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Review on R, L, C and RLC in AC circuits An AC power source is a sinusoidal voltage source which is v described as v Vmax sin t Vmax We study the current i as a function of time when a resistor, inductor, capacitor or the serial connection of the three, to the AC voltage source. When resistance is involved, i t v Vmax sin t Assuming AC voltage source: Laws to apply R Ohm’s Law C RLC Kirchhoff’s Loop Kirchhoff’s Loop Kirchhoff’s Loop In phase or Vmax 0 R i I max sin t 2 v I max Vmax XL , X L L i I max sin t 2 I max Vmax XC 1 , XC C leads, or 2 i leads, or 2 i I max sin t I max Vmax X XC tan L R Z Z R2 X L X C Phasor diag. Power pav Vrms I rms i I max sin t I max L Phase w/ v Current i 2 Vmax 2Vrms I max 2I rms No power is consumed. No power is consumed. To be discussed today Power discussion of RLC in an AC circuit I did not discuss power with L or C in AC circuit. Because there was no power consumed. With R comes into the play, we have to examine the power issue. i I max sin t The power delivered from the AC voltage source is: v Vmax sin t p v i Vmax sin t I max sin t Vrms I rms 2 sin t sin t Vrms I rms cos cos 2t Vrms 2Vmax , I max 2I rms 2 sin sin cos cos The (more useful) average power is: T T 1 1 pav pdt Vrms I rms cos cos 2t dt T0 T0 T T 1 Vrms I rms cos dt cos 2t dt T 0 0 Vrms I rms cos The final result: pav Vrms I rms cos Power and phase The average power delivered from the AC voltage source to the RLC circuit is: pav Vrms I rms cos This power depends on the phase angle. Circuits involving heavy motors (heavy inductive load) usually have capacitors to shift the phase to improve the power delivery efficiency. i I max sin t v Vmax sin t Resonance in an AC Circuit A series RLC circuit, R=0.01Ω, L= 4.34mH, C=1.00mF. ΔVmax=150 V. Complete the following table. i I max sin t v Vmax sin t (rad/sec) 440 460 480 500 520 I max (A) pav (W) Formula needed: I max Vmax Z Z R2 X L X C X L L pav Vrms I rms cos XC 1 C 2 See the spreadsheet tan X L XC R Resonance in an AC Circuit From these formulas: I max Vmax Z Z R2 X L X C 2 i I max sin t 1 1 , or 2 When L C LC We have X L X C and Z min R v Vmax sin t Under this condition, V I max max reaches maximum with a given Vmax Z X XC tan L 0, =0 and cos 1 R This frequency is called the resonant frequency: 0 1 LC Power also resonates The average power is pav Vrms I rms cos 2 Vrms Z cos Now I want to express the power as a function of the angular frequency 2 Z Vrms XL- XC pav cos Z 2 Vrms R Z2 cos 2 Vrms R 1 R L C 2 Vrms R 2 R Z R 2 2 1 R 2 2 L2 2 LC 2 Vrms R 2 R 2 2 L2 2 02 2 multiply 2 2 So when 0 the power reaches a maximum: resonates. PLAY ACTIVE FIGURE How narrow (good) is the resonance: the Q (quality) factor The sharpness of the resonance curve is usually described by a dimensionless parameter known as the quality factor, Q 0 0 L Q R Δω is the half-power width: width of the average power curve, measured between the two values of ω for which pav has half its maximum value. Because R usually comes from the resistance of the wire that is used to construct the inductor, one tries to design the inductor in such a way that it maximizes the L, and minimizes the R.