3. Fundamental Theory

8/8/2019 3. Fundamental Theory

1/14

Introduction 3.1

Vector algebra 3.2

Manipulation of complex quantities 3.3

Circuit quantities and conventions 3.4

Impedance notation 3.5

Basic circuit laws, 3.6

theorems and network reduction

References 3.7

3 F u n d a m e n t a l T h e o r y


2/14

N e t w o r k P r o t e c t i o n & A u t o m a t i o n G u i d e 1 7

3.1 INTRODUCTION

The Protection Engineer is concerned with limiting theeffects of disturbances in a power system. Thesedisturbances, if allowed to persist, may damage plantand interrupt the supply of electric energy. They aredescribed as faults (short and open circuits) or powerswings, and result from natural hazards (for instancelightning), plant failure or human error.

To facilitate rapid removal of a disturbance from a power

system, the system is divided into 'protection zones'.Relays monitor the system quantities (current, voltage)appearing in these zones; if a fault occurs inside a zone,the relays operate to isolate the zone from the remainderof the power system.

The operating characteristic of a relay depends on theenergizing quantities fed to it such as current or voltage,or various combinations of these two quantities, and onthe manner in which the relay is designed to respond tothis information. For example, a directional relaycharacteristic would be obtained by designing the relay

to compare the phase angle between voltage and currentat the relaying point. An impedance-measuringcharacteristic, on the other hand, would be obtained bydesigning the relay to divide voltage by current. Manyother more complex relay characteristics may beobtained by supplying various combinations of currentand voltage to the relay. Relays may also be designed torespond to other system quantities such as frequency,power, etc.

In order to apply protection relays, it is usually necessaryto know the limiting values of current and voltage, andtheir relative phase displacement at the relay location,for various types of short circuit and their position in thesystem. This normally requires some system analysis forfaults occurring at various points in the system.

The main components that make up a power system aregenerating sources, transmission and distributionnetworks, and loads. Many transmission and distributioncircuits radiate from key points in the system and thesecircuits are controlled by circuit breakers. For thepurpose of analysis, the power system is treated as a

network of circuit elements contained in branchesradiating from nodes to form closed loops or meshes.The system variables are current and voltage, and in

3 Fundamental Th eor y


3/14

N e t w o r k P r o t e c t i o n & A u t o m a t i o n G u i d e

3

Fundamenta

lTheory

1 8

steady state analysis, they are regarded as time varyingquantities at a single and constant frequency. Thenetwork parameters are impedance and admittance;these are assumed to be linear, bilateral (independent ofcurrent direction) and constant for a constant frequency.

3.2 VECTOR ALGEBRA

A vector represents a quantity in both magnitude anddirection. In Figure 3.1 the vector OP has a magnitude|Z| at an angle with the reference axis OX.

Figure 3.1

It may be resolved into two components at right anglesto each other, in this case xandy. The magnitude orscalar value of vector Zis known as the modulus |Z|, and

the angle is the argument, and is written as arg. Z.The conventional method of expressing a vector Z

is to

write simply |Z|.

This form completely specifies a vector for graphicalrepresentation or conversion into other forms.

For vectors to be useful, they must be expressedalgebraically. In Figure 3.1, the vector

Zis the resultant

of vectorially adding its components x and y;algebraically this vector may be written as:

Z= x+ jy Equation 3.1

where the operatorjindicates that the componentyisperpendicular to component x. In electricalnomenclature, the axis OC is the 'real' or 'in-phase' axis,and the vertical axis OY is called the 'imaginary' or'quadrature' axis. The operatorjrotates a vector anti-clockwise through 90. If a vector is made to rotate anti-clockwise through 180, then the operator j hasperformed its function twice, and since the vector hasreversed its sense, then:

jx jor j2 = -1

whence j= -1

The representation of a vector quantity algebraically interms of its rectangular co-ordinates is called a 'complexquantity'. Therefore,x+jyis a complex quantity and isthe rectangular form of the vector |Z| where:

Equation 3.2

From Equations 3.1 and 3.2:Z= |Z| (cos +jsin) Equation 3.3

and since cos and sin may be expressed inexponential form by the identities:

it follows thatZmay also be written as:

Z= |Z|ej Equation 3.4

Therefore, a vector quantity may also be representedtrigonometrically and exponentially.

3.3 MANIPULATION

OF COMPLEX QUANTIT IES

Complex quantities may be represented in any of thefour co-ordinate systems given below:

a. Polar Z

b. Rectangular x+ jy

c. Trigonometric |Z| (cos +jsin)

d. Exponential |Z|ej

The modulus |Z| and the argument are together knownas 'polar co-ordinates', and xand y are described as'cartesian co-ordinates'. Conversion between co-ordinate systems is easily achieved. As the operatorjobeys the ordinary laws of algebra, complex quantities inrectangular form can be manipulated algebraically, ascan be seen by the following:

Z1 +

Z2 = (x1+x2) +j(y1+y2) Equation 3.5

Z1 -

Z2 = (x1-x2) +j(y1-y2) Equation 3.6

(see Figure 3.2)

cos

= e ej j

2

sin

= e e

j

j j

2

Z x y

y

xx Z

y Z

= +( )

=

=

=

2 2

1

tan

cos

sin

Figure 3.1: Vector OP

0

Y

X

P

|Z|

y

x

q


4/14


3

Fundamenta

lTheory

Equation 3.7

3.3.1 Complex variables

Some complex quantities are variable with, for example,time; when manipulating such variables in differentialequations it is expedient to write the complex quantityin exponential form.

When dealing with such functions it is important toappreciate that the quantity contains real and imaginary

components. If it is required to investigate only onecomponent of the complex variable, separation intocomponents must be carried out after the mathematicaloperation has taken place.

Example: Determine the rate of change of the realcomponent of a vector |Z|wtwith time.

|Z|wt= |Z| (coswt+jsinwt)

= |Z|ejwt

The real component of the vector is |Z|coswt.

Differentiating |Z|ejwt

with respect to time:

=jw|Z| (coswt+jsinwt)

Separating into real and imaginary components:

Thus, the rate of change of the real component of avector |Z|wtis:

-|Z| w sinwt

d

dtZ e Z w wt jw wt jwt( )= +( )sin cos

d

dtZ e jw Z e jwt jwt =

Z Z Z Z

Z

Z

Z

Z

1 2 1 2 1 2

1

2

1

2

1 2

= +

=

3.3.2 Complex Numbers

A complex number may be defined as a constant thatrepresents the real and imaginary components of aphysical quantity. The impedance parameter of anelectric circuit is a complex number having real andimaginary components, which are described as resistanceand reactance respectively.

Confusion often arises between vectors and complexnumbers. A vector, as previously defined, may be acomplex number. In this context, it is simply a physicalquantity of constant magnitude acting in a constantdirection. A complex number, which, being a physicalquantity relating stimulus and response in a givenoperation, is known as a 'complex operator'. In thiscontext, it is distinguished from a vector by the fact thatit has no direction of its own.

Because complex numbers assume a passive role in anycalculation, the form taken by the variables in the

problem determines the method of representing them.

3.3.3 Mathematical Operators

Mathematical operators are complex numbers that areused to move a vector through a given angle withoutchanging the magnitude or character of the vector. Anoperator is not a physical quantity; it is dimensionless.

The symbol j, which has been compounded withquadrature components of complex quantities, is an

operator that rotates a quantity anti-clockwise through90. Another useful operator is one which moves avector anti-clockwise through 120, commonlyrepresented by the symbol a.

Operators are distinguished by one further feature; theyare the roots of unity. Using De Moivre's theorem, thenth root of unity is given by solving the expression:

11/n = (cos2m + jsin2m)1/n

where m is any integer. Hence:

where m has values 1, 2, 3, ... (n-1)

From the above expressionjis found to be the 4th rootand a the 3rd root of unity, as they have four and threedistinct values respectively. Table 3.1 gives some usefulfunctions of the a operator.

1

2 21/ cos sinn

m

n jm

n= +

Figure 3.2: Addition of vectors

0

Y

X

y1

y2

x2x1

|Z1|

|Z2|


5/14


1=1+j0 = ej0

1+ a + a2= 0

Table 3.1: Properties of the a operator

3.4 CIRCUIT QUANTIT IES

AND CONVENTIONS

Circuit analysis may be described as the study of theresponse of a circuit to an imposed condition, forexample a short circuit. The circuit variables are currentand voltage. Conventionally, current flow results fromthe application of a driving voltage, but there iscomplete duality between the variables and either maybe regarded as the cause of the other.

When a circuit exists, there is an interchange of energy;a circuit may be described as being made up of 'sources'and 'sinks' for energy. The parts of a circuit are describedas elements; a 'source' may be regarded as an 'active'element and a 'sink' as a 'passive' element. Some circuitelements are dissipative, that is, they are continuoussinks for energy, for example resistance. Other circuitelements may be alternately sources and sinks, forexample capacitance and inductance. The elements of acircuit are connected together to form a network havingnodes (terminals or junctions) and branches (seriesgroups of elements) that form closed loops (meshes).

In steady state a.c. circuit theory, the ability of a circuitto accept a current flow resulting from a given drivingvoltage is called the impedance of the circuit. Sincecurrent and voltage are duals the impedance parametermust also have a dual, called admittance.

3.4.1 Circuit Variables

As current and voltage are sinusoidal functions of time,varying at a single and constant frequency, they areregarded as rotating vectors and can be drawn as plan

vectors (that is, vectors defined by two co-ordinates) ona vector diagram.

ja a=

2

3

a a j =2 3

1 32 = a j a

1 32 =a j a

a j ej

2

4

31

2

3

2= =

a j ej

= + =1

2

3

2

2

3

For example, the instantaneous value, e, of a voltagevarying sinusoidally with time is:

e=Emsin(wt+) Equation 3.8

where:

Em is the maximum amplitude of the waveform;=2f, the angular velocity,

is the argument defining the amplitude of thevoltage at a time t=0

At t=0, the actual value of the voltage is Emsin. So ifEm is regarded as the modulus of a vector, whose

argument is , then Emsin is the imaginary componentof the vector |Em|. Figure 3.3 illustrates this quantityas a vector and as a sinusoidal function of time.

Figure 3.3

The current resulting from applying a voltage to a circuitdepends upon the circuit impedance. If the voltage is asinusoidal function at a given frequency and theimpedance is constant the current will also varyharmonically at the same frequency, so it can be shownon the same vector diagram as the voltage vector, and isgiven by the equation

Equation 3.9

where:

Equation 3.10

From Equations 3.9 and 3.10 it can be seen that theangular displacement between the current and voltagevectors and the current magnitude |Im|=|Em|/|Z| is

dependent upon the impedance

Z. In complex form theimpedance may be written Z=R+jX. The 'realcomponent', R, is the circuit resistance, and the

Z R X

X LC

XR

= +

=

=

2 2

1

1

tan

iE

Zwt

m= + ( )sin 3

Fundamenta

lTheory

2 0

Figure 3.3: Representationof a sinusoidal function

Y

X' X0

Y'

e

t= 0

t

|Em| Em


6/14


'imaginary component',X, is the circuit reactance. Whenthe circuit reactance is inductive (that is, wL>1/wC), thecurrent 'lags' the voltage by an angle , and when it iscapacitive (that is, 1/wC>wL) it 'leads' the voltage by anangle .

When drawing vector diagrams, one vector is chosen asthe 'reference vector' and all other vectors are drawn

relative to the reference vector in terms of magnitudeand angle. The circuit impedance |Z| is a complexoperator and is distinguished from a vector only by thefact that it has no direction of its own. A furtherconvention is that sinusoidally varying quantities aredescribed by their 'effective' or 'root mean square' (r.m.s.)values; these are usually written using the relevantsymbol without a suffix.

Thus:

Equation 3.11

The 'root mean square' value is that value which has thesame heating effect as a direct current quantity of thatvalue in the same circuit, and this definition applies tonon-sinusoidal as well as sinusoidal quantities.

3.4.2 Sign Conventions

In describing the electrical state of a circuit, it is oftennecessary to refer to the 'potential difference' existing

between two points in the circuit. Since wherever sucha potential difference exists, current will flow and energywill either be transferred or absorbed, it is obviouslynecessary to define a potential difference in more exactterms. For this reason, the terms voltage rise and voltagedrop are used to define more accurately the nature of thepotential difference.

Voltage rise is a rise in potential measured in thedirection of current flow between two points in a circuit.

Voltage drop is the converse. A circuit element with avoltage rise across it acts as a source of energy. A circuit

element with a voltage drop across it acts as a sink ofenergy. Voltage sources are usually active circuitelements, while sinks are usually passive circuitelements. The positive direction of energy flow is fromsources to sinks.

Kirchhoff's first law states that the sum of the drivingvoltages must equal the sum of the passive voltages in aclosed loop. This is illustrated by the fundamentalequation of an electric circuit:

Equation 3.12

where the terms on the left hand side of the equation arevoltage drops across the circuit elements. Expressed in

iR

Ldi

dt C idt e+ + =

1

I I

E E

m

m

=

=

2

2

steady state terms Equation 3.12 may be written:

Equation 3.13

and this is known as the equated-voltage equation [3.1].

It is the equation most usually adopted in electricalnetwork calculations, since it equates the drivingvoltages, which are known, to the passive voltages,

which are functions of the currents to be calculated.

In describing circuits and drawing vector diagrams, forformal analysis or calculations, it is necessary to adopt anotation which defines the positive direction of assumedcurrent flow, and establishes the direction in whichpositive voltage drops and voltage rises act. Twomethods are available; one, the double suffix method, isused for symbolic analysis, the other, the single suffix ordiagrammatic method, is used for numericalcalculations.

In the double suffix method the positive direction ofcurrent flow is assumed to be from node a to node b andthe current is designated Iab . With the diagrammaticmethod, an arrow indicates the direction of current flow.

The voltage rises are positive when acting in thedirection of current flow. It can be seen from Figure 3.4that

E1 and

Ean are positive voltage rises and

E2 and

Ebn are negative voltage rises. In the diagrammaticmethod their direction of action is simply indicated by anarrow, whereas in the double suffix method,

Ean and

Ebn

indicate that there is a potential rise in directions na andnb.

Figure 3.4 Methods or representing a circuit

E I Z =

3

Fundamenta

lTheory

(a) Diagrammatic

(b) Double suffix

a b

n

abI

Z3

Z2Z1

E1

Zan

Zab

Ean

Zbn

Ebn

E2

E1-E2=(Z1+Z2+Z3)I

Ean-Ebn=(Zan+Zab+Zbn)Iab

I

Figure 3.4 Methods of representing a circuit


7/14


Voltage drops are also positive when acting in thedirection of current flow. From Figure 3.4(a) it can beseen that (

Z1+

Z2+

Z3)

Iis the total voltage drop in the

loop in the direction of current flow, and must equate tothe total voltage rise

E1-

E2. In Figure 3.4(b), the voltage

drop between nodes a and b designatedVab indicates

that point b is at a lower potential than a, and is positivewhen current flows from a to b. Conversely

Vba is a

negative voltage drop.

Symbolically:Vab =

Van -

Vbn

Vba =

Vbn -

Van Equation 3.14

where n is a common reference point.

3.4.3 Power

The product of the potential difference across and the

current through a branch of a circuit is a measure of therate at which energy is exchanged between that branchand the remainder of the circuit. If the potentialdifference is a positive voltage drop, the branch ispassive and absorbs energy. Conversely, if the potentialdifference is a positive voltage rise, the branch is activeand supplies energy.

The rate at which energy is exchanged is known aspower, and by convention, the power is positive whenenergy is being absorbed and negative when beingsupplied. With a.c. circuits the power alternates, so, toobtain a rate at which energy is supplied or absorbed, itis necessary to take the average power over one wholecycle.Ife=Emsin(wt+) and i=Imsin(wt+-), then the powerequation is:

p=ei=P[1-cos2(wt+)]+Qsin2(wt+)

Equation 3.15where:

P=|E||I|cos and

Q=|E||I|sin

From Equation 3.15 it can be seen that the quantity Pvaries from 0 to 2Pand quantity Q varies from -Q to +Qin one cycle, and that the waveform is of twice theperiodic frequency of the current voltage waveform.

The average value of the power exchanged in one cycleis a constant, equal to quantity P, and as this quantity isthe product of the voltage and the component of currentwhich is 'in phase' with the voltage it is known as the'real' or 'active' power.

The average value of quantity Q is zero when taken over

a cycle, suggesting that energy is stored in one half-cycleand returned to the circuit in the remaining half-cycle.Q is the product of voltage and the quadrature

component of current, and is known as 'reactive power'.

As P and Q are constants which specify the powerexchange in a given circuit, and are products of thecurrent and voltage vectors, then if

S is the vector

productE

Iit follows that with

Eas the reference vector

and as the angle betweenEand

I:

S = P+ jQ Equation 3.16

The quantity S is described as the 'apparent power', andis the term used in establishing the rating of a circuit.S has units of VA.

3.4.4 Single-Phase and Polyphase Systems

A system is single or polyphase depending upon whetherthe sources feeding it are single or polyphase. A sourceis single or polyphase according to whether there are oneor several driving voltages associated with it. For

example, a three-phase source is a source containingthree alternating driving voltages that are assumed toreach a maximum in phase order, A, B, C. Each phasedriving voltage is associated with a phase branch of thesystem network as shown in Figure 3.5(a).

If a polyphase system has balanced voltages, that is,equal in magnitude and reaching a maximum at equallydisplaced time intervals, and the phase branchimpedances are identical, it is called a 'balanced' system.It will become 'unbalanced' if any of the aboveconditions are not satisfied. Calculations using a

balanced polyphase system are simplified, as it is onlynecessary to solve for a single phase, the solution for theremaining phases being obtained by symmetry.

The power system is normally operated as a three-phase,balanced, system. For this reason the phase voltages areequal in magnitude and can be represented by threevectors spaced 120 or 2/3 radians apart, as shown inFigure 3.5(b).

3

Fundamenta

lTheory

2 2

Figure 3.5: Three-phase systems

(a) Three-phase system

BC B'C'N N'

Ean

Ecn Ebn

A'A

Phase branches

Directionof rotation

(b) Balanced system of vectors

120

120

120

Ea

Ec=aEa Eb=a2Ea


8/14


Since the voltages are symmetrical, they may beexpressed in terms of one, that is:

Ea =

Ea

Eb = a2

Ea

Ec = a

Ea Equation 3.17

where a is the vector operator ej2/3. Further, if the phasebranch impedances are identical in a balanced system, itfollows that the resulting currents are also balanced.

3.5 IMPEDANCE NOTATION

It can be seen by inspection of any power systemdiagram that:

a. several voltage levels exist in a system

b. it is common practice to refer to plant MVA in

terms of per unit or percentage valuesc. transmission line and cable constants are given in

ohms/km

Before any system calculations can take place, thesystem parameters must be referred to 'base quantities'and represented as a unified system of impedances ineither ohmic, percentage, or per unit values.

The base quantities are power and voltage. Normally,they are given in terms of the three-phase power in MVAand the line voltage in kV. The base impedance resulting

from the above base quantities is:

ohms Equation 3.18

and, provided the system is balanced, the baseimpedance may be calculated using either single-phaseor three-phase quantities.

The per unit or percentage value of any impedance in thesystem is the ratio of actual to base impedance values.

Hence:

Equation 3.19

where MVAb = base MVA

kVb = base kV

Simple transposition of the above formulae will refer theohmic value of impedance to the per unit or percentage

values and base quantities.Having chosen base quantities of suitable magnitude all

Z p u Z ohmsMVA

kV

Z Z p u

b

b

. .

% . .

( )= ( )( )

( )= ( )

2

100

ZkV

MVAb =

( )2

system impedances may be converted to those basequantities by using the equations given below:

Equation 3.20

where suffix b1 denotes the value to the original base

and b2 denotes the value to new base

The choice of impedance notation depends upon thecomplexity of the system, plant impedance notation andthe nature of the system calculations envisaged.

If the system is relatively simple and contains mainlytransmission line data, given in ohms, then the ohmicmethod can be adopted with advantage. However, theper unit method of impedance notation is the mostcommon for general system studies since:

1. impedances are the same referred to either side ofa transformer if the ratio of base voltages on thetwo sides of a transformer is equal to thetransformer turns ratio

2. confusion caused by the introduction of powers of100 in percentage calculations is avoided

3. by a suitable choice of bases, the magnitudes ofthe data and results are kept within a predictablerange, and hence errors in data and computationsare easier to spot

Most power system studies are carried out usingsoftware in per unit quantities. Irrespective of themethod of calculation, the choice of base voltage, andunifying system impedances to this base, should beapproached with caution, as shown in the followingexample.

Z ZMVA

MVA

Z ZkV

kV

b bb

b

b bb

b

2 1

2

1

2 1

1

2

2

=

=

3

Fundamenta

lTheory

Figure 3.6: Selection of base voltages

11.8kV 11.8/141kV

132kVOverhead line

132/11kV

Distribution11kV

Wrong selection of base voltage

11.8kV 132kV 11kV

Right selection

11.8kV 141kV x 11=11.7kV141132


9/14


10/14


concerned with network reduction.

Of the many theorems that exist, the three mostimportant are given. These are: the SuperpositionTheorem, Thvenin's Theorem and Kennelly's Star/DeltaTheorem.

3.6.2.1 Superposition Theorem(general network theorem)

The resultant current that flows in any branch of anetwork due to the simultaneous action of severaldriving voltages is equal to the algebraic sum of thecomponent currents due to each driving voltage actingalone with the remainder short-circuited.

3.6.2.2 Thvenin's Theorem(active network reduction theorem)

Any active network that may be viewed from twoterminals can be replaced by a single driving voltageacting in series with a single impedance. The driving

voltage is the open-circuit voltage between the twoterminals and the impedance is the impedance of thenetwork viewed from the terminals with all sourcesshort-circuited.

3.6.2.3 Kennelly's Star/Delta Theorem(passive network reduction theorem)

Any three-terminal network can be replaced by a delta orstar impedance equivalent without disturbing theexternal network. The formulae relating the replacementof a delta network by the equivalent star network is as

follows (Figure 3.8):Zco =

Z13

Z23 / (

Z12 +

Z13 +

Z23)

and so on.

Figure 3.8: Star/Delta network reduction

The impedance of a delta network corresponding to andreplacing any star network is:

Z12 =

Zao +

Zbo +

Zao

Zbo

Zco

and so on.

3.6.3 Network Reduction

The aim of network reduction is to reduce a system to asimple equivalent while retaining the identity of thatpart of the system to be studied.

For example, consider the system shown in Figure 3.9.The network has two sources E and E, a line AOBshunted by an impedance, which may be regarded as the

reduction of a further network connected betweenA andB, and a load connected between O andN. The object ofthe reduction is to study the effect of opening a breakeratA or B during normal system operations, or of a faultatA or B. Thus the identity of nodesA and B must beretained together with the sources, but the branch ONcan be eliminated, simplifying the study. Proceeding,A,B, N, forms a star branch and can therefore be convertedto an equivalent delta.

Figure 3.9

= 51 ohms

=30.6 ohms

= 1.2 ohms (since ZNO>>> ZAOZBO)

Figure 3.10

Z Z ZZ Z

Z AN AO BO

AO BO

NO

= + +

= + + 0 45 18 85 0 45 18 850 75

. .. .

.

Z Z ZZ Z

Z

BN BO NOBO NO

AO

= + +

= + + 0 75 18 85 0 75 18 850 45

. .. .

.

Z Z ZZ Z

Z AN AO NO

AO NO

BO

= + +

3

Fundamenta

lTheory

Figure 3.8: Star-Delta network transformation

c

Zao

Zbo

Z12

Z23

Z13

Oa b 1 2

3

(a) Star network (b) Delta network

Zco

Figure 3.9: Typical power system network

E' E''

N

0A B

1.6

0.75 0.45

18.85

2.55

0.4


11/14


The network is now reduced as shown in Figure 3.10.

By applying Thvenin's theorem to the active loops, thesecan be replaced by a single driving voltage in series withan impedance as shown in Figure 3.11.

Figure 3.11

The network shown in Figure 3.9 is now reduced to thatshown in Figure 3.12 with the nodesA and B retainingtheir identity. Further, the load impedance has beencompletely eliminated.

The network shown in Figure 3.12 may now be used tostudy system disturbances, for example power swingswith and without faults.

Figure 3.12

Most reduction problems follow the same pattern as theexample above. The rules to apply in practical networkreduction are:

a. decide on the nature of the disturbance ordisturbances to be studied

b. decide on the information required, for examplethe branch currents in the network for a fault at a

particular location

c. reduce all passive sections of the network notdirectly involved with the section underexamination

d. reduce all active meshes to a simple equivalent,that is, to a simple source in series with a singleimpedance

With the widespread availability of computer-basedpower system simulation software, it is now usual to usesuch software on a routine basis for network calculations

without significant network reduction taking place.However, the network reduction techniques given aboveare still valid, as there will be occasions where suchsoftware is not immediately available and a handcalculation must be carried out.

In certain circuits, for example parallel lines on the sametowers, there is mutual coupling between branches.Correct circuit reduction must take account of thiscoupling.

Figure 3.13

Three cases are of interest. These are:

a. two branches connected together at their nodes

b. two branches connected together at one node only

c. two branches that remain unconnected

3

Fundamenta

lTheory

2 6

Figure 3.10: Reduction usingstar/delta transform

E'

A

51 30.6

0.4

2.5

1.21.6

N

B

E''

Figure 3.12: Reduction of typicalpower system network

N

A B

1.2

2.5

1.55

0.97E'

0.39

0.99E''

Figure 3.11: Reduction of active meshes:Thvenin's Theorem

E'

A

N

(a) Reduction of left active mesh

N

A

(b) Reduction of right active mesh

E''

N

B B

N

E''

31

30.630.6

31

0.4 x 30.6

52.6

1.6 x 51

E'52.6

5151

1.6

0.4

Figure 3.13: Reduction of two brancheswith mutual coupling

(a) Actual circuit

IP Q

P Q

(b) Equivalent when ZaaZbb

(c) Equivalent when Zaa=Zbb

P Q

21Z= (Zaa+Zbb)

Zaa

Zbb

Z=ZaaZbb-Z

2ab

Zaa+Zbb-2Zab

Zab

Ia

Ib

I

I


12/14


Considering each case in turn:

a. consider the circuit shown in Figure 3.13(a). Theapplication of a voltage Vbetween the terminals Pand Q gives:

V= IaZaa + IbZab

V= IaZab + IbZbb

where Ia and Ib are the currents in branches a andb, respectively and I= Ia + Ib , the total currententering at terminal Pand leaving at terminal Q.

Solving for Ia and Ib :

from which

and

so that the equivalent impedance of the originalcircuit is:

Equation 3.21

(Figure 3.13(b)), and, if the branch impedances areequal, the usual case, then:

Equation 3.22

(Figure 3.13(c)).

b. consider the circuit in Figure 3.14(a).

Z Z Zaa ab= +( )1

2

ZV

I

Z Z Z

Z Z Z

aa bb ab

aa bb ab

= =

+

2

2

I I IV Z Z Z

Z Z Za b

aa bb ab

aa bb ab

= + =+ ( )

2

2

IZ Z V

Z Z Z

b

aa ab

aa bb ab

=( )

2

IZ Z V

Z Z Za

bb ab

aa bb ab

=( )

2

The assumption is made that an equivalent starnetwork can replace the network shown. Frominspection with one terminal isolated in turn and avoltage Vimpressed across the remaining terminalsit can be seen that:

Za+Zc=Zaa

Zb+Zc=Zbb

Za+Zb=Zaa+Zbb-2Zab

Solving these equations gives:

Equation 3.23

-see Figure 3.14(b).

c. consider the four-terminal network given in Figure3.15(a), in which the branches 11' and 22' areelectrically separate except for a mutual link. Theequations defining the network are:

V1=Z11I1+Z12I2

V2=Z21I1+Z22I2

I1=Y11V1+Y12V2

I2=Y21V1+Y22V2

where Z12=Z21 and Y12=Y21 , if the network is

assumed to be reciprocal. Further, by solving theabove equations it can be shown that:

Equation 3.24

There are three independent coefficients, namelyZ12, Z11, Z22, so the original circuit may be

replaced by an equivalent mesh containing fourexternal terminals, each terminal being connectedto the other three by branch impedances as shownin Figure 3.15(b).

Y Z

Y Z

Y Z

Z Z Z

11 22

22 11

12 12

11 22 122

=

=

=

=

Z Z Z

Z Z Z

Z Z

a aa ab

b bb ab

c ab

=

=

=

3

Fundamenta

lTheory

Figure 3.14: Reduction of mutually-coupled brancheswith a common terminal

A

C

B

(b) Equivalent circuit

B

C

A

(a) Actual circuit

Zaa

Zbb

Zab

Za=Zaa-Zab

Zb=Zbb-Zab

Zc=Zab

Figure 3.15 : Equivalent circuits forfour terminal network with mutual coupling

(a) Actual circuit

2

1

2'

1'

2'

1'

(b) Equivalent circuit

2

1

Z11

Z22

Z11

Z22

Z12 Z12 Z12 Z12Z21


13/14


defining the equivalent mesh in Figure 3.15(b), andinserting radial branches having impedances equalto Z11 and Z22 in terminals 1 and 2, results inFigure 3.15(d).

3.7 REFERENCES

3.1Power System Analysis.

J. R. Mortlock andM. W. Humphrey Davies. Chapman & Hall.

3.2 Equivalent Circuits I. Frank M. Starr, Proc. A.I.E.E.Vol. 51. 1932, pp. 287-298.

3

Fundamenta

lTheory

2 8

Figure 3.15: Equivalent circuits forfour terminal network with mutual coupling

2'

1'

(d) Equivalent circuit

11

C 2

(c) Equivalent with all

nodes commonedexcept 1

Z11 Z12

Z11

Z12

Z12

Z12

-Z12 -Z12Z12

In order to evaluate the branches of the equivalentmesh let all points of entry of the actual circuit becommoned except node 1 of circuit 1, as shown inFigure 3.15(c). Then all impressed voltages exceptV1 will be zero and:

I1 = Y11V1

I2 = Y12V1

If the same conditions are applied to the equivalentmesh, then:

I1 = V1Z11

I2 = -V1/Z12 = -V1/Z12

These relations follow from the fact that the branchconnecting nodes 1 and 1' carries current I1 andthe branches connecting nodes 1 and 2' and 1 and2 carry current I2. This must be true since branchesbetween pairs of commoned nodes can carry no

current.By considering each node in turn with theremainder commoned, the following relationshipsare found:

Z11 = 1/Y11

Z22 = 1/Y22

Z12 = -1/Y12

Z12 = Z12 = -Z21 = -Z12

Hence:

Z11 = Z11Z22-Z212_______________Z22

Z22 = Z11Z22-Z212_______________Z11

Z12 = Z11Z22-Z212_______________Z12 Equation 3.25

A similar but equally rigorous equivalent circuit isshown in Figure 3.15(d). This circuit [3.2] followsfrom the fact that the self-impedance of any circuitis independent of all other circuits. Therefore, it

need not appear in any of the mutual branches if itis lumped as a radial branch at the terminals. Soputting Z11 and Z22 equal to zero in Equation 3.25,


14/14

Documents

3. Fundamental Theory