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Lecture 20-1 Alternating Current (AC) = Electric current that changes direction periodically ac generator is a device which creates an ac emf/current. A sinusoidally oscillating EMF is induced in a loop of wire that rotates in a uniform magnetic field. B NBA cos NBA cos t dB NBA sin t dt where 2 2 f T ac motor = ac generator run in reverse http://www.wvic.com/how-gen-works.htm http://www.pbs.org/wgbh/amex/edison/sfeature/acdc.html Lecture 20-2 Root-Mean-Square Values I I 2 I rms rms 1 2 sin t I peak 2 2 peak 2 I rms 1.41 I rms 1 2 2 Pav I peak R I rm sR 2 1 1 peak I peak R I rms R 2 2 2 2 1 I I peak 0.707 I peak 2 I Similarly, 2 peak Pav rms I rms Lecture 20-3 Capacitive vs Inductive Load I(t) leads v(t) by 90 capacitive reactance 1 XC C -- + Pav 0 VC , peak X C I peak vL vL(t) leads I(t) by 90 inductive reactance X L L VL , peak X L I peak + -- Lecture 20-4 (Ideal) LC Circuit • From Kirchhoff’s Loop Rule Q dI L 0 C dt dQ I dt d 2Q 1 Q 0 2 dt LC harmonic oscillator with angular 1 frequency 0 Q Q peak cos(0t ) dQ I 0Q peak sin( 0t ) dt LC Natural Frequency Lecture 20-5 Mechanical Analogy d 2x k k x 0 0 dt 2 m m No friction = No dissipation f 0 0 / 2 U 0, K Kmax U U max , K 0 1 2 1 2 dx U kx , K mv , v 2 2 dt U U max , K 0 E const. U K harmonic oscillator with k 0 m Lecture 20-6 LC Oscillations Q2 1 2 dQ UE , U B LI , I 2C 2 dt No Resistance = No dissipation Lecture 20-7 More on LC Oscillations Charge and current: Q Q peak cos 0t (with =0) I dQ 0Q peak sin 0t dt x 0, r1 .. r1 n Energy stored in capacitor: 1 1 2 U E (t ) Q peak cos2 0t 2C UE f( x ) 0.5 Energy stored in inductor: 1 2 2 U B (t ) L0 Q peak sin 2 0t 2 1 where 0 LC 1 2 U B (t ) Q peak sin 2 0t 2C so U E (t ) U B (t ) 2 Q peak 2C 00 0 2 tx 4 6 Period is half that of Q(t) x 0, r1 .. r1 n UB 1 f( x ) 0.5 00 0 2 t x 4 6 Lecture 20-8 Non-scored Test Quiz A LC circuit has inductance L and capacitance C, what’s the natural frequency? A. C L B. LC C. 1 LC D. L C Lecture 20-9 Series RLC Circuits The resistance R may be a separate component in the circuit, or the resistance inherent in the inductor (or other parts of the circuit) may be represented by R. Finite R Energy dissipation damped oscillation dI Q L IR 0 dt C d 2Q dQ 1 L 2 R Q0 dt dt C only if R is “small” d 2x dx m 2 b kx 0 dt dt multiply by I d 1 2 2 d Q2 LI I R 0 dt 2 dt 2C For large R Lecture 20-10 Driven Series RLC Circuit Kirchhoff’s Loop Rule: dI Q L IR 0 dt C Vapp , peak sin t d 2x dx m 2 b kx F0 sin t dt dt common current I I I peak sin(t ) must be determined Lecture 20-11 Voltage and Current in Driven Series RLC Circuit VR I peak R vR (t ) I (t ) R VR sin t dI vL (t ) L VL sin(t ) dt 2 q(t ) vC (t ) VC sin( t ) C 2 VL I peak L I peak X L I peak VC I peak X C C (t ) vL (t ) vR (t ) vC (t ) peak sin(t ) peak I peak R X L X C 2 Z 2 Phasors Lecture 20-12 Impedance in Driven Series RLC Circuit peak I peak R X L X C 2 2 Z 1 Z R L C 2 2 impedance, 1 L VL VC R C tan , cos VR R Z Lecture 20-13 Resonance For given peak, R, L, and C, the current amplitude Ipeak will be at the maximum when the impedance Z is at the minimum. peak I peak R X L X C 2 2 Z X L XC res L 1 resC i.e., load purely resistive This is called resonance. Resonance angular frequency: 1 res LC , Z R, and I peak peak R ε and I in phase Lecture 20-14 Resonance (continued) angular frequency 1 res LC (radians/s): frequency (Hz): f res 1 2 LC Phase difference between ε and I: X L XC R tan 0, cos R Z Power dissipated: • In a steady, driven RLC circuit, power dissipated = power supplied by ac source. • This power is dissipated only in R. • At resonance, this power is maximum. Lecture 20-15 Power Delivered 1 2 1 peak 1 Pav I peak R I R peak 2 2 Z 2 1 1 R peak I peak 2 2 Z rms I rms cos Pav rms I rms R R 2 rms 2 Z Z 2 rms R 2 1 2 L R C 2 rms R 2 2 2 2 2 L ( res ) 2 R2 Power factor R cos Z Lecture 20-16 Lecture 20-17 Transformer • AC voltage can be stepped up or down by using a transformer. • AC current in the primary coil creates a time-varying magnetic flux through the secondary coil via the iron core. This induces EMF in the secondary circuit. Ideal transformer (no losses and magnetic flux per turn is the same on primary and secondary). (With no load) d B V1 V2 turn dt N1 N 2 N1 N 2 V1 V2 N1 N 2 V1 V2 step-up step-down With resistive load R in secondary, current I2 flows in secondary by the induced EMF. This then induces opposing EMF back in the primary. The latter EMF must somehow be exactly cancelled because is a defined voltage source. This occurs by another current I1 which is induced on the primary side due to I2. Lecture 20-18 Transformer with a Load With switch S closed: conservation of energy I1V1 I 2V2 Imag+I1 proportional to average power I1 V2 N I2 2 I2 V1 N1 2 N 2 V2 1 N 2 V1 V 1 2 N1 R R N1 N1 / N 2 R equivalent resistance Req The generator “sees” a resistance of Req Impedance Matching: Maximum energy transfer occurs when impedance within the EMF source equals that of the load. Transformer can vary the “effective” impedance of the load. I2 S Lecture 20-19 Physics 241 –Quiz 17b – March 25, 2008 An LC circuit has a natural frequency of 141 MHz. If you want to decrease the natural frequency to 100 MHz, which of the following will accomplish that? a) Double L b) Double both L and C c) Halve L d) Halve both L and C e) Double L and halve C Lecture 20-20 Physics 241 –Quiz 17c – March 25, 2008 An LC circuit has a natural frequency of 100 MHz. If you want to decrease the natural frequency to 71 MHz, which of the following will accomplish that? a) Double C b) Double both L and C c) Halve C d) Halve both L and C e) Double L and halve C