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Download Lecture 30 Chapter 33 EM Oscillations and AC
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Lecture 30 Chapter 33 EM Oscillations and AC Review • Characterized ideal LC circuit – Charge, current and voltage vary sinusoidally • Added resistance to LC circuit – Oscillations become damped – Charge, current and voltage still vary sinusoidally but decay exponentially • Added ac generator to circuits with just a – Resistor – Capacitor – Inductor Review Element Resistor Capacito r Inductor Reactance Phase of Phase Amplitude / Current angle φ Relation Resistance VR=IRR R In phase 0° XC= 1/ωdC Leads vC (ICE) XL=ωdL Lags vL (ELI) • ELI the ICE man -90° VC=ICXC +90° VL=ILXL – Voltage or emf (E) before current (I) in an inductor (L) – Current (I) before voltage or emf (E) in capacitor (C) EM Oscillations (41) • RLC circuit – resistor, capacitor and inductor in series • Apply alternating emf E = Em sin ω d t • Elements are in series so i same current is driven through each • From the loop rule, at any time E t, the sum of the voltages across the elements must l th li d f = I sin(ωd t − φ ) = vR + vC + vL EM Oscillations (42) • Want to find amplitude I and the phase constant φ • Using phasors, represent the current at time t – Length is amplitude I – Projection on vertical axis is current i at time t – Angle of rotation is the phase at time t ωd t − φ EM Oscillations (43) • Draw phasors for voltages of R, C and L at same time t • Orient VR, VL, & VC phasors relative to current phasor • Resistor – VR and I are in phase • Inductor – (ELI) VL is ahead of I by 90° • Capacitor – (ICE) I is ahead of VC by 90° • vR, vC, & vL are projections EM Oscillations (44) • Draw phasor for applied emf E = E m sin ω d t • Length is amplitude Em • Projection is E at time t • Angle is phase of emf ω d t • From loop rule the projection E = the algebraic sum of projections vR, vL & vC E = vR + vC + vL EM Oscillations (45) E = vR + vC + vL • Phasors rotate together so equality always holds • Phasor Em = vector sum of voltage phasors r r r r Em = VR + VC + VL • Combine VL & VC to form single phasor V L − VC EM Oscillations (46) • Using Pythagorean theorem 2 2 Em = VR + (VL − VC ) 2 • From amplitude relations replace voltages with VR = IR VL = IX L VC = IX C E = ( IR ) + (IX L − IX C ) 2 m 2 • Rearrange to find amplitude I 2 I= Em R + ( X L − XC ) 2 2 EM Oscillations (47) I= Em R 2 + ( X L − X C )2 • Define impedance, Z to be Z = R 2 + ( X L − X C )2 • Using reactances rewrite current as I= X L = ωd L Em R + (ω d L − 1 / ω d C ) 2 2 1 XC = ωd C Em I= Z EM Oscillations (48) • Using trig find the phase constant φ VL − VC tan φ = VR • Using amplitude relations IX L − IX C tan φ = IR XL − XC tan φ = R • Examine 3 cases: • XL > XC • XL < XC • XL = XC EM Oscillations (49) XL − XC tan φ = R • If XL > XC the circuit is more inductive than capacitive – φ is positive – Emf is before current (ELI) • If XL < XC the circuit is more capacitive than inductive – φ is negative – Current is before emf (ICE) EM Oscillations (50) X L − XC tan φ = R • If XL = XC the circuit is in resonance – emf and current are in phase • Current amplitude I is max when impedance, Z is min X L − XC = 0 Em = I= Z Em Z=R Em = 2 2 R R + (X L − XC ) EM Oscillations (51) • When XL = XC the driving frequency is 1 ωd L = ωd C ωd = 1 LC • This is the same as the natural frequency, ω ωd = ω = 1 LC • For RLC circuit, resonance and the max current I occurs when ωd = ω EM Oscillations (52) • For small driving frequency, ωd < ω I= – XL is small but XC is large – Circuit capacitive • For large driving frequency, ωd > ω – XC is small but XL is large – Circuit inductive • For ωd = ω, circuit is in resonance Em R + (ω d L − 1 / ω d C ) 2 2 EM Oscillations (53) • Instantaneous rate which energy is dissipated in resistor is 2 P=i R • But i = I sin(ωd t − φ ) P = I R sin (ωd t − φ ) 2 2 • Want average rate, Pavg – Average over complete cycle T 2 sin θ = 1 2 EM Oscillations (53) • For alternating current circuits define rootmean-square or rms values for i, V and emf V E I V rms = E rms = I rms = 2 2 2 • Ammeters, voltmeters - give rms values • Write average power dissipated by resistor in an ac circuit is Pavg 2 I R I = = R 2 2 2 Pavg = I 2 rms R EM Oscillations (54) • Write average power in another form using I rms Erms = Z Erms R Pavg = I R = I rms R = Erms I rms Z Z 2 rms • Using phasor and amplitude relations VR IR R cos φ = = = Em IZ Z • Rewrite average power as Pavg = Erms I rms cosφ EM Oscillations (55) • If ac circuit has only resistive load R/Z = 1 Pavg = E rms I rms = I rmsVrms • Trade-off between current and voltage – For general use want low voltage – Means high current but 2 Pavg = I rms R • General energy transmission rule: Transmit at the highest possible voltage and the lowest possible current EM Oscillations (56) • Transformer – device used to raise (for transmission) and lower (for use) the ac voltage in a circuit, keeping iV constant – Has 2 coils (primary and secondary) wound on same iron core with different #s of turns EM Oscillations (57) • Alternating primary current induces alternating magnetic flux in iron core • Same core in both coils so induced flux also goes through the secondary coil • Using Faraday’s law dΦ B VP = − N P dt dΦ B VS = − N S dt VP VS = NP NS EM Oscillations (58) • Transformation of voltage is • If NS > NP called a step-up transformer step• If NS < NP called a down transformer • Conservation of energy I PVP = I SVS VP NP IS = IP = IP VS NS NS VS = VP NP EM Oscillations (59) • The current IP appears in primary circuit due to R in secondary circuit. I PVP = I SVS I S = VS / R 2 VS VS 1 V 1 NS V P = IP = V P = R VP RV R NP 2 S 2 P • Has for of IP = VP/Req where 2 R eq NP R = NS