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3.5 Resonance peak properties
Examining Fig. 10 we see that the resonance peak increases in height with higher Q (see (35) for more on Q). Below we will use (36) to determine this height and also the width of the resonance peaks and how they vary with Q. After we do this, (in the next posting) we shall examine an alternate way to view a driven resonator and gain a qualitative understanding why the Q has these effects on the height and width of the resonance peak.
Resonance curve peak height
When ω = ω_{0} , (36) becomes:
Thus we see that the amplitude at resonance is proportional to the Q as Fig. 10 suggests.
Half power points, bandwidth
As a measure of how sharp the resonance curve is, we introduce the concepts of half power points and bandwidth. The half power points are the points on either side of the resonance at which the amplitude of oscillations is 70.7% the values at the peak (i.e. 1/√2 times the peak value). Because energy is proportional to the amplitude squared (i.e. U = kx^{2} ) at these points the energy in the resonator (and the power dissipated in the resonator) is half the peak value, hence the name. These points are marked as dots on the resonance curves in Fig. 10.
The bandwidth is defined as the horizontal distance between the two half power points (in frequency units). We next derive the bandwidth in terms of the Q for resonators of large Q's.
Table 1 - derivation of bandwidth as a function of Q for resonators with large Q's. | |
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For large Q's, the bandwidth can be solved using Equation (36). |
(36) Copy of Equation (36). |
The part of (36) that changes rapidly near ω = ω_{0} is the quantity inside the large parenthesis on the right side of the equation (36). We will focus on that factor, i.e.: | (51) |
For this derivation, we will assume that the factor before the parenthesis can be approximated as constant for ω's near the resonance peak, i.e.: | (52) |
At the peak ω = ω_{0} and thus equals 0 and the quantity inside the large parenthesis on the right side of the equation is equal to 1, i.e.: | (53) |
In the next few boxes, we will show that the half power points occur when: | (54) |
At these values of , the quantity inside the large parenthesis on the right side of equation(36) becomes: | (55) |
The energy in the resonator is proportional to the magnitude of this quantity squared: | (56) |
Thus we have shown that when (54) holds, the energy in the resonator is half its peak value. We next calculate the difference between two values of ω for which (54) holds. We call these two values ω_{1} and ω_{2} (the left and the right half power points). We rearrange (54) as: |
and (57) |
We manipulate the left sides of the two equations of (57): |
(58) |
Equating this modified left side to the right side of (57) for our two half power points ω_{1} and ω_{2}, we have: |
and (59) |
Subtracting the two equations (59), one from the other and dividing by 2, we have: | (61) |
Rearranging a little, we have the difference in frequencies i.e. the bandwidth Δω_{½} (in angular frequency units, i.e. radians per second): | (62) |
In summary, we have shown that:
for large Q values, i.e. Q >> 1 . Thus bandwidth and Q are inversely related. The larger the Q the more narrow the resonance peak.
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