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Lab. 7 RLC 회로

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이 실험의 목표

RLC 회로의 동작 특성 이해 Damping Factor에 따른 Overdamping과 Underdamping 특성 Resonance Frequency란 무엇인가? 정현파 입력 신호에 대한 Stead-State Response 특성

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RLC Circuit

Applying Kirchhoff’s voltage law, we have( ) ( ) ( ) ( )C s

di tL i t R v t v tdt

2

2

( ) ( ) ( )C CC s

d v t dvLC RC v t v tdt dt

Since( )( ) Cdv ti t C

dt

RLC circuits are those that consist of resistors, capacitors, and inductors.

We only consider one example here, where a resistor, a capacitor, and a inductor are connected in series.

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The non-homogeneous equation may be rewritten as

Defining the damping factor and the resonance frequency , we have

where and

Then, the characteristic equation is

The two roots are so that the homogeneous solution becomes

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RLC Circuit

2

2

( ) 1 1( ) ( )C CC s

d v t dvR v t v tdt L dt LC LC

02

2 20 02

( ) 2 ( ) ( )C CC s

d v t dv v t v tdt dt

2RL

01LC

2 2

02 0s s

2 20s

2 2 2 20 0

1 2( )t t

hv t K e K e

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The roots of C.E. assume three possible conditions:

(along with the variation of dependent on variable resistor)

Overdamping: Two real and distinct roots when

Critical damping: Two real equal roots when

Underdamping: Two complex roots when

In overdamped circuit, the homogeneous solution can be expressed as

In critical damped circuit, the double root is , then the H.S. for two simultaneous equal roots becomes

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Natural Response of an RLC Circuit

2 20

2 20

2 20

1 2( ) d dt t

hv t K e K e

1 2( ) thv t K K t e

s

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In underdamped circuit, the H.S. is

where

Since and must be complex conjugates, the equation becomes

Choosing , it can be recasted as

If the unknown constants determine by applying the initial conditions, we can have the natural response of the RLC circuit

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1 2( ) d dj t j t

hv t K e K e

2 20d

1K 2K

1 1( ) d dj t j tthv t e K e K e

1jK Ke

( )

2 cos

d dj t j tth

td

v t e Ke Ke

Ke t

Natural Response of an RLC Circuit

( )nv t

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a) an underdamped oscillatorb) a critically damped oscillatorc) an overdamped oscillator

For critically damped and overdamped circuits, there is no periodic motion For underdamped circuit, the system oscillates with decaying amplitude

Natural Response of an RLC Circuit

dT

022d d

d

fT

Thus, the resonance frequency is

2 20 0d

If 0 , [Underdamped case]

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Variable Resistor

A variable resistor consists of a track which provides the resistance path.

Two terminals of the device are connected to both the ends of the track.

The third terminal is connected to a wiper that decides the motion of the track.

The motion of the wiper through the track helps in increasing and decreasing the resistance.

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Forced Response of an RLC Circuit

Assume that a sinusoidal signal , which is called the forcing function, is applied in the circuit Then, expressing the non-homogeneous equation as Phasor

form gives

where

Rearranging to , we have

Thus, the forced response is defined as

( ) coss sv t V t

2 2 20 02j j j

C C C sj V e j V e V e V

jCV e

2 210 0

2 2 2 22 22 20 00

2tan2 2

j s sC

V VV ej

( ) j tc Cv t V e

210

2 22 22 2 00

2cos tan2

sf

Vv t t

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Complete Response of an RLC Circuit

Finally, the complete response of an RLC circuit is given by the sum of the natural response and the forced response as follows:

The RLC circuit starts oscillations due to the sinusoidal forcing function

The largest amplitude oscillations occur at or near RESONANCE ( )

210

2 22 22 2 00

2cos tan2

C n f

sn

v t v t v t

Vv t t

0

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Resonance of an RLC Circuit

A damping becomes weaker, the resonance sharpens (it becomes narrow and tall)

0 /

Band-Pass Filter

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The most common application of resonance in RF circuits is called tuning. In the figure, the LC circuit is resonant at 1000 kHz. The result is maximum output at 1000 kHz, compared with lower or higher

frequencies.

Series Resonant Circuit

Resonance of an RLC Circuit

022 r

r

fT

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A parallel resonant circuit stores energy in the magnetic field of the coil and the electric field of the capacitor. The energy is transferred back and forth between the coil and capacitor.

If there is no lost of energy by resistor potential energy equal to magnetic energy: , it will oscillate for a long time.

Resonance of an RLC Circuit

The coil deenergizes as the capacitor charges.

The capacitor discharges as the coil energizes.

2 21 12 2

CV LI

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Resonance of an RLC Circuit

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Resonance

Forced resonant torsional oscillations due to wind - Tacoma Narrows Bridge

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Resonance

Roadway collapse - Tacoma Narrows Bridge