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Review z Characterized ideal LC circuit z z Added resistance to LC circuit z z z Charge, current and voltage vary sinusoidally Oscillations become damped Charge, current and voltage still vary sinusoidally but decay exponentially Added ac generator to circuits with just a z z z Resistor Capacitor Inductor Review Element Resistor Reactance/ Phase of Phase Amplitude Resistance Current angle φ Relation R In phase 0° VR=IRR Capacitor XC= 1/ωdC Leads vC Inductor XL=ωdL Lags vL -90° VC=ICXC +90° VL=ILXL EM Oscillations z z RLC circuit – resistor, capacitor and inductor in series Apply alternating emf E = E m sin ωd t z z Elements are in series so same current is driven through each i = I sin ωd t − φ From the loop rule, at any time t, the sum of the voltages across the elements must E = vR + vC + vL equal the applied emf ( ) EM Oscillations z Want to find amplitude I and the phase constant φ z Using phasors, represent the current at time t z z z 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 z z z z z z 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 – VL is ahead of I by 90° Capacitor – I is ahead of VC by 90° vR, vC, & vL are projections EM Oscillations z Draw phasor for applied emf E = E m sin ωd t z z z z 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 E = VR + VC + VL z z Phasors rotate together so equality always holds Phasor Em = vector sum of voltage phasors r r r r Em = VR + VC + VL z Combine VL & VC to form single phasor V L − VC EM Oscillations z Using Pythagorean theorem E = V + (VL − VC ) 2 m z 2 R 2 From amplitude relations replace voltages with VR = IR VL = IX L VC = IX C E = ( IR) + (IX L − IX C ) 2 m z 2 Rearrange to find amplitude I I= 2 Em R + (X L − XC ) 2 2 EM Oscillations I= z Em R 2 + ( X L − X C )2 Define impedance, Z to be Z = R 2 + ( X L − X C )2 z Using reactances rewrite current as I= X L = ωd L Em R + (ωd L − 1 / ωd C ) 2 2 1 XC = ωd C Em I= Z EM Oscillations z Using trig find the phase constant φ VL − VC tan φ = VR z Using amplitude relations IX L − IX C tan φ = IR X L − XC tan φ = R z Examine 3 cases: z z z XL > XC XL < XC XL = XC EM Oscillations X L − XC tan φ = R z z If XL > XC the circuit is more inductive than capacitive z φ is positive z Emf is before current If XL < XC the circuit is more capacitive than inductive z φ is negative z Current is before emf EM Oscillations X L − XC tan φ = R z z 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 z When XL = XC the driving frequency is 1 ωd L = ωd C z This is the same as the natural frequency, ω ωd = ω = z ωd = 1 LC 1 LC For RLC circuit, resonance and the max current I occurs when ωd = ω EM Oscillations z z z For small driving frequency, ωd < ω I= z XL is small but XC is large z Circuit capacitive For large driving frequency, ωd > ω z XC is small but XL is large z Circuit inductive For ωd = ω, circuit is in resonance Em R + (ωd L − 1 / ωd C ) 2 2