Difference between revisions of "TF EIM Chapt3"

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=== Phase shift===
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=== gain===
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Loop Theorem
  
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<math>X_C = \frac{-i}{\omega C} \Rightarrow</math> Voltage lags current for a capacitor
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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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<math>X_L = i \omega L \Rightarrow</math> Voltage leads current for Inductor
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or
  
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[[File:TF_EIM_RLC_Phasor.png | 300 px]]
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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} = \frac{1}{LC}</math> then the AC signal is attenuated.
  
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:<math>\tan(\phi) = \frac{X_{out}}{R_L} = \frac{\omega L - \frac{1}{\omega C}}{R}= \frac{\omega^2 - \omega_o^2}{\omega_0^2R}</math>
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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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If <math>\omega < \omega_0</math> then the phase shift is negative
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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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If <math>\omega > \omega_0</math> then the phase shift is positive
 
  
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::<math> = \sqrt{ \left [ \frac{X_{tot} }{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} \right ]  \left [ \frac{X_{tot} }{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} \right ]^*}</math>
  
 
[[Forest_Electronic_Instrumentation_and_Measurement]]
 
[[Forest_Electronic_Instrumentation_and_Measurement]]

Revision as of 05:02, 2 February 2011

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} = \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{X_{tot} }{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} \right ] \left [ \frac{X_{tot} }{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} \right ]^*}[/math]

Forest_Electronic_Instrumentation_and_Measurement

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