CIRCUITS and SYSTEMS – part II

Prof. dr hab. Stanisław Osowski

Electrical Engineering (B.Sc.) Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

Lecture 9

Transient states in electrical circuits – differential equation

approach

3

Basic notionsSteady state – the sdtate in the circuit when the response of circuit is of the same shape as the excitation.

Transient state – the response of the circuit following the commutation in the circuit. In this state the response is of different character than the excitation. Transient response is the superposition of steady state and natural response.

Natural response - response of circuit deprived of external excitation, following the nonzero initial conditions)

Commutation – arbitrary change in the circuit.0t - the time point prior to commutation (left side limit)

0t - the time point directly after commutation (right side limit)

4

Commutation laws

Commutation law for capacitors

Determination of initial conditions: Calculate the steady state response of circuit before commutation Write the response in time form Calculate the currents of inductors and voltages of capacitors at the

time t0 of commutation

Commutation law for inductors

)0()0( )0()0( CCi

ii

i uuorqq

)0()0( )0()0( LLi

ii

i iior

5

Example

Determine the initial conditions in the circuit. Assume: L=1H, C=0,5F, R=1, AttiVtte oo )45sin(2)( ,)45sin(210)(

Solution:

Complex represenation of elements:

2/ ,1 Z,1 ,2

2 ,10 L

4545 jCjZjLjeIeE Cjj oo

6

Initial conditions

Circuit equations in steady state

Initial conditions

)135sin(4)( ),31,11sin(221,7)(

solution form Time

2

4 ,21,7 , 13531,11

oC

oL

ojCC

oj

LLLLL

ttutti

eIZUeZR

RIEIIIRIZE

22)0( ,2)0( CL ui

7

State space decription of the circuitThe general differential form description of the linear circuit

)(...

..................

)(...

)(...

2211

222221212

112121111

tfxaxaxadt

dx

tfxaxaxadt

dx

tfxaxaxadtdx

nnnnnnn

nn

nn

The variables x of the minimal quantity form the state variables.

8

Matrix form of state description

The normal state description

)()()(

ttdt

tdBuAx

x

A, B – state matrices

Response matrix equation y(t)

)()()( ttt DuCxy

C, D – output matrices.

9

Example

Determine the state description of the circuit in normal form

From Kirchhof laws and definition of elements

CLLCC iiiuuRie ,dt

duCi

dtdi

Lu CC

LL ,

iidt

duCu

dtdi

LiiRe LC

CL

L ,)(

we get

10

Example (cont.)Matrix form of state equations

State vector x and excitation vector u

Assuming: R=2, L=1H, C=1F we get

i

e

C

LR

Lu

i

C

LLR

dtdudtdi

C

L

C

L

10

1

01

1

i

e

u

i

C

L ux ,

10

21 ,

01

12

BA

11

Solution of transient state using classical method

In the first step we transform the system of n first order state space equations into one nth order differential equation of one variable x.

The solution of it is composed of two components: the steady state xu and natural response xp. The steady state corresponds to the external excitation and natural response to nonzero initial conditions only.

)(... 012

2

21

1

1 tfxadtdx

adt

xda

dtxd

adt

xda n

n

nn

n

nn

n

n

12

Natural responseThe natural response corresponds to the solution of the homogenous differential equation (zero excitation)

Characteristic equation

The roots of this equation si (i=1, 2, ..., n) are the poles of the system.

0... 012

2

21

1

1

pp

np

n

nnp

n

nnp

n

n xadt

dxa

dt

xda

dt

xda

dt

xda

0... 012

21

1

asasasasa n

nn

nn

n

13

Final solutionThe general solution of the homogenous differential equation of nth order is in the form

Ai – constants of integration calculated on the basis of initial conditions (solution of system of linear equations).

The final solution of the nonhomogenous differential equation is the sum of steady state and natural response solutions

This method is called the classical method of solution of the differential equations. It is very easy in application to the first order differential equations only.

tisn

iip eAtx

1

)(

)()()( txtxtx pu

14

Consider the transient response in RL circuit at DC excitation.

The steady state current in the circuit

RE

tiLu )(

Transient in RL circuit at DC excitation

15

Solution of transient stateThe homogenous differential equation

Characteristic equation

0 LpLp Ri

dt

diL

0 RLs

RL

t

Lp eAiL

Rs /

11 ,

General form of solution of natural response

The final (general) form of transient

RL

t

LpLuL eARE

tititi /1)()()(

16

Solution of transient state (cont.)

Commutation law 10)0()0( ARE

ii LL

Hence REA /1

Current of inductor

RL

t

L eRE

ti /1)(

The current in RL circuit at different time constants

RL Time constant

of RL circuit

17

Voltage of inductor

Transient voltage of the inductor

RL

tL

L Eedt

tdiLtu /)(

)(

The voltage of the inductor in RL circuit at different time constants

18

Transient in RC circuit at DC excitation

Consider the transient response in RC circuit at DC excitation

The voltage of capacitor in steady state

EtuCu )(

19

Solution of transient state

After eliminating the source we get the homogenous equation

0 CpCp u

dt

duRC

Characteristic equation

General solution of natural response

Final general form of solution of transient

01RCs

RC

tts

Cp eAeAuRCs

11

11 ),/(1

RC

t

CpCuC eAEtututu

1)()()(

20

Solution of transient state (cont.)

Commutation law

Hence

Final solution

10)0()0( AEuu CC

EA 1

RC

t

C eEtu 1)(

Graphical presentation of capacitor voltage at different time constants

RCTime constant

of RC circuit

21

Current of the capacitor

Current of capacitor in transient form

RC

tc

C eRE

dttdu

Cti

)()(

Graphical presentation of capacitor current at different time constants