The Modified Nodal Analysis (MNA) Method
Dr. José Ernesto Rayas-Sánchez
February 10, 2016
The Modified Nodal Analysis (MNA)
Method
Dr. José Ernesto Rayas-Sánchez
1
Nodal Formulation

Matrix T in the Tableau formulation is very sparse

The Nodal formulation emerges as a block elimination
process on the Tableau formulation

It yields a very compact formulation, but it is not general
Dr. J. E. Rayas-Sánchez
2
1
The Modified Nodal Analysis (MNA) Method
Dr. José Ernesto Rayas-Sánchez
February 10, 2016
Nodal Formulation (cont.)
From the Tableau equations,
YVb  ZI  W
ATVn  Vb  0
AI  0
Substituting first equation into the second equation,
YATVn  ZI  W
Combining last two equations,
YAT
 0

Z  Vn  W 

an (b+n) × (b+n) matrix is achieved
A  I   0 
Dr. J. E. Rayas-Sánchez
3
Nodal Formulation (cont.)
If every element in the network can be represented by
 YVb  I  J
Using ATVn  Vb  0
 YATVn  I  J
Since AI  0
AI  A( J  YATVn )  0
AYATVn   AJ an n × n matrix is achieved
Yn = AYAT is the Nodal Admittance Matrix, and Jn = AJ
YnVn  J n
Dr. J. E. Rayas-Sánchez
4
2
The Modified Nodal Analysis (MNA) Method
Dr. José Ernesto Rayas-Sánchez
February 10, 2016
Nodal Formulation – Final Remarks

It requires solving an n by n system of equations

It is restricted to elements that have an admittance
representation:
 YVb  I  J

Independent voltage sources, ideal CCVS, ideal CCCS,
transformers, etc, can not be represented in this manner
Dr. J. E. Rayas-Sánchez
5
Modified Nodal Analysis (MNA)
It splits the circuit elements into groups: one group for
elements that accept an admittance description, and one
group for those which do not
 Y1V1b  I1  J1
Y2V2b  Z 2 I 2  W2
If A1 and A2 are the corresponding incidence matrices for
each group of elements, since AI = 0 then
A1 I1  A2 I 2  0
Since ATVn = Vb, then
A1TVn  V1b
A2TVn  V2b
Dr. J. E. Rayas-Sánchez
6
3
The Modified Nodal Analysis (MNA) Method
Dr. José Ernesto Rayas-Sánchez
February 10, 2016
Modified Nodal Analysis, MNA (cont.)
Substituting last two equations into first two,
 Y1 A1TVn  I1  J1
Y2 A2TVn  Z 2 I 2  W2
Pre-multiplying by A1 first previous equation
 A1Y1 A1TVn  A1 I1  A1J1
Using A1 I1  A2 I 2  0
 A1Y1 A1TVn  A2 I 2  A1J1
Then
 A1Y1 A1T
 Y AT
 2 2
A2  Vn   A1J1 

Z 2   I 2   W2 
HX  W
Dr. J. E. Rayas-Sánchez
7
MNA – Final Remarks

Once the MNA equation is solved,
 A1Y1 A1T
 Y AT
 2 2

A2  Vn   A1J1 

Z 2   I 2   W2 
The rest of the unknowns are calculated using
I1  J1  Y1 A1TVn
V1b  A1TVn
V2 b  A2TVn

MNA retains the advantages of both the Nodal and the
Tableau methods

The above formulation of MNA can be applied to linear
circuits only, but can be extended to nonlinear
Dr. J. E. Rayas-Sánchez
8
4
The Modified Nodal Analysis (MNA) Method
Dr. José Ernesto Rayas-Sánchez
February 10, 2016
A Nodal Formulation to Nonlinear Circuits
Suppose that the branch equations of all elements in a circuit
accept a non-linear admittance representation,
I  g (Vb )
From KVL and KCL: Vb  ATVn
AI  0
Then
A( g ( ATVn ))  0
(a system of n nonlinear equations with n unknowns)
Dr. J. E. Rayas-Sánchez
9
MNA for Linear and Nonlinear Circuits
Split the circuit branches into three groups:
 Y1V1b  I1  J1
Y2V2b  Z 2 I 2  W2
I 3  g (V3b )
If A1, A2, and A3 are the corresponding incidence matrices
for each group of elements, from KCL
A1 I1  A2 I 2  A3 I 3  0
From KVL,
V1b  A1TVn
V2b  A2TVn
V3b  A3TVn
Substituting KVL into first three equations,
 Y1 A1TVn  I1  J1
Y2 A2TVn  Z 2 I 2  W2
Dr. J. E. Rayas-Sánchez
I 3  g ( A3TVn )
10
5
The Modified Nodal Analysis (MNA) Method
Dr. José Ernesto Rayas-Sánchez
February 10, 2016
MNA for Linear and Nonlinear Circuits (cont.)
 Y1 A1TVn  I1  J1
Y2 A2TVn  Z 2 I 2  W2
I 3  g ( A3TVn )
Pre-multiplying by A1 first above equation
 A1Y1 A1TVn  A1 I1  A1J1
Using A1I1 + A2I2 + A3I3 = 0,
A1Y1 A1TVn  A2 I 2  A3 g ( A3TVn )   A1J1
Solving Z 2 I 2  W2  Y2 A2TVn for I2 at each iteration and
substituting into the above equation yields a system of n
nonlinear equations that can be solved for Vn
Dr. J. E. Rayas-Sánchez
11
MNA Formulation Using Stamps

The MNA equation
 A1Y1 A1T A2  Vn   A1J1 

HX  W
 Y AT
Z 2   I 2   W2 
 2 2
can be formulated without using oriented graphs or
incidence matrices A1 and A2

H and W can be directly formulated by inspection, using
stamps

Further, the L and C elements can be separated so that
( H 1  sH 2 ) X  W
can also be written directly
Dr. J. E. Rayas-Sánchez
12
6
The Modified Nodal Analysis (MNA) Method
Dr. José Ernesto Rayas-Sánchez
February 10, 2016
MNA Formulation Using Stamps (cont.)

Start by entering by inspection in a matrix H all elements
which have an admittance description

Initially H has an n by n size, where n is the number of
ungrounded nodes

Increase the size of H whenever we enter an element
which does not have an admittance description

The constitutive voltage equations of an element without
admittance description are attached as an extra row of H,
while the current flowing into that element is attached as
an extra column in H

W is generated according to the independent sources
Dr. J. E. Rayas-Sánchez
13
MNA Formulation Using Stamps (cont.)
Admittance
j
Vj
i
Vj'
I j  y (V j  V j ' )
j  y  y
j '  y y 
I j '   y (V j  V j ' )
k
Vj
Vj'
I j  I j'  0
 g
g 
I k  g (V j  V j ' )
k'
k g
k '  g
y v
j'
VCCS
j
v
j'
gv
Dr. J. E. Rayas-Sánchez
I k '   g (V j  V j ' )
14
7
The Modified Nodal Analysis (MNA) Method
Dr. José Ernesto Rayas-Sánchez
February 10, 2016
MNA Formulation Using Stamps (cont.)
Impedance
j
i
Vj Vj'
I
j
1
j' 
 1


1  1  z 
z v
j'
Ij  I
I j'  I
V j  V j '  zI  0
Dr. J. E. Rayas-Sánchez
15
MNA Formulation Using Stamps (cont.)
VCVS
V j V j ' Vk
I
j
j
j' 

k
k'

 
k
v
v
j'
k'
Ij 0
Ik  I
I j'  0
Ik '  I
Vk '
I



1
 1

 1 1

 V j  V j '  Vk  Vk '  0
Dr. J. E. Rayas-Sánchez
16
8
The Modified Nodal Analysis (MNA) Method
Dr. José Ernesto Rayas-Sánchez
February 10, 2016
MNA Formulation Using Stamps (cont.)
CCCS
V j V j ' Vk Vk '
I
j
k
I
j'
k'
Ij  I
I k  I
I j'  I
I k '  I
I
1 
1 

 
 


j
j' 

k
k'

1  1
Vj  Vj'  0
Dr. J. E. Rayas-Sánchez
17
MNA Formulation Using Stamps (cont.)
CCVS
I1
V j V j ' Vk Vk '
I2
j
k
rI1
j'
k'
I j  I1
Ik  I2
I j '   I1
Ik '  I2
I1
I2
j
1



j'
1


k
1
k'
 1


1 1  r




1  1

Vj Vj'  0
Vk  Vk '  rI1  0
Dr. J. E. Rayas-Sánchez
18
9
The Modified Nodal Analysis (MNA) Method
Dr. José Ernesto Rayas-Sánchez
February 10, 2016
MNA Formulation Using Stamps (cont.)
Transformer
I1
V j V j ' Vk Vk '
M
I2
j
k
V1
L1
L2
V2
j'
k'
I j  I1
Ik  I2
I j '   I1
Ik '  I2
I1
I2
1
j
1
j' 

k
k'

1  1
 sL1

1  1  sM




1 
1 

 sM 

 sL2 
V j  V j '  sL1 I1  sMI 2  0
Vk  Vk '  sL2 I 2  sMI1  0
Dr. J. E. Rayas-Sánchez
19
MNA Formulation Using Stamps (cont.)
Ideal Op-Amp
j
j'
I
V j V j ' Vk
k
k'
Ij 0
Ik  I
I j'  0
Ik '  I
j
j' 

k
k'

1  1
Vk '
I



1
 1


Vj  Vj'  0
Dr. J. E. Rayas-Sánchez
20
10
The Modified Nodal Analysis (MNA) Method
Dr. José Ernesto Rayas-Sánchez
February 10, 2016
MNA Formulation Using Stamps (cont.)
Current Source
j
(source vector)
j J 
j '  J 
J
I j  J
I j'  J
j'
Voltage Source
j
Vj Vj'
E
I
j'
I
(source vector)
 
 
 
 E 
j
1
j' 
 1


1  1

Dr. J. E. Rayas-Sánchez
Ij  I
I j'  I
Vj Vj'  E
21
11