Prozorov_22 - PHYSICS 222 Introduction to Classical Physics...

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Unformatted text preview: PHYSICS 222 Introduction to Classical Physics II Prof. Ruslan Prozorov Iowa State University Fall 2011 LECTURE 22 Alternating current. Driven RLC circuit. a driven RLC circuit Analogy between circuits and simple harmonic motion: Circuits Mechanical Oscillations. No energy dissipation. LC Spring/mass (SHM) Oscillations with decreasing amplitude. Energy dissipation. RLC Spring/mass with damping Driven oscillations. Energy dissipation and supply RLC with AC generator Spring/mass with damping and driving force PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 2 Preview – Source: ~ produces an oscillating voltage (supplies whatever current the rest of the circuit “requires”) – Resistor: causes a voltage drop when a current flows through. As soon as the voltage changes, so does the current always in phase. – Capacitor: resists change in charge Q resists change in voltage. Voltage across capacitor lags behind the current by 90°. – Inductor: resists change in magnetic flux resists change in current. Voltage across inductor is ahead of the current by 90˚. what does this mean? PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 3 an AC generator Generic problem: we are given an AC voltage source that “drives” a circuit t E cos t ~ Supplies a given emf that depends on time (usually as a sine of cosine) Goal: Determine current that flows: i I cos t • Current must be the same through all elements if they are in series. • It is reasonable to expect another sinusoidal function with frequency (and possibly a phase ). Notations: upper case for amplitude, lower case for t-dependent quantity. t PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 4 alternating current (AC) I I cos t PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 5 driven resistor L E cos t Power source: i I cos t Assume solution is C R Voltage across resistor: vR Ri vR RI cos t E cos t , 0 x 0, r1 .. r1 x n 0, r1 .. r1 I1 n IR 1 vR i 0 f( x ) 0 IR 1 Voltage across resistor is in phase with current. 0 f( x ) 0 0 0 I 2 tx 4 6 1 0 0 PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 2 tx 4 6 14 October 2011 6 driven capacitor L vC C R q C sin cos 2 dq i I cos t dt q I cos t dt I I sin t cos t 2 Voltage across capacitor lags current by v I cos t X I cos t C C a quarter of a cycle (90°). C 2 2 r1 r1 x 0, x .. r1 n I 1 C I1 vC XC i 0 I 0 C 10 .. r1 n 0 f( x ) 0 f( x ) 0 0, 2 t x 4 6 I1 1 C Capacitive reactance 0 0 2 PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University tx 4 6 14 October 2011 7 driven inductor L di dt i I cos t vL L C R sin cos 2 vL L di LI sin t LI cos t dt 2 Voltage across inductor leads current vL XLI cos t VL cos t by a quarter of a cycle (90°). 2 2 r1 x 6 4 2 0 1 LI x 0, .. r1 n I1 vL 0 ) x (f 0 0 f( x ) 0 I1 1 LI 0 XL L i t Inductive reactance 0 0 2 tx 1r .. n 1r PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 4 6 14 October 2011 8 ,0 x comparing AC circuit elements PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 9 phasor PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 10 PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 11 phasors math E cos t i I cos t vR VR cos t VR RI vC VC cos t 2 VR X C I vL VL cos t 2 VL X L I We can make it easier if we think that these are components of vectors called phasors. XC 1 C X L L Magnitude of the phasors: VL VR VR RI VC XC I VL X LI I t The entire thing rotates CCW. VC Real voltages = horizontal projections of the phasors. PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 12 impedance L Kirchhoff’s loop rule: vR vC vL As vectors (phasors): vR vC vL C R and then take projections. VL VL−VC E VR VC E I V VC L VR RI VC XC I VL X LI t Z X L XC 2 VR2 E I 2 X L XC 2 R2 R2 Impedance PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 13 phase angle VL XL E I VL−VC XL−XC VR t VC tan But also: tan Z VL VC VR XC XL XC R sin XL XC Z R cos R Z Etc. ALWAYS DRAW THE DIAGRAM!!! PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 14 resistors in an AC circuit PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 15 inductors in an AC circuit PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 16 capacitance in an AC circuit PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 17 reactance is frequency-dependent resistance of inductors and capacitors XC 1 C For high , XC ~ 0 Capacitor looks like a wire (“short”) For low , XC ∞ Capacitor looks like a break XL L For low , XL ~ 0 Inductor looks like a wire (“short”) For high , XL ∞ Inductor looks like a break " XR " R No frequency dependence. PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 18 The loudspeaker, a useful application PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 19 equal reactance XC X L 1 LC 1 L C - the natural frequency of the circuit! Z X L XC tan XL XC 0 R 2 R2 R =0 PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 20 example – driven capacitor A capacitor is connected to an AC generator as shown. If the frequency of the generator is doubled, the amplitude of the current in the circuit will C A. increase by a factor of 2 B. not change C. decrease by a factor of 2 E I EC Z Z XL XC R 2 XC 2 1 C PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 21 impedance networks Z eq Z1 Z 2 ... Z n 1 1 1 1 ... Z eq Z1 Z 2 Zn Z R iX only if used complex math: i 1 i 2 1 PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 22 without it, the notation is complicated imagine just TWO impedances in parallel PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 23 power in the circuits PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 24 power in a resistor instantaneous power is v2 t p t v t i t Ri 2 t R t Average power (over the cycle): 1 P 2 VRMS 2 0 V2 1 p d R 2 2 2 V 2 VRMS 2 cos d RI RMS 2R R 0 2 V I , I RMS 2 2 PHYS222 - Lecture 22 - Prof. Ruslan Prozorov - Iowa State University 14 October 2011 25 ...
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