lecture23 - 24. LRC series circuit 1) Impedance R Current...

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Unformatted text preview: 24. LRC series circuit 1) Impedance R Current i(t) is the same in all elements of the series circuit. L v( t ) C Note! There is no current inside the capacitor, but we can apply Kirchhoff's rules taking into account displacement current v R = VR cos t i ( t ) = I cos t v L = V L cos( t + / 2) = -VL sin t v = V cos( t - / 2 ) = V sin t C C C V R = IR V L = IX L VC = IX C v( t ) = v R + v L + vC = V R cos t - (V L - VC ) sin t = V cos( t + ) V = IZ Impedance: Z R 2 + ( X L - X C ) = R 2 + ( L - 1 C ) 2 2 X L - X C L - 1 C tan = = R R Z = R cos Calculations: Trigonometry: A cos x - B sin x = A B A 2 + B 2 = cos A 2 + B 2 = sin A 2 + B 2 cos( x + ) , tan = where B A v( t ) = v R + v L + vC = V R cos t + (VC - V L ) sin t = V cos( t + ) , where V = V R2 + (V L - VC ) = I R 2 + ( X L - X C ) V = IZ 2 2 Z R2 + ( X L - X C ) 2 V L - VC IX L - IX C tan = = VR IR XL - XC tan = R cos = R z XL - XC sin = Z 2) Impedance and phasor v R = VR cos t i ( t ) = I cos t v L = V L cos( t + / 2 ) v = V cos( t - / 2 ) C C I V = V R2 + (VL - VC ) VL 2 V L - VC VR tan = t VC V L - VC VR 3) Power in AC circuit cos( x + y ) = cos x cos y - sin x sin y cos 2 x = 1 / 2; sin x cos x = 0 = IV ( cos t cos + sin sin t ) cos t p = iv = I cos( t + )V cos t = IV cos 2 t cos + sin sin t cos t ( ) p = 1 IV cos = I rmsVrms cos 2 Recall that cos = R z R=0 p =0 4) Resonance in AC circuit V I= = Z V R + ( XL - XC ) 2 2 = V R 2 + ( L - 1 C ) 2 I = I max Z = Z min L - 1 C = 1 LC 0 = 1 LC 1 LC 1 f0 = 2 / 0 4) Q factor (Quality factor) a) Question: How wide is the resonance curve? Z R + ( L - 1 C ) 2 2 1 = R 1+ R 2 L LC - 1 LC C ( ) 2 1 = R 1+ R L 1 0 - C 0 Q factor: b) Definition 1 L L X L0 X C 0 Q= = 0 = = R C R R R Energy storied Q 2 Energy dissipated per period LI 2 2 L L 1 L Q = 2 1 2 = = 0 = T R R R C 2 RI T 1 2 ...
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lecture23 - 24. LRC series circuit 1) Impedance R Current...

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