E E 2415

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# E E 2415 - PowerPoint PPT Presentation

E E 2415. Lecture 7 Natural and Step Responses of RL and RC Circuits. Conservation of Charge (1/4). Energy transferred if v 10  v 20 Total system charge is conserved. Conservation of Charge (2/4). Initial stored energy:. At equilibrium:. Conservation of Charge (3/4). Initial Charge:.

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### E E2415

Lecture 7

Natural 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:

• Energy transferred if i10 i20
• Total system flux linkage is conserved.

Initial stored energy:

At equilibrium:

Since

Final stored energy:

Energy consumed in R:

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)

Separate

the

variables:

Integrate:

KVL:

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:

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
• transient
• Use initial voltage to determine transient

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 (4/5)