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| === gain===
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| Loop Theorem
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| :<math>\Rightarrow V= I(R+X_{tot}) = I \left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )</math>
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| or
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| :<math> I= \frac{V_0 e^{i \omega t}}{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )}</math>
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| ;Notice
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| :When <math>\omega \approx \omega_{LC} = \sqrt{\frac{1}{LC}}</math> then the AC signal is attenuated.
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| Looking at the Voltage divider aspect of the circuit
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| :<math>V_{AB}=V_{out} = \frac{X_{tot} }{R + X_{tot}}V_{in}</math>
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| :<math>\left |\frac{ V_{out}} {V_{in}}\right | = \sqrt{ \left [ \frac{X_{tot} }{R + X_{tot}} \right ] \left [ \frac{X_{tot} }{R + X_{tot}} \right ]^*}</math>
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| ::<math> = \sqrt{ \left [ \frac{\frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} }{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} \right ] \left [ \frac{\frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} }{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} \right ]^*}</math>
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| ::<math> = \sqrt{ \frac{ \omega^2 L^2 \omega_{LC}^4}{R^2(\omega_{LC}^2 - \omega^2)^2 - \omega^2L^2 \omega_{LC}^4}}</math>
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| ::<math> = \sqrt{ \frac{ \omega^2 }{C^2R^2(\omega_{LC}^2 - \omega^2)^2 - \omega^2}}</math>
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| ::<math> = \sqrt{ \frac{ \omega^2 }{\frac{1}{\omega_{RC}^2}(\omega_{LC}^2 - \omega^2)^2 - \omega^2}}</math>
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| {| border="1" |cellpadding="20" cellspacing="0
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| |<math>L = 33 \mu H</math> and <math>C = 1 \mu F </math> and R=200 <math>\Omega</math>
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| |[[File:TF_EIM_LC_paral_Gain.png | 200 px]]
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| |<math>\Rightarrow \omega = 174077.66</math> rad/s or <math>\nu = \frac{\omega}{2 \pi} = 27705</math> Hz
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| |}
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| ==== Q and Bandwidth====
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| In the above circuit
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| : <math>\nu = \omega/2 \pi = \frac{1}{2 \pi \sqrt{LC}}= 27705 Hz</math>
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| The inductors reactance at this resonance frequency is
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| :<math>X_{L} = \omega L = 2 \pi \nu L = \frac{L}{\sqrt{LC}} = \sqrt{\frac{L}{C}} = \sqrt{\frac{33 \times 10^{-6} H}{1 \times 10^{-6}F}} = 5.7 \Omega</math>
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| ;Bandwith: The most common definition for the Bandwidth of this circuit is the frequency range over which the output decreases by 3 dB. This correspond to the frequency at which the circuits power is cut in half from the resonance frequency.
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| :<math>P = I^2 R</math>
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| :<math>P_{1/2} = (\sqrt{2}I)^2 R</math>
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| [[File:TF_EIM_BandWidthDef_LC.gif | 200 px]]
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| ;Quality Factor
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| :The Quality factor for this circuit may be expressed in term of the bandwidth such that
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| ::<math>Q=\frac{\omega_0}{BW}=\frac{\omega_0}{\Delta \omega}</math>
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| [[Forest_Electronic_Instrumentation_and_Measurement]] | | [[Forest_Electronic_Instrumentation_and_Measurement]] |