The equation describing the circuit is one for which there is a constant applied voltage: d2 q dq q + R + = V0 , (1) 2 dt dt C where L is the inductance, R the resistance, C the capacitance, q the charge on the capacitor, and V0 the (constant) applied voltage. L Making the substitution Q = q − CV0 , the equation becomes: L d2 Q dQ Q +R + =0 2 dt dt C (2) This equation could also be written dx d2 x + b + kx = 0, 2 dt dt where we have substituted x for Q, m for L, b for R and k for 1/C. This is an equation for a simple harmonic oscillator with a frictional damping force proportional to the velocity. So we may expect this LRC circuit to have behavior similar to a damped simple harmonic oscillator. A derivation of the solution is given in the Appendix to be found on the course website. In this laboratory, we will measure the transient waveforms, and compare the measurements with predictions. m In this experiment, we use a signal generator with a rectangular output waveform, an example of which is shown in Fig. 1. At time t = 0, the voltage has been at -V0 for a “long” time; any current from the previous transition has decayed to zero, and the voltage on the capacitor is -V0 . With these two initial conditions, the charge can be shown to be: α q = CV0 − 2CV0 e−αt (cos ωt + sin ωt), (3) ω where q is the instantaneous charge, t is time, and ω is the “angular frequency” of the oscillation.1 The constant α is R α= . (4) 2L 1 Recall that the frequency, f and the angular frequency, ω are related by ω = 2πf . 1 Figure 1: Output of a square-wave generator. For no damping, α = 0, the oscillation is at the “natural” angular frequency s ω0 = 1 . LC (5) With damping, the angular frequency is less: s ω= or ω= 1 R 2 − , LC 2L q ω02 − α2 . (6) (7) Equation 3 describes a sinusoidal oscillation, as long as ω is real. (Remember that the sum of sinusoids, with different amplitudes and phases but the same frequency, is still sinusoidal.) In this experiment we will not measure q, but the current i = dq , as determined by dt measuring the voltage drop across the discrete resistor Rload in the circuit, Fig. 4, and Ohm’s law. Therefore from Eq. 3 ” i = 2CV0 e−αt α2 + ω 2 sin ωt. ω # (8) By fitting the measured current as a function of time we will measure α, ω, and the amplitude of the decay, 2CV0 , and compare these to the those values predicted by Eqs. 4-7. 2 Figure 2: Damped oscillations. To keep ω real, Equation 7 tells us that the damping cannot be too large. We must have α < ω0 . For this case we have damped oscillations; an example is shown in Figure 2. If α > ω0 , the system is overdamped, and the oscillations disappear. The special case α = ω0 is called critical damping. Examples of these cases are shown in Fig. 3. It can be shown that for critical damping, the current goes to zero faster than in the overdamped case. Figure 3: Critical and overdamped oscillations. 3 2 Experimental Setup and Data-taking Ch 1 Ch 2 62.5Ω 50Ω 0.8H .047µ F 100Ω Figure 4: Schematic diagram of the apparatus. The 50 and 62.5 Ω resistors are the internal resistances of the generator and the inductor. The values above are representative. You will want to measure as many of them as you can. Set up the circuit shown in Figure 4. The square-wave generator has the same effect as a battery or power supply being switched on and off. In calculating the decay constant α, the resistance R is the total resistance in the circuit, including the resistance of the signal generator, as well as those of the variable load resistor and the inductor. But the voltage drop across the load resistor is just iRload . 2.1 Damped Oscillations 1. Set the signal generator square wave amplitude to 5.0 V or greater. Measure and record its value. Trigger the oscilloscope on Channel 1 (the signal generator output) and look at Channel 2, which is the voltage across the load resistor. 2. Adjust the frequency of the signal generator so that it is low enough that oscillations decay to zero before the voltage makes a transition to the opposite sign. 3. Adjust the timing and vertical gain to fill the display with about 15 cycles of the oscillation. Then read out the scope with the “Open Choice Desktop” program. Making sure you have set the tabular display to include time. Save the data to the desktop using the button “Save as”. Choose file type CSV (comma separated values). 4. Using the matlab code, fit the voltage across the load resistor, VRload , using Equations 8, and the fact that VRload = iRload . Don’t include any negative times in order to make correct predictions. Make a plot which shows the data and fit together on the same graph. Compare fit parameters, α and ω, to the expected values. 4 5. When fitting the data, the Matlab code uses the lsqnonlin function, http://www.mathworks.com/help/optim/ug/lsqnonlin.html: a = lsqnonlin(fun,a0) starts at the point a0 and finds a minimum of the sum of squares of the functions described in “fun” In the code what is minimized is the value χ2 2 χ = n X 1 observed value − expected value(a) uncertainty !2 , (9) where the sum is over the number of points n. χ2 is an indicator of the agreement between the observed value and the expected value as a function of the variable a. If the agreement is good χ2 will be on the order of n and if it is not good it will be much greater than n. A somewhat better procedure is to compare χ2 to the number of degrees of freedom, d = n − (number of fit parameters). We define a quantity called the reduced “chi-squared” χ̃2 χ̃2 = χ2 /d. (10) Now if the agreement is good χ̃2 will be on the order of unity and if it is not good it will be much greater than one. Record χ̃2 from your fit and comment on it. The code takes the uncertainty to be 1% of the maximum amplitude. 2.2 Critical Damping Replace the load resistor with a variable resistor. Increase this resistance until critical damping is reached. Check to see if (R/2L) = q (1/LC). Remember to disconnect the variable resistor from the circuit when measuring its resistance. Again, R in the above equation is the total series resistance in the circuit. 5Do you similar assignment and would want someone to complete it for you? Click on the ORDER NOW option to get instant services at essayloop.com
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