Second Order Circuits

ES-3

Download:http://www.ece.tufts.edu/~hopwood/downloads/es3/SecondOrderCircuits.ppt

Second Order Circuits

• Contain two independent reactive components

• This results in a second order differential equation containing d2i/dt2 or d2v/dt2

Example: Series RLC circuit

+-

VS(t)

L

C

Ri(t)

KVL:VL + VR + VC = Vs

0)(2

0:

.1

2...

1)(2

1)(

0)(

)()0()(1

)()()()(

202

2

0

202

2

2

2

2

2

0

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Three Cases

• Case 1 - Overdamped: large R:

LCL

R 1

2

tstsn eKeKti

s

s

2121

20

22

20

21

)(

:lsexponentia decaying twois response natural The

0

0

Three Cases

• Case 2 - Underdamped: small R:

LCL

R 1

2

220

)(2

)(1

21

220

20

22

220

220

20

21

)...cos(

)(

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Three Cases

• Case 3 – Critically damped:

LCL

R 1

2

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20

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isresponsenaturalThe

00

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Examples

+-

VS(t)

L

C

Ri(t)

L = 10uH

C = 10nF

o = 3.16x106 rad/sec (504 kHz)

R = 10, 63.2, and 1000 ohms

= 5x105, 3.16x106, and 5x107 s-1

UnderdampedCritically Damped

Overdamped

t

vs

Overdamped: R=1000

Time

0s 0.2us 0.4us 0.6us 0.8us 1.0us 1.2us 1.4us 1.6us 1.8us 2.0usV(R1:2)

0V

5V

10V

Underdamped: R=10

Time

0s 1us 2us 3us 4us 5us 6us 7us 8us 9us 10usV(R1:2)

-10V

-5V

0V

5V

10V

Critically Damped: R=63.2

Time

0s 1us 2us 3us 4us 5us 6us 7us 8us 9us 10usV(R1:2)

-5V

0V

5V

10V

Mechanical Analogy

Automobile: Mass inductorSuspension: Spring capacitorShock Absorber: Damper resistor

Force voltageVelocity current

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