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• 1

U1. Transient circuits

response

Circuit Analysis, Grado en Ingeniería de Comunicaciones

Curso 2016-2017

Philip Siegmann (philip.siegmann@uah.es)

Departamento de Teoría de la Señal y Comunicaciones

mailto:philip.siegmann@uah.es

• Index

• Recall

• Goals and motivation

• Linear differential equation (Recall)

• The homogeneous solution

• 1st and 2nd order linear differential equations

• Examples of 2nd order circuits

• Initial conditions

• Transient responses (example on 2nd order circuit)

• Cause of the transient responses

• Analysis using Laplace transform

• Recall

• Relation between i(t) and v(t) for the

passive elements R,L,C

– For R:

– For C:

– For L:

• Energy in these elements:

– Dissipated in R:

– Stored in C and L:

Circuit Analysis / Transient circuits response / Recall

)()( tRitv 

dt

tdv CtitCvtq CCC

)( )()()( 

dt

tdi Ltv LL

)( )( 

)(tiL

L vL(t)

)(tiC

C vC(t)

)(tiR

R vR(t)

 2)( 2

1 )( tvCtW CC   

2 )(

2

1 )( tiLtW LL 

 dttiRtW RR )()( 2

dt

tdq ti

dq

tdW tv

)( )(,

)( )( 

• Goals

• We want to solve circuits for whatever applied

source (not only DC and Sinusoidal Steady

State)

– Direct resolution in the time domain

– Resolution using Laplace transforms

• In particular we want to understand what

happens when an abrupt change takes place in

the circuit, which will produce the transient

response.

4

Circuit Analysis / Transient circuits response / Goals

• Motivation

• Signal transmission

Transient response

Circuit Analysis / Transient circuits response / Motivation

S.S.S.

• 6

Examples of 2nd order circuits

• RLC-serial (http://en.wikipedia.org/wiki/RLC_circuit )

• RLC-parallel

dt

tde

L ti

LCdt

tdi

L

R

dt

tid

tetq Cdt

tdi LtRi

tetvtvtv

g

g

gCLR

)(1 )(

1)()(

)()( 1)(

)(

)()()()(

2

2







 

dt

tdi

C tv

LCdt

tdv

RCdt

tvd

dt

tdi

dt

tvd Ctv

Ldt

tdv

R

titititi dt

d

g

g

gCLR

)(1 )(

1)(1)(

)()( )(

1)(1

)()()()(

2

2

2

2







(Energy conservation)

(Charge conservation)

Circuit Analysis / Transient circuits response / Example 2nd order circuits

http://en.wikipedia.org/wiki/RLC_circuit

• 7

Transient response

• Response of a circuit (voltage or current) when

an abrupt change happens (e.g. switching)

• Time evolution until achieving a new equilibrium

• The transition function follows exponential

variations (decreasing or increasing, fluctuating

or no fluctuating)

• They are solutions of linear differential equations

Circuit Analysis / Transient circuits response / Introduction

• 8

General solution

),()()(

),()(... )()(

01

1

1

tytyty

tgtya dt

tyd a

dt

tyd

ph

n

n

nn

n



 

yh(t) is the solution for g(t)=0: the Complementary, natural or homogeneous

solution. Gives the transient behavior of the circuit due to the passive elements.

It is dependent of the initial conditions.

yp(t) is a particular solution for the given source or forcing function g(t). The

particular solution looks like the forcing function, e.g.:

- If g(t) is constant, then yp(t) is constant

- If g(t) is sinusoidal, then yp(t) is sinusoidal (i.e. the S.S.S.)

The homogeneous solution decreases exponentially so that

y(t)  yp(t)

Linear differential equation of order n:

Circuit Analysis / Transient circuits response / Linear differential equation

• 9

The homogeneous solution

yh(t) is the solution for g(t)=0 (external energy

supply =0)

The solution has the form: yh(t)=A·exp(st) (“Ansatz”)

Circuit Analysis / Transient circuits response / The homogeneous solution

  stkst k

k

AsA dt

d ee :since 

 







n

k

ts

k

ts

n

tsts

h

n

n

n

n

kn AAAAty

ssss

asas

1

21

21

0

1

1

ee...ee)(

},...,,{

equation polynomial sticcharacteri ,0...

21

A1, A2, … are obtained with the initial (and/or boundary) conditions

characteristic polynomial equation

(“Eigenwerte”)

• 10

1st and 2nd order linear

differential equations

2

0

1

012

2

2

),()( )()(

n

n

a

a

tgtya dt

tdy a

dt

tyd





 is called the damping ratio (accounts for the energy loss) n is called the natural frequency (maximum energy storage) (http://en.wikipedia.org/wiki/Damping)

1

),()( )(

0

0



a

tgtya dt

tdy

 is a time constant (how fast yh(t) decreases)

Circuit Analysis / Transient circuits response / 1st nd 2nd order differential equations

1st order

2nd order

For circuits containing

one energy storage

element (C or L)

For circuits containing

two independent energy

storage element

http://en.wikipedia.org/wiki/Damping

• 11

Example of 2º orden circuits

• RLC-serial (http://en.wikipedia.org/wiki/RLC_circuit )

• RLC-parallel

L

R

LC

dt

tde

L ti

LCdt

tdi

L

R

dt

tid

n

n

g

 

2 ,

1

)(1 )(

1)()( 2

2





CRLC

dt

tdi

C tv

LCdt

tdv

RCdt

tvd

n

n

g

 

2

1 ,

1

)(1 )(

1)(1)( 2

2





Circuit Analysis / Transient circuits response / Ejemplos circuitos 2º orden

http://en.wikipedia.org/wiki/RLC_circuit

• 12

Initial conditions

• For each energy storage element we need an

initial condition (at t=t0):

– For C: vC(t=t0) = V0

– For L: iL(t=t0) = I0

Circuit Analysis / Transient circuits response / Initial conditions

 2)( 2

1 )( tiLtW LL 

 2)( 2

1 )( tvCtW CL 

vC(t) iL(t)

(t)

E(t)

• 13

Condiciones iniciales

• When an abrupt change happens at t=t0 in a

circuit, there is always continuity in the

variation of the energies in C and L 

– There is continuity in the voltage at C:

– There is continuity in the current trough L:

Circuit Analysis / Transient circuits response / Initial conditions

)()( 00 

 tvtv CC

)()( 00 

 titi LL

Justo después de t0 Justo antes de t0

(not so for the current iC(t))

(not so for the voltage vL(t))

• 14

Example of transient response

0 1

:equationsticCharacteri

,e)(

form theof solutionsth equaion wi free) (source Homogeneus

,0)( 1)()(

,0 )(

)( 1

)(

0For

)0(

0)0(

:0atconditionsInitial

2

2







  

LC s

L

R s

Ati

ti LCdt

tdi

L

R

dt

tid

dt

tdi Ltq

C tRi

t

Ev

i

t

2

st

gC

Discharge of the capacitor

Characteristic equation:

Circuit Analysis / Transient circuits response / Transient responses

dt

tdi LtRitvtvtv LRC

)( )()()()( 

Once i(t) is known we can obtain vC(t):

For t ≥0

• 15

Type of solutions of

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