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E E 2315 Lecture 10 Natural and Step Responses of RL and RC Circuits

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Conservation of Charge (1/4) Energy transferred if v 10 v 20 Total system charge is conserved

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Conservation of Charge (2/4) Initial stored energy: At equilibrium:

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Conservation of Charge (3/4) Initial Charge: Final Charge: Since

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Conservation of Charge (4/4) Final stored energy: Energy consumed in R:

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Conservation of Flux Linkage (1/3) Energy transferred if i 10 i 20 Total system flux linkage is conserved. Initial stored energy: At equilibrium:

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Conservation of Flux Linkage (2/3) Initial flux linkage: Final flux linkage: Since

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Final stored energy: Energy consumed in R: Conservation of Flux Linkage (3/3)

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Natural RL Response (1/2) Inductor has initial current, i o. Switch opens at t = 0 Inductor current can’t change instantaneously

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Natural RL Response (2/2) KVL: Separate the variables: Integrate: Exponential of both sides:

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Natural RC Response (1/2) Capacitor has initial voltage, v o. Switch closes at t = 0. Capacitor voltage can’t change instantaneously KCL: Separate the variables:

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Natural RC Response (2/2) Integrate: Exponential of both sides:

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RL Step Response (1/4) Make-before-break switch changes from position a to b at t = 0. For t < 0, I o circulates unchanged through inductor.

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RL Step Response (2/4) For t > 0, circuit is as below. Initial value of inductor current, i, is I o. The KVL equation provides the differential equation.

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RL Step Response (3/4) Solution has two parts: Steady State Response Transient Response Determine k by initial conditions:

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RL Step Response (4/4) Inductor behaves as a short circuit to DC in steady state mode

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RC Step Response (1/3) Switch closes at t = 0. Capacitor has initial voltage, V o. v-i relationship: By KVL & Ohm’s Law:

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RC Step Response (2/3) Response has two parts –steady state –transient Use initial voltage to determine transient Steady State ResponseTransient Response

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RC Step Response (3/3) Capacitor becomes an open circuit to DC after the transient response has decayed.

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Unbounded Response (1/5) Need Thévenin equivalent circuit from terminal pair connected to inductor Let initial current = 0A in this example.

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Unbounded Response (2/5) Voltage divider to get v x : Then Thévenin voltage

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Unbounded Response (3/5) Therefore:

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Unbounded Response (4/5) Steady state: Transient:

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Unbounded Response (5/5) Use initial conditions to determine k. Complete response is unbounded:

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