circuit analysis 2

first and second order circuits

α > ω₀

Roots are real and unequal —> overdamped

α = ω₀

Roots are real and equal —> Critically Damped Case

α < ω₀

Roots are complex conjugates —> Underdamped Case

Overdamped: α > ω₀

x(t) = Ae^(s₁t) + Be^(s₂t)

Critically Damped: α = ω₀

x(t) = (At + B)e^(-αt)

Underdamped: α < ω₀

x(t) = e^(-αt)[Acos(ω_dt) + Bsin(ω_dt)], where ω_d=√ (ω₀² - α²)

for a parallel RLC

α = 1/(2RC), ω₀ = 1/√(LC)

for a series RLC

α = R/2L, ω₀ = 1/√(LC)

Step Response of Series RLC ( in voltage)

LC(d²v/dt²) + RC(dv/dt) + v = Vs

Step Response of Parallel RLC (in current)

LC(d²i/dt²) + (L/R)(di/dt) + i = Is

Series RLC

solve for capacitor voltage: V_C

Parallel RLC

solve for inductor current i(t): I_L

Capacitor Voltage

V_C(0⁺)=V_C(0⁻). Always continuous, 0⁻ denotes the time just before a switching event and 0⁺ denotes the time just after, assuming that the switching event takes place at t=0.

Inductor Current

I_L(0⁺)=I_L(0⁻). Always continuous, 0⁻ denotes the time just before a switching event and 0⁺ denotes the time just after, assuming that the switching event takes place at t=0.

Roots of a Second-Order RLC Circuit

s₁,s₂ = −α ± √(α² − ω₀²)