NETWORK ANALYSIS. UNIT – I INTRODUCTION TO ELECTRICAL CIRCUITS: Circuit concept – R-L-C parameters Voltage and current sources Independent and dependent.

NETWORK ANALYSIS

UNIT – I INTRODUCTION TO ELECTRICAL CIRCUITS:

Circuit concept – R-L-C parameters

Voltage and current sources

Independent and dependent sources

Source transformations

Kirchhoff’s laws

network reduction techniques

series, parallel, series parallel

star – to –delta and or delta – to – star transformation

Mesh Analysis

Nodal analysis

Super mesh

super node concept

• Network: The interconnection of two or more circuit elements (voltage sources ,resistors , inductors and capacitors) is called an electrical network. If the network contains at least one closed path is called circuit.

• Every circuit is a network , but all the networks are not circuit.

4

Active Components (have directionality)Voltage and current sources

Passive Components (Have no directionality)Resistors, capacitors, inductors(with all the initial conditions are zero)

Ohm’s Law

I = V / R

Georg Simon Ohm (1787-1854)

I = Current (Amperes) (amps)

V = Voltage (Volts)

R = Resistance (ohms)

How you should be thinking about electric circuits:Voltage: a force that pushes the current through the circuit (in this picture it would be equivalent to gravity)

Resistance: friction that impedes flow of current through the circuit (rocks in the river)

How you should be thinking about electric circuits:

Current: the actual “substance” that is flowing through the wires of the circuit (electrons!)

How you should be thinking about electric circuits:

Lect1 EEE 202 9

Basic Electrical Quantities

• Basic quantities: current, voltage and power– Current: time rate of change of electric charge

I = dq/dt1 Amp = 1 Coulomb/sec

– Voltage: electromotive force or potential, V 1 Volt = 1 Joule/Coulomb = 1 N·m/coulomb

– Power: P = I V1 Watt = 1 Volt·Amp = 1 Joule/sec

Overview of Circuit Theory

• Power is the rate at which energy is being absorbed or supplied.

• Power is computed as the product of voltage and current:

• Sign convention: positive power means that energy is being absorbed; negative power means that power is being supplied.

VIPtitvtp or

Lect1 EEE 202 11

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)

Energy Storage Elements

• Capacitors store energy in an electric field.• Inductors store energy in a magnetic field.• Capacitors and inductors are passive

elements:– Can store energy supplied by circuit– Can return stored energy to circuit– Cannot supply more energy to circuit than is

stored.

Independent sources :

1. Voltage source

2. Current source

Dependent sources:

3. Voltage dependent voltage source

4. Voltage dependent current source

5. current dependent voltage source

6. current dependent current source

Types of sources

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 resistors

would be 10V but the currents would be different.

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

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.

Practical current source:

Practical current source has an internal resistance, but we treat this internal resistance as being connected in parallel with an ideal current source. An ideal current source has infinite internal resistance.

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.

A dependent source has a value that depends on another

voltage or current in the circuit.

Source transformation

Another circuit simplifying technique

It is the process of replacing a voltage source vS in series with a resistor 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

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.

+-sv

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

+-sv

a

b

R

LR si

a

b

R LRLi Li

s

L

v

R R

+LiL

RR R

+L si i

Equating we have ,

s

L L

v RR R R R

+ + si s

s

v i

R OR s s v Ri

Simple Circuits • Series circuit

– All in a row– 1 path for electricity– 1 light goes out and the

circuit is broken

• Parallel circuit– Many paths for electricity– 1 light goes out and the

others stay on

Resistors in Series

• A single loop circuit is one which has only a single loop.

• The same current flows through each element of the circuit - the elements are in series.

Resistors in Series

Two elements are in series if the current that flows through one must also flow through the other.

R1 R2

Series

Resistors in SeriesConsider two resistors in series with a voltage v(t) across them:

v1(t)

v2(t)

21

11 )()(

RR

Rtvtv

+

21

22 )()(

RR

Rtvtv

+

R1

R2

-+

+

-

+

-

v(t)

i(t)Voltage division:

Resistors in Series

• If we wish to replace the two series resistors with a single equivalent resistor whose voltage-current relationship is the same, the equivalent resistor has a value given by

21 RRReq +

Resistors in Series• For N resistors in series, the equivalent resistor has a value given by

Neq RRRRR ++++ 321

R1

R3

R2 Req

Resistors in Parallel

• When the terminals of two or more circuit elements are connected to the same two nodes, the circuit elements are said to be in parallel.

Resistors in ParallelConsider two resistors in parallel with a voltage v(t) across them:

21

21 )()(

RR

Rtiti

+

21

12 )()(

RR

Rtiti

+

R1 R2

+

-

v(t)

i(t)Current division:

i1(t) i2(t)

Resistors in Parallel

• If we wish to replace the two parallel resistors with a single equivalent resistor whose voltage-current relationship is the same, the equivalent resistor has a value given by

21

21

RR

RRReq +

Resistors in Parallel• For N resistors in parallel, the equivalent resistor has a value given by

N

eq

RRRR

R1111

1

321

++++

ReqR3R2R1

Lect1 EEE 202 33

Parallel

Two elements are in parallel if they are connected between (share) the same two (distinct) end nodes.

Parallel Not Parallel

R1

R2

R1

R2

ECE 201 Circuit Theory I 34

Series-Parallel Combinations of Inductance and Capacitance

• Inductors in Series– All have the same current

1 1

div L

dt

2 2

div L

dt 3 3

div L

dt

1 2 3v v v v + +


1 2 3

1 2 3

1 2 3

1 2 3

( )

eq

eq

v v v v

di di div L L L

dt dt dtdi

v L L Ldt

div L

dtL L L L

+ +

+ +

+ +

+ +

To Determine the Equivalent Inductance


The Equivalent Inductance


Inductors in Parallel

All Inductors have the same voltage across their terminals.


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 tL

i vd i tL

i vd i tL

+

+

+


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 tL L L

i vd i tL

L L L L

i t i t i t i t

+ +

+ + + + +

+

+ +

+ +


Summary for Inductors in Parallel


Capacitors in SeriesProblem # 6.30


Capacitors in ParallelProblem # 6.31

Ch06 Capacitors and Inductors 43

6.3 Series and Parallel Capacitors

• The equivalent capacitance of N parallel-connected capacitors is the sum of the individual capacitance.

Niiiii ++++ ...321

dtdv

Cdtdv

Cdtdv

Cdtdv

Ci N++++ ...321

dtdv

Cdtdv

C eq

N

kK

1

Neq CCCCC ++++ ....321


Fig 6.15

Neq CCCCC

1...

1111

321

++++


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

111CCCeq

+21

21

CCCC

Ceq +


Table 6.1

)(1cba

cb

RRR

RRR

++

)(2cba

ac

RRR

RRR

++

)(3cba

ba

RRR

RRR

++

1

133221

R

RRRRRRRa

++

2

133221

R

RRRRRRRb

++

3

133221

R

RRRRRRRc

++

Delta -> Star Star -> Delta

Y transformation

Star delta transformation

Lect1 EEE 202 49

Kirchhoff’s Laws

• Kirchhoff’s 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• Kirchhoff’s Voltage Law (KVL)

– sum of voltages around any loop in a circuit is zero

Lect1 EEE 202 50

KCL (Kirchhoff’s 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

jj ti

1

0)(

Lect1 EEE 202 51

Open Circuit

• What if R = ?

• i(t) = v(t)/R = 0

v(t)

The Rest of the Circuit

i(t)=0

+

–

i(t)=0

Lect1 EEE 202 52

Short Circuit

• What if R = 0 ?

• v(t) = R i(t) = 0

The Rest of the Circuit v(t)=0

i(t)

+

–

Lect1 EEE 202 53

Resistors

• A resistor is a circuit element that dissipates electrical energy (usually as heat)

• Real-world devices that are modeled by resistors: incandescent light bulbs, heating elements (stoves, heaters, etc.), long wires

• Resistance is measured in Ohms (Ω)

Overview of Circuit Theory

• Basic quantities are voltage, current, and power.

• The sign convention is important in computing power supplied by or absorbed by a circuit element.

• Circuit elements can be active or passive; active elements are sources.

KCL and KVL

• Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL) are the fundamental laws of circuit analysis.• KCL is the basis of nodal analysis – in which the unknowns are the voltages at each of the nodes of the circuit.• KVL is the basis of mesh analysis – in which the unknowns are the currents flowing in each of the meshes of the circuit.

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

jj ti

1

0)(

KCL and KVL

• KVL– The sum of voltages

around any loop in a circuit is zero.

0)(1

n

jj tv

+

-

v1(t)

++

-

-

v2(t)

v3(t)

KCL and KVL

• In KVL:– A voltage encountered + to - is positive.– A voltage encountered - to + is negative.

• Arrows are sometimes used to represent voltage differences; they point from low to high voltage.

+

-

v(t) v(t)≡

NETWORK ANALYSIS. UNIT – I INTRODUCTION TO ELECTRICAL CIRCUITS: Circuit concept – R-L-C parameters Voltage and current sources Independent and dependent.

Documents

constant voltage

product of voltage

practical voltage source

current sources independent

voltage volts r

c parameters voltage

flow of current

inductors store energy