To find the step response of an RC circuit

0

)]()([)()( 0

tt

evtvvtv

The final capacitor voltage v()

The initial capacitor voltage v(t0)

The time constant = RC

To find the step response of an RL circuit

0

)]()([)()( 0

tt

eitiiti

The final capacitor voltage v()

The initial capacitor voltage v(t0)

The time constant = L/R

2

Lecture 6 DC Circuits Transient Circuits – 2nd Order circuits

Contents

• Examples of 2nd order RLC circuit

• The source-free series RLC circuit

• The source-free parallel RLC circuit

• Step response of a series RLC circuit

• Step response of a parallel RLC

3

Examples of Second Order RLC circuits

What is a 2nd order circuit?

4

A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements of different type or the same type.

Source-Free Series RLC Circuits

5

• The solution of the source-free series RLC circuit is called as the natural response of the circuit.

• The circuit is excited by the energy initially stored in the capacitor and inductor.

02

2

LC

i

dt

di

L

R

dt

idThe 2nd order of expression

How to derive and how to solve?

Source-Free Series RLC Circuits: to derive the equation

vR=Ri

KVL:

To eliminate the integral

02

2

C

i

dt

idL

dt

diR

dt

diLvL

t

C idtC

v1

01

t

idtCdt

diLRi

Source-Free Series RLC Circuits: to solve the equation

02

2

LC

i

dt

di

L

R

dt

id

Assume: the solution has exponential form, stAei

To be determined

To be determined

012

LCs

L

RsAe st

Define: L

R

2

LC

10

Neper frequency/ damping factor

Resonant frequency/ undamped natural factor 02 2

0

2 ss

Characteristic equation

Solution: 2

0

2

2,1 s Natural Frequencies

Linear equation: tstseAeAti 21

21)(

Still solving… tsts

eAeAti 21

21)( How to get A?

Initial conditions:

The inductor: 0)0( Ii The capacitor: 0

01

)0( VidtC

v

021)0( IAAi 0)0( 0

0

Vdt

diLRi

t

tstsesAesA

dt

tdi21

2211

)(

0022110 VsAsALRIA1 and A2

What if s1 = s2 (when ) 0 Ls

VRIAA 00

21

I0 and V0

not independent

02 2

2

2

idt

di

dt

id 0

i

dt

dii

dt

di

dt

d teAi

dt

di 1

tetAAti 12)(To get Ai

02 2

02

2

idt

di

dt

id

9

There are three possible solutions for the following 2nd order differential equation:

1. If > o, over-damped case

tstseAeAti 21

21)( 2

0

2

2,1 swhere

2. If = o, critical damped case

tetAAti )()( 122,1swhere

3. If < o, under-damped case

)sincos()( 21 tBtBeti dd

t where

22

0 d

Source-Free Series: RLC Circuits

Damped natural frequency

Finding initial and final values

• Initial values: – Passive convention

• The polarity of voltage

• The direction of current

– Variables that cannot change abruptly • The capacitor:

• The inductor:

10

00

)(,

)(),0(),0(

tt dt

tdi

dt

tdviv

)(),( vi

)0()0( vv

)0()0( ii Focus on these variables

• Final values: – DC steady state

• The capacitor: Open circuit

• The inductor: short circuit

The time just before a switching event which

takes place at t = 0

Example

If R = 10 Ω, L = 5 H, and C = 2 mF in 8.8, find α, ω0, s1 and s2.

What type of natural response will the circuit have?

11

Answer: underdamped

12

L

R 10

10

LC 95.912

0

2

2,1 js

Example

The circuit shown below has reached steady state at t = 0-.

If the make-before-break switch moves to position b at t = 0, calculate i(t) for

t > 0.

12

Answer: i(t) = e–2.5t[5cos1.6583t – 7.538sin1.6583t] A

Step-Response Series: RLC Circuits

13

• 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 s2

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.

Step-Response Series: RLC Circuits

14

The solution of the equation should have two components: the transient response vt(t) & the steady-state response vss(t):

)()()( tvtvtv sst

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

The steady-state response is the final value of v(t).

• vss(t) = v(∞)

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

• v(0) and dv(0)/dt.

tsts

t eAeAtv 21

21)( (over-damped)

t

t etAAtv )()( 21 (critically damped)

)sincos()( 21 tAtAetv dd

t

t

(under-damped)

Example

Having been in position for a long time, the switch in the circuit below is moved to position b at t = 0. Find v(t) and vR(t) for t > 0.

15

Answer: v(t) = {10 + [(–2cos3.464t – 1.1547sin3.464t)e–2t]} V

vR(t)= [2.31sin3.464t]e–2t V

Lecture 7 AC Circuits Sinusoids and Phasors

Appendix

Source-Free Parallel: RLC Circuits

18

The 2nd order of expression

011

2

2

vLCdt

dv

RCdt

vd

0

0 )(1

)0( dttvL

IiLet

v(0) = V0

Apply KCL to the top node:

t

dt

dvCvdt

LR

v0

1

Taking the derivative with

respect to t and dividing by C

Source-Free Parallel: RLC Circuits

LCRCv

dt

dv

dt

vd 1and

2

1 where0 2 0

2

02

2

19

There are three possible solutions for the following 2nd order differential equation:

1. If > o, over-damped case

tstseAeAtv 21 )( 21 2

0

2

2,1 swhere

2. If = o, critical damped case

tetAAtv )( )( 12 2,1swhere

3. If < o, under-damped case

)sincos()( 21 tBtBetv dd

t where

22

0 d

Example

Refer to the circuit shown below. Find v(t) for t > 0.

20

• Please refer to lecture or textbook for more detail elaboration.

Answer: v(t) = 66.67(e–10t – e–2.5t) V

Step-Response Parallel: RLC Circuits

21

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

The 2nd order of expression

It has 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.

LC

I

LC

i

dt

di

RCdt

id s1

2

2

Step-Response Parallel : RLC Circuits

22

The solution of the equation should have two components: the transient response vt(t) & the steady-state response vss(t):

)()()( tititi sst

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

The steady-state response is the final value of i(t).

iss(t) = i(∞) = Is

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

i(0) and di(0)/dt.

tsts

t eAeAti 21

21)( (over-damped)

t

t etAAti )()( 21 (critical damped)

)sincos()( 21 tAtAeti dd

t

t

(under-damped)

Example

Find i(t) and v(t) for t > 0 in the circuit shown in circuit shown below:

23

• Please refer to lecture or textbook for more detail elaboration.

Answer: v(t) = Ldi/dt = 5x20sint = 100sint V