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3.2.5.     Alternating Electric Currents   <CJ chap  23 >

3.2.5.1. Discussion

3.2.5.2. Mathematical

3.2.5.3. Advanced

3.2.5.3.1.     Solve the general RCLV circuit:   L d²q/dt² + R dq/dt  +  (1/C) q  = V₀  Use q(t) = q₀ e^at  Find a

3.2.5.3.2.     Define   g = -R/2L   w₀² =  1/LC  then a = - g     

3.2.5.3.3.     Three cases result from the square root:

3.2.5.3.3.1. Over damped g  > w₀ then q(t) = A e^-gt- t  +  B e^-gt- t  

3.2.5.3.3.2. Critically damped   g  = w₀    then q(t) = A e ^-gt  +  B t e ^-gt  (degenerate case)

3.2.5.3.3.3. Underdamped       g < w₀    then q(t) = A e ^-gt+w₁^t  +  B e ^-gt-w₁^t  where w₁² =  w₀² - g²