Experiment 08: RLC Circuits and Resonance Dr. Pezzaglia

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1 Mar9 RLC Circuit Page Experiment 8: RLC Circuits and Resonance Dr. Pezzaglia Theory When a system at a stable equilibrium is displaced, it will tend to oscillate. An Inductor combined with Capacitor will tend to oscillate if hit with an electrical impulse (i.e. a square wave).. Mechanical Oscillations: The mechanical analogy would be a mass on a spring. If you hit the mass, it will oscillate. The equation of motion for that system would be, v m kx t x v () t k f m where k is the Spring constant, m the mass, v the velocity, x the displacement and is the resonant angular frequency (radians per second) as opposed to the cyclic frequency f (units of Hertz or cycles per second).. Electrical Oscillations: Lord Kelvin in 853 first provided a good theory of the electrical oscillations of RLC circuits. The role of the spring (potential energy) is played by the capacitor, as it stores charge. The role of kinetic energy is played by the inductor, as it has inertia of flowing charge (current). Hence replace the mechanical quantities by their corresponding electrical ones (Voltage plays role of Force, Inductance as mass, /C as spring constant, charge Q as displacement, current I as velocity): I L Q t C Q I () t f LC The square wave signal generator shocks the system regularly when the voltage abruptly changes polarity. The system responds by oscillating (imagine hitting a pendulum with a hammer, it will respond by oscillating).

2 Mar9 RLC Circuit Page 3. Damped Oscillations: All real systems will not oscillate forever as energy will be dissipated due to friction. Oscillations tend to die out over time exponentially. At the right, the oscillation period is second, with a time constant of =.5 seconds for the decay. x( t) e t cos t (3) The constant is called the decay constant or damping constant (the inverse of the time constant) with units of inverse time. Note that the presence of damping makes the oscillating frequency to be less than the resonant frequency. If the friction in the system is higher increases (system decays faster). If then oscillations will not occur. This might be what you want; for example, if you drive your car fast over a speed bump, you don t want the car to bouncy-bounce. The shocks in your car have springs in them, so to prevent oscillation they also have a BIG mechanical resistance. In an electrical circuit, the analogy of friction is resistance. There is always resistance in a circuit, if nothing else, then in the wires themselves. Real inductors often have significant resistance (R L ) because they contain many meters of wire in their coils. The signal generator as well has some resistance inside it (R s is approximately 5 ohms). In our circuit, Rd represents a decade resistor. In the formulas below, the resistance R represents the total resistance in the circuit. R R d R L R L Q R s (4) L L Q R C R The symbol Q is called the quality factor of the system. It is a measure of the average energy stored in the system divided by the energy lost per cycle. A system with a big Q will oscillate for a long time (relative to its period of oscillation). In order for oscillations to occur, Q.5, which is equivalent to saying. In terms of building a circuit, its perhaps more convenient to say that the total resistance in the circuit must be less than the critical value: L R C (5) C

3 Mar9 RLC Circuit Page 3 4. Resonance (Forced Response) Now lets use a sine wave to force the system to oscillate at the frequency we set. When you are close to resonance, the system will oscillate at its maximum amplitude. As you move either above or below resonance, the amplitude will drop off. The fatness (i.e. the bandwidth ) of the curve is described by the Q factor. For a big Q the resonance is very narrow. For a low Q value its quite broad. The exact shape of the curve is quite messy: V V ( f ) (6) f f Q f f This is so messy that there is no way that you can get Excel to fit a curve to it. Instead, to determine the Q factor from a resonant curve, you determine the frequencies below (f ) and above (f ) where the amplitude drops to 7% of the value at resonance (aka as the half power points ). The bandwidth f is defined to be the difference of these frequencies, and it is simply related to the Q factor. f f f f (7) Q f Note in making the graph above we used a Log scale for the frequency. At resonance, the AC voltage across the inductor should be a factor of Q bigger than across the resistor. The same can be said for the capacitor. If you replace the signal generator with an antennae, then all different frequencies will be received from radio stations simultaneously. However, only the station whose frequency is near resonance will create a big signal, and if you take the signal off of either the inductor or capacitor, it will be amplified by a factor of Q.

4 Mar9 RLC Circuit Page 4 Experiments A. Setup: Inductor: Use the big inductor (around 5 H?) inductor. Before using it, be sure to measure its internal resistance. Capacitor: Smaller is better, otherwise the critical resistance would be smaller than the 5 ohms already in the circuit (due to signal generator) and we won t be able to get it to oscillate. Start with perhaps nf (nanofarads). Resistor: Set the decade resistor box to around ohm. In the circuit shown, G is the ground (black wire of signal generator and oscilloscope). Measure the voltage across the resistor using the oscilloscope (point V and G to channel input). Use a square wave at around Hertz to start with. Vary frequency as need to get the full exponential decay. In addition, it might be helpful to run a wire from point H to the External Trigger input of the oscilloscope. In trigger menu select external. Adjust decade resistor (and frequency) as necessary to get a good case of ringing Question : Inductor Report the value of the inductance used Measure and report the resistance of the inductor R L. Question : Capacitor Report the value of the capacitor used Calculate the theoretical resonant (angular) frequency. [Equation c] Calculate resonant frequency f. Question 3: Resistance Report the value decade resistor used (when you have a good picture of ringing). Calculate the total resistance in the circuit. How much smaller is your circuit resistance than the critical resistance (equation 5). Question 4: Theoretical decay constant Calculate the expected decay constant (equation 4b) Calculate the expected oscillation angular frequency (equation 3c) Calculate the expected oscillation frequency in Hertz: f=/().

5 Mar9 RLC Circuit Page 5 B. Measurements of Ringing (Natural Response) Measure the period of the oscillation. From this determine the frequency. Measure the voltages (and times) of 5 successive peaks. Plot these peaks in excel, and fit with a decaying exponential Determine the decay constant from your graph. Question 5: Decay Constant Measurement Does your graph show a good exponential decay? (value of Rsquared?) What is your measures decay constant from the exponential fit? Compare to your theoretical calculation (question 4). Question 6: Oscillation Frequency What is your measured oscillation frequency? Compare to your theoretical calculation (question 4). C. Forced Response: Resonance Change signal generator to sine wave. Put signal generator at (near) the resonant frequency f (see question ) Tune the frequency so that the signal has the maximum amplitude. Measure the (peak) amplitude of the signal as a function of frequency above and below the resonance. In particular, try to find the frequencies below (f ) and above (f ) resonance where the amplitude drops to 7% of the maximum value (at resonance). Plot signal amplitude vs frequency. Label the resonance Label the frequencies f and f, and determine the bandwidth. At resonance, using a multimeter, measure the AC voltage across the resistor, the inductor, the capacitor, the inductor plus capacitor and across all three. Question 7: Resonant Curve Does your curve show the expected shape of a resonance? What is your observed resonant frequency (where amplitude is maximum)? Compare to the theoretical value (equation c) Question 8: Bandwidth What are your lower and upper half power points (f and f )? What is the bandwidth: f= (f - f )? From bandwidth determine Q (equation 7b) Compare to your theory value (equation 4c). Question 9: AC voltages at resonance What is the AC voltage across the resistor at resonance? What is the AC voltage across the inductor? Is it approximately a factor of Q bigger than across the resistor? What is the AC voltage across the Capacitor? Is it approximately a factor of Q bigger than across the resistor? What is the AC voltage across the Capacitor plus Inductor? Comment on why it is so small when the voltages across the individual components is so large.