Chapter 1 DC Circuit

7/25/2019 Chapter 1 DC Circuit

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Chapter 1

Direct-Current Circuits


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DEFINATIONS

Linear elements : In an electric circuit, a linear element is an electricalelement with a linear relationship between current and voltage. Resistors arethe most common example of a linear element; other examples includecapacitors, inductors, and transformers.

Nonlinear Elements :A nonlinear element is one which does not have alinear input/output relation. In a diode, for example, the current is a non-linear function of the voltage . Most semiconductor devices have non-linearcharacteristics.

Active Elements :The elements which generates or produces electrical

energy are called active elements. Some of the examples are batteries,generators , transistors, operational amplifiers , vacuum tubes etc.

Passive Elements :All elements which consume rather than produceenergy are called passive elements, like resistors, Inductors and capacitors.


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In unilateral element, voltage current relation is not same for boththe direction. Example: Diode, Transistors.

In bilateral element, voltage current relation is same for both thedirection. Example: Resistor

Ideal Voltage Source: The voltage generated by the source does not

vary with any circuit quantity. It is only a function of time. Such a sourceis called an ideal voltage Source.

Ideal Current Source: The current generated by the source does notvary with any circuit quantity. It is only a function of time. Such a sourceis called as an ideal current source.

Resistance : It is the property of a substance which opposes the flow ofcurrent through it. The resistance of element is denoted by the symbolR. It is measured in Ohms.


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Electric Current

Resistance and Ohms Law

Energy and Power in Electric Circuits

Resistors in Series and Parallel

Kirchhoffs Rules

Circuits Containing CapacitorsRCCircuits

Ammeters and Voltmeters


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Electric Current

Electric current is the flow of electric charge from

one place to another.

A closed path through which charge can flow,returning to its starting point, is called an

electric circuit.


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Electric Current

A battery uses chemical reactions to produce a

potential difference between its terminals. Itcauses current to flow through the flashlight

bulb similar to the way the person lifting the

water causes the water to flow through the

paddle wheel.


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Electric Current

A battery that is disconnected from any circuithas an electric potential difference between its

terminals that is called the electromotive force or

emf:

Rememberdespite its name, the emfis an

electric potential, not a force.

The amount of work it takes to move a chargeQfrom one terminal to the other is:


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Electric Current

The direction of current flowfrom the positive

terminal to the negative onewas decidedbefore it was realized that electrons are

negatively charged. Therefore, current flows

around a circuit in the direction a positive chargewould move;

electrons move

the other way.

However, thisdoes not matter

in most circuits.


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Electric Current

Finally, the actual motion of electrons along a

wire is quite slow; the electrons spend most of

their time bouncing around randomly, and have

only a small velocity component opposite to

the direction of the current. (The electric signalpropagates much more quickly!)


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2 Resistance and Ohms Law

Under normal circumstances, wires present

some resistance to the motion of electrons.Ohms law relates the voltage to the current:

Be carefulOhms law is not a universal lawand is only useful for certain materials

(which include most metallic conductors).


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Solving for the resistance, we find

The units of resistance, volts per ampere,are called ohms:


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Two wires of the same length and diameter will

have different resistances if they are made of

different materials. This property of a material is

called the resistivity.


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The difference between

insulators,

semiconductors, andconductors can be clearly

seen in their resistivities:


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In general, the resistance of materials goes up

as the temperature goes up, due to thermaleffects. This property can be used in

thermometers.

Resistivity decreases as the temperaturedecreases, but there is a certain class of

materials called superconductors in which the

resistivity drops suddenly to zero at a finite

temperature, called the critical temperature TC.


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3 Energy and Power in Electric Circuits

When a charge moves across a potential

difference, its potential energy changes:

Therefore, the power it takes to do this is


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In materials for which Ohms law holds, thepower can also be written:

This power mostly becomes heat inside the

resistive material.


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When the electric company sends you a bill,

your usage is quoted in kilowatt-hours (kWh).

They are charging you for energy use, and kWh

are a measure of energy.


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4 Resistors in Series and Parallel

Resistors connected end to end are said to be in

series. They can be replaced by a single

equivalent resistance without changing the

current in the circuit.


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Since the current through the series resistors

must be the same in each, and the total potential

difference is the sum of the potential differences

across each resistor, we find that the equivalent

resistance is:


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Resistors are in parallel

when they are across the

same potential

difference; they can

again be replaced by a

single equivalent

resistance:


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Using the fact that the potential difference

across each resistor is the same, and the total

current is the sum of the currents in each

resistor, we find:

Note that this equation gives you the inverse of

the resistance, not the resistance itself!


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If a circuit is more complex, start with

combinations of resistors that are either purely

in series or in parallel. Replace these with their

equivalent resistances; as you go on you will be

able to replace more and more of them.


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5 Kirchhoffs Rules

More complex circuits cannot be broken down

into series and parallel pieces.For these circuits, Kirchhoffs rules are useful.

The junction rule is a consequence of charge

conservation; the loop rule is a consequenceof energy conservation.


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5 Kirchhoffs Rules

The junction rule: At any junction, the current

entering the junction must equal the current

leaving it.


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5 Kirchhoffs Rules

The loop rule: The algebraic sum of the potential

differences around a closed loop must be zero (it

must return to its original value at the original

point).


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5 Kirchhoffs Rules

Using Kirchhoffs rules:The variables for which you are solving are the

currents through the resistors.

You need as many independent equations asyou have variables to solve for.

You will need both loop and junction rules.


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6 Circuits Containing Capacitors

Capacitors can also be connected in series or in

parallel.

When capacitors are

connected in parallel,the potential difference

across each one is the

same.


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Therefore, the equivalent capacitance is thesum of the individual capacitances:


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Capacitors connected in

series do not have the

same potential difference

across them, but they doall carry the same charge.

The total potential

difference is the sum of the

potential differencesacross each one.


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Therefore, the equivalent capacitance is

Capacitors in series combine like resistors in

parallel, and vice versa.

Note that this equation gives you the inverse of

the capacitance, not the capacitance itself!


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7 RCCircuits

In a circuit containing

only batteries and

capacitors, charge

appears almostinstantaneously on the

capacitors when the

circuit is connected.

However, if the circuitcontains resistors as

well, this is not the case.


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7 RCCircuits

Using calculus, it can be shown that the charge

on the capacitor increases as:

Here, is the time constant of the circuit:

And is the final charge on the capacitor, Q.


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7 RCCircuits

Here is the charge vs. time for an RCcircuit:


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7 RCCircuits

It can be shown that the current in the circuit

has a related behavior:


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8 Ammeters and Voltmeters

An ammeter is a device for measuring current,

and a voltmeter measures voltages.The current in the circuit must flow through the

ammeter; therefore the ammeter should have

as low a resistance as possible, for the least

disturbance.


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8 Ammeters and Voltmeters

A voltmeter measures the potential

drop between two points in a circuit.It therefore is connected in parallel;

in order to minimize the effect on

the circuit, it should have as large a

resistance as possible.


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Transients Analysis


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Solution to First Order Differential Equation

)()()(

tfKtxdt

tdxs

Consider the general Equation

Let the initial condition bex(t = 0) = x( 0 ),

then we solve the differential equation:

)()()(

tfKtxdt

tdxs

The complete solution consists of two parts:the homogeneous solution (natural solution)

the particular solution (forced solution)


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The Natural Response

/)(,)(

)(

)()(,)()(0)()(

t

N

N

N

N

NNNN

N

etxdt

tx

tdx

dttxtdxtx

dttdxortx

dttdx


Setting the excitation f (t) equal to zero,

)()()( tfKtxdt

tdxs

It is called the natural response.


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The Forced Response

0)(

)(

)(

tforFKtx

FKtxdt

tdx

SF

SFF


Setting the excitation f (t) equal to F, a

constant for t 0

)()()( tfKtxdt

tdxs

It is called the forced response.


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The Complete Response

)(

)()(

/

/

xe

FKe

txtxx

t

St

FN


The complete

response is:

the natural response+the forced response

)()()( tfKtxdt

tdxs

Solve for ,

)()0(

)()0()0(

0

xx

xxtx

tfor

The Completesolution:)()]()0([)( / xexxtx t

/

)]()0([

t

exx

called transient respon

)(x called steady state response


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WHAT IS TRANSIENT RESPONSE

Figure 5.1


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Transients Analysis

1. Solve first-order

RC

or

RL

circuits.

2. Understand the concepts of transient

response and steady-state response.

3. Relate the transient response of first-

order

circuits to the time constant.


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Transients

The solution of the differential equation

represents are response of the circuit. It

is called natural response

The response must eventually die out,

and therefore referred to as transient

response

source free response)


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Discharge of a Capacitance through a

Resistance

ic iR 0,0 RC iii

0

R

tv

dt

tdv

C

CC

Solving the above equa

with the initial conditio

Vc(0) = Vi

Di h f C it th h


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Discharge of a Capacitance through a

Resistance

0 R

tv

dt

tdv

C

CC

0 tvdttdv

RC CC

stC Ketv

0 stst KeRCKse

RC

s 1

RCtC Ketv

K

Ke

VvRC

iC

/0

)0(

RCtiC eVtv


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RCtiC eVtv Exponential decay waveform

RC is called the time constant.

At time constant, the voltage is 36.8%

of the initial voltage.

)1( RCtiC eVtv

Exponential rising

waveform

RC is called the time

constant.

At time constant, the

voltage is 63.2% of the


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RC CIRCUIT

for t= 0-, i(t) = 0

u(t) is voltage-step function

Vu(t)

R

C

+

VC

-

i(t)t = 0

+

_V

R

C

+

VC

-

i(t)t = 0

+

_Vu(t)


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RC CIRCUIT

Vu(t)

0)(,

,)(

tforVtuvVvdt

dv

RC

dt

dvCi

R

vtvui

ii

CC

CC

CR

CR

Solving the differential equation


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Complete Response

Complete response

= natural response + forced response

Natural response (source free response) is

due to the initial condition

Forced response is the due to the external

excitation.


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Figure

5.17,5.18

5-8

a). Complete, transient and

steady state response

b). Complete, natural, andforced responses of the

circuit


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Circuit Analysis for RC Circuit

Vs

+

Vc

-

+ VR-

R

C

iR

iC

sRC

CC

RsR

CR

vRC

vRCdt

dv

dt

dvCi

R

vvi

ii

11

,

Apply KCL

v

s

is the source applied.


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Solution to First Order Differential Equation

)()()(

tfKtxdt

tdxs


Let the initial condition bex(t = 0) = x( 0 ),

then we solve the differential equation:

)()()(

tfKtxdt

tdxs

The complete solution consits of two parts:the homogeneous solution (natural solution)

the particular solution (forced solution)


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The Natural Response

/)(

)()(

0)(

)(

tN

NNN

N

etx

tx

dt

tdx

ortxdt

tdx


Setting the excitation f (t) equal to zero,

)()()( tfKtxdttdx

s

It is called the natural response.


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The Forced Response

0)(

)(

)(

tforFKtx

FKtxdt

tdx

SF

SF

F


Setting the excitation f (t) equal to F, a

constant for t 0

)()()( tfKtxdttdx

s

It is called the forced response.


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The Complete Response

)(

)()(

/

/

xe

FKe

txtxx

t

St

FN


The complete

response is:

the natural

response +the forced

response

)()()( tfKtxdttdx

s

Solve for ,

)()0(

)()0()0(

0

xx

xxtx

tfor

The Completesolution:)()]()0([)( / xexxtx t

/

)]()0([ t

exx

called transient respons)(x called steady state response

E l


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Example

+

Vc

-

+ VR-

100 k

ohms

0.01

microF

iR

iC

00V

Initial condition Vc(0) = 0V

sCC

CC

CsR

CR

vvdt

dvRC

dt

dvCi

R

vvi

ii

,

10010

1001001.010

3

65

CC

CC

vdt

dv

vdt

dv

E l


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Example

+

Vc

-

+ VR-

100 k

ohms

0.01

microF

iR

iC

00V

Initial condition Vc(0) = 0V

and

3

3

10

10

100100

100

1000,0)0(100

t

c

c

t

c

ev

A

AvAsAev

)()()(

tfKtx

dt

tdxs

)(

)()(

/

/

xe

FKe

txtxx

t

St

FN

10010 3 CC vdt

dv

Energy stored in capacitor


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Energy stored in capacitor

22 )()(21 o

tt

tt

tt

tvtvC

vdvCdtdt

dvCvpdt

dt

dvCvvip

ooo

If the zero-energy reference is selected atto, i

capacitor voltage is also zero at that instant, th 2

2

1)( Cvtwc

RC CIRCUIT


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R C

Power dissipation in the

resistor is:pR= V

2/R = (Vo2/R)e -2 t/RC

2

0/22

0/22

0

2

1

|)2

1(

o

RCto

RCto

RR

CV

eRC

RV

R

dteV

dtpW

RC CIRCUIT

Total energy turned into heat in

the resistor


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RL CIRCUITS

LR

-

VR

+

+

VL

-

i(t)

Initial condition

i(t= 0) = Io

equationaldifferentitheSolving

idtdi

RL

dt

diLRivv LR

0

0

CIRCUITS


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RL CIRCUITS

LR-VR

+

+VL

-

i(t)

Initial conditioni(t= 0) = Io LRt

o

o

to

iI

t

o

ti

I

eIti

tL

R

Ii

tL

Ri

dtL

R

i

didt

L

R

i

di

iL

R

dt

di

o

o

/

)(

)(

lnln

||ln

,

0

CIRCUIT


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RL CIRCUIT

LR-VR

+

+VL

-

i(t)

Power dissipation in the

resistor is:pR= i

2R = Io2e-2Rt/LRTotal energy turned into heat in

the resistor

2

0/22

0

/22

0

2

1

|)2

(

o

LRto

LRtoRR

LI

eR

LRI

dteRIdtpW

It is expected as the energy stored in the inductor2

2

1

oLI

V (t)

i(t)


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RL CIRCUITVu(t)

RL

+

VL

-

+

_Vu(t)

ktRiVR

L

sidesbothgIntegratin

dtRiV

Ldi

Vdt

diLRi

)ln(

,

0,

]ln)[ln(

ln,0)0(

/

/

tforeR

V

R

V

i

oreV

RiV

tVRiV

R

L

VRLkthusi

LRt

LRt

where L/R is the time const

Chapter 1 DC Circuit

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