The name is absent

124

Solve the characteristic equation s² + 2as + ωθ

LCR_m

1 R₁

O' — ^-———— -f∙ ---

2CR_m 2L

-2a ± ʌ/4cv² - 4ω²

^ 1

³~~2CR_m

ʌʃɪ
2L V ^v 2CR_m

-^¹)2 _ /_!_ ₊

2L^{} y}LC LCR_m

Now sɪ and S2 are defined, along with ω₀ the resonant frequency (in radians∕sec) and a the
damping factor

We look for a solution of the form υ(t) = K₁e^s1t + K₂e^s2t + K₃

We need to solve for K₁, K₂ and K₃ from the final and initial conditions

K₁e~^o° + K₂e °° + K₃ = i_s(R₁∖∖R_m)

■ ⅛[K₁e∙>< + ¾_e∙=< + ¾] = ⅛∣_ho = ≈√C_m

Kie⁰ + K₂e⁰ + K₃ = υ(0) = 0

K₃ - ⅞ ( .Ri I ∣,R_to ) - i_s β₁¹₊‰

■ s₁K₁ + s₂K₂ = ^∖^ = i_s∣C_m

K₁+ K₂ = -K₃

Now the system is fully defined