普通物理 General Physics 31 - Alternating Fields and Current 郭艷光Yen-Kuang Kuo 國立彰化師大物理系暨光電科技研究所 電子郵件: ykuo@cc.ncue.edu.tw 網頁: http://ykuo.ncue.edu.tw

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Outline 31-1 What Is Physics? 31-2 LC Oscillations, Qualitatively 31-3 The Electrical–Mechanical Analogy 31-4 LC Oscillations, Quantitatively 31-5 Damped Oscillations in an RLC Circuit 31-6 Alternating Current 31-7 Forced Oscillations 31-8 Three Simple Circuits 31-9 The Series RLC Circuit 31-10 Power in Alternating–Current Circuits 31-11 Transformers 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 31-1 What Is Physics? We next turn to the associated applied physics, in which the energy stored in one location can be transferred to another location so that it can be put to use. In most parts of the world, electrical energy is transferred not as a direct current but as a sinusoidally oscillating current (alternating current, or ac). 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-2 LC Oscillations, Qualitatively 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-1 A 1.5 μF capacitor is charged to 57 V. The charging battery is then disconnected, and a 12 mH coil is connected in series with the capacitor so that LC oscillations occur. What is the maximum current in the coil? Assume that the circuit contains no resistance. 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-1 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-3 The Electrical–Mechanical Analogy This is a homogeneous, second order, linear differential equation which we have encountered previously. We used it to describe the simple harmonic oscillator (SHO) with solution: 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-4 LC Oscillations, Quantitatively The energy of the electric field has a maximum value of The energy of the magnetic field has a maximum Note: When UE is maximum UB is zero, and vice versa 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-4 LC Oscillations, Quantitatively The energy of the electric field has a maximum value of The energy of the magnetic field has a Note: When UE is maximum UB is zero, and vice versa. 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-4 LC Oscillations, Quantitatively 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-2 For the situation described in Sample Problem 31-1, let the coil (inductor) be connected to the charged capacitor at time t = 0. The result is an LC circuit like that in Fig. 31-1. (a) What is the potential difference vL(t) across the inductor as a function of time? (b) What is the maximum rate (di/dt)max at which the current i changes in the circuit? 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-2 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-5 Damped Oscillations in an RLC Circuit 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-5 Damped Oscillations in an RLC Circuit 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-3 A series RLC circuit has inductance L = 12 mH, capacitance C = 1.6 μF, and resistance R = 1.5 . (a) At what time t will the amplitude of the charge oscillations in the circuit be 50% of its initial value? (b) How many oscillations are completed within this time? 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-3 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 31-6 Alternating Current “direct current” or “dc” “alternating current ” or “ac” 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 31-7 Forced Oscillations Whatever the natural angular frequency ω of a circuit may be, forced oscillations of charge, current, and potential difference in the circuit always occur at the driving angular frequency ωd. 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-8 Three Simple Circuits A convention 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-8 Three Simple Circuits A resistive load Current amplitude: 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-8 Three Simple Circuits 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-4 Purely resistive load. In Fig. 31-8a, resistance R is 200 Ω and the sinusoidal alternating emf device operates at amplitude εm = 36.0 V and frequency fd = 60.0 Hz. (a) What is the potential difference vR(t) across the resistance as a function of time t, and what is the amplitude VR of vR(t)? (b) What are the current iR(t) in the resistance and the amplitude IR of iR(t)? 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-4 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-8 Three Simple Circuits A capacitive load O 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-8 Three Simple Circuits The current leads the voltage by a quarter of a period. The voltage and current are out of phase by . 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-5 Purely capacitive load. In Fig. 31-9a, capacitance C is 15.0 μF and the sinusoidal alternating emf device operates at amplitude εm = 36.0 V and frequency fd = 60.0 Hz. (a) What are the potential difference vC(t) across the capacitance and the amplitude VC of vC(t)? (b) What are the current iC(t) in the circuit as a function of time and the amplitude IC of iC(t)? 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-5 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-8 Three Simple Circuits An inductive load O 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-8 Three Simple Circuits The current lags behind the voltage by a quarter of a period. The voltage and current are out of phase by 90o. 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-6 Purely inductive load. In Fig. 31-10a, inductance L is 230 mH and the sinusoidal alternating emf device operates at amplitude εm = 36.0 V and frequency fd = 60.0 Hz. (a) What are the potential difference vL(t) across the inductance and the amplitude VL of vL(t)? (b) What are the current iL(t) in the circuit as a function of time and the amplitude IL of iL(t)? 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-6 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-8 Three Simple Circuits Circuit element Average power Reactance Phase of current Voltage amplitude Resistor R Current is in phase with the voltage. Capacitor C Current leads voltage by a quarter of a period. Inductor L Current lags behind voltage by a quarter of a period. 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-9 The Series RLC Circuit 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-9 The Series RLC Circuit 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-9 The Series RLC Circuit XL > XC → ψ > 0 The current phasor lags behind the generator phasor. The circuit is more inductive than capacitive. XC > XL → ψ < 0 The current phasor leads ahead of the generator phasor. The circuit is more capacitive than inductive. XC = XL → ψ = 0 The current phasor and the generator phasor are in phase. 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-9 The Series RLC Circuit Resonance 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-7 (a) Figure shows that R = 200 , C = 15.0 μF, L = 230 mH, fd = 60.0 Hz, and εm = 36.0 V. What is the current amplitude I? 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-7 (a) Solutions: 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-7 (b) What is the phase constant of the current of the circuit in the circuit relative to the driving emf? 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-7 (b) Solutions: 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

31-10 Power in Alternating-Current Circuits 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-8 A series RLC circuit, driven with εrms = 120 V at frequency fd = 60.0 Hz, contains a resistance R = 200 , an inductance with XL = 80.0 , and a capacitance with XC = 150 . (a) What are the power factor cos ψ and phase constant ψ of the circuit? (b) What is the average rate Pavg at which energy is dissipated in the resistance? (c) What new capacitance Cnew is needed to maximize Pavg if the other parameters of the circuit are not changed? 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-8 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-8 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 31-11 Transformers Power Station Transmission lines Erms =735 kV , I rms = 500 A Home 110 V T1 T2 Step-up transformer Step-down transformer R = 220Ω 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 31-11 Transformers If , we have what is known a “step up” transformer. If , we have what is known a “step down” transformer. 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-9 A transformer on a utility pole operates at Vp = 8.5 kV on the primary side and supplies electrical energy to a number of nearby houses at Vs = 120 V, both quantities being rms values. Assume an ideal step-down transformer, a purely resistive load, and a power factor of unity. 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-9 (a) What is the turns ratio Np /Ns of the transformer? (b) The average rate of energy consumption (or dissipation) in the houses served by the transformer is 78 kW. What are the rms currents in the primary and secondary of the transformer? (c) What is the resistive load Rs in the secondary circuit ?What is the corresponding resistive load Rp in the primary circuit? 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 Example 31-9 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授 End of chapter 31! 2018/12/27 普通物理講義-31 / 國立彰化師範大學物理系/ 郭艷光教授