Ch4_First Order Circuits

Chapter 4: First Order Circuits

BEE1133 : Circuit Analysis I

First Order Circuits : Syllabus

4.1 Introduction to energy storage elements4.2 The source-free RC circuit4.3 The source-free RL circuit4.4 Unit step function4.5 Step response of an RC circuit4.6 Step response of an RL circuit

First Order Circuits:Lesson Outcomes

• Understand the characteristics of capacitor and inductor

• Differentiate the behaviour of first-order circuit with dc circuit

• Demonstrate the behaviour of natural responses of source-free RC and RL circuits

Energy Storage Elements :• Capacitor and Inductor

Energy Storage Elements: Capacitor• A capacitor is a passive element that stores energy

in its electric field.• It consists of two conducting plates separated by

an insulator.• Capacitance is the ratio of the charge on one plate

of a capacitor to the voltage difference between the two plates

• Measured in farads (F)• A capacitor is an open circuit to dc.• The voltage on a capacitor cannot change abruptly.

v

qC

Energy Storage Elements: Capacitor• The capacitance value depends on the physical

dimensions of the capacitor:

• Current voltage relationship of the capacitor

• Energy stored in the capacitor

d

AC

dt

dvCic

t

t

c tvidtC

v0

)(1

0

2

2

1Cvwc

Series and Parallel Capacitors• Series connection of capacitors

• Parallel connection of capacitors

Energy Storage Elements: Inductor• An inductor is a passive element that stores energy

in its magnetic field.• It consists of a coil of conducting wire.• Inductance is the property whereby an inductor

exhibits opposition to the change of current flowing through it.

• Measured in henrys (H)• An inductor acts like a short circuit to dc.• The current through an inductor cannot change instantaneously.

Energy Storage Elements: Inductor• The inductance value depends on the physical

dimensions and construction of the inductor:

• Current voltage relationship of the inductor

• Energy stored in the inductor

AN

L2

dt

diLvL

t

t

L tidttvL

i0

)()(1

0

2

2

1LiwL

where is the permeability of the core

Series and Parallel Inductors• Series connection of inductors

• Parallel connection of inductors

Energy Storage Elements: Summary

Series and Parallel Capacitor/InductorsExercise 1Find equivalent capacitance, Ceq for circuit below.

Series and Parallel Capacitor/InductorsExercise 2Find equivalent inductance, Leq for circuit below.

Source free RC circuits :• Natural response

Source free RC circuits• Source free RC circuit occurs when its DC source is

suddenly disconnected.• The energy already stored in the capacitor is

released to the resistors.

A source free RC circuit

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

• Apply Kirchhoff’s voltage laws to circuit above results in algebraic equations.

• Apply the laws to RC circuits to produce differential equations.

Capacitor law Ohm law

0 R

v

dt

dvC0 RC ii

By KCL

Source free RC circuits

Source free RC circuits• Voltage response of the RC circuit is solved below:

• This is a 1st order differential equation (only 1st derivative of v is involved:

• Integrating both sides

• Taking power of e (V0 is initial voltage at t =0)

0R

v

dt

dvC 0

RC

v

dt

dvor

dtRCv

dv 1

ARC

tv lnln

RC

t

A

vlnbecomes

RCtAetv )( orRCteVtv 0)(

• The natural response of a circuit refers to the behavior (i.e voltages & currents) of the circuit itself, with no external sources of excitation.

• 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.

• v decays faster for small t and slower for large t.

CRTime constantDecays more slowly

Decays faster

Source free RC circuits

• The key to working with a source-free RC circuit is finding:

1. The initial voltage v(0) = V0 across the capacitor.

2. The time constant = RC.

/0)( teVtv CRwhere

Source free RC circuits

Source free RC circuitsExample 1Refer to RC circuit below. Let vc(0) = 30 V. For t 0,

determine vc , vx , io , and wc

Source free RC circuitsSolution for Example 1• Since the voltage across capacitor cannot change

instantly, the voltage across capacitor at t = 0- is the same at t = 0:

• Solve for the time constant

• Find Req to solve for

VVvc 30)0( 0

CReq

1286//12eqR

s43

112

Source free RC circuitsSolution for Example 1 (cont…)• Now solve for vc and wc

• Now solve for vx and io

tttcc eeevtv 25.04 3030)0()(

JCvw cc 150303

1

2

1)0(

2

1)0( 22

tttxx eeevtv 25.025.0 10

84

430)0()(

ttto eee

vti 25.025.08 5.2

8

20

8)(

Source free RC circuitsExample 2If the switch opens at t = 0, find v(t) and wc(0) for t 0

Source free RC circuitsSolution for Example 2• Voltage across capacitor at t = 0- is the same at t = 0.

To calculate v(0), capacitor is open (due to dc source)

• Solve for the time constant

• Find Req to solve for

Vvc 82436

324

)4//12(6

4//12)0(

CReq

34//12eqR

s5.06

13

Source free RC circuitsSolution for Example 2 (cont…)• Now solve for vc and wc

tttcc eeevtv 25.0 88)0()(

JCvw cc 33.586

1

2

1)0(

2

1)0( 22

Source free RC circuitsExercise 3If the switch opens at t = 0, find :i) , vc(t), wc(0)and iR(t) for t 0

ii) Value of vc at t = 3 sec.

t = 0

+VC

_

iR10

2

5

7 +_20 V

12

1 F

Source free RC circuitsExercise 4If the switch opens at t = 0, find :i) , vc(t), wc(0)and iR(t) for t 0

ii) Value of vc at t = 4 sec.

t = 0

+VC

_

iR

+_15 V

5

10 10

5

F5

1

Source free RL circuits :• Natural Response

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

0 RL vv 0 iRdt

diL

Inductors law Ohms law

Source free RL circuits

By KVL

• Apply Kirchhoff’s voltage laws to circuit above.• Apply the laws to RL circuits to produce

differential equations.

Source free RL circuits• Current response of the RL circuit is solved below:

• This is a 1st order differential equation (only 1st derivative of i is involved:

• After Integrating both sides

• Taking power of e (V0 is initial voltage at t =0)

or

L

Rt

I

ti

0

)(lnor

LRteIti 0)(

0 iRdt

diL 0 i

L

R

dt

di

dtL

R

i

di

tti

I

dtL

R

i

di

0

)(

0

0 Iln i(t)ln 0 L

Rt

• 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 t and slower for large t.

/0)( teIti

R

L

A general form representing a RL

where

Source free RL circuits

/0)( teIti

R

L

A RL source-free circuit

where /0)( teVtv RC

A RC source-free circuit

where

Comparison between a RL and RC circuit

Source free RL circuits

• The key to working with a source-free RL circuit is finding:

1. The initial voltage i(0) = I0 through the inductor.

2. The time constant = L/R.

/0)( teIti

R

Lwhere

Source free RL circuits

Source free RL circuitsExample 3If the switch opens at t = 0, find iL(t) and wL(0) for t 0

25 V +_ 6 6

2 3 H

Source free RL circuitsSolution for Example 3• Current through inductor at t = 0- is the same at t = 0.

To calculate i(0), inductor is open (due to dc source)

• Solve for the time constant

• Find Req to solve for

AiL 5.26

125

32

3

6

125

)6//6(2

6//6)0(

eqRL

1266eqR

s25.012/3

2

6 6 6 6

3 H

25 V

Source free RL circuitsSolution for Example 3 (cont…)• Now solve for iL and wL

tttLL eeeiti 425.0 5.25.2)0()(

JLiw cL 375.95.232

1)0(

2

1)0( 22

Source free RL circuitsExercise 5If the switch opens at t = 0, find :i) , iL(t), wL(0)and iR(t) for t 0

ii) Value of iL at t = 5 sec.

t = 0

15 A 5 10

10

5

iL(t) 5 H

Source free RL circuitsExercise 6If the switch opens at t = 0, find :i) , iL(t), wL(0)and iR(t) for t 0

ii) Value of iL at t = 5 sec.

t = 0iL(t)

15 A10 10 5 5 H

First Order Circuits:Lesson Outcomes

• Understand the concept of singularity functions and relate with application in electric circuits

• Formulate the step response of an RC and RL circuits

Unit step function

Unit Step Function• Unit step function is one type of singularity function,

useful in circuit analysis.• It can be used as a good approximation to the

switching signals.• Three most widely used singularity functions are unit

step, unit impulse and unit ramp functions.

0

0

,1

,0)(

t

ttu

• The unit step function u(t) is 0 for negative values of t and 1 for positive values of t.

• In mathematical terms:

Unit Step Function• Below are the images for delayed unit step

function (a) and advanced (b) by t0

0

00 ,1

,0)(

tt

ttttu

0

00 ,1

,0)(

tt

ttttu

Step Response of an RC circuit

Step Response of an RC circuit• The step response of a circuit is its behavior when

the excitation is the step function (a voltage or a current source).

• In other words, it is the response of the circuit due to a sudden application of dc voltage or current source.

• For finding the step response of an RC circuit, we break the v(t) into natural response and forced response:

Complete response = natural response + forced responsestored energy independent source

Step Response of an RC circuit• In mathematical form:

• Where

• A forced response is produced by the circuit when an external force is applied.

• A natural response refers to the behavior of the circuit itself with no external source of excitation.

• The natural response eventually dies out along with the transient component of the forced response.

fn vvv t

n eVv 0

)1( tsf eVv

Step Response of an RC circuit• Expending the equation for complete response:

• If the switch changes position at time, t = t0 (with time delay), the response becomes:

fn vvv )1(0

ts

t eVeVv t

sst eVVeV 0

ts

ts eVeVV 0

ttss evvveVVV ))()0(()()( 0

)(00

0))()(()()( ttevtVvtv

Step Response of an RC circuitExample 4Refer to RC circuit below. Assume the switch has

been open for a long time and is closed at t =0. Calculate v(t) at t = 0.5 s

+_

_+

2 6

5 V10 V 1/3 F+v_

Step Response of an RC circuitSolution for Example 4For t < 0, switch is closed. Capacitor act as o/c. v (0-) = 10 V = v(0).

At t = 0, switch is closed. = ReqC = (2//6)*1/3 = 3/2*1/3 = 0.5 s

v() =

+__+

2 6

5 V10 V 1/3 F+v_

Step Response of an RL circuit

Conclusion

• Nodal

The End