Chapter 31 Electromagnetic Oscillations and Alternating Current LC Oscillations, Qualitatively

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1 Chapter 3 Electromagnetic Oscillations and Alternating Current LC Oscillations, Qualitatively In the LC circuit the charge, current, and potential difference vary sinusoidally (with period T and angular frequency ). The resulting oscillations of the capacitor s electric field and the inductor s magnetic field are said to be electromagnetic oscillations. oscillations the energy stored in the electric field of the capacitor is q UE C L i the energy stored in the magnetic field of the inductor is U B To determine the charge q(t) on the capacitor,put in a voltmeter to measure the potential difference (or voltage) vc that exists across the capacitor C: vc q / C To measure the current, connect a small resistance in series in the circuit and measure the potential difference v across it: v i

2 In an actual LC circuit, the oscillations will not continue indefinitely because there is always some resistance present that will drain energy from the electric and magnetic fields and dissipate it as thermal energy (the circuit may become warmer).

3 The Electrical Mechanical Analogy the analogy between the oscillating LC system and an oscillating block spring system: q corresponds to x, /C correspond to k i corresponds to v, L correspond to m These correspondences suggest that in an LC oscillator, the capacitor is mathematically like the spring in a block spring system and the inductor is like the block. In a block spring system: k block spring system m The correspondences suggest that to find the angular frequency of oscillation for an ideal LC circuit, k Block Spring System LC Oscillator should be replaced by /C and m by L, Element Energy Element Energy L C LC circuit Spring Potential, k x / Block Kinetic, m v / vd x/d t Capacitor Electrical, q / C Inductor Magnetic, L i / i d q/ d t

4 LC Oscillations, Quantitatively The Block Spring Oscillator the total energy of a block spring oscillator: U U b U s m v k x Energy conservation, no friction: du d dv dx 0 m v k x m v k x m d x k x 0 x t X cos t displacement X is the amplitude of the mechanical oscillations, is the angular frequency of the oscillations, and is a phase constant. The LC Oscillator the total energy in an oscillating q LC circuit: U U B U E L i C UB is the energy stored in the magnetic field of the inductor and UE is the energy stored in the electric field of the capacitor. Energy conservation, no resistance: du d di q dq q 0 Li L i C C q t Q cos t charge d q L q 0 LC oscillation C

5 The current of the i t LC oscillator: dq Q sin t I sin t current The angular frequency of the d q I Q LC oscillator: Q cos t L Q cos t Q C cos t L d q q C 0 / LC The phase is determined by the initial conditions. The electrical energy stored in the LC circuit q Q UE cos t C C The magnetic energy stored in the Q UB sin t C UB LC circuit L i L Q sin t Note: U U Q / C E max B max U E U B Q / C constant 3 U E U E max when U B 0 ; U B U B max when U E 0 problem 3-

6 Damped Oscillations in an LC Circuit A circuit containing resistance, inductance, and capacitance is called an LC circuit. We shall here discuss only series LC circuits With a resistance present, the total EM energy of the circuit is no longer constant; it decreases with time as energy is transferred to thermal energy in the resistance. Because of this loss of energy, the oscillations of charge, current, and potential difference decrease in amplitude, and the oscillations are damped. du i The rate of energy transferred to thermal energy: d Li q di q dq d q dq L i i L q 0 LC curcuit C d t C C ' / L / L C q Q t / L The electrical energy: U e cos ' t E C C Q t / L U e C Li Q t / L the magnetic energy: U B problem 3- e sin ' t C q Q e t / L cos ' t

7 Alternating Current If the energy is supplied via oscillating emfs and currents, the current is said to be an alternating current, current or ac for short. The nonoscillating current from a battery is said to be a direct current, current or dc. dc These oscillating emfs and currents vary sinusoidally with time, reversing direction (in North America) 0 times per second and thus having frequency f 60 Hz. As the current alternates, so does the magnetic field that surrounds the conductor. This makes possible the use of The advantage of alternating current: Faraday s law of induction. In a generator: ℰ ℰ m sin d t, i I sin d t where d is called the driving angular frequency. frequency the current may not be in phase with the emf. the driving frequency f d d /

8 Forced Oscillations An undamped LC circuits or a damped LC circuits (with small enough ) without any external emf are said to be free oscillations, and the angular frequency / LC is said to be the circuit s natural angular frequency. frequency When the external alternating emf is connected to an LC circuit, the oscillations of charge, potential difference, and current are said to be driven oscillations or forced oscillations., with the driving angular frequency d : Whatever the natural angular frequency of a circuit may be, forced oscillations of charge, current, and potential difference in the circuit always occur at the driving angular frequency d. Three Simple Circuits A esistive Load By the loop rule: ℰ v 0 v ℰ m sin d t v V sin d t V ℰ m i I sin d t 0, v V i sin d t V I resistor v and i are in phase, which means that their corresponding maxima (and minima) occur at the same times.

9 problem 3-3 A Capacitive Load the potential difference across the capacitor vc V C sin d t qc C v C C V C sin d t d qc d C V C cos d t The current: ic capacitive reactance: X capacitive reactance reactance C d C the SI unit of XC is the ohm, just as for resistance. cos d t sin d t / i C I C sin d t V C IC XC capacitor VC sin d t / XC true for any capacitance in any circuit

10 Problem 3-4 An Inductive Load the potential difference across the inductance v L V L sin d t L d il d il VL sin d t L VL cos d t The current: i L d i L sin d t d t L d L capacitive reactance: reactance X L d L inductive reactance the SI unit of VL XL is the ohm, just as for XC an for. cos d t sin d t / i L I L sin d t V L I L XL inductor VL sin d t / XL true for any inductance in any circuit

11 Problem 3-5 Phase and Amplitude elations for Alternating Currents and Voltages Circuit esistance Phase of Phase Constant Amplitude or ecactance the Current or Angle elation Element Symbol esistor In phase with v 0 Capacitor C X C / d C Leads vc by / / V C I C XC Inductor L X L d L Lags v L by / / VL IL XL VI

12 The Series LC Circuit Apply a LC circuit the alternating emf ℰ ℰ m sin d t applied emf i I sin d t The Current Amplitude For the loop rule:ℰ v v C v L ℰ m V V L V C ℰ I I X L I X C where Z X L X C I ℰ X L X C impedance I ℰ Z ℰ [ L / C ] d d I depends on the difference between dl and / dc or, equivalently, the difference between XL and XC. The value of current amplitude

13 The current that we have been describing in this section is the steady state current that occurs after the alternating emf has been applied for some time. When the emf is first applied to a circuit, a brief transient current occurs. Its duration is determined by the time constants LL/ and CC as the inductive and capacitive elements turn on. The Phase Constant V L VC I XL I XC From the plot: tan V I tan X L XC phase constant

14 Three different results for the phase constant XL>XC: The circuit is said to be more inductive than capacitive. XC>XL: The circuit is said to be more capacitive than inductive. XCXL: The circuit is said to be in resonance. purely inductive circuit, where XL is nonzero and XC0, then / (the greatest value of ). In the purely capacitive circuit, where XC is nonzero and XL0, then - / (the least value of ). In the esonance For a given resistance, that amplitude is a maximum when the quantity dl-/ dc in the denominator is zero d L d C d L C maximum I the natural angular frequency of the LC circuit is also equal to / L C, the maximum value of I occurs when the driving angular frequency matches the natural angular frequency that is, at resonance. d L C resonance

15 resonance curves peak at their maximum current amplitude I (ℰm/) when d, but the maximum value of I decreases with increasing. The curves also increase in wih (measuring at half the maximum value of I) with increasing. The XL( dl) is small and XC(/ dc) is large. Thus, the circuit is mainly capacitive and the impedance is dominated by the large XC, which keeps For small d, the current low. XC remains dominant but decreases while XL increases. The decrease in XC decreases the impedance, allowing the current I to increase. When the increasing XL and the decreasing XC reach equal values, the current I is As d increases, greatest and the circuit is in resonance, with d. XL becomes more dominant over the decreasing XC. The impedance increases because of XL and the current decreases. As d continue to increase, the increasing In summary: The low-angular-frequency side of a resonance curve is dominated by the capacitor s reactance, the high-angular frequency side is dominated by the inductor s reactance, and resonance occurs in the middle. Problem 3-6

16 Power in Alternating-Current Circuits In steady-state operation the average energy stored in the capacitor and inductor together remains constant. The net transfer of energy is thus from the generator to the resistor, where EM energy is dissipated as thermal energy. The instantaneous rate at which energy is dissipated in the resistor P i [ I sin d t ] I sin d t The average rate at which energy is dissipated Pavg T 0 P d t T I T I I rms T 0 sin I d t d t I rms current Pavg I rms average power if we switch to the rms current, we can compute the average rate of energy dissipation for alternatingcurrent circuits just as for direct-current circuits. V rms V, ℰ rms ℰ rms voltage; rms emf

17 Alternating-current instruments, such as ammeters and voltmeters, are usually calibrated to read Irms, Vrms, and ℰrms. plug an alternating-current voltmeter into a electrical outlet and it reads 0 V, that is an rms voltage. The maximum value of the potential difference at the outlet is 0 V or 70 V. I rms ℰ rms Z ℰ rms X L X C Pavg Pavg ℰ rms I rms cos average power ℰ rms Z I rms ℰ rms I rms cos V ℰm Z I IZ power Z factor The equation is independent of the sign of the phase constant cos cos( ). To maximize the rate at which energy is supplied to a resistive load in an LC circuit, we should keep the power factor cos as close to as possible 0. Problem 3-7

18 Transformers Energy Transmission equirements an ac circuit with only a resistive load, the power factor cos 0, Pavg ℰ I I V A range of choices of I and of V provided only that the product IV is as required. in the transmission of electrical energy from the generating plant to the consumer, we want the lowest practical current (hence the largest practical voltage) to minimize I losses (often called ohmic losses) in the transmission line. consider a 735 kv line to transmit electrical energy for 000 km. If the current is 500 A and the power factor ~ unity. Thensupply Pavg ℰ I V 500 A 368 MW The resistance of the transmission line is about 0. /km 0 Energy is dissipated due to that resistance at a rate total Pavg I 500 A 0 55 MW ~ 5 % Psupply avg In the other case: I ' I, ℰ ' ℰ / P supply ℰ ' I ' ℰ I avg Pavg I ' 000 A 0 0 MW ~ 65 % Psupply avg the general energy transmission rule:transmit at the highest possible voltage and the lowest possible current.

19 The Ideal Transformer need a device with which we can raise (for transmission) and lower (for use) the ac voltage in a circuit, keeping the product current voltage essentially constant the transformer. The ideal transformer consists of two coils, with different numbers of turns, wound around an iron core. the primary winding, of Np turns, is connected to an AC generator whose emf is The secondary winding, of ℰ ℰ m sin t Ns turns, is connected to load resistance. magnetizing current Imag, lags the primary voltage Vp by 90 (no power is delivered). The sinusoidally changing primary current Imag the primary current, the produces a sinusoidally changing magnetic flux B in the iron core. The core strengthens the flux and to bring it through the secondary winding. Because B varies, it induces an emf ℰturn (d B/) in each turn of the secondary. the emf per turn ℰturn is the same in the primary and the secondary ℰ turn Vp Np Vs Ns V s V p Ns Np transformation of voltage

20 Ns>Np: step up transformer because Vs>Vp ; Ns<Np: step down transformer because Vs<Vp. connect the secondary to the resistive load the generator:, now energy is transferred from An AC Is appears in the secondary circuit, with corresponding energy dissipation rate Is(Vs/) in the resistive load. Is produces its own alternating magnetic flux in the iron core, and this flux induces an opposing emf in the primary windings. 3 Vp of the primary cannot change in response to this opposing emf because it must always be equal to the emf that is provided by the generator. Vp, the generator now produces (in addition to Imag) an AC Ip in the primary circuit; the emf induced by Ip in the primary will exactly cancel the emf induced there by Is. Because the phase constant of Ip is not 90 like that of Imag, this current Ip can transfer energy to the primary. 4 To maintain

21 Assume no energy is lost, conservation of energy requires that I p V p Is Vs I pis Ns Np Ns Vs Np Ns Np Vp Vp eq eq Np Ns Impedance Matching For maximum transfer of energy from an emf device to a load, the impedance of the emf device must equal the impedance of the load. We can match the impedances of the two devices by coupling them through a transformer that has a suitable turns ratio. Solar Activity and Power-Grid Systems problem 3-8 Selected problems:, 6, 36,50, 58

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