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

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Unformatted text preview: 24. LRC series circuit 1) Impedance Current i(t) is the same in all elements of the series circuit. R L v( t ) Note! There is no current inside the capacitor, but we can apply Kirchhoff’s rules taking into account displacement current C v R = VR cos ωt i ( t ) = I cos ωt ⇒ v L = V L cos( ωt + π / 2) = −VL sin ωt v = V cos( ωt − π / 2 ) = V sin ωt C C C V R = IR V L = IX L VC = IX C v( t ) = v R + v L + vC = V R cos ωt − (V L − VC ) sin ωt = V cos( ωt + φ ) ⇒ V = IZ Impedance: Z ≡ R 2 + ( X L − X C ) = R 2 + ( ωL − 1 ωC ) X L − X C ωL − 1 ωC tan φ = = R R 2 Z = R cos φ 2 Calculations: Trigonometry: A cos x − B sin x = A A 2 + B 2 = cos φ B A 2 + B 2 cos( x + φ ) , A 2 + B 2 = sin φ tan φ = where B A v( t ) = v R + v L + vC = V R cos ωt + (VC − V L ) sin ωt = V cos( ωt + φ ) , where V = V R2 + (V L − VC ) = I R 2 + ( X L − X C ) ⇒ V = IZ 2 Z ≡ R2 + ( X L − X C ) 2 2 V L − VC IX L − IX C tan φ = = ⇒ VR IR XL − XC tan φ = R cos φ = R z XL − XC sin φ = Z 2) Impedance and phasor v R = VR cos ωt i ( t ) = I cos ωt ⇒ v L = V L cos( ωt + π / 2 ) v = V cos( ωt − π / 2 ) C C V = V R2 + (VL − VC ) I 2 VL φ V L − VC ωt VC VR tan φ = V L − VC VR cos( x + y ) = cos x cos y − sin x sin y 3) Power in AC circuit cos 2 x = 1 / 2; sin x cos x = 0 p = iv = I cos( ωt + φ )V cos ωt = IV ( cos ωt cos φ + sin φ sin ωt ) cos ωt ( ) = IV cos 2 ωt cos φ + sin φ sin ωt cos ωt ⇒ p = 1 IV cos φ = I rmsVrms cos φ 2 Recall that cos φ = R z R=0⇒ p =0 4) Resonance in AC circuit V I= = Z V R + ( XL − XC ) 2 2 = V R 2 + ( ωL − 1 ωC ) 2 I = I max ⇔ Z = Z min ⇔ ωL − 1 ωC ⇔ ω = 1 LC ω0 = 1 f0 = 2π 1 LC 1 LC ω / ω0 4) Q factor (Quality factor) a) Question: How wide is the resonance curve? Z ≡ R + ( ωL − 1 ωC ) 2 2 1 = R 1+ R L 1 ω ω0 − C ω ω0 1 = R 1+ R ( L ω LC − 1 ω LC C 2 Q factor: 1L L X L0 X C 0 Q= = ω0 = = RC R R R b) Definition Energy storied Q ≡ 2π Energy dissipated per period LI 2 2π L L1L Q = 2π 1 2 = = ω0 = TR R RC 2 RI T 1 2 ) 2 ...
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This note was uploaded on 12/10/2011 for the course PHYSICS 222 taught by Professor Frishman during the Spring '11 term at Iowa Central Community College.

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

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