Difference between revisions of "TF EIM Chapt3"

Revision as of 20:18, 7 February 2011

Forest_Electronic_Instrumentation_and_Measurement

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

Difference between revisions of "TF EIM Chapt3"

Revision as of 20:18, 7 February 2011

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