M2 Electric Circuitswebstaff.kmutt.ac.th/~werapon.chi/M2_1/1_2013/M2_1_Class... · 2013. 10....

LECTURE 8

TIME RESPONSES

M2Electric Circuits

1

Agenda2

� Time response

� First-order circuits

� Natural response

� Forced responses

� Second-order circuits

� Natural response

� Forced responses

Responses in Time Domain

3

A first-order circuit is characterized by a first-order differential equation

��

N. Mohan, Power Electronics: Converters, Applications and Design, 3 rd edition, Wiley

�� 1��

�� 1 � ��

�

First-Order Circuits4

• Source-free RC circuits• Source-free RL circuits

• Step-response RC circuits• Step-response RL circuits

Homogenous differential equations (i.e. natural response)

Nonhomogenous differential equations (i.e. forced response)

Pre-Charging the Capacitor

• For this particular case, the capacitor voltage is equal to the voltage across the resistor R2

• At t = 0 (the beginning of our interest), the switch is opened and the capacitor is discharging

• The switch has been in the “closed” position (i.e. the switch is on) for a long while

• The capacitor acts as an open circuit path

C.K. Alexander and M.N.O. Sadiku, Fundamentals of Electric Circuits, 5th edition, McGraw Hill

5

6

Source-Free RC Circuit

� The capacitor is initially charged

(energy is already stored v(0) � V0)

� Solving the above equation

6

0 =+dt

dvC

R

v0 =+ CR iiBy KCL

1dvdt

v RC= −∫ ∫

0 0( )t / RC t /v t V e V e− − τ= =

Ohms lawCapacitor



7

CR=τTime constant

10 0 00 368

t /V e V e . V− τ −= =

� t = �

t v(t)/V0

τ 0.36788

2τ 0.13534

3τ 0.04979

4τ 0.01832

5τ 0.00674

Decays more slowly

Decays faster



8

� The natural response of a circuit = the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation

� The time constant τ of a circuit is the time required for the response to decay by a factor of 1/e or 36.8% of its initial value.

� � decays faster for small �and slower for large �� The key to working with a source-free RC circuit is finding:

� The initial voltage � 0 � �� across the capacitor.� The time constant � � �

Energizing the Inductor9

• For this particular case, the inductor current is equal to the current through the resistor ��

• At � � 0(the beginning of our interest), the switch is opened and the inductor is de-energized

• The switch has been in the “closed” position (i.e. the switch is on) for a long while

• The inductor acts as a short-circuited path


Source-Free RL Circuit

10

� A first-order RL circuit consists of an inductor L (or its equivalent) and a resistor R (or its equivalent)

0 =+ RL vvBy KVL

0 =+ iRdt

diL

Inductor relationship

Ohms law



11

Solving for i(t)( )

(0) 0

i t t

i

di Rdt

i L= −∫ ∫

ln ( ) ln (0) 0Rt

i t iL

− = − +

τ−= /0)(teIti

R

L=τ

A general representation

where



12

� The time constant � of a circuit is the time required for the response to decay by a factor of 1/e or 36.8% of its initial value.

� i(t) decays faster for small τ and slower for large �� The general form is very similar to a RC source-free

circuit.

� Thy key: �� and τ

τ/0)(

teIti −=R

L=τwhere


Summary: Natural Responses

13

A RL source-free circuit

where

A RC source-free circuit

where

� � � �� ⁄� � �

� � � �� ⁄� � � ��


Initial Conditions

14

� The initial capacitor voltage could be either zero or nonzero

� When the switch is left at either position for a long time, the capacitor voltage builds up to a specific value


1

2

15

Step Response: RC Circuit

� The step response of a circuit is its behavior when the excitation is the step function, which may be a voltage or a current source.

• Initial condition:

• Applying KCL,

• Rearrange and integrate both sides

� 0� =� 0� � ��

�� 0

0

( )

0

v t t

Vs

dv dt

v V RC= −

−∫ ∫C.K. Alexander and M.N.O. Sadiku, Fundamentals of Electric Circuits, 5th edition, McGraw Hill

Step Response: RC Circuit

16

• Integration gives

• Combine and take the exponential

• Complete response

/

0

, t ts RC

s

v Ve e RC

V Vτ τ

− −− = = =−

0

/0

, 0

( ) , 0ts s

V tv

V V V e tτ−

( ) ( )0ln lns st

v V V VRC

− − − = −


Step Response: RC Circuit17

Initially uncharged capacitor Initially charged capacitor

0

/0

, 0

( ) , 0ts s

V tv

V V V e tτ−


Complete Response: RC

18

>−+

<=

− 0)(

0)(

/0

0

teVVV

tVtv

tss

τ

Final value

at t -> ∞

Initial value

at t = 0

Source-free

Response

Complete Response = Natural response + Forced Response

(stored energy) (independent source)

= V0e–t/τ + Vs(1–e

–t/τ)

Complete Response = transient response + steady-state response

(temporary) (permanent)

= ( V0 - Vs)e–t/τ + Vs

Summary: RC Step Response

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� Typical form

� To determine the step response of an RC circuit:

� The initial capacitor voltage v(0)

� The final capacitor voltage v(∞) — DC voltage across C

� The time constant τ.

τ/ )]( )0( [ )( )( tevvvtv −∞−++∞=

Step Response: RL Circuit

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The step response of a circuit is its behavior when the excitation is the step function, which may be a voltage or a current source

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• Initial current

• Final inductor current

• Time constant τ = L/R

� 0� =� 0� � ��

� ∞ � ��

� � �!"#�$%# � ��%"&'��"%C.K. Alexander and M.N.O. Sadiku, Fundamentals of Electric Circuits, 5th edition, McGraw Hill

21

/L Rτ =

0sVI A

R= +

/0

ts sV Vi I eR R

τ− = − +


Will try this by inspection

� � �!"#�$%# � ��%"&'��"%� � (�� ⁄ � ��

The initial condition gives


22


� Three steps in finding out the step response of an RL circuit:

� The initial inductor current i(0) at t = 0+

� The final inductor current i(∞)

� The time constant τ

Note: This technique applies only for the step responses

τ/ )]( )0( [ )( )( teiiiti −∞−++∞=

/0

ts sV Vi I eR R

τ− = − +


Second-Order Circuits23

• Source-free series RLC circuits• Source-free parallel RLC circuits

• Step-response series RLC circuits• Step-response series RLC circuits

What is a 2nd order circuit?

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• A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalence of two energy storage elements.

Series RLC circuit

Parallel RLC circuit

RL T-configuration

RC Pi-configuration


Source-Free Series RLC Circuits

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0

0 0

1(0) , (0)

−∞= = =∫v idt V i IC

10

tdiRi L idt

dt C −∞+ + =∫2

20

d i R di i

dt L dt LC+ + =

• KVL

• 2 initial conditions needed


�� 0 � � ��)*� � �� 0


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• Our experience with the first-order circuit suggests

sti Ae=

2 0st st stAR A

As e se eL LC

+ + = 2 1 0stR

Ae s sL LC

+ + =

(A & s are to be determined)

2

20

d i R di i

dt L dt LC+ + =

• Plugging in the differential equation

Characteristic equation


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� Roots of the characteristic equation

� 2 possible solutions

� Linear combination

21

2 2

R Rs

L L LC = − ± −

2 21 0

2 22 0

s

s

α α ω

α α ω

= − + −

= − − −

02 202

2

=++ idt

di

dt

idωα LC

andL

R 1

2 0== ωα

General 2nd order Form

where

(damping factor)

(undamped natural frequency)

1 21 1 2 2 &

s t s ti A e i A e= =1 2

1 2( )s t s ti t A e A e= +

The types of solutions for i(t) depend on the relative values of α and ω.

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• 3 possible solutions1. If α > ωo, over-damped case

tsts eAeAti 21 21)( +=2

02

2,1 ωαα −±−=swhere

2. If α = ωo, critically damped case tetAAti α−+= )()( 12 1 2s s α= = −where

3. If α < ωo, under-damped case )sincos()( 21 tBtBeti dd

t ωωα += −

where 220 αωω −=d



Critically Damped Response

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1 21 2( )

s t s ti t A e A e= + 1 2 3( )t t ti t A e A e A eα α α− − −= + =

22

22 0

d i dii

dt dtα α+ + = 0

d di dii i

dt dt dtα α α + + + =

0df

fdt

α+ =

1tf A e α−=

1tdi i A e

dtαα −+ = ( ) 1t t tdi de e i e i Adt dt

α α αα+ = =

( )1 2 ti A t A e α−= +1 2te i A t Aα = +

� Critically damped case (repeated roots; α = ωo)

� Recall

� Something wrong!! 2 ICs cannot satisfy a single constant

� Go back to the general form

� Thus,

� Integrating both sides

Underdamped Response

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� Underdamped case (α < ωo)

� Euler’s identities

2 21 0

2 22 0

d

d

s j

s j

α α ω α ω

α α ω α ω

= − + − = − +

= − − − = − −2 20dω ω α= −

Damping frequency

( )( ) ( )1 2 1 2( ) d d d dj t j t j t j tti t A e A e e A e A eα ω α ω ω ωα− − − + −−= + = +

[ ]( )

1 2

1 2 1 2

( ) (cos sin ) (cos sin )

cos ( )sin

td d d d

td d

i t e A t j t A t j t

e A A t j A A t

α

α

ω ω ω ω

ω ω

−

−

= + + −

= + + −

[ ]1 2( ) cos sint d di t e B t B tα ω ω−= +

Source-Free Parallel RLC Circuits

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� The damping effect is due to the presence of ‘R’

� If R = 0, the response is not only undamped but also oscillatory

� By adjusting R, the response can be varied

� Oscillatory response is possible due to the storage elements.

� It is difficult to tell the difference between overdamped and critically damped responses.

� Critically damped response approaches the final values most rapidly.

LCand

L

R 1

2 0== ωα


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0

0

1(0) ( )i I v t dt

L −∞= = ∫Let

v(0) = V0Apply KCL to the top node:

∫∞−

=++t

dt

dvCvdt

LR

v0

1

011

2

2

=++ vLCdt

dv

RCdt

vdThe 2nd order expression

Characteristic equation

2 2 20

1 12s s s s

RC LCα ω+ + = + +

0

1 1

2and

RC LCα ω= =



33

The solutions of the characteristic equation are

2 21,2 0= − ± −s α α ω

1. If a > wo, overdamped casetsts eAeAtv 21 )( 21 +=

20

22,1 ωαα −±−=swhere

2. If a = wo, critically damped case

tetAAtv α−+= )( )( 12 α−= 2,1swhere

3. If a < wo, underdamped case

)sincos()( 21 tBtBetv ddt ωωα += − where 220 αωω −=d

1,2 ds jα ω= − ±

Step-Response: Series RLC Circuits

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� The step response is obtained by the sudden application of a dc source.

The 2nd order of expression LC

v

LC

v

dt

dv

L

R

dt

vd s=++2

2

The above equation has the same form as the equation for source-free series RLC circuit. • The same coefficients (important in determining

the frequency parameters). • Different circuit variable in the equation.


� The solution of the equation have 2 components:

� Transient response �!+�,� Steady-state response ��+�,

� The transient response vt is the same as that of the source-free case

� The steady-state response is the final value of � �� +∞,


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(over-damped)

(critically damped)

(under-damped)

� � � �! � � ��

�! � � (��- � (.��/�! � � +(��(.�,��0�! � � ��0 (�1234&� � (.3�54&�


36

� The complete response becomes:

(over-damped)

(critically damped)

(under-damped)

� The values of A1 and A2 are obtained from the initial conditions:

� � � �� (��- � (.��/� � � �� +(��(.�,��0� � � �� 0 (�1234&� � (.3�54&�

� 0 , ��)*�

Step-Response: Parallel RLC Circuits

37

• The step response is obtained by the sudden application of a dc source.

The 2nd order of expression LC

I

LC

i

dt

di

RCdt

id s=++1

2

2

This is the same form as the equation for source-free parallel RLC circuit.

• The same coefficients (important in determining the frequency parameters).

• Different circuit variable in the equation.



38

• The solution of the equation have two components:o Transient response: �!+�,o Steady-state response: ��+�,

• The transient response it is the same as the source-free case

(over-damped)

(critical damped)

(under-damped)

• The steady-state response is the final value of i(t) o iss(t) = i(∞) = Is

� � � �! � � ��

�! � � (��- � (.��/�! � � +(��(.�,��0�! � � ��0 (�1234&� � (.3�54&�


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� The complete response:

tstss eAeAIti 21 21)( ++= (over-damped)

ts etAAIti

α−++= )()( 21 (critically damped)

)sincos()( 21 tAtAeIti ddt

s ωωα ++= − (under-damped)

� The values of A1 and A2 are obtained from the initial conditions:

� 0 , ��)*�

M2 Electric Circuitswebstaff.kmutt.ac.th/~werapon.chi/M2_1/1_2013/M2_1_Class... · 2013. 10....

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Transcript of M2 Electric Circuitswebstaff.kmutt.ac.th/~werapon.chi/M2_1/1_2013/M2_1_Class... · 2013. 10....