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 Relation Resistance VR IRR R In phase 0 XC 1 dC Leads vC ICE XL dL 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 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 d t vR vC vL EM Oscillations 42 Want to find amplitude I and the phase constant 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 d t 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 d t Length is amplitude Em Projection is E at time t Angle is phase of emf 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 d L Em R d L 1 d C 2 2 1 XC d C Em I Z EM Oscillations 48 Using trig find the phase constant VL VC tan VR Using amplitude relations IX L IX C tan IR XL XC tan R Examine 3 cases XL XC XL XC XL XC EM Oscillations 49 XL XC tan R If XL XC the circuit is more inductive than capacitive is positive Emf is before current ELI If XL XC the circuit is more capacitive than inductive is negative Current is before emf ICE EM Oscillations 50 X L XC tan 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 d L d C d 1 LC This is the same as the natural frequency d 1 LC For RLC circuit resonance and the max current I occurs when d EM Oscillations 52 For small driving frequency d I XL is small but XC is large Circuit capacitive For large driving frequency d XC is small but XL is large Circuit inductive For d circuit is in resonance Em R d L 1 d C 2 2 EM Oscillations 53 Instantaneous rate which energy is dissipated in resistor is 2 P i R But i I sin d t P I R sin d t 2 2 Want average rate Pavg Average over complete cycle T 2 sin 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 Em IZ Z Rewrite average power as Pavg Erms I rms cos 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 d B VP N P dt d 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
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