Time-domain analysis of first-order RL and RC circuits, analysis of first-order RL and RC...Time-domain analysis of first-order RL and RC circuits • Analysis of response of circuit

Time-domain analysis of first-order RL and RC circuits,

Prof. N. M. Deshkar

Associate Professor

Department of Electrical Engineering

RCoEM

Prof. P. R. Sawarkar

Assistant Professor

Department of Electrical Engineering

RCoEM

Time-domain analysis of first-order RL and RC circuits

• Analysis of response of circuit consisting of R, L, C voltage source , current source

& switches to sudden application of voltage or current is called as Time domain

Analysis & Transient Response.

• When A.C. or D.C. voltage source is connected to circuit, a steady current can be

calculated by many methods , already discussed . (Ohm’s law).

• It is also assumed that circuit elements R, L, C are constant and source is very strong

to absorb any disturbances.

• Amongst basic circuit elements Resistor is energy dissipating component & Inductor ,

Capacitor are energy storing elements. (electro magnetic & Electro static)

• Response of these elements to nature of source and disturbance varies from source to

source.

• Transients (current or voltage lasting for short duration) in circuit is due to energy

storing elements.

• For source free circuit transients response is called as Natural Response .

• For circuit with source transient response is called as Forced Response.

• Disturbance in steady operation of circuit is unavoidable & can be of any type

as below

I. Any circuit suddenly connected to source or disconnected from

source.

II. Sudden change in applied voltage from one level tom another

III. Faults like short circuit or open circuit.

•After disturbance current or voltage shall have two components


I. Final steady state component (t� ∞ )

II. Transient component lasting for short duration that may settle down to

zero or final value

R L

Source Free RL Network

R

L

RL Network

DC. Source+

_

S


Steady System


During Disturbance


Continuous Disturbance

• Equations for these circuits, formed using KVL & KCL, consisting of basic

elements contain derivatives & integrals of Currents / Voltages .

• Due to above facts equations are not algebraic but are differential in nature.

• Solutions of differential equations are functions of time & not constant as in case


of purely resistive circuits.

Series RL Circuit


R

L

Fig. 1 Series RL Circuit

Vs+

_

SFig. 1 shows a series RL circuit connected across a DC source

through a switch S. When switch ‘S’ is close at t>0 the as per

KVL network equation will be …

Vsdt

tdiLtRi =+

)()(

1

Above equation is non homogenous equation linear differential equation of first order.

The solution of Eq. 1 will give i(t) which consists of two components i.e.

i) Complimentary function (i (t))

…………………. Eq. 1

)()()( tititi Jn +=

i) Complimentary function (in(t))

Which will satisfy

ii) Particular integral (if(t))

Which will satisfy

0)()(

=+ tiL

R

dt

tdi

Vsdt

tdiLtRi =+

)()(

Thus complete solution may be written as …

…………………. Eq. 2

Series RL Circuit


)/()(

)(τt

KeKetit

L

R

n

−==

−

2

)(

0

)(

0)( τ

tt

L

R

eIeIti−−

==

Where ζ = L/R time constant of RL circuit

…………………. Eq. 3

Eq. 3 provides the natural reproduced and is reproduced below….

…………………. Eq. 4)( KeKetin ==

Vsdt

dILRI =+

0=

dt

dIL

R

VsIti f ==)(

Eq. 1 can be written with ( i(t) = I = constant )

Since I = Constant

…………………. Eq. 5

…………………. Eq. 6

Substitute Eq. 4 & Eq.6 in Eq.2 yields the solution of Eq. 1

Series RL Circuit


IKeR

VsKeti

tt+=+=

−− ττ //)(

IR

VsK −=−=

3

We get..

…………………. Eq. 7

K is determined from initial condition i.e. t=0 Eq. 7 will be …

…………………. Eq. 8

)1()(

1)(

/

)(

τt

tL

R

eIti

eR

Vsti

−

−

−=

−=

Hence complete solution of Eq. 1 is given by

For t >0

Series RL Circuit

Time-domain analysis of first-order RL and RC circuits4

Series RL Circuit


Series RC Circuit


C

Fig. 2 Series RC Circuit

Vs+

_

SR

0)()( =−− tvtRiV Cs

tdv )(

Fig. 2 shows a series RC circuit connected across a DC source

through a switch S. It is assumed that capacitor voltage id V0

When switch ‘S’ is close at t>0 the as per KVL network

equation will be …

…………………. Eq. 9

For analysis of circuit of Fig. 2 the capacitor voltage Vc(t) is chosen as variable.

dt

tdvcti c )(

)( =

scc Vtvdt

tdvRC =+ )(

)(

s

t

c Vketv +=− τ/)(

Substituting in Eq. 9 We get.

For t >0) …………………. Eq. 10

Above Eq.10 is like Eq. 1 it is also non homogenous equation linear differential equation of

first order. Therefore solution, solution is also similar to Eq. 1. i.e.

…………………. Eq. 11

Series RC Circuit


RC=τ

sVVK −= 0

VeVeVtv

t

S

t

C

−+=

−−

ττ 1)()(

0

In Eq. 11 the time constant is

By substituting initial condition Eq. 11 i.e. Vc = V0 it leads to

By substituting value of K in Eq. 11 and after simplification we get …

For t > 0 …………………. Eq. 12

The expression for the current in the circuit is given by….

+

−=

=

−−

ττ

ττ

1

0

11)(

)()(

eVeVCti

dt

tdvCti

S

t

C

The expression for the current in the circuit is given by….

τ

t

S eVVRC

Cti

−

−= )()( 0

τ

t

S eR

VVti

−−=

)()( 0

…………………. Eq. 13

Series RC Circuit


Series RC Circuit


Reference :-

[1] Basic Electrical Engineering , Second Edition, T. K. Nagsarkar & M. S. Sukhija.

OXFORD University Press. Pp 96 – 122.

Documents

Time-domain analysis of first-order RL and RC circuits, analysis of first-order RL and RC...Time-domain analysis of first-order RL and RC circuits • Analysis of response of circuit