PHYS 164 LAB #18 RLC Circuits and Resonance - Part I DATA PRELIMINARY DATA NOTE: All components have a tolerance of ± 10% ± 10 R (W) 1000 L (mH) 27 ± 2,7 C (pF)*** 100 ± 10 Part 1: Underdamped conditions Resonance Frequency and Half-life fo (kHz) 82,5 ± 0,6 w0 (rad/s) 518362,7878 ± 3769,911 t1/2 (ms) 29,2 ± 0,4 t (s) 4,21267E-05 ± 5,77E-07 Part 2: Critical Resistance Rcrit (W) 22000 ± 220 ***Must include the oscilloscope capacitance of 15 pF ANALYSIS Part 1 1 2 Quantity w0 t Rcrit Observed 518362,7878 4,21267E-05 22000 Expected   3769,911184 567504,4 53520,45 5,77078E-07 5,40E-05 5,94E-06 220 % diff 8,7 22,0 Agree? Yes No 27 #18 RLC Circuits and Resonance - Part I Objectives orth 1. To investigate the behavior of RLC circuits. 2. To become familiar with the experimental aspects of resonance. Introduction and Theory In Experiment #17 we studied the exponential discharge of a capacitor through a re- sistor and the response of the resistor-capacitor network to a sinusoidal driving voltage. In this experiment we add an inductor to the circuit. An inductor is simply a coil of wire (ideally with zero resistance) which is wound around a frame (usually cylindrical in more shape). Experiment shows that the voltage across an inductor is proportional to the rate of change of current through it; i.e., wonlove dI V = L (18.1) dt . where L is a constant characteristic of the device called its inductance. As we shall see, this so-called RLC circuit is the electrical analog of a damped harmonic oscillator in me- chanics. To introduce the basic ideas to be investigated, consider first the circuit shown in Fig. 18.1. and SRL S A B is work Volt I L y from Fig. 18.1. An oscillating LC circuit Suppose the capacitor has previously been given an initial charge Qo. If the switch ishon closed at time t=0 the capacitor begins to discharge through the inductor. As in Experi- cool ment #17, suppose we wish to find the charge Q on the capacitor as a function of time. 99 Since the sum of the voltage changes around a complete circuit = 0, we have L dl Q + dt С = 0. (18.2) Since I = dQ/dt, di/dt = d+Q/di. Substituting for dI/dt in Eq. 18.1 and rearranging gives 28 1 -one, d’Q dt? L ON ce. (18.3) This equation has exactly the same form as the equation of motion of a mass oscil- lating on a spring in the absence of friction. Applying Newton's second law to the latter gives the following: : d’x F = ma → - kx dt? = m or dx ma = - kx. (18.4) dt? A factor-by-factor comparison of Eqs. 18.3 and 18.4 shows that the mass m corre- sponds to the inductance L, the spring constant k of the spring corresponds to the recipro- cal of the capacitance 17C, and the position x of the mass corresponds to the charge Q on the capacitor. From mechanics, we know the solution of Eq. 18.4 is given by X = XOCOS Wot, (18.5) where xo is the amplitude and 60 (=26/) is the angular frequency. Recall that Eq. 18.5 is a solution to Eq. 18.4 only if k 00 = (18.6) m We use the zero subscript to distinguish the natural frequency of oscillation, 00, from other frequencies o at which we may choose to drive the system. By analogy, we see that in the electric circuit the charge on the capacitor also oscillates with time according to the equation Q = Qocoswot, (18.7) with an angular frequency wo given by 00 1 V LC VEC (18.8) In the harmonic oscillator the energy is continually transformed from potential to ki- netic and back again during the motion. At the points of maximum displacement the ve- locity is zero, so the energy is entirely potential. Conversely, at the points of zero dis- placement the velocity is a maximum, so the energy is entirely kinetic. At intermediate points the energy is a combination of potential and kinetic. Similarly, in the LC circuit, at times when the capacitor is charged to its maximum value the current is zero. All of the energy is stored as potential energy and none is kinetic (zero current means no moving charges). Conversely, when the current is a maximum the charge on the capacitor is zero. In this case the entire energy of the network is kinetic. At this point it does not take very much imagination to see that the electrical analogai of a damped harmonic oscillator is an electrical circuit containing an inductor (“mass”), a 29 capacitor (“spring”) and a resistor (“friction/damping component”). The harmonic oscilla- tor equation (Eq. 18.4) must be modified by the addition of a term -yv = -y(dx/dt) corre- sponding to the damping force, assumed to be proportional to velocity but in the opposite direction. The constant y is called the damping constant. Thus, the differential equation for the motion of the damped harmonic oscillator is d²x dx +y+ kx = 0. dt? (18.9) dt m- [ dt? Likewise, for the circuit shown in Fig. 18.2, we must modify Eq. 18.2 by the addi- tion of a term representing the voltage across the resistor = RI = R(dQ/dt). The result is d? Q dQ L +7Q=0 (18.10) dt +R + . Note that this has exactly the same form as Eq. 18.9. AS before, L is analogous to m, 1/C to k, and the resistance R to the damping constant y. R C tQ -Q .. variation prole 2 ozonomistiwa svo apimtqmosos i Ils Lane to szol set a word consigaibov o lo sbutions to Spact bus tallema smo Fig. 18.2. An RLC circuit 1600 mannsbo Another aspect of the analogy between the damped harmonic oscillator and the RLC circuit is seen by considering energy relations in the two systems. As already observed, the total mechanical energy of an undamped harmonic oscillator is constant. The effect of the damping force is to continuously decrease the energy, since at each instant its direction is opposite to that of the velocity, and it therefore always does negative work on the sys- tem. Similarly, the total energy of an LC circuit without resistance is constant. The induc- tor and the capacitor store energy but do not remove electrical energy from the circuit. The addition of the resistance provides a means for the system to lose energy through îR power loss, which continuously decreases the electrical energy in the circuit, converting it into heat in the resistor. Of course, a harmonic oscillator which is completely undamped is an idealization which cannot be realized. For example, in experiments with a linear air track one finds that the viscosity of the air layer which supports the glider provides a small but not com- pletely negligible damping force which is approximately proportional to velocity. In the same way, an LC circuit with no resistance is an idealization. Even if no resistor is includ- ed in the circuit, the resistance of the inductor coil wire and the connecting wires is never completely negligible. STO 30 1/2Q. t1/2 + t Fig. 18.3. Charge Q vs. time t for an RLC Circuit Experience with harmonic oscillators (either on the air track or with a simple pen- dulum) also shows that the loss of energy due to the damping force is accompanied by a steady decrease in the amplitude of the oscillations, so the successive displacements from equilibrium become smaller and smaller. By analogy, we expect the electrical os- cillations in the RLC circuit to decrease as shown in Fig. 18.3, the peak charge on the capacitor for any cycle being somewhat smaller than that of the previous cycle. This is just what is meant by the term damped oscillation. How rapidly the oscillations are damped depends on the magnitude of the damping constant y or the resistor R. The larg- er y or R, the more rapid the decay of the oscillations. We can most easily solve Eq. 18.9 (or Eq. 18.10) via energy considerations. To use the energy approach we ask the following question: if the maximum displacement (ampli- tude) for a given cycle is xo, how much energy does the system lose during that cycle? Since power = Fv, the instantaneous rate of loss of energy is the rate of doing work against the damping force which is simply the magnitude of the force (yv) times the veloc- ity v, or yv? This quantity varies during the cycle, but the total energy loss is still given approximately by the average rate of energy loss during the cycle (the average value of yu?) multiplied by the time required for a cycle, which is 1 21 т T= 21 f k (18.11) ave To find the average value of v? we note that the average kinetic energy
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