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Electric and electronic circuit analysis with Millman theorem
VAHÉ NERGUIZIAN1, MUSTAPHA RAFAF1 AND CHAHÉ NERGUIZIAN 2
1 Electrical
Engineering
École de Technologie Supérieure
1100 Notre Dame West, Montreal, Quebec, H3C-1K3
CANADA
2 Electrical
Engineering
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 laws and several theorems such as Thévenin, Norton and superposition.
Moreover, to enhance the efficiency of the analysis, some methods and techniques such as voltage and current
divider and mesh 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 containing 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 circuit 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, theorems,
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
thoroughly 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)
 Laplace 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, the theorem of Millman 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 classes 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 them 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, recommendations and a
conclusion.
2 Millman theorem
2.1 Statement of the theorem
Consider a node A connected to K branches as shown
in Figure 1.
V3
3 Millman theorem for simple circuits
I3
Z3
I2
IK
VA
V ..K
ZK
V2
A
Z2
Z1
I1
V1
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 VA at node A, and V1 to VK
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.
 Vi Yi
i 1
K
Ii
Zi
A
Vi
K
VA 
The example of Figure 2 is a simple operational
amplifier inverter with input resistive impedance and
feedback capacitive impedance. For analysis
simplicity, the operational amplifiers 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 amplifiers is
analyzed a good and efficient approach uses the
following major steps:
 Identify the topologies of each sub-section in
the schematic or in the circuit model
(e.g. inverter, non inverter, follower,
summer,
substractor,
integrator,
differentiator, instrumentation and other
topologies)
 Use the known gain of each topology
 Analyze 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.
Zf
Vo
(1)
 Yi
i 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.
K
 Ii  I1  I 2  ...  I K  0
Fig. 2 Basic Operational amplifier inverter
3.1 Classical analysis
Using Kirchhoff’s current law at node A and Ohm’s
law to calculate the gain of the amplifier, we can
write equations 4 and 5 to obtain the equation 6.
Ii 
Vi  0
Zi
(4)
Ii 
0  Vo
Zf
(5)
(2)
i 1
V1  VA V2  VA
V  VA
(3)

 ...  K
0
Z1
Z2
ZK
Separating VA terms gives the statement of Millman
theorem identified by equation 1.
Av
Vo
Z
 f
Vi
Zi
(6)
V1  V2
R
3.2 Millman theorem analysis
Using Millman theorem at node A, we can write
equation 7 to obtain equation 8.
I
1
1
 Vo
Zi
Zf
VA 
 V  V  0
Yi  Yf
V
Z
Av o   f
Vi
Zi
V3  V2 V3  V4

0
ZC
R
(10)
V5  V4 V5  0

0
R
R
(11)
Vi
(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 analysis
of this circuit.
With V1  V3  V5  V ,
(9)
ZAB
R2

.
ZC
1
2
therefore ZAB  R Cs  Ls
Cs
As an example, for R  10 K, C  10 nF , then
Since Z C 
L 1H .
4 Millman theorem for complex
Bruton circuit
The example of Figure 3 is a complex Bruton
circuit [6] representing a gyrator composed of
operational amplifiers, resistive and capacitive
impedances. The impedance seen at the inputs
A and 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.
4.2 Millman theorem analysis
Similarly, using Millman theorem at nodes 3 and 5,
we can write directly equations 12 and 13 to obtain
the same impedance between A and B as with
classical approach.
V2 V4

ZC R
V3 
1 1

ZC R
(12)
V4 0

R
R
V5 
1 1

R R
(13)
ZAB  R 2 Cs
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 of results obtained with
other circuit analysis methods.
Fig. 3 Bruton circuit
4.1 Classical 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 ZAB between A and B.
5 Millman theorem for complex filter
circuit
The circuit of Figure 4 represents a Rauch filter
considered as complex circuit [7]. Classical analysis
of this circuit would require, after identification 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 equations 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.
Z4
Z5
Z1
A
Z3
Vin
B
Vout
Furthermore, when the Millman theorem is applied 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 electric and electronic
complex circuits
 Fast verification and validation tool
improving examination results of students
 Broadening circuit analysis tools based on
general standard circuit laws
 Elimination of writing several independent
equations of Ohm’s and Kirchhoff’s current
laws solving and obtaining electrical
parameters (such as voltage gain and others).
Z2
7 Conclusion
Fig. 4 Rauch Filter
Vi 0
0 V0



Z1 Z2 Z3 Z4
VA 
1
1
1
1



Z1 Z2 Z3 Z4
(14)
VA Vo

Z3 Z5
VB 
0
1
1

Z3 Z5
(15)
H(s) 
Vo
1

Vi Z3  Z1  Z1Z3  Z1Z3
Z5 Z 5 Z 2 Z 5 Z 4 Z 5
(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.
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 with
correct answers and solutions, compared with
traditional classical approach.
Millman theorem is a good pedagogical and
educational tool to be used in electric 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 1st of June 2005.
[2] P. Bildstein, Filtres actifs, McGraw-Hill, 1989.
[3]Robert F. Coughlin and Frederick F. Driscoll,
Operational Amplifiers and Linear Integrated
Circuits, Prentice –Hall, Englewood Cliffs, NJ, 3rd
edition 1987.
[4] Jacob Millman and Arvin Grabel, Micro
électronique, McGraw-Hill, 1988.
[5] http://encyclopedie.cc/Théorème_de_Millman,
last visited 1st 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 1st of June 2005.