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E E 2415

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E E2415

Lecture 7

Natural and Step Responses of RL and RC Circuits

- Energy transferred if v10 v20
- Total system charge is conserved

Initial stored energy:

At equilibrium:

Initial Charge:

Final Charge:

Since

Final stored energy:

Energy consumed in R:

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

Initial stored energy:

At equilibrium:

Initial flux linkage:

Final flux linkage:

Since

Final stored energy:

Energy consumed in R:

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

Separate

the

variables:

Integrate:

KVL:

Exponential of both

sides:

- Capacitor has initial voltage, vo.
- Switch closes at t = 0.
- Capacitor voltage can’t change instantaneously

KCL:

Separate the variables:

Integrate:

Exponential of both

sides:

- Make-before-break switch changes from position a to b at t = 0.
- For t < 0, Io circulates unchanged through inductor.

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

Solution has two parts:

Steady State Response

Transient Response

Determine k by initial conditions:

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

- Switch closes at t = 0.
- Capacitor has initial voltage, Vo.

v-i relationship:

By KVL & Ohm’s Law:

- Response has two parts
- steady state
- transient

- Use initial voltage to determine transient

Steady State Response

Transient Response

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

- Need Thévenin equivalent circuit from terminal pair connected to inductor
- Let initial current = 0A in this example.

Voltage divider to get vx:

Then

Thévenin voltage

Therefore:

Steady state:

Transient:

Use initial conditions to determine k.

Complete response is unbounded: