Interconnection Equations

4. INTERCONNECTION EQUATIONS

4.1 Branch Current and Branch Voltage

Consider the branch shown in Fig. 4.1.1a. The current i(t) flowing through the branch and the
voltage v(t) across it are called the branch current and the branch voltage, respectively.

(a) (b) (c) (d) (e)

There are two possible branch current directions (see Figs. 4.1.1b, c) and two possible branch
voltage directions (see Figs. 4.1.1d, e). Hence, there are four possible combinations of branch
current and branch voltage directions as shown in Fig. 4.1.2. When the current direction is from
the terminal marked "+" towards the terminal marked "-" as in Figs. 4.1.2a, d, the two directions
are said to be associated.

(a) (b) (c) (d)

Energy and Power

Consider the directions shown in Fig. 4.1.2a. Suppose that on the interval [t, t+dt] the net amount
of charge flowing from terminal "a" through the branch to terminal "b" is dq coulombs. The energy
released by this flow is dw = v(t)dq joules, and this much energy is delivered to the branch.
The instantaneous power p is the time rate change of energy. It is measured in watts (W) where
watt = joul/sec {= (joul/coul) x (coul/sec) = volt x amp}. The value of p at time t is
p(t) = dw/dt = (dw/dq)∙(dq/dt)
= v(t)∙i(t). (4.1.1)
p(t) can be positive, zero, or negative. If it is negative, this means actually the branch supplies
- p(t) watts of power at time t.

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The net amount of energy delivered to the branch on the interval (t1, t2) is the area under p(t) on
this interval, i.e.,
t
W(t1, t2) = ∫t12 p(t)dt . {W(t1, t2) can be positive, zero, or negative.} (4.1.2)
{The above discussion also holds for the directions shown in Fig. 4.1.2d.}

4.2 Independent Current and Independent Voltage Equations

Consider a circuit with b branches, nt nodes, and p parts. Assign the branch currents and voltages
(with associated directions), and orient the branches of the corresponding graph yielding the
oriented (directed) circuit graph (see Fig. 4.2.1).

(a) (b)
Figure 4.2.1 (a) A lumped circuit and (b) the corresponding oriented circuit graph.

The branch current vector i and the branch voltage vector v are defined by
İ = (i1 i2 … ib)T, v = (v1 v2 … vb)T.
In a lumped circuit one can write, in general, many current and voltage equations. But these
equations are, in general, not all independent. As we will see below, the maximum number of
independent current equations is n = nt - p, and the maximum number of independent voltage
equations is m = b - n. Hence, the maximum number of independent interconnection (i.e., current
and voltage) equations is b = n + m.

4.2.1 Reduced Incidence Matrix

The incidence matrix for the oriented graph of Fig. 4.2.1b is given below, where the nonzero
entries in each column are +1 and -1.
1 2 3 4 5
a 1 1 1 0 0
b  -1 -1 0 0 1 
At   .
c 0 0 -1 1 0
 
d 0 0 0 -1 - 1

Branch 1 is directed from node a to node b; then in the first column of the incidence matrix
(the one corresponding to branch 1), the entry at the row position corresponding to node a is +1,
and the entry at the row position corresponding to node b is -1, etc. Since there are exactly one

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+1 and one -1 in each column of At, the rows of At are linearly dependent, and the rank of At is less
than nt.
The columns of At corresponding to the branches of a subgraph containing a loop are linearly
dependent, and those corresponding to the branches of a loop free subgraph are linearly
independent. {For example, for the above incidence matrix, the columns 1, 2, the columns 2, 3, 4,
5, and the columns 1, 3, 4, 5 are linearly dependent; whereas the columns 1, 3, 4 are linearly
independent.} Since a tree is a maximal loop free subgraph and the number of tree-branches is n,
there are (at most) n linearly independent columns of At implying that the rank of At is n.
For a connected graph, let us arbitrarily pick any one of the nodes as a reference node. Deleting
the corresponding row from At, we obtain the n 𝑥 b reduced incidence matrix A. Since the rank of
A is n, the rows of A are linearly independent. As an example, picking node d as a reference node,
the reduced incidence matrix A for the graph of Fig. 4.2.1b follows as

1 2 3 4 5
a  1 1 1 0 0
A  b  - 1 - 1 0 0 1 .
c  0 0 - 1 1 0

Given a reduced incidence matrix A, inserting the row for the reference node, we can easily obtain
At and draw the corresponding oriented graph.
For an unconnected graph with p parts, we pick a reference node in each part. In this case, there
are p reference nodes and n nonreference nodes. .
The incidence and reduced incidence matrices for the graph of Fig. 4.2.2 are given below where
nodes c and g are picked as reference nodes.

2 7
a b e f
1 3 6 4 8

c d g
5

Figure 4.2.2 An oriented unconnected graph.

1 2 3 5 6 4 7 8
a 1 1 0 0 0 0 0 0
1 2 3 5 6 4 7 8
b  0 -1 1 0 1 0 0 0  a 1 1 0 0 0 0 0 0
c  -1 0 -1 1 0 0 0 0 b  0 -1 1 0 1 0 0 0 
 
A t  d 0 0 0 - 1 - 1 0 0 0 , A  d 0 0 0 -1 -1 0 0 0 .
 
e0 0 0 0 0 1 1 0 e0 0 0 0 0 1 1 0
 
f 0 0 0 0 0 0 -1 1  f  0 0 0 0 0 0 -1 1 
g 0 0 0 0 0 -1 0 -1

For the circuit of Fig. 4.2.1a, the current equations written at the nodes (using KCL) are
node a: i1 + i2 + i3 = 0,
node b: - i1 - i2 + i5 = 0,

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node c: - i3 + i4 = 0,
node d: - i4 - i5 = 0.
In compact form,
At i = 0. (4.2.1)
The four equations above are not independent (since the rank of At is not 4); but the equations
written at the nonreference nodes are independent, and this set of n independent current
equations can be compactly expressed in the form
Ai = 0. (4.2.2)
The voltage between a nonreference node and the (appropriate) reference node, where the non-
reference node side is marked "+" and the reference node side is marked "-", is called a node
voltage. The vector en of n node voltages is called the node voltage vector. For the circuit of
Fig. 4.2.1a (or equivalently for the graph of Fig. 4.2.1b), picking node d as a reference node, we
have (see Fig. 4.2.3) en = (ea eb ec)T; and for the graph of Fig. 4.2.2, picking nodes c and g as
reference nodes, we have en = (ea eb ed ee ef)T.

Figure 4.2.3 Node voltages for the circuit of Fig. 4.2.1a.

Branch voltages can be expressed in terms of node voltages using KVL. For example, for the circuit
of Fig. 4.2.3,
v1 = ea - eb, v2 = ea - eb, v3 = ea - ec, v4 = ec, v5 = eb,
or in compact form
v = ATen. (4.2.3)
In summary, for any circuit, a maximal set of independent current equations can be written as in
(4.2.2); and branch and node voltages are related as in (4.2.3).
In the following, unless otherwise stated, we will refer to A simply as the incidence matrix.

4.2.2 Tellegen's Theorem

Consider two circuits having the same oriented circuit graph with the incidence matrix A. Let for
s = 1, 2, is(ts), vs(ts), en,s(ts) be the branch current, the branch voltage, and the node voltage vectors
for the sth circuit at time ts. Since
Ai1 = 0 , v1 = ATen,1 ,
Ai2 = 0 , v2 = ATen,2 ,

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we have
v1Ti2 = (ATen,1)Ti2 = en,1TAi1
= 0. (4.2.4a)
Similarly,
v2Ti1 = 0, (4.2.4b)
v1Ti1 = 0, (4.2.4c)
v2Ti2 = 0. (4.2.4d)
As a special case, we conclude that for any circuit
v(t)Ti(t) = 0 {or i(t)Tv(t) = 0}. (4.2.5)
Since pk(t) = vk(t)∙ik(t) is the instantaneous power delivered to the kth branch, k = 1, 2, … , b,
(4.2.5) implies
p1(t) + p2(t) + … + pb(t) = 0; (4.2.6)
that is, the instantaneous power is conserved.

Example 4.2.1 The four circuits Ca, Cb, Cc, and Cd have the same oriented circuit graph with five
branches. The following measurements are recorded.
Ca: v1a = 2 V, v2a = - 3 V, v3a = 5 V, v4a = 2 V, v5a = 3 V;
Cb: v1b = - 1 V, v2b = ?, v3b = - 3 V, v4b = - 5 V, v5b = 2 V;
Cc: i1c = 3 A, i2c = - 3 A, i3c = ?, i4c = 1 A, i5c = 1 A;
Cd: i1d = - 4 A, i2d = ?, i3d = - 1 A, i4d = ?, i5d = 5 A.

Using the above information, we want to determine v2b, i3c, i2d, and i4d.
Applying Tellegen’s Theorem, from the measurements on Ca and Cc
6 + 9 + 5i3c + 2 + 3 = 0 ⇒ i3c = - 4 A;
from the measurements on Cb and Cc
- 3 - 3v2b + 12 - 5 + 2 = 0 ⇒ v2b = 2 V;
from the measurements on Ca and Cd
- 8 - 3i2d - 5 + 2i4d + 15 = 0 ⇒ - 3i2d + 2i4d = - 2;
and from the measurements on Cb and Cd
4 + 2i2d + 3 - 5i4d + 10 = 0 ⇒ 2i2d - 5i4d = - 17.
Solving the last two equations,
−3 2 i2d −2 i 1 −5 −2 −2 4
[ ][ ] = [ ] ⇒ [ 2d ] = [ ][ ] = [ ],
2 −5 i4d −17 i4d 11 −2 −3 −17 5
we have i2d = 4 A and i4d = 5 A. ###

4.2.3 Mesh Matrix

Mesh matrix M for a planar nonseparable oriented circuit graph is an m x b matrix where there
corresponds a row to each mesh and a column to each branch.

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The graph of Fig. 4.2.1b is planar connected and nonseparable. It has two meshes labeled A and B,
and the outer mesh labeled C as shown in Fig. 4.2.4a. {The capital letter A stands for a reduced
incidence matrix, a mesh label, etc. The actual meaning will be clear from the context.}

(a) (b)
Figure 4.2.4 (a) Meshes and (b) mesh currents for the circuit of Fig. 4.2.1a.

The mesh matrix M for this graph is

There are two possible mesh orientations: clockwise or counterclockwise. The clockwise orientation
is taken as reference. The construction of M is as follows: The branches on mesh A are 1 and 2; the
orientation of 2 is positive (i.e., clockwise) and that of 1 is negative. Then in the first row (the one
corresponding to mesh A), the entry at the column position 1 is -1 and the entry at the column
position 2 is +1. Since the other branches do not appear on mesh A, the remaining entries are all
zero. The second row of M (corresponding to mesh B) is formed similarly. As we will show later, the
rank of M is m.
By augmenting M with the row corresponding to the outer mesh, the matrix Mt is obtained. For
the outer mesh, the positive orientation is the counterclockwise orientation. Then for the graph
of Fig. 4.2.4a,
1 2 3 4 5
A −1 1 0 0 0
Mt = B [ 0 −1 1 1 −1].
C 1 0 −1 −1 1
.
Note that in Mt, there are exactly one +1 and one -1 in each column. Branch 1 is shared by meshes
A and C; its orientation is positive for mesh C and negative for mesh A. Then in the first column,
we have -1 at the row position corresponding to A and +1 at the row position corresponding to C,
etc. {Compare the properties of At and Mt, and A and M.}
The mesh matrix M for the planar unconnected nonseparable graph of Fig. 4.2.5 is given below.

Figure 4.2.5 Meshes of a planar unconnected graph.

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1 2 3 4 5 6 7
A −1 1 0 0 0 0 0
M= B[ 0 −1 0 0 1 1 0].
C 0 0 1 −1 0 0 1

For the circuit of Fig. 4.2.4b (also refer to Fig. 4.2.1a), the voltage equations written on the meshes
(using KVL) are
mesh A: - v1 + v2 = 0,
mesh B: - v2 + v3 + v4 - v5 = 0.
In compact form,
Mv = 0. (4.2.7)
Since the rank of M is 2, the two equations above are independent. {Note that Mtv = 0. The rank
of 3 x 5 matrix Mt is 2, and the three voltage equations implied are not independent.}
Current circulating in a mesh is called a mesh current. The mesh currents iA and iB are indicated in
Fig. 4.2.4b.
The vector im of m mesh currents is called the mesh current vector. For the circuit of Fig. 4.2.4b (or
equivalently for the graph of Fig. 4.2.4a),
im = (iA iB)T,
and for the graph of Fig. 4.2.5,
im = (iA iB iC)T.
For planar nonseparable circuits, branch currents can be expressed in terms of mesh currents
using KCL. For example, for the circuit of Fig. 4.2.4b,
i1 = - iA, i2 = iA - iB, i3 = iB, i4 = iB, i5 = - iB,
or in compact form
i = MTim. (4.2.8)
In summary, for any planar nonseparable circuit, a maximal set of independent voltage equations
can be written as in (4.2.7); and branch and mesh currents are related as in (4.2.8).

4.2.4 Fundamental Loop and Fundamental Cutset Matrices

Consider a loop and a cutset of an oriented circuit graph, for example the loop {4, 3, 8, 7, 6, 5} and
the cutset {9, 8, 5} shown in Fig. 4.2.6a. Arbitrarily set the loop and cutset directions as shown in
the figure. {For the loop the clockwise orientation and for the cutset the direction from inside of
the gaussian surface to the outside are chosen.} The directions of branches 4, 3, 7 coincide with
the loop direction, i.e., positive, and those of branches 8, 6, 5 are negative. The directions of
branches 9 and 8 coincide with the cutset direction, i.e., positive, and that of branch 5 is negative.
The loop vector r and the cutset vector x (for the loop and the cutset under consideration) are
given below.
1 2 3 4 5 6 7 8 9 10
r = [0 0 1 1 -1 -1 1 -1 0 0], (4.2.9a)

1 2 3 4 5 6 7 8 9 10
x = [0 0 0 0 -1 0 0 1 1 0]. (4.2.9b)

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(a) (b)

Figure 4.2.6 (a) A loop and a cutset of an oriented circuit graph, and (b) a tree.

In r, the entries corresponding to branches 3, 4, 7 (with positive direction) are +1, and the entries
corresponding to branches 5, 6, 8 (with negative direction) are -1. The remaining entries are zero
since the corresponding branches do not appear on the loop.
In x, the entries corresponding to branches 8 and 9 (with positive direction) are +1, and the entry
corresponding to branch 5 (with negative direction) is -1. The remaining entries are zero since the
corresponding branches do not appear on the cutset.
A loop and a cutset are orthogonal; i.e., if r is a loop vector and x is a cutset vector, then
rTx = 0 {or xTr = 0}. (4.2.10)
This is so, because the number of branches common to a loop and a cutset is even; and a half of
the common branches have the same directions (both positive or both negative) on the loop and
on the cutset, and the remaining half have the opposite directions (positive on the loop and
negative on the cutset or vice versa).
Let T be a tree for an oriented graph. Order the branches so that the cotree-branches follow the
tree-branches. For example let T = {4, 5 ,6 ,2 ,8 ,10} be a tree for the graph of Fig. 4.2.6a (see
Fig. 4.2.6b). An ordering of branches is {4, 5, 6, 2, 8, 10; 1, 3, 9, 7}. {The entries of branch current and
branch voltage vectors are ordered accordingly.}
Each cotree-branch defines a fundamental loop (f-loop) whose direction is set by the direction of
the defining cotree-branch. The fundamental loop matrix B with respect to T is an m x b matrix
where there corresponds a row to each cotree-branch and a column to each branch. The row
vectors of B are the fundamental loop vectors.
The B matrix with respect to the chosen tree T for the graph of Fig. 4.2.6b is given below.

4 5 6 2 8 10 1 3 9 7
1 -1 1 0 -1 1 -1 1 0 0 0
 
3 1 -1 0 0 -1 1 0 1 0 0
B .
9 0 0 0 1 -1 0 0 0 1 0
 
7  0 0 -1 0 0 -1 0 0 0 1

B is of the form B = [B1 U] where B1 is an m x n matrix and U is the m x m identity matrix. Hence,
the m rows of B are linearly independent, and the rank of B is m.

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Any loop can be generated from the fundamental loops. For example, consider the loop {4, 3, 8, 7,
6, 5} in Fig. 4.2.6a and the tree T of Fig. 4.2.6b. Branches 3 and 7 on the loop are cotree-branches.
According to the chosen loop direction, the directions of these branches are both positive. Let rk
denote the fundamental loop vector defined by the cotree-branch k and define

4 5 6 2 8 10 1 3 9 7
r’ = r3 + r7 = [1 -1 -1 0 -1 0 0 1 0 1].

r' is the loop vector for the loop {4, 3, 8, 7, 6, 5}. {Compare the r of (4.2.9a) and r'. Note that
the branch orderings are different.}
Each tree-branch defines a fundamental cutset (f-cutset) whose direction is set by the direction of
the defining tree-branch. The fundamental cutset matrix Q with respect to T is an n x b matrix
where there corresponds a row to each tree-branch and a column to each branch. The row vectors
of Q are the fundamental cutset vectors.
The Q matrix with respect to the chosen tree T for the graph of Fig. 4.2.6b is given below.

4 5 6 2 8 10 1 3 9 7
4 1 0 0 0 0 0 1 -1 0 0
 
5 0 1 0 0 0 0 -1 1 0 0
6 0 0 1 0 0 0 0 0 0 1
Q  .
2 0 0 0 1 0 0 1 0 -1 0
8 0 0 0 0 1 0 -1 1 1 0
 
10 0 0 0 0 0 1 1 -1 0 1 

Q is of the form Q = [U Q1] where U is the n x n identitiy matrix and Q1 is an n x m matrix. Hence,
the n rows of Q are linearly independent, and the rank of Q is n.
Any cutset can be generated from the fundamental cutsets. For example, consider the cutset {9,
8, 5} in Fig. 4.2.6a and the tree T of Fig. 4.2.6b. Branches 8 and 5 on the cutset are tree-branches.
According to the chosen cutset direction, the direction of branch 8 is positive and that of branch
5 is negative. Let xk denote the fundamental cutset vector defined by the tree-branch k and define

4 5 6 2 8 10 1 3 9 7
x’ = x8 - x5 = [0 -1 0 0 1 0 0 0 1 0].

x' is the cutset vector for the cutset {9, 8, 5}. {Compare the x of (4.2.9b) and x'. Note that the branch
orderings are different.}
Since loops and cutsets are orthogonal, denoting a zero matrix by O,
BQT = O {or QBT = O}. (4.2.11)
In open form
U
[B1 U][QT ] = O ⇒ B1 + Q1T = O,
1
implying
B1 = - Q1T {or Q1 = - B1T}. (4.2.12)
From (4.2.12) we conclude that given B, we can obtain Q and vice versa.
In Fig. 4.2.6b, the voltage equations written on the fundamental loops (using KVL) are

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f-loop 1: - v4 + v5 - v2 + v8 - v10 + v1 = 0,
f-loop 3: v4 - v5 - v8 + v10 + v3 = 0,
f-loop 9: v2 - v8 + v9 = 0,
f-loop 7: - v6 - v10 + v7 = 0.
In compact form,
Bv = 0. (4.2.13)
The four equations above are independent. The voltage equations written on the remaining loops
can be derived from these equations.
The current equations written on the fundamental cutsets (using KCL) are
f-cutset 4: i4 + i1 - i3 = 0,
f-cutset 5: i5 - i1 + i3 = 0,
f-cutset 6: i6 + i7 = 0,
f-cutset 2: i2 + i1 - i9 = 0,
f-cutset 8: i8 - i1 + i3 + i9 = 0,
f-cutset 10: i10 + i1 - i3 + i7 = 0.
In compact form
Qi = 0. (4.2.14)
The six equations above are independent. The current equations written on the remaining cutsets
can be derived from these equations.
The vector of tree-branch voltages, denoted by ex, is referred to as the cutset voltage vector; and
the vector of cotree-branch currents, denoted by il is referred to as the loop current vector. For
example, for the graph of Fig. 4.2.6b
ex = (v4 v5 v6 v2 v8 v10)T , il = (i1 i3 i9 i7)T.
Denoting the vector of cotree-branch voltages by el, from (4.2.13) it follows that
ex
[B1 U][ e ] = 0 ⇒ el = - B1ex = Q1Tex.
l
Hence,
v = QTex. (4.2.15)
That is, the branch voltages can be determined once the cutset (tree-branch) voltages are known.
Denoting the vector of tree-branch currents by ix, from (4.2.14) it follows that
i
[U Q1][ ix ] = 0 ⇒ ix = - Q1il = B1Til.
l

Hence,
i = BT il. (4.2.16)
That is, the branch currents can be determined once the loop (cotree-branch) currents are known.
In summary, for any circuit, a maximal set of independent voltage equations can be written as in
(4.2.13); and branch and loop currents are related as in (4.2.16). For any circuit, a maximal set of
independent current equations can be written as in (4.2.14); and branch and cutset voltages are
related as in (4.2.15).

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4.3 Dual Oriented Graphs

Dual of an oriented planar connected nonseparable graph G is another oriented planar connected
nonseparable graph Ĝ.
The construction of dual of the graph of Fig. 4.3.1a is explained in Fig. 4.3.1b, and the dual graph
is shown in Fig. 4.3.1c. The structure of Ĝ follows as if G is a nonoriented graph.
{Refer to Fig. 3.2.13.}

(a) (b) (c)

To determine the branch directions in Ĝ, we proceed as shown in Fig. 4.3.2. In Fig. 4.3.2a, α1 is a
branch of G, and α1' is the corresponding branch of Ĝ. The direction of α1' is determined from that
of α1 by rotating α1 90 degrees in the clockwise direction. Similarly, in Fig. 4.3.2b, α2 and α2' are
the corresponding branches of G and Ĝ, respectively. The direction of α2' follows from that of α2
by rotating α2 90 degrees in the clockwise direction. {The dual of Ĝ is G provided that the above-
mentioned rotations are in the counterclockwise sense.}

(a) (b)
Figure 4.3.2 Determination of the directions of dual graph branches.

The construction of the dual graph is such that the matrix At of G is the matrix M ̂ t of Ĝ, and the
̂ t) is n, and the rank of Ât
matrix Mt of G is the negative of the matrix Ât of Ĝ. The rank of At (and M
(and Mt) is m. The fundamental loops of G with respect to a tree T are the fundamental cutsets of
Ĝ with respect to the tree T ̂=T ̅ (T̅ is the cotree corresponding to the tree T of G), and the
fundamental cutsets of G with respect to T are the fundamental loops of Ĝ with respect to T ̂.
̂ 1 and B
{That is, B1 = Q ̂ 1 = Q1 where B = [B1 U] and Q = [U Q1] are the fundamental loop and the funda-
mental cutset matrices with respect to T for G, respectively; and B ̂ = [B
̂ 1 U] and Q ̂ = [U Q
̂ 1] are the
fundamental loop and the fundamental cutset matrices with respect to T ̂ for Ĝ, respectively.}

As an example, for the dual oriented graphs of Fig. 4.3.1,

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1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
b  1 1 0 1 0 0 0 0  A  1 0 0 1 1 0 0 0 
c   1 0 0 0  1 1 0 0 
 B  0  1  1  1 0 0 0 0 

M̂ t  A t  d  0 0  1 1 1 0  1 1 , Â t  - M t  C  0 0 0 0 1 1 1 0 
   
e  0 0 0 0 0 1 1 1  D  0 0 0 0 0 0 1 1 
a  0  1 1 0 0 0 0 0  E   1 1 1 0 0  1 0 1 

̂ = {4, 5, 6, 8} is a tree for Ĝ),

1 2 3 7 4 5 6 8
4 0 1 1 0 1 0 1 -1 0 
5 -1 1 1 0  2 - 1 - 1 1 0 
B 1  Q̂ 1   , B̂ 1  Q 1   .
6  1 -1 -1 1 3 - 1 - 1 1 0
   
8 0 0 0 1 7 0 0 - 1 - 1

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