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Unformatted text preview: Chapter 31 Electromagnetic Oscillations and Alternating Current In this chapter we will cover the following topics:Electromagnetic oscillations in an LC circuit Alternating current (AC) circuits with resistors, capacitors, and inductors in series ( RCL circuits) Resonance in RCL circuits Power in ACcircuits Transformers, AC power transmission (31  1) L C The circuit shown in the figure consists of a capacitor and an inductor . We give the capacitor an initial charge and then observe how the circuit behaves. The capacitor will d LC C L Q Oscillations ischarge through the inductor resulting in a time dependent current . i We will show that the charge on the capacitor plates as well as the current 1 in the inductor oscillate with constant amplitude at an angular frequency The total energy in the circuit is t q i LC U = 2 2 he sum of the energy stored in the electric field of the capacitor and the energy stored the magnetic field of the inductor. . 2 2 The total energy of the circuit does not chan E B E B U U q Li U U U C = + = + 2 2 2 2 ge with time. Thus 1 0. dU dt dU q dq di dq di d q d q Li i L q dt C dt dt dt dt dt dt C = = + = = = + = (31  2) 2 2 1 d q L q dt C + = L C 2 2 2 2 1 ( ) This is a homogeneous, second order, linear differential equation which we have encountered previously. We 1 0 use d it to describe the simple harmonic d q q dt L d q L q d C t C + = + = eqs.1 2 2 2 oscillator (SHO) ( ) 0 with solution: ( ) cos( ) d x x dt x t X t + = = + eqs.2 ( 29 If we compare eqs.1 with eqs.2 we find that the solution to the differential equation that describes the circuit (eqs.1) is: 1 ( ) cos where , and is the phase angle. The current LC q t Q t LC = + = ( 29 sin Q and depend on the inital conditions: (0) and (0) dq i Q t dt Q i = =  + Note : ( 29 ( ) co s q t Q t = + 1 LC = (31  3) L C ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 2 The energy stored in the electric field of the capacitor cos 2 2 The energy stored in the magnetic field of the inductor sin sin 2 2 2 The total energy 2 E B E B q Q U t C C Li L Q Q U t t C U U U Q U = = + = = + = + = + = ( 29 ( 29 2 2 2 cos sin 2 The total energy is constant; Q t t C C + + + = energy is conserved 2 2 3 The energy of the has a value of at 0, , , ,... 2 2 2 3 5 The energy of the has a value of at , , ,... 2 4 4 4 When is maximum is ze E B Q T T t T C Q T T T t C U U = = electric field maximum magnetic field maximum Note : ro, and vice versa (31  4) t = 1 2 /8 t T = 3 / 4 t T = 4 3 /8 t T = 5 5 / 2 t T = 4 3 2 1 6 6 5 /8 t T = 3 / 4 t T = 7 /8 t T = 7 8 7 8 (31  5) 2 2 If we add a resistor in an RL cicuit (see figure) we must modify the energy equation because now energy is being dissipated on the resistor....
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 Fall '08
 IASHVILI
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