Electricity and Magnetism 59 Chapter 6 ALTERNATING CURRENT CIRCUIT 6.1 RLC Circuit The storage energy (Fig. 6.1) U = U E + U B = 2 Li 2 1 + 2 c Cv 2 1 [J] (6.1) 1) Undamped Oscillation Consider the circuit in Fig. 6.1. At t < 0, the switch K is at 1. At t > 0, the switch K is at 2. If the circuit is lossless (there is no resistance) dt dU = Li dt di + Cv c dt dv c = 0 (6.2) i = -C dt dv c ⇒ dt di = -C 2 c 2 dt vd ⇒ LC 2 c 2 dt vd + v c = 0 (6.3) ⇒ v c (t) = Vcos( ω t) (6.4) and i(t) = V ω Csin( ω t) (6.5) where LC 1 =ω (6.6) Fig. 6.1 Fig. 6.2 2) Damped Oscillation Consider the circuit in Fig. 6.2. At t < 0, the switch K is at 1. At t > 0, the switch K is at 2. If a dissipative element R is present dt dU = Li dt di + Cv c dt dv c = -Ri 2 (6.7) ⇒ LC 2 c 2 dt vd + RC dt dv c + v c = 0 (6.8) ⇒ v c (t) = V o e -Rt/2L cos( ω t+ ϕ o ) (6.9) Electricity and Magnetism 60 where 2 ) L2 R ( LC 1 −=ω 6.2 Alternating current circuit 1) Resitive load : (Fig. 6.3) the current i and the voltage e across the resistor are in phase. The impedance of the resistor R I V z m m == I m , V m : amplitude of i and e, respectively. Fig. 6.3 Fig. 6.4 2) Inductive load : (Fig. 6.4) the current in the inductor lags the voltage by 90 ° . The impedance of the inductor ωL I V z m m == I m , V m : amplitude of i and e, respectively. 3) Capacitive load : (Fig. 6.5) the current in the capacitor leads the voltage by 90 ° . The impedance of the capacitor ωC 1 == m m I V z I m , V m : amplitude of i and e, respectively. Fig. 6.5 Fig. 6.6 4) The series RLC circuit (Fig. 6.6) The impedance of the circuit Electricity and Magnetism 61 2 ω ω       +== C 1 -LR I V z 2 m m The phase constant R C 1 -L ω ω )tan( = ϕ C 1 L ω ω > : the circuit is more inductive than capacitive, the current i lags the voltage e. C 1 L ω ω < : the circuit is more capacitive than inductive, the current i leads the voltage e. C 1 L ω ω = : the circuit is in resonance, the current i and the voltage e are in phase. The resonance frequency LC 1 = o ω 6.3 Phasor The sinusoidal quantity i = I m cos( ω t+ ϕ ) is represented by a vector of length I m which rotates around the origin with the angular speed ω (Fig. 6.7). At time t = 0 this vector is the phasor I m ∠ ϕ of the sinusoidal quantity. Fig. 6.7 Fig. 6.8 6.4 Transformer (Fig. 6.8) 2 1 2 1 n n u u = 1 2 2 1 n n i i −= Electricity and Magnetism 62 Problems 6.1) Consider the circuit in Fig. P6.1 with e(t) = 12sin(120 π t) V. When S 1 and S 2 are open, i leads e by 30 ° . When S 1 is closed and S 2 is open, i lags e by 30 ° . When S 1 and S 2 are closed, i has amplitude 0.5A. What are R, L and C ? Fig. P6.1 Fig. P6.2 Fig. P6.3 6.2) Consider the circuit in Fig. P6.2 with e(t) = 12sin(120 π t) V, r = 10 Ω . Find the value of R such that the power in R is maximized ? 6.3) Consider the circuit in Fig. P6.3 with e(t) = 12sin(120 π t) V, L = 0.0265mH. Find the value of R such that the power in R is maximized ? 6.4) Consider the circuits in Fig. P6.4 where R = 100 Ω , L = 100mH, C = 10 µ F, e = 100sin( ω t) volts. Find i R (t), i L (t), i C (t), V(t), the storage energy of the capacitor, the storage energy of the inductor, and the total storage energy in 3 cases : a) ω = 500 rad/s, b) ω = 1000 rad/s, c) ω = 2000 rad/s Fig. P6.4 6.5) Consider the circuit in Fig. P6.5 where e = 100sin( ω t) volts, R = 100 Ω , L = 100mH, C = 10 µ F. Determine i(t), v R (t), v L (t), v C (t), the storage energy of the capacitor U C (t), the storage energy of the inductor U L (t), the average power of the resistor P R , the average power of the source P e in 3 cases : a) ω = 500 rad/s, b) ω = 100 rad/s, c) ω = 1000 rad/s Fig. P6.5 Fig. P6.6 6.6) Consider the circuit in Fig. P6.6 where R = 100 Ω , C = 10 µ F, e = 100sin(1000t) volts. The capacitor C has circular plates of radius a, the space between the two plates is d = 0.1mm. a) Find the voltage v and the current i. Electricity and Magnetism 63 b) Find the electric field E, the magnetic field B and the displacement current i d between the capacitor plates. 6.7) A typical “light dimmer” used to dim the stage lights in a theater consist of a variable inductor L connected in series with the light bulb B as shown in the figure P6.7. The power supply is 220 V (rms) at 60 Hz; the light bulb is marked “220 V, 1000W” a) What maximum inductance L is required if the power in the light bulb is to be varied by a factor of five? Assume that the resistance of the light bulb is independent of its temperature? b) Could one use a variable resistor instead of an inductor? If so, what maximum resistance is required? Why isn’t this done? Fig. P6.7 Homeworks 6 H6.1 Consider the circuits in Fig. H6.1 where e = 100sin(1000t) volts. Find i R (t), i L (t), i C (t), V(t), the storage energy of the capacitor, the storage energy of the inductor, and the total storage energy (R in Ω , L in mH, C in µ F). Fig. H6.1 Fig. H6.2 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 R 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 L 5 10 20 40 60 80 100 120 150 175 200 225 250 275 300 350 C 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 R 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 L 5 10 20 40 60 80 100 120 150 175 200 225 250 275 300 350 C 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 n 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 R 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 L 5 10 20 40 60 80 100 120 150 175 200 225 250 275 300 350 C 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 Electricity and Magnetism 64 n 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 R 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 L 5 10 20 40 60 80 100 120 150 175 200 225 250 275 300 350 C 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 H6.2 Consider the circuits in Fig. H6.2 where e = 100sin(1000t) volts. Find i(t), the storage energy of the capacitor, the storage energy of the inductor, and the total storage energy (R in Ω , L in mH, C in µ F). n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 R 100 100 100 100 100 100 100 100 200 200 200 200 200 200 200 200 L 25 50 75 100 125 150 175 200 25 50 75 100 125 150 175 200 C 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 R 100 100 100 100 100 100 100 100 200 200 200 200 200 200 200 200 L 25 50 75 100 125 150 175 200 25 50 75 100 125 150 175 200 C 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 n 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 R 100 100 100 100 100 100 100 100 200 200 200 200 200 200 200 200 L 25 50 75 100 125 150 175 200 25 50 75 100 125 150 175 200 C 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 n 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 R 100 100 100 100 100 100 100 100 200 200 200 200 200 200 200 200 L 25 50 75 100 125 150 175 200 25 50 75 100 125 150 175 200 C 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 . 150 150 150 L 5 10 20 40 60 80 100 120 150 175 200 225 250 275 30 0 35 0 C 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 H6.2 Consider the circuits in Fig. H6.2 where e = 100sin(1000t). 200 200 200 200 200 L 5 10 20 40 60 80 100 120 150 175 200 225 250 275 30 0 35 0 C 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 n 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 R 50 50 50 50 50 50 50 50. 20 40 60 80 100 120 150 175 200 225 250 275 30 0 35 0 C 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 Electricity and Magnetism 64 n 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 R