WIXLIB

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- Transformers

1. Phasors and Alternating CurrentsEx. source of ac: coil of wire rotating withconstant ω in a magnetic field sinusoidalalternating emf.v V cos ωti I cos ωtv, 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 ahorizontal axis.- Phasor: vector that rotates counterclockwisewith constant ω.

- Diode (rectifier): device that conducts better inone direction than in the other. If ideal, R 0 inone direction and R in other.full wave rectifier circuitRectified average current (Irav): during any wholenumber of cycles, the total charge that flows is sameas if current were constant (Irav).irav 2πIaverage valueof Іcos ωtІ orІsin ωtІ

Root-Mean Square (rms) values:2irms (i ) avI 2Vrmsi 2 I 2 cos 2 ωtcos 2 ωt 0.5 (1 cos 2ωt )i 2 0.5 I 2 0.5 I 2 cos(2ωt )V 2

2. Resistance and ReactanceResistor in an ac circuitvR iR ( IR) cos ωt VR cos ωt (instantaneouspotential)VR IR (amplitude –max- of voltage across R)- Current in phase with voltage phasors rotate together

Inductor in an ac Circuit- Current varies with time self-induced emf di/dt 0 ε 0diε LdtVa Vb Vab Va-Vb VL L di/dt 0didvL L L ( I cos ωt )dtdt(vL IωL sin ωt IωL cos ωt 90 vL has 90º “head start” with respect to i.)

Inductor in an ac circuiti I cos ωtvL IωL cos(ωt 90 )VLv V cos(ωt ϕ )φ phase angle phase of voltage relative to currentPure resistor:φ 0Pure inductor:φ 90ºInductive reactance:Voltage amplitude:X L ωLVL IX L IωLVLI ωLHigh ω low ILow ω high IInductors used to block high ω

Capacitor in an ac circuitAs the capacitor charges and discharges at each t,there is “i” in each plate, and equal displacement currentbetween the plates, as though charge was conductedthrough C.dqi I cos ωtdtq Iω dq I cos ωtdtsin ωtqIIvc sin ωt cos(ωt 90 )ωCC ωCIVC ωCPure capacitor: φ 90ºvc lags current by 90º.C q / vC

Capacitive reactance:VC IX CI VCωCXC 1ωC(amplitude of voltage across C)High ω high ILow ω low ICapacitors used to block low ω (or low f) high-pass filterCapacitor in an ac circuit

Comparing ac circuit elements:- R is independent of ω.- XL and 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 nocharge 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 Circuit- 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 ontohorizontal axis at t give rise to instantaneousi and v.VC IRVL IX LVC IX C(amplitudes maximumvalues)

-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 andinstantaneous voltage vad across series of elements.V VR2 (VL Vc ) 2 ( IR) 2 ( IX L IX c ) 2 I R 2 ( X L X c ) 2Impedance:Z R 2 ( X L X c )2V IZZ R 2 [ωL (1 / ωC )]2Impedance of R-L-C series circuit

V VI (X L X C ) X L X Ctan ϕ L C VRIRRtan ϕ ωL 1 / ωCRi I cos ωtv V cos(ωt ϕ )Vrms I rms ZVI Z22Phase angle of the sourcevoltage with respect tocurrent

Example 31.5

4. Power in Alternating-Current Circuits1P VI2V IVrms2Pav Vrms I rms I rms R R2 21P VI22

Power in a General CircuitP vi [V cos(ωt ϕ )][ I cos ωt ] [V (cos ωt cos ϕ sin ωt sin ϕ )][ I cos ωt ] VI cos ϕ cos 2 ωt VI sin ϕ cos ωt sin ωt1Pav VI cos ϕ Vrms I rms cos ϕ2

5. Resonance in Alternating-Current CircuitsX L XC1ω 0L ω0 Cω0 1LC

6. Transformersε 1 N1dΦ Bdtε 2 N2 ε 1 N1V2 N 2 V1 N1V2R I1 ( N 2 / N1 )ε 2 N2dΦ Bdt

1. Phasors and Alternating Currents v V cos ωt Ex. source of ac: coil of wire rotating with constant ωin a magnetic field sinusoidal alternating emf. i I co