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NETWORK ANALYSIS

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UNIT I INTRODUCTION TO ELECTRICAL CIRCUITS:

Circuit concept R-L-C parameters

Voltage and current sources

Independent and dependent sources

Source transformations

Kirchhoffs lawsnetwork reduction techniques

series, parallel, series parallel

star todelta and or delta to star transformation

Mesh AnalysisNodal analysis

Super mesh

super node concept

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3

CIRCUITInput Output

An Electronic Circuit is a combination of electronic components

and conductive wires interconnected in a way as to achieve an

outcome: - Achieve a current /voltage of a certain value (signal)

- Amplify a signal

- Transfer data

The purpose of an electronic component is to allow the designer tocontrol the flow of current as to achieve a specified result/output.

(Resistors, Capacitors, Inductors, Diodes, Transistors )

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4

Active Components (have directionality)

Voltage and current sources

Passive Components (Have no directionality)Resistors, capacitors, inductors

(with all the initial conditions are zero)

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Ohms Law

I = V / R

Georg Simon Ohm (1787-1854)

I = Current (Amperes) (amps)

V = Voltage (Volts)

R = Resistance (ohms)

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How you should be thinkingabout electric circuits:

Voltage: a force that

pushes the current

through the circuit (in

this picture it would beequivalent to gravity)

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Resistance: friction that

impedes flow of current

through the circuit(rocks in the river)

How you should be thinkingabout electric circuits:

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Current: the actual

substance that is

flowing through the

wires of the circuit

(electrons!)

How you should be thinkingabout electric circuits:

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Lect1 EEE 202 9

Basic Electrical Quantities

Basic quantities: current,voltage and power

Current: time rate of change of electric charge

I = dq/dt

1 Amp = 1 Coulomb/sec

Voltage: electromotive force or potential, V 1 Volt= 1 Joule/Coulomb = 1 Nm/coulomb

Power: P = IV1 Watt = 1 VoltAmp = 1 Joule/sec

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Overview of Circuit Theory

Power is the rate at which energy is beingabsorbed or supplied.

Poweris computed as the product of voltage

and current:

Sign convention: positive power means that

energy is being absorbed; negative power

means that power is beingsupplied.

VIPtitvtp or

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11

Resistors are passive elements that oppose/restrict

the flow of current.

A voltage is developed across its terminal,

proportional to the current through the resistor.

V = IR

Units: Ohms ()

A. Resistors:

Electronic Components (cont.)

http://www.clker.com/clipart-resistor-symbol.html

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Inductors

Inductors are the passive element that opposes sudden change in voltage

Inductor stores certain energy in the presence of magnetic field

The voltage across an inductor is proportional to the rate of change of current

V=L(di/dt)Units for inductance is : henrys(H)

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Inductors

Inductance occurs when current flows through a(real) conductor.

The current flowing through the conductor sets up a

magnetic field that is proportional to the current. The voltage difference across the conductor isproportional to the rate of change of the magneticflux.

The proportionality constant is called the inductance,denotedL.

Inductance is measured in Henrys (H).

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Inductors

dt

tdiLtv

)()( i(t)

+

-

v(t)

The

rest

of

the

circuit

L

t

dxxvL

ti )(1

)(

t

t

dxxvL

titi

0

)(1

)()( 0

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Inductors

The current through an inductor cannot

change instantaneously.

The energy stored in the inductor is given by

)(2

1)( 2 tLitw

L

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Ch06 Capacitors and Inductors 16

6.4 Inductors

An inductor is made of a coil of conducting wire

l

ANL

2

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Ch06 Capacitors and Inductors 17

Fig 6.22

(H/m)104 70

0

2

r

lANL

turns.ofnumber:N

length.:l

area.sectionalcross: A coretheoftypermeabili:

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Ch06 Capacitors and Inductors 18

Fig 6.23

(a) air-core

(b) iron-core

(c) variable iron-core

l ( )

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19

B. Capacitors:

behave like a tiny rechargeable battery.

(store energy and release it later. ) are made of two parallel conductors separated by a

dielectric.

The ability of a capacitor to store charge is called

Capacitance

C = Q/V (amount of charge stored/applied voltage)

The unit of capacitance is the Farad.

Commonly used capacitances are much smaller than 1

Farad, micro-Farads (10-6 Farad, F),

nano-Farads (10-9 Farad, nF),

pico-Farads (10-12 Farad, pF).

Electronic Components (cont.)

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Capacitors

Capacitance occurs when two conductors are

separated by a dielectric (insulator).

Charge on the two conductors creates an electric

field that stores energy. The voltage difference between the two conductors

is proportional to the charge.

The proportionality constant Cis called capacitance.

Capacitance is measured in Farads (F).

tvCtq

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Capacitors

i(t)+

-

v(t)

The

rest

of

the

circuit

dt

tdvCti

)()(

t

dxxiC

tv )(1

)(

t

t

dxxiC

tvtv

0

)(1)()( 0

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Capacitors

The voltage across a capacitor cannot change

instantaneously.

The energy stored in the capacitors is given by

)(

2

1)( 2 tCvtwC

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Ch06 Capacitors and Inductors 23

6.2 Capacitors

A capacitor consists of two conducting platesseparated by an insulator (or dielectric).

(F/m)10854.8

12

0

0

r

dAC

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Ch06 Capacitors and Inductors 24

Three factors affecting the value ofcapacitance:

1. Area: the larger the area, the greater thecapacitance.

2. Spacing between the plates: the smaller the

spacing, the greater the capacitance.3. Material permittivity: the higher the permittivity,

the greater the capacitance.

dAC

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Ch06 Capacitors and Inductors 25

Fig 6.4

(a) Polyester capacitor, (b) Ceramic capacitor, (c) Electrolytic capacitor

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Ch06 Capacitors and Inductors 26

Fig 6.5

Variable capacitors

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Ch06 Capacitors and Inductors 27

Fig 6.3

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Ch06 Capacitors and Inductors 28

Fig 6.2

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Lect1 EEE 202 29

Active vs. Passive Elements

Active elements can generate energy

Voltage and current sources

Batteries

Passive elements cannot generate energy

Resistors

Capacitors and Inductors (but CAN store energy)

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Energy Storage Elements

Capacitorsstore energy in an electric field.

Inductorsstore energy in a magnetic field.

Capacitors and inductors are passiveelements:

Can store energy supplied by circuit

Can return stored energy to circuit

Cannot supply more energy to circuit than is

stored.

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Independent sources :

1. Voltage source

2. Current source

Dependent sources:

1. Voltage dependent voltage source

2. Voltage dependent current source

3. current dependent voltage source

4. current dependent current source

Types of sources

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Ideal voltage source:

An ideal voltage source has zero internal resistance so that changes in load

resistance will not change the voltage supplied.

An ideal voltage source gives a constant voltage, whatever the current is.

A simple example is a 10V battery. For example, a 1ohm resistor or a

10ohm resistor could be connected to it; the voltage across both resistorswould be 10V but the currents would be different.

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Practical voltage source:

Practical voltage source has an internal resistance (greater than zero),

but we treat this internal resistance as being connected in series with

an ideal voltage source.

An ideal voltage source has zero internal resistance

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Ideal current source:

An ideal current source is a circuit element that maintains a prescribed

current through its terminals regardless of the voltage across those

terminals.

A ideal current source gives a constant current whatever the load is.

If you have a 2A current source for example:

-with a 3 ohm resistor it would automatically change the voltage to 6V

-with a 30 ohm resistor it would automatically change the voltage to 60V

but the current would be 2A whichever resistor was connected.

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Practical current source:

Practical current source has an internal resistance, but we treat thisinternal resistance as being connected in parallel with an ideal current

source.

An ideal current source has infinite internal resistance.

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Dependent sources :

Dependent sources behave just like independent voltage and current

sources, except their values are dependent in some way on another

voltage or current in the circuit.

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A dependent sourcehas a value that depends on another

voltage or current in the circuit.

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Source transformation

Another circuit simplifying technique

Itis the process of replacing a voltage source vS in series with aresistor R by a current source iS in parallel with a resistor R, or vice

versa

+

R

vs

a

b

Terminal a-b sees:

Open circuit voltage: vs

Short circuit current: vs/R

For this circuit to be equivalent, it

must have the same terminal

charateristics

Ris

a

b

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Source Transformations

A method called Source Transformations will allow the transformations of a voltage

source in series with a resistor to a current source in parallel with resistor.

s

a

b

R

The double arrow indicate that the transformation is bilateral , that we can start with either

configuration and drive the other

si

a

b

R

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s

a

b

R

LRsi

a

b

R LRLi

Li

s

L

v

R R

Li

L

R

R R

L si i

Equating we have ,

s

L L

v RR R R R

s

i ssviR

OR s sv Ri

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Simple Circuits Series circuit

All in a row

1 path for electricity

1 light goes out and thecircuit is broken

Parallel circuit Many paths for electricity

1 light goes out and theothers stay on

d

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circuit diagram

cell switchlamp wires

Scientists usually draw electric circuits using symbols;

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Series resistors & voltage division

Series: Two or more elements are in series if they are cascaded or

connected sequentially and consequently carry the same current.

The equivalent resistance of any number of resistors connected in a

series is the sum of the individual resistances

N

n

nNeq RRRRR1

21

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Series resistors & voltage division

Lets say we want to find v2

v2 = iR2 where,

i vR

1R

2

v

2

R2

R1 R2

v- Voltage Division Rule

- Principle of Voltage Division

Note that if R2 >> R1, then v2 v

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Series resistors & voltage division

R1

v

R = i = 0

+ v1 + v2

vv

2

v

21

22

RR

Rv

If R2 is replaced with open circuit,

the resistance would be

0v1

v21

11

RR

Rv

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Resistors in Series

Consider two resistors in series with a voltagev(t) across them:

v1(t)

v2(t)

21

11 )()(

RR

Rtvtv

21

22 )()(

RRRtvtv

R1

R2

-

+

+

-

+

-

v(t)

i(t)

Voltage division:

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Parallel resistors & current division

Parallel: Two or more elements are in parallel if they are connected

to the same two nodes and consequently have the same voltageacross them.

The equivalent resistance of a circuit with N resistors in parallel is:

Neq RRRR1111

21

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Parallel resistors & current division

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Lets say we want to find i2

where,

21

21eq

RR

RRiiRv

iRR

Ri

21

12

- Current Division Rule

2

2R

vi

Parallel resistors & current division

- Principle of Current Division

+

v

+

v

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Parallel resistors & current division

iiiRR

Ri 2

21

12

0iiRR

Ri 1

21

21

0iiRR

Ri 2

21

12

iiiRR

Ri 1

21

21

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Resistors in Parallel

Consider two resistors in parallelwith a voltagev(t) across them:

21

21 )()(

RR

Rtiti

21

12 )()(

RR

Rtiti

R1 R2

+

-

v(t)

i(t)

Current division:

i1(t) i2(t)

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ECE 201 Circuit Theory I 52

Series-Parallel Combinations of Inductance and

Capacitance

Inductors in Series

All have the same current

1 1

di

v L dt 2 2di

v L dt3 3

di

v L dt

1 2 3v v v v

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ECE 201 Circuit Theory I 53

1 2 3

1 2 3

1 2 3

1 2 3

( )

eq

eq

v v v vdi di di

v L L Ldt dt dt

div L L L

dt

div L dt

L L L

To Determine the Equivalent Inductance

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ECE 201 Circuit Theory I 54

The Equivalent Inductance

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ECE 201 Circuit Theory I 55

Inductors in Parallel

All Inductors have the same voltage acrosstheir terminals.

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ECE 201 Circuit Theory I 56

0

0

0

1 1 0

1

2 2 0

2

3 3 0

3

1( )

1( )

1( )

t

t

t

t

t

t

i vd i t

L

i vd i t L

i vd i t L

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ECE 201 Circuit Theory I 57

00

0

1 2 3

1 2 0 3 0

1 2 3

0

1 2 3

0 1 0 2 0 3 0

1 1 1

( ) ( ) ( )

1( )

1 1 1 1

( ) ( ) ( ) ( )

t

t

t

t

eq

eq

i i i i

i vd i t i t i t L L L

i vd i t L

L L L L

i t i t i t i t

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ECE 201 Circuit Theory I 58

Summary for Inductors in Parallel

C i i S i

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ECE 201 Circuit Theory I 59

Capacitors in Series

Problem # 6.30

C i i P ll l

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ECE 201 Circuit Theory I 60

Capacitors in Parallel

Problem # 6.31

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Ch06 Capacitors and Inductors 61

6.3 Series and Parallel Capacitors

The equivalent capacitance ofN parallel-

connected capacitors is the sum of the

individual capacitance.

Niiiii ...321

dt

dvC

dt

dvC

dt

dvC

dt

dvCi N ...321

dtdvC

dtdvC eq

N

kK

1

Neq CCCCC ....321

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Ch06 Capacitors and Inductors 62

Fig 6.15

Neq CCCCC

1...

1111

321

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Ch06 Capacitors and Inductors 63

Series Capacitors

The equivalent capacitance of series-connected capacitors is the reciprocal of the

sum of the reciprocals of the individual

capacitances.

Neq

t

N

t

eq

C

tq

C

tq

C

tq

C

tq

idCCCC

idC

)()()()(

)1...111(1

21

321

)(...)()()( 21 tvtvtvtv N

21

111

CCCeq

21

21

CC

CCCeq

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Ch06 Capacitors and Inductors 64

These results enable us to look the capacitor

in this way: 1/C has the equivalent effect as

the resistance. The equivalent capacitor of

capacitors connected in parallel or series canbe obtained via this point of view, so is the Y-

connection and its transformation

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Ch06 Capacitors and Inductors 65

Table 6.1

Y transformation

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Y transformationStar delta transformation

How can we combine R1 to R7 ?

Y transformation

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)(1

cba

cb

RRR

RRR

)(2 cba

ac

RRR

RR

R

)(3

cba

ba

RRR

RRR

1

133221

R

RRRRRRRa

2

133221

R

RRRRRRR

b

3

133221

R

RRRRRRRc

Delta -> Star Star -> Delta

Y transformationStar delta transformation

Y transformation

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Y transformationStar delta transformationexample

Y transformation

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Y transformationStar delta transformationexample

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Lect1 EEE 202 72

Kirchhoffs Laws

Kirchhoffs Current Law (KCL)

sum of all currents entering a node is zero

sum of currents entering node is equal to sum of

currents leaving node

Kirchhoffs Voltage Law (KVL)

sum of voltages around any loop in a circuit is zero

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Lect1 EEE 202 73

KCL (Kirchhoffs Current Law)

The sum of currents entering the node is zero:

Analogy: mass flow at pipe junction

i1(t)

i2(t) i4(t)

i5(t)

i3(t)

n

j

j ti1

0)(

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Lect1 EEE 202 74

Open Circuit

What if R = ?

i(t) = v(t)/R = 0

v(t)

The Rest

of the

Circuit

i(t)=0

+

i(t)=0

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Lect1 EEE 202 75

Short Circuit

What if R = 0 ?

v(t) = R i(t) = 0

The Rest

of the

Circuitv(t)=0

i(t)

+

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Overview of Circuit Theory

Basic quantities are voltage, current, andpower.

The sign convention is important in

computing power supplied by or absorbed bya circuit element.

Circuit elementscan be active or passive;

active elements are sources.

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KCL and KVL

Kirchhoffs Current Law (KCL) and KirchhoffsVoltage Law (KVL) are the fundamental laws ofcircuit analysis.

KCL is the basis ofnodal analysis in whichthe unknowns are the voltages at each of thenodes of the circuit.

KVL is the basis ofmesh analysis in whichthe unknowns are the currents flowing in eachof the meshes of the circuit.

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KCL and KVL

KCL

The sum of all currents

entering a node is

zero, or The sum of currents

entering node is equal

to sum of currents

leaving node.

i1(t)

i2(t) i4(t)

i5(t)

i3(t)

n

j

j ti

1

0)(

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Nodal Analysis

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Nodal Analysis

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Nodal Analysis

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Nodal Analysis

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