Electric and electronic circuit analysis with Millman theorem

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Electric and electronic c
ircuit analysis with Millman theorem


VAHÉ NERGUIZIAN
1
, MUSTAPHA RAFAF
1

AND CHAHÉ NERGUIZIAN

2



1
Electrical Engineering

École de Technologie Supérieure

1100 Notre Dame West, Montreal, Quebec, H3C
-
1K3

CANADA


2
Electrical Enginee
ring

École Polytechnique de Montréal

2500 Chemin de Polytechnique, Montreal, Quebec, H3C
-
3A7

CANADA


http://www.etsmtl.ca



Abstract:
-

In the electrical engineering curriculum, after circuit modeling, the analysis
of a circuit
is done using
voltage
-
current Ohm law, Kirchhoff’s law
s

and several theorems such as
Th
é
venin, Norton and superposition.
Moreover, to enhance the efficiency of the analysis, some methods and techniques such as voltage and current
divider and m
esh or node methods are used. In this paper the theorem of Millman is introduced in order to ease
the analysis of some circuits and to verify the results with a second mean or tool. The application of this theorem
is specifically interesting for circuits c
ontaining operational amplifiers. The pedagogical goal of this analysis
approach is to give the students another tool to improve and verify the analysis
of electric

and electronic circuits.


Key
-
Words:
-

Analysis method, Analysis verification,
Electric and

electronic c
ircuit analysis, Millman theorem
.


1

Introduction

Electrical engineering basic courses introduce to
students the conversion mechanism from the physical
electrical
network

to electrical circuit model for
analysis. In parallel, different laws, th
eorems,
analysis methods and techniques are thought to help
students in finding the appropriate electrical
parameters such as currents, voltages, powers and
energies. The basic Ohm and Kirchhoff’s laws,
Thévenin, Norton and superposition theorems are
thoro
ughly explained to students since they present
strong circuit analysis tools. Usually, when the
students are analyzing electrical circuits, all these
studied tools can be applied in 3 different domains
such as:



Time domain (with differential equations)



Lap
lace domain (with algebraic equations)



Frequency domain (with phasor notions)


Although the differential equations analysis provides
the complete transient and steady state responses of
the system, they become very cumbersome for
complex circuits. Laplace

domain analysis is the
most efficient, specifically for complex circuits, and
it gives the complete natural and forced responses.
The phasor analysis is a special case for sinusoidal
input and a steady state response only.

Moreover, t
he theorem of Millma
n can be used with
any of these domain analyses and can provide a
perfect
efficient
tool to analyze complex circuits

[1]
,

[2]
,
[3]

and [4]
,

and to verify results with other basic
means
. Unfortunately, this theorem is not often
elaborated or thought in clas
ses and its utility is not
properly identified. The Millman theorem is basically
a derivative of Kirchhoff’s current law and is very
simple to be used in circuit analysis. It can act as
complementary or supplementary analysis tool for
students permitting t
hem to analyze and verify the
circuits
with different methods
. In
section 2 the
Millman theorem is
described

in details
. Sections 3,
4
and 5
show typical simple and complex circuit

examples

analyzed with Millman theorem
. Sections

6
and 7

give
some
hints
, r
ecommendations

and

a
conclusion.



2

Millman theorem



2.1 Statement of the theorem

Consider a node A connected to K branches as shown
in
F
igure 1.


Fig. 1 Circuit example with
a common
node
A
connected to K branches


Each branch can contain impedances of
different
nature (combination of resistors, inductors and
capacitors). The voltage V
A

at node A, and V
1

to V
K

at all other ends of the branches are identified by
potentials with respect to a reference voltage

(a reference voltage that can be 0 volt). The
theorem
identified by the equation 1 states that the voltage at
A is the sum of branch voltages multiplied by their
associated admittances, divided by the total
admittance

[
5
]
. For Laplace domain, the voltages and
currents represent the Laplace values and
the
impedances are operational impedances. For the
phasor domain, the voltages and current are
phasors
and

the impedances are complex impedances.







(1)


2.2 Proof of the theorem

Equation 2 is obtained by applying the Kirchhoff’s
current law at node A. From generalized Ohms law
we can write each current by its equivalent potential
difference
divided
by the
impedance

as per

equation 3.





(2)




(3)

Separating V
A

terms gives the stat
ement of Millman
theorem identified by equation 1.


3

Millman theorem for simple circuits

The example of
F
igure 2 is a simple operational
amplifier inverter with input
resistive impedance
and
feedback capacitive impedance. For analysis
simplicity, the oper
ational amplifier
s

are

considered
ideal
in this article
,

and therefore the voltages at
positive and negative inputs
of the operational
amplifiers
are virtually identical.

When a circuit containing operational amplifier
s

is
analyzed
a

good
and
efficient app
roach

uses the
following major steps:



I
dentify the topolog
ies

o
f each sub
-
section in

the schematic or
in
the circuit model

(e.g. inverter, non inverter, follower,
summer, substractor, integrator,
differentiator, instru
mentation and other
topologies)



U
se
t
he
known
gain of each topology



A
nalyze
and solve
the complete

cascaded
circuit
using

superposition theorem.

With Millman theorem, the topology identification is
not necessary,
and
similar to

Kirchhoff’s current law,
Millman theorem is
applied

at different
specific
nodes.


Fig. 2 Basic Operational amplifier inverter


3.1 Classical analysis

Using Kirchhoff’s current
law at node A and Ohm
’s

law
t
o calculate the gain of the amplifier, we can
write equations 4

and

5
to obtain the equ
ation
6.








(4)







(5)






(6)



3.2 Millman theorem analysis

U
sing Millman theorem

at node A
, we can
write
equation

7
to
obtain
equation
8.





(7)





(8)


3.3 Comparison

It is obvious with this
simple
example that Millman
theorem is
a
direct derivative of Kirchhoff’s current
law and the Millman theorem approach does not
present any advantage
or difference
for
the analy
sis
of
this circuit
.



4

Millman theorem for complex
Bruton circuit

The example of
F
igure 3 is a complex Bruton

circuit
[
6
]
representing a gyrator
composed

of
operational amplifiers, resistive and capacitive
impedances. The impedance seen at the input
s

A an
d B
of this circuit presents
an impedance

that is
purely
inductive. In fact this circuit is very useful to
simulate high value inductance in a small package
using
two

operational amplifiers,
four

resistors and
one

capacitor.


Fig. 3 Bruton circuit


4.1 Cl
assical analysis

Using
Ohm’s law
and

Kirchhoff’s current law
at

nodes 3 and 5
we can write equations 9
, 10 and 11 to
calculate the impedance Z
AB

between A and B
.







(9
)





(
10
)








(11)

With
,

.

S
ince

therefore



As an example,

for
,

then
.


4.2 Millman theorem ana
lysis

Similarly, using Millman theorem

at nodes
3 and 5
,
we can write
directly
equations
1
2

and 13 to obtain
the same impedance between A and B

as with
classical approach
.









(12)










(1
3
)





4.3 Comparison

It is obvious with this example that Millman theorem
gives

much
simpler and
efficient
equations
and
presents faster analysis than the classical method.

It
also permits the validation o
f results
obtained
with
other circuit analysis methods.




5

Millman theorem for complex
f
ilter
circuit

The circuit of Figure 4 represents a
Rauch

filter
considered as complex circuit

[
7
]
. Classical analysis
of this circuit would require
, after identificatio
n
of
dependant

equations,

the writing of
two

independent
equations using Kirchhoff’s laws or other analysis
methods, such as node
-
voltage

method. Compared to
this classical method, Millman theorem requires the
writing of
Millman
simple equation
s at nodes

A and B

given by
equations
14

and
15
. Solving these
equations gives

the
filter response or voltage gain
given
by equation

16
.




Fig. 4
Rauch

Filter





(
14
)









(
15
)





(
16
)



6

Helpful hints and recommendations

The Millman theorem is mostly applied for circuits
with several operational amplifiers representing
complex circuit topology. With too many nodes in
the complex circuit
the analysis with classical
approach would lead for cumbersome and
complicated equations that students can easily make
mistakes. Therefore in these situations it is strongly
recommended to use Millman theorem.

Furthermore, w
hen the Millman theorem is appl
ied in
the analysis of a circuit, students shall be very careful
in
identifying the common node and to apply
adequately equation 1
based on the
circuit of

Figure 1.

The pedagogical and practical advantages in using
Millman theorem are:



Fast analysis of e
lectric and electronic
complex circuits



Fast verification and validation tool
improving
examination results of students



Broaden
ing

circuit analysis tools based on
general standard circuit laws



Elimination of writing several independent
equations
of Ohm’s a
nd Kirchhoff’s current
laws
solv
ing

and obtain
ing

electrical
parameters
(
such as voltage gain and others
)
.



7

Conclusion

The Millman theorem permits fast
and efficient
resolution of complex circuits when applied
correctly. Teaching experience and students
statistical
evaluation results had shown that students
using this theorem were responding faster wit
h
correct answers and solutions, compared with
traditional

classical
approach.

Millman theorem is a good pedagogical and
educational tool to be used in elec
tric and electronic
courses to enhance student’s knowledge and their
ability to analyze
and to validate the results of
complex circuit efficiently.


References:


[1]



http://wwwmathlabo.univ
-
poitiers.fr/

enseignement/caplp/documents/2003.pdf
,

page 44, last visited 1
st

of June 2005.

[
2
]

P.
B
ildstein
,

Filtres actifs, McGraw
-
Hill
,
1989
.

[3]Robert F. Coughlin and Frederick F. Driscoll,
Operational Amplifiers and Linear

Integrated
Circuits, Prentice

Hall,
Englewood Cliffs, NJ,
3
rd

edition
19
87
.

[4] Jacob Millman and Arvin Grabel, Micro
électronique, M
cGraw
-
Hill
, 1988.

[
5
]


http://encyclopedie.cc/Théorème_de_Mill
man
,
last visited 1
st

of June 2005.

[
6
]
L.T.
Bruton, RC
-
Active Circuits Theory and
Design, Prentice
-
Hall, Englewood Cliffs, NJ, 1980.

[
7
]


http://www.upf.pf/~guarino/virt_lab/electro/

rauch.pdf
, last visited 1
st

of June 2005.