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Lab. 6 RC 및 RL 회로

Dept. Electronics and Information Eng. 2

이 실험의 목표

RC 및 RL 회로의 동작 특성 이해 Transient Response Steady-State Response Capacitor 및 Indiuctor에서 Time-Constant의 의미 정현파 입력 신호에 대한 출력 특성

Dept. Electronics and Information Eng. 3

Capacitor

sin , cosCC m C m

dvv V t i C CV tdt

iC leads VC by 90o

Capacitors in AC Circuits

Impedance of a capacitor :

Sinusoidal response of capacitor :

,j t CC m C C

dVV V e I C j CVdt

(Ohm’s law for capacitor)1C

CC

VZI j C

Dept. Electronics and Information Eng.

The time-varying currents and voltages resulting from the sudden application of sources, usually due to switching, are called transients

By writing circuit equations, we obtain homogeneous (no input signal !) differential equation as follows:

Then, the characteristic equation is

The homogeneous solution becomes Applying the initial condition , the natural response is

defined as

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RC Circuit

0R

tvdt

tdvC CC

1 0sRC

The root is

1sRC

/( ) t RChv t Ae

iC Vv 0/( ) t RC

n iv t V e

Dept. Electronics and Information Eng. 5

RC Circuit

(a) Electrical circuit(b) Fluid-flow analogy: a filled water tank discharged through a small pipe

A capacitance discharging through a resistance and its fluid-flow analogy.

The capacitor is charged to Vi prior to t = 0 (by circuitry that is not shown).

At t = 0, the switch closes and the capacitor discharges through the resistor.

Dept. Electronics and Information Eng. 6

RC Circuit

The time interval =RC is called the time constant of the circuit

Voltage versus time for the circuit of figure (a). When the switch is closed, the voltage across the capacitor

exponentially to zero. At one time constant, the voltage is equal to 36.8%(e-1) of its

initial value.

Dept. Electronics and Information Eng. 7

RC Circuit

When connecting a DC source, the non-homogeneous (with an input signal !) differential equation is

Since the input signal is not zero, there exists a particular solution

( ) ( )s CV Ri t v t ( )Cs C

dv tV RC v t

dt

The switch closes at t = 0connecting the DC source Vs to the circuit.

Capacitance charging through a resistance.

RC circuit with a DC source

Dept. Electronics and Information Eng.8

RC Circuit The particular solution must have the same form as the input

signal ! Thus, the solution must be Substituting this into differential equation, unknown constant A is

determined

Thus, the non-homogeneous solution becomes

Consequently, applying the initial condition the complete response of RC circuit is given by

( )pv t A

( )Cs C

dv tV RC v t

dt 0sV A

/( ) ( ) ( ) t RCnh h p sv t v t v t Ke V

0 0Cv

/( ) t RCs C s sK V v t V V e

Stead-State Response Transient Response vanishing as t

Dept. Electronics and Information Eng.9

RC Circuit

The charging transient for the RC circuit.

1 1( ) 0.6321s s sC sV V e Vv e V at time constant

Dept. Electronics and Information Eng.10

RC Circuit

( Transient response of the RC circuit when a rectangular pulse is applied )

Dept. Electronics and Information Eng.

RC circuit with a sinusoidal source

The non-homogeneous equation is

It turns out to be easier to use as the input signal rather than and then take the real part of the solution

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RC Circuit

cos ( )C

M C

dv tV t RC v t

dt

Re Re cos sin cosj tM M M MV e V t jV t V t Note that

j tMV e

cosMV t

(Time Domain) (Phasor Domain)

0MV

RV

CV1

j CcosMV t

( )Rv t

( )Cv t

Dept. Electronics and Information Eng. 12

RC Circuit

j t j td e j edt

This is because

which will allow us to convert the differential equation to an algebraic equation

Rearranging to , Recalling that

we have

0M C CV j RCV V

1M

CVVj RC

CV

2 11 1 tanj RC RC RC

1

2tan

1M

CVV RC

RC

Dept. Electronics and Information Eng. 13

RC Circuit

Thus, the voltage taking the real part is given by

1

2( ) cos tan

1M

CVv t t RC

RC

Phase Difference between the input signal and the capacitor voltage

Dept. Electronics and Information Eng. 14

Inductor

When sine-wave variations of current produce an induced voltage, the current lags its induced voltage by exactly 90°, as shown in Figure

The phasors in Fig. (c) show the 90° phase angle between iL and vL. The 90° phase relationship between iL and vL is true in any sine-

wave AC circuit, whether L is in series or parallel.

Inductors in AC Circuits

Dept. Electronics and Information Eng.

Reactance of Inductor ( XL ) For AC circuits

Reactance: Voltage leads current by At low frequency (small ), inductor short-circuit At low frequency (large ), inductor short-circuit

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Inductor

LX L

LVZ j LI

(Impedance of a Inductor)

90

,j tL m

LL L

I I edIV L j LIdt

Dept. Electronics and Information Eng. 16

RL Circuit

When connecting a DC source, the non-homogeneous is

Then, the particular solution is And, the homogeneous solution is

( )s

di tV L Ri t

dt

( ) /p si t A V R /( ) R L t

hi t Ke

RL circuit with a DC source

Dept. Electronics and Information Eng. 17

RL Circuit Thus, the non-homogeneous solution becomes

Consequently, applying the initial condition the complete response of RL circuit is given by

/( ) ( ) ( ) /R L tnh h p si t i t i t Ke V R

0 0i

/ /

/

( ) 1 1

( )

R L t ts s

ts

V Vi t e eR R

v t V e

Time constant isLR

0.368 36.8( )sV V

Dept. Electronics and Information Eng. 18

RL Circuit(Square wave applied to a RL circuit)

Dept. Electronics and Information Eng. 19

RL Circuit

RL circuit with a sinusoidal source

The non-homogeneous equation is

Using the phasor form, we have

cos ( )M

di tV t Ri t L

dt

0MV RI j LI R j L I

cosMV t

( )Rv t

( )Lv t 0MV

RV

LV j L

(Time Domain) (Phasor Domain)

Dept. Electronics and Information Eng. 20

RL Circuit Rearranging to , it becomes

Then, since , we have

Thus, the voltage taking the real part is given by

I0MVI

R j L

LV j LI

1tan

20 0

1

RjL

L M Mj L eV V V

R j L RL

1

2( ) cos tan

1

ML

V Rv t tLR

L

Phase Difference between the input signal and the Inductor voltage

Dept. Electronics and Information Eng. 21

RL Circuit

T

360t

T

22t ft tT

Using

, we have

(Method to calculate Phase-Shift without using Lissajous)