Chapter 31 – Alternating Current- Phasors and Alternating Currents- Resistance and Reactance- Magnetic-Field Energy- The L-R-C Series Circuit- Power in Alternating-Current Circuits- Resonance in Alternating-Current Circuits- Transformers1. Phasors and Alternating CurrentstVvωcos=Ex. source of ac: coil of wire rotating withconstant ω in a magnetic field sinusoidalalternating emf.tIiωcos=v, i = instantaneous potential difference / current.V, I = maximum potential difference / current voltage/current amplitude.ω = 2πfPhasor Diagrams- Represent sinusoidally varying voltages / currents through the projection of a vector,with length equal to the amplitude, onto a horizontal axis.- Phasor: vector that rotates counterclockwisewith constant ω.Iiravπ2=Rectified average current (Irav): during any whole number of cycles, the total charge that flows is sameas if current were constant (Irav).- Diode (rectifier): device that conducts better inone direction than in the other. If ideal, R = 0 in one direction and R = ∞ in other. full wave rectifier circuitaverage value of Іcos ωtІ orІsin ωtІ)2cos1(5.0cos2ttωω+⋅=()tIIiω2cos5.05.0222+=2)(2Iiiavrms==2VVrms=Root-Mean Square (rms) values:tIiω222cos=2. Resistance and ReactancetVtIRiRvRRωωcoscos)(===IRVR=Resistor in an ac circuit(instantaneouspotential)(amplitude –max- of voltage across R)- Current in phase with voltage phasors rotate togetherInductor in an ac CircuitdtdiL−=ε- Current varies with time self-induced emf di/dt > 0 ε < 0 Va> Vb Vab= Va-Vb= VL= L di/dt > 0 ( )90cossin)cos(+=−===tLItLIvtIdtdLdtdiLvLLωωωωωvLhas 90º “head start” with respect to i.Inductor in an ac circuit)90cos(+= tLIvLωωtIiωcos=)cos(ϕω+=tVvLXLω=φ = phase angle = phase of voltage relative to current VLPure resistor: φ = 0Pure inductor: φ = 90ºLIIXVLLω==Inductive reactance:Voltage amplitude:LVILω=High ω low ILow ω high IInductors used to block high ωtIdtdqiωcos==)90cos(sin−=== tCItCICqvcωωωωCapacitor in an ac circuittIqωωsin=CIVCω=As the capacitor charges and discharges at each t,there is “i” in each plate, and equal displacement currentbetween the plates, as though charge was conducted through C.∫∫=→ tdtIdqωcosC = q / vCPure capacitor: φ = 90ºvclags current by 90º.CXCω1=CCIXV=Capacitor in an ac circuitCapacitive reactance:CVICω=High ω high ILow ω low I(amplitude of voltage across C)Capacitors used to block low ω (or low f) high-pass filterComparing ac circuit elements:- R is independent of ω.- XLand XC depend on ω.- If ω = 0 (dc circuit) Xc= 1/ωC ∞ ic= 0XL= ωL = 0 - If ω ∞, XL ∞ iL= 0XC = 0 VC = 0 current changes direction so rapidly that no charge can build up on each plate.Example: amplifier C in tweeter branch blocks low-f components of soundbut passes high-f; L in woofer branch does the opposite.3. The L-R-C Series CircuitCCLLCIXVIXVIRV===- Instantaneous v across L, C, R = vad= v source- Total voltage phasor = vector sum of phasors ofindividual voltages.- C, R, L in series same current, i = I cosωt only one phasor (I) for three circuit elements, amplitude I.- The projections of I and V phasors onto horizontal axis at t give rise to instantaneousi and v.(amplitudes = maximumvalues)222222)()()()(cLcLcLRXXRIIXIXIRVVVV −+=−+=−+=IZV=-The instantaneous potential difference between terminals a,d == algebraic sum of vR, vC, vL(instantaneous voltages) == sum of projections of phasors VR, VC, VL= projection of their vector sum (V) that represents the source voltage v and instantaneous voltage vadacross series of elements.Impedance:22)]/1([ CLRZωω−+=22)(cLXXRZ −+=Impedance of R-L-C series circuit()RXXIRXXIVVVCLCLRCL−=−=−=ϕtanRCLωωϕ/1tan−=ZIV22=ZIVrmsrms=Phase angle of the source voltage with respect to current()ϕω+=tVv costIiωcos=Example 31.54. Power in Alternating-Current CircuitsVIP21=RVRIIVIVPrmsrmsrmsrmsav2222====VIP21=ttVItVItIttVtItVviPωωϕωϕωϕωϕωωϕωsincossincoscos]cos)][sinsincos(cos[]cos)][cos([2−=−=+==Power in a General Circuitϕϕcoscos21rmsrmsavIVVIP ==5. Resonance in Alternating-Current CircuitsLCCLXXCL11000===ωωω6.
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