In
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, modified nodal analysis or MNA is an extension of
nodal analysis
In electric circuit analysis, nodal analysis (also referred to as node-voltage analysis or the branch current method) is a method of determining the voltage between nodes (points where elements or branches connect) in an electrical circuit in ter ...
which not only determines the circuit's node voltages (as in classical nodal analysis), but also ''some'' branch currents. Modified nodal analysis was developed as a formalism to mitigate the difficulty of representing voltage-defined components in nodal analysis (e.g. voltage-controlled voltage sources). It is one such formalism. Others, such as sparse tableau formulation, are equally general and related via matrix transformations.
Method
The MNA uses the element's ''branch constitutive equations'' or BCE, i.e., their
voltage
Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...
-
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (hydr ...
characteristic and the
Kirchhoff's circuit laws
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirc ...
. The method is often done in four steps, but it can be reduced to three:
Step 1
Write the
KCL equations of the circuit. At each node of an
electric circuit
An electrical network is an interconnection of electrical components (e.g., battery (electricity), batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e. ...
, write the currents coming into and out of the node. Take care, however, in the MNA method, the current of the independent voltage sources is taken from the "plus" to the "minus" (see Figure 1). Also, note that the right hand side of each equation is ''always'' equal to zero, so that the branch currents that come into the node are given a negative sign and those that go out are given a positive sign.
Step 2
Use the BCEs in terms of the node voltages of the circuit to eliminate as many branch currents as possible. Writing the BCEs in terms of the node voltages saves one step. If the BCEs were written in terms of the branch voltages, one more step, i.e., replacing the branches voltages for the node ones, would be necessary. In this article the letter "e" is used to name the node voltages, while the letter "v" is used to name the branch voltages.
Step 3
Finally, write down the unused equations.
Example
The figure shows a RC series circuit and the table shows the BCE of a linear resistor and a linear capacitor. Note that in the case of the resistor the
admittance
In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the multiplicative inverse, reciprocal of Electrical impedance, impedance, analogous to how Electrical resistanc ...
i,
, is used instead of
. We now proceed as explained above.

Step 1
In this case there are two nodes,
and
. Also there are ''three'' currents:
,
and
.
At node ''e1'' the KCL yields:
and at node ''e2'':
Step 2
With the provided BCEs in the table and observing that:
the following equations are the result:
Step 3
Note that at this point there are two equations but three unknowns. The missing equation comes from the fact that
and so finally we have three equations and three unknowns, that leads to a solvable linear system.
Modified Nodal Analysis and DAEs
If the vector
is defined, then the above equations can be put in the form
where
,
and
.
This is a linear
differential algebraic equation
In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system.
The set of the solutions of such a system is a ...
(DAE), since
is singular. It can be proved that such a DAE coming from the Modified Nodal Analysis will have
differentiation index less or equal than two as long as only passive RLC components are used.
[Tischendorf C. Topological index of DAEs in the Circuit Simulation.] When using active components, such as
operational amplifiers
An operational amplifier (often op amp or opamp) is a DC-coupled electronic voltage amplifier with a differential input, a (usually) single-ended output, and an extremely high gain. Its name comes from its original use of performing mathem ...
, the differentiation index can be arbitrarily high.
Non-smooth analysis
DAEs assume
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
characteristics for individual components; for example, a
diode
A diode is a two-Terminal (electronics), terminal electronic component that conducts electric current primarily in One-way traffic, one direction (asymmetric electrical conductance, conductance). It has low (ideally zero) Electrical resistance ...
can be modeled/represented in a MNA with DAEs via the
Shockley equation, but one cannot use an apparently simpler (more ideal) model where the
sharply exponential forward and breakdown conduction regions of the curve are just straight vertical lines. Circuit analysis (including MNA) with the latter kind of equations is actually more involved (than using DAEs) and is the topic of
non-smooth dynamical systems (NSDS) analysis, which relies on the theory of
differential inclusion
In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form
:\frac(t)\in F(t,x(t)),
where ''F'' is a multivalued map, i.e. ''F''(''t'', ''x'') is a ''set'' rather than a single point ...
s.
References
External links
* {{cite web, url=http://qucs.sourceforge.net/tech/node14.html , title=Modified Nodal Analysis (DC algorithm description in Qucs technical documentation) , access-date=22 December 2012
Electronic circuits