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Lecture 7Natural and Step Responses of RL and RC Circuits

Conservation of Charge (1/4)

• Energy transferred if v10 v20

• Total system charge is conserved

Conservation of Charge (2/4)

Initial stored energy:

At equilibrium:

Conservation of Charge (3/4)

Initial Charge:Final Charge:

Since

Conservation of Charge (4/4)

Final stored energy:

Energy consumed in R:

Conservation of Flux Linkage (1/3)

• Energy transferred if i10 i20

• Total system flux linkage is conserved.

Initial stored energy:

At equilibrium:

Conservation of Flux Linkage (2/3)

Initial flux linkage:

Final flux linkage:

Since

Final stored energy:

Energy consumed in R:

Conservation of Flux Linkage (3/3)

Natural RL Response (1/2)

• Inductor has initial current, io.• Switch opens at t = 0• Inductor current can’t change

instantaneously

Natural RL Response (2/2)

KVL:

Separatethe variables:

Integrate:

Exponential of both sides:

Natural RC Response (1/2)• Capacitor has initial voltage, vo.

• Switch closes at t = 0.• Capacitor voltage can’t change

instantaneously

KCL:

Separate the variables:

Natural RC Response (2/2)

Integrate:

Exponential of both sides:

RL Step Response (1/4)

• Make-before-break switch changes from position a to b at t = 0.

• For t < 0, Io circulates unchanged through inductor.

RL Step Response (2/4)

• For t > 0, circuit is as below.• Initial value of inductor current, i, is Io.• The KVL equation provides the

differential equation.

RL Step Response (3/4)

Solution has two parts:

Steady State Response

Transient Response

Determine k by initial conditions:

RL Step Response (4/4)

• Inductor behaves as a short circuit to DC in steady state mode

RC Step Response (1/3)• Switch closes at t = 0.

• Capacitor has initial voltage, Vo.

v-i relationship:

By KVL & Ohm’s Law:

RC Step Response (2/3)

• Response has two parts– steady state– transient

• Use initial voltage to determine transient

Steady State Response Transient Response

RC Step Response (3/3)

• Capacitor becomes an open circuit to DC after the transient response has decayed.

Unbounded Response (1/5)

• Need Thévenin equivalent circuit from terminal pair connected to inductor

• Let initial current = 0A in this example.

Unbounded Response (2/5)

Voltage divider to get vx:

Then Thévenin voltage

Unbounded Response (3/5)

Therefore:

Unbounded Response (4/5)

Steady state:

Transient:

Unbounded Response (5/5)

Use initial conditions to determine k.

Complete response is unbounded: