I .1A V 50V P o fo = Q p212c32: 15 LRC series circuit example (and more) R 160 C .1F L .6 H o fo = Q p212c32: 16 Parallel L-R-C Circuit iL(t)= ILcos(t-90° ) L iR(t)= IRcos(t) R iC(t)= ICcos(t ) C v(t)= Vcos(t) i = I cos(t+) = IL cos(t-90° ) + IR cos(t) + IC cos(t) = V/XLcos(t-90° ) + V/R cos(t) + V/XC cos(t) p212c32: 17 IL i = I cos(t+) = IL cos(t-90° ) + IR cos(t) + IC cos(t) I2 = IR2+ (ICIL)2 tan() = (ICIL)/IR IC I C- I L V I=V/Z IR IC IL I=V/Z IR I C- I L V p212c32: 18 I 1 V Z 1 2 2 (1 R) (1 X C 1 X L ) Z (1 R) (C 1 L) 2 2 IC I L tan R(C 1 L) IR i I cos(t ) current leads (lags) voltage if X L XC ( X L XC ) p212c32: 19 LRC parallel circuit example R 300 C .5F V 50V L 60mH f 1591 Hz XL XC Z I IR IL IC P p212c32: 20 Parallel “Resonance” Z XL XC R log() Z I log() phi p212c32: 21 V I Z V VC R o 1 LC p212c32: 22 Transformers: Ferromagnetic Materials Strengthen Flux B1 B 2 all field lines go through both areas d B1 V1 N1 dt d B 2 V2 N 2 dt V1 N1 V2 N 2 N1 V1, I1 Primary B N2 V2, I2 Secondary p212c32: 23 Transformers V1 N1 V2 N 2 P1 P2 conservati on of energy I1V1 I 2V2 B N1 V1, I1 Primary N2 V2, I2 Secondary (rms values) V1 N1 I 2 V2 N 2 I1 N2 >N1 => V2 > V1 Step-Up Transformer N2 <N1 => V2 < V1 Step-Down Transformer p212c32: 24 A coffee maker from Europe is designed to operate on a 240-V line (rms) to obtain 960W of power. (a) Determine what characteristics are needed by a transformer so that the proper delivery voltage be obtained from the US standard voltage of 120 V (rms)? (b) What current is drawn at the secondary? (c) What is the resistance of the coffee maker? (d) What current is drawn from the 120 V outlet by the primary? (e) What is the power delivered by the 120 V source? p212c32: 25