Unbounded Response of an RC Circuit Learning Goal: To analyze an RC circuit with a dependent source that exhibits an unbounded response, and to determine the time when a component such as the capacitor breaks down or goes into a saturation state. Sometimes when an RC circuit includes a dependent source, it may exhibit an unbounded response and grow exponentially with time. In this situation, the dependent source may result in a negative Thevenin equivalent resistance with respect to the capacitor. This negative resistance creates a negative time constant which causes the unbounded response. This response may cause the capacitor to break down when the voltage reaches a limiting value. (Figure 1) Figure 1 of 1 R₂ www b t=0 Part A - Find the initial voltage across the capacitor at t = 0 before the switch moves. V₁ = 4.0 V, R₁ = 6.0 N, R2 = 7.0 N, R3 = 8.0 N, R₁ = 3.0 N, C = 5.0 F, α = 90.0 Express your answer to three significant figures and include the appropriate units. v(0) = v(0+): == Value Submit Units Hints My Answers Give Up Review Part ? Part B - Find the Thevenin equivalent resistance seen by the capacitor at t=0+. Express your answer to three significant figures and include the appropriate units. ? < RTH = Value Units Submit Hints My Answers Give Up Review Part 0 +7 1 V R₁ αἰ R3 w Part C - Find the time differential equation that describes the circuit at t=0*. V V₁ = 4.0 V, R₁ = 6.0, R2 = 7.0 N, R3 = 8.0 N, R₁ = 3.0, C = 5.0 F, α = 90.0 dv = Express your answer as an algebraic equation of the form, 0 du + Kv, where K is a constant. dt 0 = VO ΑΣΦ Π vec ? Part D - Write an expression for the capacitor voltage, v(t), for t≥ 0. V₁ = 4.0 V, R₁ = 6.0 N, R₂ = 7.0, R3 = 8.0 N, R₁ = 3.0 2, C = 5.0 F, α = 90.0 4 Express your answer as an algebraic expression as a function of time. ΜΕ ΑΣΦ ΑΣΦ Π vec v(t) = Submit Hints My Answers Give Up Review Part ? V Part E - Assume that the capacitor breaks down when the terminal voltage reaches Vx = 90.0 V, how many seconds elapse before the capacitor breaks down. V₁ = 4.0 V, R₁ = 6.0 N, R2 = 7.0 N, R3 = 8.0 N, R₁ = 3.0 N, C = 5.0 F, α = 90.0 ✗. Express your answer to three significant figures and include the appropriate units. t = HA Value ? Units Submit Hints My Answers Give Up Review Part Provide Feedback Continue