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Chapter 4: Oscillations 19 1. Series connection: V = IZ , Z tot = ± i Z i ,L tot = ± i L i , 1 C tot = ± i 1 C i ,Q = Z 0 R ,Z = R (1 + iQ δ ) 2. parallel connection: V = IZ , 1 Z tot = ± i 1 Z i , 1 L tot = ± i 1 L i ,C tot = ± i C i ,Q = R Z 0 ,Z = R 1+ iQ δ Here, Z 0 = ² L C and ω 0 = 1 LC . The power given by a source is given by P ( t )= V ( t ) · I ( t ) , so ± P ² t = ˆ V e f ˆ I e f cos( Δ φ ) = 1 2 ˆ V ˆ I cos( φ v - φ i )= 1 2 ˆ I 2 Re( Z )= 1 2 ˆ V 2 Re(1 /Z ) , where cos( Δ φ ) is the work factor. 4.4 Waves in long conductors These cables are in use for signal transfer, e.g. coax cable. For them holds: Z 0 = ² dL dx dx dC . The transmission velocity is given by v = ² dx dL dx dC . 4.5 Coupled conductors and transformers For two coils enclosing each others ±ux holds: if Φ 12 is the part of the ±ux originating from I 2 through coil 2 which is enclosed by coil 1, than holds Φ 12 = M 12 I 2 , Φ 21 = M 21 I 1 . For the coef²cients of mutual induction M ij holds: M 12 = M 21 := M = k ³ L 1 L 2 = N 1 Φ 1 I 2 = N 2 Φ 2 I 1 N 1 N 2 where 0 k 1 is the coupling factor . For a transformer is
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Unformatted text preview: k ≈ 1 . At full load holds: V 1 V 2 = I 2 I 1 =-i ω M i ω L 2 + R load ≈ -² L 1 L 2 =-N 1 N 2 4.6 Pendulums The oscillation time T = 1 /f , and for different types of pendulums is given by: • Oscillating spring: T = 2 π ³ m/C if the spring force is given by F = C · Δ l . • Physical pendulum: T = 2 π ³ I/ τ with τ the moment of force and I the moment of inertia. • Torsion pendulum: T = 2 π ³ I/ κ with κ = 2 lm π r 4 Δ ϕ the constant of torsion and I the moment of inertia. • Mathematical pendulum: T = 2 π ³ l/g with g the acceleration of gravity and l the length of the pendu-lum....
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This note was uploaded on 11/16/2010 for the course ENGR 201 taught by Professor Elder during the Spring '10 term at Blinn College.

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