Download E E 2315 Circuits I Lecture 9

E E 2315
Lecture 10
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:
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
Conservation of Flux Linkage (3/3)
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)
Integrate:
KVL:
Separate the variables:
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: