V; t=0 R Vo Transfer function : LS 71,2 2RC R2C52 +25+R 3 + 5 + 1 하는 RC radis ± LC 2R 1-4R²C 2RC V L
At time t=0 the switch is clsoed and DC voltage Vi(t) is applied as input and Vo(t) is the voltage across the inductor and the capacitor. Using this information, find the possible general solutions for Vo(t). This includes finding general solutions based on the nature of the poles of this system.
1. **Overdamped (Distinct Real Poles):** Vo(t)\=A1eλ1t+A2eλ2tV\_o(t) = A\_1 e^{\\lambda\_1 t} + A\_2 e^{\\lambda\_2 t}Vo(t)\=A1eλ1t+A2eλ2t where λ1\\lambda\_1λ1 and λ2\\lambda\_2λ2 are the distinct real poles.
2. **Critically Damped (Repeated Real Poles):** Vo(t)\=(A1+A2t)eλtV\_o(t) = (A\_1 + A\_2 t) e^{\\lambda t}Vo(t)\=(A1+A2t)eλt where λ\\lambdaλ is the repeated real pole.
3. **Underdamped (Complex Conjugate Poles):** Vo(t)\=e−ζωnt(A1cos(ωdt)+A2sin(ωdt))V\_o(t) = e^{-\\zeta \\omega\_n t} (A\_1 \\cos(\\omega\_d t) + A\_2 \\sin(\\omega\_d t))Vo(t)\=e−ζωnt(A1cos(ωdt)+A2sin(ωdt)) where:
* ωn\=1LC\\omega\_n = \\frac{1}{\\sqrt{LC}}ωn\=LC1 is the natural frequency.
* ζ\=R2CL\\zeta = \\frac{R}{2} \\sqrt{\\frac{C}{L}}ζ\=2RLC is the damping factor.
* ωd\=ωn1−ζ2\\omega\_d = \\omega\_n \\sqrt{1 - \\zeta^2}ωd\=ωn1−ζ2 is the damped natural frequency.
(note: do not use laplace transformations)
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