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Lab. 7 RLC 회로
Dept. of Information and Communication Eng. 2
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
Dept. of Information and Communication Eng. 3
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.
Dept. of Information and Communication Eng.
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
4
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
Dept. of Information and Communication Eng.
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
5
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
Dept. of Information and Communication Eng.
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
6
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
Dept. of Information and Communication Eng. 7
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]
Dept. of Information and Communication Eng. 8
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.
Dept. of Information and Communication Eng. 9
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
Dept. of Information and Communication Eng. 10
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
Dept. of Information and Communication Eng. 11
Resonance of an RLC Circuit
A damping becomes weaker, the resonance sharpens (it becomes narrow and tall)
0 /
Band-Pass Filter
Dept. of Information and Communication Eng. 12
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
Dept. of Information and Communication Eng. 13
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
Dept. of Information and Communication Eng. 14
Resonance of an RLC Circuit
Dept. of Information and Communication Eng. 15
Resonance
Forced resonant torsional oscillations due to wind - Tacoma Narrows Bridge
Dept. of Information and Communication Eng. 16
Resonance
Roadway collapse - Tacoma Narrows Bridge
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