Complete Response of RC and RL Circuits

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EE101 EE CIRCUITS 1

1T SY 2014-2015

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Complete Response of RC and RLCircuits

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LEARNING OUTCOMES

Solve for the complete response of RC circuits usingLaplace transformation and by getting the natural

response and forced response.

Solve for the complete response of RL circuits using

Laplace transformation and by getting the sum of thenatural response and forced response.

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FIRST-ORDER CIRCUITS

A first-order circuit is characterized by a first-orderdifferential equation.

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= +

() = + −

= +

() =

+

−

= +

=

+

= +

= +

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COMPLETE RESPONSE OF RC CIRCUITS

Three steps to find out the step response of an RC circuit:

1. The initial capacitor voltage, (0)

2. The final capacitor voltage, (∞) 3. The time constant, = RC

() = + −

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COMPLETE RESPONSE OF RL CIRCUITS

Three steps to find out the step response of an RL circuit:

1. The initial inductor current, (0)

2. The final inductor current, (∞)

3. The time constant, = L/R

() = +

−

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Natural response: behavior of the circuit itself, with no external

sources of excitation; eventually dies out

Forced response: produced

by the circuit when an

external “force’’ is applied

Complete Response in Terms of Source of the

Responses

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Complete Response in Terms of Permanency

of Responses

Transient response: temporary response that will die out with

time/decay to zero as time approaches infinity

Steady-state response:

behavior of the circuit a long

time after an external

excitation is applied/portionthat remains after the

transient response has died

out

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Time Constant

The rate at which capacitor voltage charges by 63.2% ordischarges by 36.8% of its initial value.

The rate at which inductor current builds-up by 63.2% or

decays by 36.8% of its initial value.

= = /

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Time Constant

Transients last for only five time constants 5 Capacitor is fully discharged (or charged) after five time constants (same

as in the case of inductor).

In other words, it takes for the circuit to reach its final state or

steady-state when no changes take place with time.

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Universal Time Constant Chart

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Time Constant

At any rate, whether the time constant is small or large, the circuit reaches

steady state in five time constants

Smaller the time constant:

• Shorter duration of the transient

• Fast response

• Reaches its steady state/final state

quickly

Larger the time constant:

• Longer duration of transient

• Slow response

• Takes longer to reach its steady

state/final state

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Capacitor Voltage and Current Response

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Inductor Voltage and Current Response

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ILLUSTRATION 1

If the switch in the circuit below opens at t = 0, find v(t)for t>0 and w(0).

= 8−

V, 0 = 5.33 J MCBLOYOLA

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ILLUSTRATION 2

The switch in the circuit below has been closed for a longtime. At t = 0, the switch is opened. Calculate i(t) for t>0.

What about the voltage

across the 16-Ω resistor? = 6− A MCBLOYOLA

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ILLUSTRATION 3

The switch in the figure below has been in position A fora long time. At t = 0, the switch moves to B. Determine

v(t) for t>0 and calculate its value at t =1 s and t = 4 s.

= 30 15−. V, 1 = 20.9 V, 4 = 27.97 V MCBLOYOLA

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ILLUSTRATION 4

The switch in the circuit below has been closed for a longtime. It opens at t=0. Find i(t) for t>0.

= 6 + 3− A MCBLOYOLA

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MORE ILLUSTRATIONS

5.How long does it take a 20-μF capacitor charged to 150V to discharge through a 3-MΩ resistor, and what is the

total energy dissipated in the resistor?

6. A 2-μF capacitor, initially charged to 300 V, is discharged

through a 270-k Ω

resistor. What is the capacitor voltageat 0.25 s after the capacitor starts to discharge?

7. Closing a switch connects in series a 200-V source, a 2-

MΩ resistor, and an uncharged 0.1-μF capacitor. Find the

time required for the capacitor voltage to reach 50 V.

300 , 0.225

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189

57.5

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MORE ILLUSTRATIONS

8. A coil for a relay has a resistance of 30 Ω and an

inductance of 2 H. If the relay requires 250 mA tooperate, how soon will it operate after 12 V is appliedto the coil?

9. How long after a short circuit is placed across a coil

carrying a current of 2 A does the current go to zero ifthe coil has 1.2 H of inductance and 40 Ω resistance?Also, how much energy is dissipated?

10. A short is placed across a coil that at the time is

carrying 0.5 A. If the coil has an inductance of 0.5 H anda resistance of 2 Ω, what is the coil current 0.1 s afterthe short is applied?

65.4

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0.15 ,2.4

0.335

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UNIT-STEP FUNCTION

Unit step function delayed by

Unit step function advanced by

Unit step function

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UNIT-STEP FUNCTION

Represent an abrupt change for:

1. for voltage source.

2. for current source:

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ILLUSTRATION 11

In the figure below, the switch has been closed for a longtime and is opened at t=0. Find i and v for t>0.

= (20 10−.) V ; = (1 + −.) A MCBLOYOLA

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REFERENCES

Please refer to course syllabus.

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