LC Circuit

10 Key excerpts on "LC Circuit"

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  eBook - ePub

  Complete Electronics Self-Teaching Guide with Projects

  • Earl Boysen, Harry Kybett(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 7 Resonant Circuits You have seen how the inductor and the capacitor each present an opposition to the flow of an AC current, and how the magnitude of this reactance depends upon the frequency of the applied signal.

  When inductors and capacitors are used together in a circuit (referred to as an LC Circuit), a useful phenomenon called resonance occurs. Resonance is the frequency at which the reactance of the capacitor and the inductor is equal.

  In this chapter, you learn about some of the properties of resonant circuits, and concentrate on those properties that lead to the study of oscillators (which is touched upon in the last few problems in this chapter and covered in more depth in Chapter 9, “Oscillators”).

  After completing this chapter, you will be able to do the following:

  • Find the impedance of a series LC Circuit.
  • Calculate the series LC Circuit's resonant frequency.
  • Sketch a graph of the series LC Circuit's output voltage.
  • Find the impedance of a parallel LC Circuit.
  • Calculate the parallel LC Circuit's resonant frequency.
  • Sketch a graph of the parallel LC Circuit's output voltage.
  • Calculate the bandwidth and the quality factor (Q) of simple series and parallel LC Circuits.
  • Calculate the frequency of an oscillator.

  The Capacitor and Inductor in Series

   1  Many electronic circuits contain a capacitor and an inductor placed in series, as shown in Figure 7.1 .

  Figure 7.1

  You can combine a capacitor and an inductor in series with a resistor to form voltage divider circuits, such as the two circuits shown in Figure 7.2 . A circuit that contains resistance (R), inductance (L), and capacitance (C) is referred to as an RLC Circuit. Although the order of the capacitor and inductor differs in the two circuits shown in Figure 7.2

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  Halliday's Fundamentals of Physics, 1st Australian & New Zealand Edition

  • David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Here, we begin the study of generating alternating currents, first using only capacitors and inductors, and then using inductors, capacitors and resistors. LC oscillations, qualitatively Of our three circuit elements (resistance R, capacitance C, and inductance L), we have so far discussed the series combinations RC (in module 27.3) and RL (in module 30.6). In these two kinds of circuit we found that the charge, current, and potential difference grow and decay exponentially. The time scale of the growth or decay is given by a time constant  , which is either capacitive or inductive. We now examine the remaining two‐element circuit combination LC. The combination will be ideal (meaning that it lacks any resistance), and we will charge the capacitor and then connect it to the inductor. As we will discuss, the charge, current, and potential difference in such an LC combination do not decay exponentially with time but 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, and the circuit is said to oscillate. Recall from module 25.4 that the energy stored in the electric field of the capacitor at any time is U E = q 2 2C , (31.1) where q is the charge on the capacitor at that time. Also, recall from module 30.7 that the energy stored in the magnetic field of the inductor at any time is U B = Li 2 2 , (31.2) where i is the current through the inductor at that time. Thus, the energy stored in the electric field and the energy stored in the magnetic field also oscillate. However, the total energy is conserved because the circuit is ideal, meaning that it does not have any resistance where energy can be dissipated. Parts a through h of figure 31.1 show the succeeding stages of the oscillations in a simple LC Circuit for one full clockwise cycle of the oscillations.

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  First and Second Order Circuits and Equations

  Technical Background and Insights

  • Robert O'Rourke(Author)
  • 2024(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  Inductors and capacitors are energy storage elements. Connected together, they can exchange energy back and forth between one another. Sinusoidal current is alternating current (AC). At resonance, the back-and-forth energy exchange between capacitor and inductor matches the back-and-forth alternating amplitude of the sinusoidal source. This match – this resonance – makes the series LC combination transparent to the sinusoidal source. This resonance is connected in series along one current path. The exchange of energy between the inductor and capacitor is in the same path as the current. Energy in the capacitor comes from stored charge which creates a voltage. Energy in the inductor comes from magnetic flux and appears as current. At resonance, the rise and fall of voltage across the capacitor, due to the exchange of energy between the inductor and the capacitor, matches the rise and fall of the sinusoidal source voltage. 13.1.3 Series RLC Impedance Magnitude and Resonant Frequency Example This section explores the magnitude of the input impedance in Equation 13.17 and calculates the resonant fre- quency for an RLC Circuit in Figure 13.8. |Z RLC-S | = √ R 2 + ( L − 1 C ) 2 (13.17) Equation 13.17 shows the magnitude of the impedance of a series RLC Circuit, as shown in Figures 13.8 and 13.9. R = 3 Ohms L = 10 nH C = 10 nF Z RLC-S Figure 13.8 Series RLC impedance schematic. Impedance is voltage divided by current. To analyze the input impedance Z RLC-S , in the circuit schematic in Figure 13.8, look at the voltage, across the left-hand-side terminal, divided by the current going into the terminal. 224 13 Second-Order RLC Frequency Response V 1 V 2 V 3 U = 1 V Pr1 + – L 2 L = 10 nH C 2 C = 10 nF R 3 R = 3 Ohm + – Figure 13.9 Series RLC simulation schematic with sinusoidal voltage source. Figure 13.9 shows a sinusoidal steady-state circuit sim- ulation schematic for this series RLC input impedance analysis.

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  Principles of Physics: Extended, International Adaptation

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  31.1.2. All four quantities vary sinusoidally. 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). The oscillations, once started, will die away as Fig. 31.1.3 sug- gests. Compare this figure with Fig. 15.5.2, which shows the decay of mechanical oscillations caused by frictional damping in a block–spring system. The Electrical–Mechanical Analogy Let us look a little closer at the analogy between the oscillating LC system of Fig. 31.1.1 and an oscillating block–spring system. Two kinds of energy are involved in the block–spring system. One is potential energy of the compressed or extended spring; the other is kinetic energy of the moving block. These two energies are given by the formulas in the first energy column in Table 31.1.1. FIGURE 31.1.3 An oscilloscope trace showing how the oscillations in an RLC Circuit actually die away because energy is dissipated in the resistor as thermal energy. Courtesy of Agilent Technologies CHECKPOINT 31.1.1 A charged capacitor and an inductor are connected in series at time t = 0. In terms of the period T of the resulting oscillations, determine how much later the following reach their maximum value: (a) the charge on the capacitor; (b) the voltage across the capacitor, with its original polarity; (c) the energy stored in the electric field; and (d) the current. 31.1 LC Oscillations 911 The table also shows, in the second energy column, the two kinds of energy involved in LC oscillations. By looking across the table, we can see an analogy between the forms of the two pairs of energies—the mechanical energies of the block–spring system and the electromagnetic energies of the LC oscillator. The equations for v and i at the bottom of the table help us see the details of the analogy.

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  Electricity and Magnetism

  • Edward Purcell(Author)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  8.1 A Resonant Circuit 8.2 Alternating Current 8.3 Alternating-Current Networks 8.4 Admittance and Impedance 8.5 Power and Energy in Alternating-Current Circuits Problems ALTERNATING· CURRENT CIRCUITS 298 303 310 313 315 318 298 FIGURE 8.1 A mechanical damped harmonic oscillator. FIGURE 8.2 A "series RLC" circuit. 1- c V L R CHAPTER EIGHT A RESONANT CIRCUIT 8.1 A mass attached to a spring is a familiar example of an oscil- lator. If the amplitude of oscillation is not too large, the motion will be a sinusoidal function of the time. In that case, we call it a harmonic oscillator. The characteristic feature of any mechanical harmonic oscillator is a restoring force proportional to the displacement of a mass m from its position of equilibrium, F = - kx (Fig. 8.1). In the absence of other external forces the mass, if initially displaced, will oscillate with unchanging amplitude at the angular frequency, w = V kj m. But usually some kind of friction will bring it eventually to rest. The simplest case is that of a retarding force proportional to the velocity of the mass, dxj dt. Motion in a viscous ft.uid provides an example. A system in which the restoring force is proportional to some displacement x and the retarding force is proportional to the time derivative dxj dt is called a damped harmonic oscillator. An electric circuit containing capacitance and inductance has the essentials of a harmonic oscillator. Ohmic resistance makes it a damped harmonic oscillator. Indeed, thanks to the extraordinary lin- earity of actual electric circuit elements, the electrical damped har- monic oscillator is more nearly ideal than most mechanical oscillators. The system we shall study first is the "series RLC" circuit dia- grammed in Fig. 8.2. Let Q be the charge, at time t. on the capacitor in this circuit. The potential difference, .or voltage across the capacitor, is V, which obviously is the same as the voltage across the series combination of inductor L and resistor R.

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  Cutnell & Johnson Physics, P-eBK

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Once again, the energy is stored in the electric field between the plates, and no energy resides in the magnetic field of the inductor. Part d of the cycle repeats part b, but with reversed directions CHAPTER 23 Alternating current circuits 659 of current and magnetic field. Thus, an ac circuit can have a resonant frequency because there is a natural tendency for energy to shuttle back and forth between the electric field of the capacitor and the magnetic field of the inductor. FIGURE 23.16 The oscillation of an object on a spring is analogous to the oscillation of the electric and magnetic fields that occur, respectively, in a capacitor and in an inductor. (PE, potential energy; KE, kinetic energy) + + + + – – – – E + + + + – – – – E I max B I max B max = 0 m/s max υ υ υ υ = 0 m/s Position when spring is unstretched Position when spring is unstretched PE (a) PE (c) KE (b) KE (d ) FIGURE 23.17 In a series RCL circuit the impedance is a minimum, and the current is a maximum, when the frequency f equals the resonant frequency f 0 of the circuit. f Impedance Z rms current f 0 To determine the resonant frequency at which energy shuttles back and forth between the capacitor and the inductor, we note that the current in a series RCL circuit is I rms = V rms /Z (equation 23.6). In this expression Z is the impedance of the circuit and is given by Z = √ R 2 + ( X L - X C ) 2 (equation 23.7). As figure 23.17 illustrates, the rms current is a max- imum when the impedance is a minimum, assuming a given generator voltage. The minimum impedance of Z = R occurs when the frequency is f 0 , such that X L = X C or 2f 0 L = 1/(2f 0 C). This result can be solved for f 0 , which is the resonant frequency: f 0 = 1 2 √ LC (23.10) The resonant frequency is determined by the inductance and the capacitance, but not the resistance.

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  Electronic Control & Digital Electronics NQF4 SB

  TVET FIRST

  • Jowaheer Consulting and Technologies, RBJ van Heerden, R Jonker, MWH Smit(Authors)
  • 2014(Publication Date)
  • Macmillan

    (Publisher)

  Overview In Modules 1 and 2 you learned about RC and RL circuits. In this module you will learn about resistor-inductor-capacitor ( RLC ) circuits and their applications. At the end of this module you will be able to: • Analyse RLC Circuits. • Determine the impedance and phase angle in series and parallel RLC Circuits. • Analyse the circuit for resonance. • Analyse the operation of resonant filters. • List practical applications for resonant circuits. 30 Module 3: RLC Circuits and resonance Module 3: RLC Circuits and resonance Units in this module Unit 3.1: RLC series circuits Unit 3.2: RLC parallel circuits Unit 3.1: RLC series circuits RLC series circuit Figure 3.1 shows a series RLC Circuit. It consists of a resistor, an inductor and a capacitor in series with an AC source. R L C V R V L V C I V Figure 3.1: Series RLC Circuit 31 Module 3: RLC Circuits and resonance Waveform and phasor diagrams In this type of circuit there are three possible phasor diagrams: • X L > X C : The circuit is inductive and has a lagging phase angle as shown in Figure 3.2. • X C > X L : The circuit is capacitive and has a leading phase angle as shown in Figure 3.3. • X L = X C : The applied voltage ( V ) and the current ( I ) are in phase as shown in Figure 3.4. This is called series resonance. ø ø V C – IX C V L = IX L V R = IR I V = IZ V L = IX L V C – V L V C – IX C V R = I R V = IZ V = IR I V L = IX L V C = IX C Impedance According to Pythagoras’ theorem and because the impedance ( Z ) is the phasor sum of R , X L and X C : When X L > X C then impedance Z in ohms (uni03A9) = R 2 + ( X L – X C ) 2 and tan ø = X L – X C R When X C > X L then impedance Z in ohms (uni03A9) = R 2 + ( X C – X L ) 2 tan ø = X C – X L R Example 3.1 An RLC series circuit consists of a 10 uni03A9 resistor, an inductor of 0,2 H and a capacitor of 45 uni03BCF. The circuit is connected across a 240 V, 50 Hz supply. Draw the circuit diagram and then calculate: 1. The impedance of the circuit.

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  Fundamentals of Physics

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  The total energy U (= U E + U B ) remains constant. LC Charge and Current Oscillations The principle of conservation of energy leads to L d 2 q ____ dt 2 + 1 __ C q = 0 (LC oscillations) (31.1.11) Review & Summary as the differential equation of LC oscillations (with no resis- tance). The solution of Eq. 31.1.11 is q = Q cos(ωt + ϕ) (charge), (31.1.12) in which Q is the charge amplitude (maximum charge on the capacitor) and the angular frequency ω of the oscillations is ω = 1 ______ √ _ LC . (31.1.4) The phase constant ϕ in Eq. 31.1.12 is determined by the initial conditions (at t = 0) of the system. The current i in the system at any time t is i = −ωQ sin(ωt + ϕ) (current), (31.1.13) in which ωQ is the current amplitude I. Additional examples, video, and practice available at WileyPLUS 990 CHAPTER 31 ELECTROMAGNETIC OSCILLATIONS AND ALTERNATING CURRENT Damped Oscillations Oscillations in an LC Circuit are damped when a dissipative element R is also present in the cir- cuit. Then L d 2 q _____ dt 2 + R dq ____ dt + 1 ___ C q = 0 (RLC Circuit). (31.2.3) The solution of this differential equation is q = Qe −Rt/2L cos(ω′t + ϕ), (31.2.4) where ω′ = √ ____________ ω 2 − (R/2L) 2 . (31.2.5) We consider only situations with small R and thus small damp- ing; then ω′ ≈ ω. Alternating Currents; Forced Oscillations A series RLC Circuit may be set into forced oscillation at a driving angu- lar frequency ω d by an external alternating emf ℰ = ℰ m sin ω d t. (31.3.1) The current driven in the circuit is i = I sin(ω d t − ϕ), (31.3.2) where ϕ is the phase constant of the current. Resonance The current amplitude I in a series RLC Circuit driven by a sinusoidal external emf is a maximum (I = ℰ m /R) when the driving angular frequency ω d equals the natural angular frequency ω of the circuit (that is, at resonance). Then X C = X L , ϕ = 0, and the current is in phase with the emf.

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  Electronics and Electronic Systems

  • George H. Olsen(Author)
  • 2013(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Conversely, when the frequency is zero, i.e. when a steady voltage is applied to the capacitor, X c is infinitely large and no current passes. This accords with our previous knowledge that direct current is prevented from flowing through a capacitor by the dielectric. When/is in hertz and C is in farads X c is measured in ohms: for example, if an r.m.s. voltage of 10 V at a frequency 500 Hz were applied to a capacitance of 20 μΡ then the r.m.s. current that would flow in the lead to the capacitor would be V/X c , i.e. 10/[1/(2TT 500 x 20 x 10 6 )] = 2π x 10 1 = 0.628 A (r.m.s.) The application of a sinusoidal voltage to an inductor results in a steady-state current that lags the voltage by 90° (see Figure 4.5(c)). If the current through the inductor is given by / = / max sin ωί, then T di T d(/ max sin ωή T T T r . , n χ v = L — = L max di '-= ü)ZJ max cosotf = coL/ max sin (ωί + —) Once again we can regard these statements as representing the alternating current version of Ohm's Law. The voltage, v, is equal to the product of a current term, / max sin((Di + nil) and a term that is a measure of the opposition to current flow in the circuit. This latter term, ooL, is known as the inductive reactance and is given the symbol X^ (The subscript L is often omitted when there is no ambiguity.) We see that the opposition to current flow is proportional to the frequency and to the inductance involved. When resistors, capacitors and inductors are interconnected to form a required network, the analysis of the network's performance appears at first sight to be complicated. Consider a simple series arrangement as shown in Figure 4.6. The sum of the instantaneous voltages across each component must be equal to the 40 Passive component networks I v s = VSinojf ί ZD ~™ 1| Figure 4.6 A simple series circuit; v s is the driving voltage, assumed to be sinusoidal instantaneous supply voltage, i.e.

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  PSpice for Circuit Theory and Electronic Devices

  • Paul Tobin(Author)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  65 C H A P T E R 6 Series and Parallel-tuned Resonance 6.1 RESONANCE The resonance phenomenon occurs in a second-order system containing capacitors and induc- tors. A circuit containing resistance, capacitance, and inductance will become purely resistive at the resonant frequency when the inductive reactance equals the equal capacitive reactance. Selectivity is the ability of the resonant circuit to extract the resonant frequency and attenuate other frequencies. Selectivity is measured by a Q-factor, which, for a series-tuned circuit, is the ratio of the inductive (or capacitive reactance) to the total resistance in the circuit (this includes source and load resistances). It is a measure of how well the circuit extracts a band of frequencies with little attenuation but rejects other frequencies outside this band. Selectivity is also dependent on the inductor-capacitor ratio. A high Q-factor value means high selectivity and a low Q-factor means low selectivity. For good series-tuned selectivity, the circuit must be fed from a voltage source because the source resistance feeding the circuit has minimum resistance and hence has minimum effect on the Q-factor (Ideal voltage source impedance is zero). Any series resistance added will reduce the overall selectivity. A parallel-tuned circuit is fed from a current source because it has a high source resistance and hence minimum loading. Any external resistive loading reduces the overall selectivity of the parallel-tuned circuit. 6.2 SERIES-TUNED CIRCUIT Fig. 6.1 shows a series-tuned circuit with inductance, capacitance, and resistance R representing source and coil resistances. The total circuit impedance is Z = E S / l S = R + j ( X L − X C ) (6.1) The total reactance is zero at the resonant frequency f 0 ( j terms are zero). ( X L − X C ) = 0 ⇒ X L = X C or ω L = 1/ωC (6.2)

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