4. Quality factor, 4.2 in simple circuits The quality factor, Q, is extremely us

Question

4. Quality factor, 4.2 in simple circuits The quality factor, Q, is extremely useful in many disciplines. Use the general definition of O above for the following In the lecture notes, Q was approximated by the ratio of a centre frequency and a 3dB (a) Find the Q for a series RC circuit in terms of the reactance, Xc and resistance, R. bandwidth, which is accurate for large Q (b) Find the Q for a parallel RL circuit in terms of the reactance, X, and resistance, R (c) Find the Q for a series RL circuit in terms of the reactance, X^ and resistance, R (d) Find the Q for a parallel RC circuit in terms of the reactance, Xc and resistance, R. (e) Explain what is meant by resonance in terms of the impedance or admittance of a circuit. For a general alternating harmonic system, can be defined by the ratio of the total energy stored to the dissipated power, and the dependency (dropped below) is Q(o), so Q as a frequency dependent quantity, is often called a "narrow band" parameter. In an LC circuit at resonance, the energy is being completely swapped between the capacitor and the inductor every cycle. So the total stored energy corresponds to the maximum energy in either the L or the C. O refers to the so-called "unloaded because the addition of an extra resistor at the load 4.3 0 elsewhere example or source changes the to a "loaded Q" value (a) A Superball manufacturer claims a Q of 100 for their ball. If you test this by bouncing it on a hard surface and in a vacuum (so there is no air resistance in the experiment), what proportion of its height should the ball return to after each bounce cycle? Q an be as low as 0.5 for audio type signals and circuits; is reported to be about 1000 for a tuning fork (giving a sufficiently well-defined frequency for a musician to identify pitch accurately); a crystal for a standard crystal oscillator is about 2(10)6 to give sufficiently high purity sinusoid for frequency shifting and demodulation in a radio; and for extremely thin linewidth (ie spectral delta-like function) lasers apparently can have a Q of (10)12 (b) Owing to your assignments being too short, you have time compare this with a ball made of fake flubber, badly copied from professor Brainard's original invention. The O of the fake flubber ball is known from Industry Canada testing, to be 103dB, well short of the original invention. Calculate how many bounce cycles are required for the fake flubber ball for it to be bouncing at half of its original height The definition can be written: stored disspated energy per radian cycle 4.4 The return of Dirac (a) Using the relationship re, what is the Q for a Dirac delta function? (b) What is this Q value in dB? (c) We saw in the last Assignment that the Dirac family products can be dodgy to deal with - 2xstored ener disspated energy per cycle 3dB stored energy power los.s 4.1 Simple Q Show that [u()S()dt--3dB. (Hint: use integration by parts) Express the maximum spread of these Os (maximum Q to minimum Q, from the 6h bullet point above) in dB. You should not need a calculator!

Explanation / Answer

4.1) From the given data,

Q1=0.5 for audio type signals

Q2= 1000 for tunnig fork

Q3=2(10)6

Q4=(10)12

The formula to express Q in DB is Q= 20 * log10 Q

By solving it Q1= 20 * log10 0.5 =20 * log10 (5*10-1)= 20 * (log105 -log1010) = 20 * (log105-1)=

20 * (0.69 -1)= 20* -0.31 =-6.2 dB

Q2= 20 * log10103 =3* 20 * log10= 60 dB

Q3=20 * log(2.106) =20 * (log2+ 6log10) = 20 * (0.3 + 6) = 126 dB

Q4 = 20 * log(1012) = 12 * 20= 240 dB

From the values it’s obvious that Q4>Q3>Q2>Q1

4.2)

For series RC circuit Q = XC / R

For series RL circuit Q = XL / R

Q for parallel is converse of series

For parallel RC circuit Q = R / XC

For parallel RL circuit Q = R / XL

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