theory of electricity -professor j r lucas- level 2-ee201_7_matrix_analysis.pdf

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Matrix Analysis of Networks – Professor J R Lucas 2 November 2001

Only the first diagram would fully satisfy the requirements of a normal tree. The second

diagram has branches closing on itself (only a tree like Nuga might appear to close on itself).

The third has branches which are in mid air not joined to the main tree. Thus we see that

there are certain properties that we associate with trees. These are

1. All branches must be part of the tree

2. There cannot be closed loops formed from branches

3. There cannot be branches which are isolated from the tree

We apply these same properties when we define a tree of a given network. However, unlike

the graph, there can be many trees associated with a given network. Also unlike in the case

of a natural tree, the tree of a network need not have a trunk coming from the ground and

branches coming from the trunk.

A tree of a network is a graph of the network with some of the links removed in such a way so

as to leave all the nodes connected together by the graph, but so as not to have any loop left in

the network. For each network graph, there are a number of possible trees. Some of the trees

are shown below.

When a tree of the network is removed from the graph, what remains is called the co-tree of

the network. It is the graph of the removed links and is the compliment of the tree. Unlike a

tree of a network, a co-tree may contain closed loops, and disconnected branches. Thecorresponding co-trees of the above trees are shown below.

Let b = number of branches in the network

n = number of nodes in the network

l = number of independent loops

A single branch is required to join two nodes. Joining each additional node would require an

additional branch. Thus the number of branches in a tree would be one less than the number

of nodes.

number of branches in tree = n – 1

number of links removed = b – (n – 1) = b – n + 1

If we add any one of the removed links to a tree, then a loop is formed. Since this involves

the link that was added this is part of an independent loop as it could not have formed part of

any of the loops that were already there.

Therefore the number of links removed from the graph to form the tree is equal to the numberof independent loops.

i.e. l = b – n + 1

Graph of Network Some of the possible trees




In analysing network, if the branches are not numbered and

they are not assigned directions, it would not be possible to

formulate the equations governing the circuit. The numbering

and the directions assigned to a graph is not unique, unlike the

graph itself.

An oriented graph for the earlier problem is shown in the

figure. The direction marked is that of the current, and the

voltage is considered to increase in the direction opposite to the

flow of current and is marked as such in the oriented graph.

Matrix Analysis of Networks

When we solve circuit problems, we need to write the equations corresponding to the Ohm’s

Law and the Kirchoff ’s Laws. These same equations need to be used even when there are a

large number of branches, and as such when using Matrix analysis of networks.

Kirchoff’s current Law in matrix form

Let I j be the current in the jth

branch.

Consider any node k in the network, as shown in the

figure. The jth

branch may either be directed away

from k th

node, may be directed towards k th

node, or

may not be incident on the k th

node.

From Kirchoff ’s current Law,

− i1 − i2 − i4 + i6 + i7 = 0 or i1 + i2 + i4 = i6 + i7 or

i1 + i2 + i4 – i6 – i7 = 0 or +1 . i1 + 1 . i2 + 0 .i3 + 1 .i4 + 0 . i5 – 1 .i6 – 1 .i7 = 0[These are different forms of the same equation].

The last form is preferred for matrix implementation, as all the currents in the network are

represented in the equation with different coefficients. The equation could also have been

written with all the coefficients negated. However, if it is to be used for computer

implementation, there must be a unique method of obtaining the coefficients. The convention

used is as follows.

Branch currents directed away from the node are associated with a coefficient of +1, while

current directed towards the node are associated with a coefficient –1. Branches not

connected to (or incident on) the node are obviously associated with a coefficient of 0.

These coefficients are defined as a jk so that Kirchoff ’s current law may thus be written as

a1k . i1 + a2k . i2 + a3k . i3 + a4k . i4 + ...... ....... ..... a7k . i7 = 0 for the k th

node

or ∑=

=⋅b

j j jk ia

1

0 at k th

node, for all k; where a jk = −1, 0, or +1

The collection of such equations for each node k, would give the matrix equation

[ ])1()1()(

0×××

=⋅bb

b

bn

I t A

In the matrix [A]t, the row vectors are dependant, since their sum is zero. For this reason, thematrix [A]

tis written with one row less, usually the last row for convenience, so that there is

only (n-1) rows.

Oriented Graph

1

3 2

4

6 5

k -1

-1

+1 +1

+1

0

0

i1

i2

i3

i4

i6

i7

i5




The matrix [A]tof dimension (n-1)×b is referred to as the node-branch incidence matrix.

The matrix [A] of dimension b× (n-1) is then referred to as the branch-node incidence matrix.

The elements of the matrix [A] are

a jk = +1 if jth

current is directed away from the k th

node

a jk = −1 if jth

current is directed towards the k th

node

a jk = 0 if jth

current is not incident on the k th

node

Kirchoff ’ s voltage Law in matrix form

Similarly, the Kirchoff ’s voltage law may

be applied for the sth

loop.

The coefficients brs of the voltages are

defined as +1, −1 or 0 depending onwhether the loop direction is the same as

the branch direction or not.

Thus ∑=

=⋅b

r r rs vb

1

0 for sth

node, for all s;

where brs = −1, 0, or +1

In matrix form this becomes

[ ])1()1(

)(

0×

××

=⋅lb

b

bl

V t B

The matrix [B]tof dimension l× b is referred to as the mesh-branch incidence matrix.

The matrix [B] of dimension b× l is then referred to as the branch-mesh incidence matrix.

The elements of the matrix [B] are

brs = +1 if rth

current is in the same direction as the sth

loop

brs = −1 if rth

current is in the opposite direction to the sth

loop

brs = 0 if rth current is in the not part of the sth loop

Ohm’ s Law in matrix form

A general branch may contain a voltage source and/or a current source in addition to the

branch impedance/admittance.

0

0

0 +1

-1

-1

0

-1

+1

s

egk

igk

Zk ik + igk ik

vk




Applying Ohm’s Law

vk = - egk + Zk igk + Zk ik for all branches k = 1, 2, .... ... b

However, using Thevenin’s Theorem or Norton’s Theorem, a current generator may be

converted to a voltage generator or vice versa, except when ideal. Thus analysis can be donewith only either a voltage source or a current source.

Thus we may either write

vk = - egk + Zk ik for all branches k = 1, 2, .... ... b

which may be written in matrix form as

[ ] bbgbb I Z E V +−=

or we may write

vk = Zk igk + Zk ik or Yk vk = Yk Zk igk + Yk Zk ik i.e. ik = Yk vk - igk for all branches k = 1, 2, .... ... b

or in matrix form as

[ ] bbgbb V Y I I +−= , where [Yb] = [Zb]-1

In Summary

From Kirchoff ’s Laws we have

[ ])1()1()(

0×××

=⋅bb

b

bn

I t A (1) (n-1) independent equations

[ ])1()1()(

0×××

=⋅lb

b

bl

V t B (2) l independent equations

and from Ohm’s Law

[ ] bbgbb I Z E V +−= (3) b independent equations

or [ ]bbgbb

V Y I I

+−=(3)* b independent equations

Thus the total number of independent equations is n – 1 + l + b = b + b = 2 b

It is seen that there is a total of 2b independent equations and 2b unknowns (corresponding tob branch currents and b branch voltages). However, it is not usual to solve the equations for

both current and voltage simultaneously. So let us see how these may be reduced for

solution. The reductions can be done in one of two ways. The first is to eliminate the

voltages and solve for currents. The second method is to eliminate the currents and solve for

the voltages. These two methods are known as mesh analysis and nodal analysis respectively.

egk

Zk

vk

ik

igk

Zk ik

vk




Mesh Analysis

In mesh analysis, we eliminate the branch voltages from the equations. We further reduce the

number of remaining currents to the minimum using Kirchoff ’s current law and apply

Kirchoff ’s voltage law for solution.

Let us define a set of mesh currents, m I .

The branch currents b I can be seen to be related to the mesh currents m

I by an algebraic

summation. In fact, if an example is considered, it will be easily seen that the relation

corresponds to the matrix [B].

[ ] mb I B I = (4)

We can eliminate Vb from the equations, by pre-multiplying equation (3) by [B]t .

i.e. [ ] [ ] [ ] [ ] bb

t

gb

t

b

t I Z B E BV B +−=

from equation (2), it can be seen that [B]tVb = 0. Also [ ] mb I B I =

∴ [ ] [ ] [ ][ ] mb

t

gb

t I B Z B E B =

[B]tVb = 0 corresponds to the sum of the voltages around a loop is zero.

∴ [B]tVb must correspond to the sum of the voltages around a loop.

Since [B]tis constant, and Egb is the branch source voltage vector, [B]

tEgb must correspond

to the sum of the source voltages around a loop. This is defined as the mesh source voltage

vector Egm .

i.e. Egm = [B]t Egb

∴ E gm = [ ] [ ][ ] [ ] mmmb

t I Z I B Z B =

where [Zm] = [ ] [ ][ ] B Z B b

t

This corresponds to only l equations, and the only unknowns in that equation is I m which

corresponds to l unknowns.

Thus the original problem of 2b equations and 2b unknowns has been reduced to that of l

equations and l unknowns.

The elements of the matrix [Zm] can be obtained either by going through the above

mathematical considerations, or by inspection in the following manner for the simple

problems.

z jj = self impedance of mesh j

= sum of all branch impedances in mesh j

z jk = mutual impedance between mesh j and mesh k

= sum of all branch impedances common to mesh j and mesh k and traversed in

mesh direction − sum of all branch impedances common to mesh j and mesh k

and traversed in opposite direction

e j = algebraic sum of the branch voltage sources in mesh j in mesh direction.

Note: Matrix [B] is also known as the tie-set matrix (as its elements tie the loop together).




Example 1

Solve the circuit using Mesh matrix analysis. Work from first principles.

Solution

Let us first number the branches and the loops.

You will notice that I have used capital letters for the loop currents and simple letters for the

branch currents. This is for convenience in not having to write a suffix m for mesh and b for

branch.

Let us write the loop currents in terms of the branch currents.

i1 = I1 i2 = - I3

i3 = I1 – I2

i4 = I2

i5 = I2 – I3

i6 = I3

This gives us the [B] matrix or the Branch-Mesh incidence matrix.

We can also write the Mesh–Branch incidence matrix [B]tmatrix as follows, independent of

the above, by writing the relation between the mesh direction and the branch direction.

[ ]

−−−=

110010

011100

000101t B

You will notice that this corresponds to the transpose of the matrix written earlier.

The vector of branch currents Egb can be written as follows.

We now need to write the branch impedance matrix.

Then we can write expressions for the mesh voltage vector and the mesh impedance matrix

using the derived equations.

Egm = [B]tEgb , and [Zm] = [B]

t[Zb] [B]

E1

100∠0

0

V

20 Ω 6 Ω

-j120 Ω

E2 100∠30

0V

10 Ω

20 Ω

10 Ω

E1

100∠00

V

20 Ω 6 Ω

-j120 Ω

E2

100∠36.870

10 Ω

20 Ω

10

Ω

I1 I2I3

i1 i4

i3i5

i6

i2

or in matrix form

∠∠

0

0

0

0

87.36100

01000

0

−

−−

=

3

2

1

6

5

4

3

2

1

100

110

010

011

100001

I

I

I

i

i

i

i

ii




As in the case of mesh analysis, the source nodal current vector IgN and the nodal admittance

matrix [YN] could be written by inspection as follows.

yii = sum of all branch admittances incident at node i

yij = negative of the sum of all branch admittances connecting node i and nodej .

The reason why the negative sign appears can be understood as follows.

Consider the simple circuit shown.

ik = yk vk = yk (Vi – V j)

At any node i, from Kirchoff ’s current law,

nodal injected current at the node i is equal to the sum of the branch currents going out from

the node

Igi = Σ ik which gives the required expression for the nodal current vector

Also,

Igi = Σ ik = Σ yk (Vi – V j)

i.e. ∑∑≠=

≠=

−= N

i j j

jk

N

i j j

ik gi V yV y I 11

at node i for all j (different k actually correspond to different j)

Since Vi is a constant for a given i,

∑∑∑∑≠=

≠=

≠=

≠=

−+

=−=

N

i j j

jk i

N

i j j

k

N

i j j

jk

N

i j j

k igi V yV yV y yV I 1

)111

(

∑≠=

+= N

i j j

jiiiiigi V yV y I 1

= ∑=

N

j jijV y

1

Thus we see that the final equation derived in this manner actually corresponds to the nodal

equation, and that the diagonal term of the matrix actually corresponds to the branch

admittances connected to the node, and that the off diagonal terms actually correspond to the

negative of the branch admittance.

As in the case of mesh analysis, the nodal equation IgN = [YN]VN is first solved to give VN

and the branch voltages and branch currents can then be obtained using the matrix equations.

Example 2

Example 1 has been reformulated as a problem with current sources rather than with voltage

sources. [If voltage sources are present, they would first have to be converted to currentsources].

i

yk ik

vk

5∠-900

A

20 Ω

6 Ω

-j120 Ω 8.575∠5.91

0A

10 Ω

20 Ω

10 Ω

i1 i4

i3i5

i2V1 V2




V1 = 81.6∠-19.84o

V

∴branch current i1 =20

68.273.23

20

84.196.81100

j

j

j

o +=

−∠−

i1 A j o09.40809.1165.1384.1

−∠=−=

which is the same answer (to calculation accuracy) that was obtained in example 1.

Conversion of Ideal sources

When voltage sources with no series impedance occurs or current sources with no shunt

admittance occurs in a network, then no conversion of source type can be made directly.

However, a circuit will always have other impedances/admittances so that the following

procedure can be adopted, where a single source is replaced by a number of equal sources

distributed in the network.

Consider the following circuit where no impedance is directly in series with the voltagesource.

In the case of an ideal voltage source, it is distributed to the branches connected to one of the

nodes of the original ideal source. Of course with this type of distribution, where there was a

common star point earlier, there would be a number of corresponding points in the new

network.

Consider the following circuit where no admittance appears directly in parallel with a current

source.

The ideal current source has been distributed around a loop connecting the two points of the

original source. Since the current coming into one of the new nodes is the same as the current

going out of the node, there is no overall change.

E

E

E E E E ≡ or

Z1

Z3

Z2

Z5Z4

Z1

Z3

Z2

Z5Z4

Z1

Z3

Z2

Z5

Z4

≡

Is

Is

Is

Is




Two-Port Theory

It is convenient to develop special methods for the systematic treatment of networks. In the

case of a single port linear active network, we obtained the Thevenin’s equivalent circuit and

the Norton’s equivalent circuit. When a linear passive network is considered, it is convenient

to study its behaviour relative to a pair of designated nodes.

Let us learn about a few terms before we proceed with the analysis.

A Port is a pair of nodes across which a device can be connected. The voltage is measured

across the pair of nodes and the current going into one node is the same as the current coming

out of the other node in the pair. These pairs are entry (or exit) points of the network.

[Compare with an Airport or a Sea Port. These are entry and exit points to a country. The

planes that enter at a given port are the ones that take off from the same port].

The Driving point impedance is defined as the ratio of the applied voltage (driving point

voltage) across a node-pair to the current entering at the same port. [This also corresponds to

the input impedance of the network seen from the particular port.]

Driving point impedance at Port 1 = V1 /I1

Driving point impedance at Port 2 = V2 /I2

The Driving point admittance is similarly defined as the ratio of the current entering at a port

to the applied voltage across the same node-pair.Driving point admittance at Port 1 = I1 /V1

Driving point admittance at Port 2 = I2 /V2

Note: The term Immittance may sometimes be used to represent either an impedance or an

admittance

The Transfer impedance is defined as the ratio of the applied voltage across a node-pair to

the current entering at the other port.

Transfer impedance = V1 /I2 , V1 /I2

The Driving point admittance is similarly defined as the ratio of the current entering at a port

to the voltage appearing across the other node-pair.

Transfer admittance = I1 /V2 , I2 /V1

The Transfer Voltage gain (or ratio) is defined as the ratio of the voltage at a node pair to the

voltage appearing at the other node-pair.

Transfer voltage = V1 /V2 , V2 /V1

The Transfer Current gain (or ratio) is similarly defined as the ratio of the current at a port

to the current at the other port.

Transfer current = I1 /I2 , I2 /I1

The external conditions of a two-port network can be completely defined by the currents andvoltages at the two ports. Hence a general two port network can be characterised by four

parameters, which may be derived from the network elements.

Linear

Passive

Network

I1I2

V1 V2Port 1 Port 2




The admittance parameter matrix may be written as

=

2

1

2221

1211

2

1

V

V

y y

y y

I

I

The parameters y11

, y

12, y

21, y

22can be defined in a similar manner, with either V

1or V

2on

short circuit.

It can be seen that the y-parameters correspond to the driving point and transfer admittances

at each port with the other port having zero voltage (i.e. short circuit). Thus these parameters

are also referred to as the short circuit parameters.

Example 4

Find the impedance parameters of the 2 port – network

shown in the figure.

For this case, with port 2 on short circuit

y11 =021

1

=V V

I = Ya + Yb

y21 =021

2

=V V

I = - Yb

similarly with port 1 shorted, y12 = - Yb , y22 = Yb + Yc

Thus the admittance parameter matrix is written as

[Y] =

+−

−+

cbb

bba

Y Y Y

Y Y Y

(c) Transmission Line Parameters (ABCD-parameters)

The ABCD parameters represent the relation between the input quantities and the output

quantities in the two port network. They are thus voltage-current pairs.

However, as the quantities are defined as an

input-output relation, the output current is

marked as going out rather than as coming

into the port.

The impedance parameter matrix may be written as

=

2

2

1

1

I

V

DC

B A

I

V

The parameters A, B, C, D can be defined in a similar manner with either port 2 on short

circuit or port 2 on open circuit. These parameters are known as transmission parameters as

in a transmission line, the currents enter at one end and leaves at the other end, and we need to

know a relation between the sending end quantities and the receiving end quantities.

In the case of symmetrical system, such as a transmission line, where the properties from one

end are the same as the properties from the other end, parameter A = D.

Also in the case of a reciprocal system, it can be shown that A.D – B.C = 1

I1 I2

V1 V2

Yb

Ya Yc

I1 I2

V1 V2=0

Yb

Ya Yc

Linear

Passive

Network

I1 I2

V1 V2Port 1 Port 2




Example 5

Find the ABCD parameters of the 2 port – network

shown in the figure.

For this case, it can be shown that

A =c

cb

Y

Y Y + , B =cY

1, C =

c

accbba

Y

Y Y Y Y Y Y ++ ,

and D =c

ca

Y

Y Y +

[It can be seen that for a symmetrical network, where Ya = Yc , A = D].

Also,

A.D – B.C =

c

accbba

cc

ca

c

cb

Y

Y Y Y Y Y Y

Y Y

Y Y

Y

Y Y ++⋅−

+⋅

+ 1

=2

2)(

c

accbbaccbacab

Y

Y Y Y Y Y Y Y Y Y Y Y Y Y ++−+++=1

(d) Hybrid Parameters (h-parameters)

The hybrid parameters represents a mixed or hybrid relation between the voltages and the

currents in the two port network.

The hybrid parameter matrix may be written

as

=

2

1

2221

1211

2

1

V

I

hh

hh

I

V

The h-parameters can be defined in a similar manner and are commonly used in some

electronic circuit analysis. The method of obtaining the parameters is very similar to the

earlier cases.

Interconnection of two-port networks

In certain applications, it becomes necessary to connected the two-port networks together.

The common connections are (a) series, (b) parallel and (c) cascade.

(a) Series connection of two-port networksAs in the case of elements, a series connection is defined when the currents in the series

elements are equal and the voltages add up to give the resultant voltage.

In the case of two-port networks, this property must be applied individually to each of the

ports.

Thus, if we consider 2 networks r and s connected in series

at port 1, Ir1 = Is1 = I1, and Vr1 + Vs1 = V1

similarly, at port 2 I r2 = Is2 = I2 and Vr2 + Vs2 = V2

The two networks, r and s can be connected in following manner to be in series with each

other.

I1 I2

V1 V2

Yc

Ya Yb

Linear

Passive

Network

I1 I2

V1 V2Port 1 Port 2




Under these conditions, it can be

easily seen that if impedance

parameters are used, then the

resultant impedance parameter

matrix for the series combination

is the addition of the twoindividual impedance matrices.

[Z] = [Zr] + [Zs]

(b) Parallel connection of two-port networks

As in the case of elements, a parallel connection is defined when the voltages in the parallel

elements are equal and the currents add up to give the resultant current.

In the case of two-port networks, this property must be applied individually to each of the

ports.

Thus, if we consider 2 networks r and s connected in parallel

at port 1, Ir1 + Is1 = I1, and Vr1 = Vs1 = V1

similarly, at port 2 I r2 + Is2 = I2 and Vr1 = Vs1 = V1

The two networks, r and s can be connected in following manner to be in parallel with each

other.

Under these conditions, it can be

easily seen that if admittance

parameters are used, then the

resultant admittance parameter matrix

for the parallel combination is the

addition of the two individualadmittance matrices.

[Y] = [Yr] + [Ys]

(c) Cascade connection of two-port networks

A cascade connection is defined when the output of one network becomes the input to the

next network.

It can be easily seen that Ir2 = Is1 and Vr2 = Vs1

Therefore it can easily be seen that the ABCD parameters are the most suitable to be used for

this connection.

=

2

2

1

1

r

r

r r

r r

r

r

I

V

DC

B A

I

V ,

=

2

2

1

1

s

s

ss

ss

s

s

I

V

DC

B A

I

V

If the intermediate voltage-current pairs are eliminated, the matrix equation can be re-written

in the following form, as a product matrix.

Linear

Passive

Network

r

Ir1 Ir2

Vr1 Vr2Port r1 Port r2

Linear

PassiveNetwork

s

Is1 Is2

Vs1 Vs2Port s1 Port s2

V2V1

Linear

Passive

Network

r

Ir1 Ir2

Vr1Vr2

Linear

Passive

Network

s

Is1 Is2

Vs1Vs2

V2V1

I1 I2

Linear

Passive

Network

Is1 Is2

Vs1 Vs2Port s1 Port s2

Linear

Passive

Network

Ir1 Ir2

Vr1 Vr2Port r1 Port r2

theory of electricity -professor j r lucas- level 2-ee201_7_matrix_analysis.pdf

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