Civil Engineering Reference
In-Depth Information
the device equations that relate the current and voltage parameters for in-
dividual elements.
The KCL and KVL relations are
linear
equations that are purely topological
in that they depend only on the connectivity of the network‚ and not on the
type of elements in the circuit. The element characteristics are wholly modeled
by the device equations‚ and these may‚ in general be
linear
or
nonlinear
or
differential
equations. Circuit simulation involves the solution of this system
of equations to find the currents and voltages everywhere in the circuit. For
practical purposes‚ the results of circuit simulation are often considered to be
the exact solution to any circuit‚ although it is useful to temper this statement
with the observation that any such solution is only as exact as the models that
are used for the devices.
For a circuit with
nodes (plus a ground node) and
branches‚ there are
independent KCL equations‚
independent KVL equations‚ and
device
equations. The circuit variables consist of
node voltages‚
branch currents
and
branch voltages‚ which yields a total of
equations in an equal
number of variables.
There are several ways of combining these equations into a more compact set‚
but we will focus our attention on the
modified nodal analysis
(MNA) method
since it is arguably the most widely-used formulation.
2.2
FORMULATION OF CIRCUIT EQUATIONS
We first introduce a formal manner for writing the equations for a circuit. To
write the topological KCL and KVL equations‚ we will consider a circuit to
be represented by a directed graph
G = (V‚E)‚
where the vertex set
V
has a
one-to-one correspondence with the set of nodes in the circuit‚ and the elements
in the edge set
E
correspond to the branches. The directions on the edges may
be chosen arbitrarily
1
. Given any such graph‚ we can define the notion of an
incidence matrix‚
on the graph
G.
This is an matrix‚ with the
rows corresponding to the nonground vertices and one vertex corresponding
to the ground node. Each of the columns corresponds to a directed edge and
has two nonzero entries: a “+1” for the source of a directed edge‚ and a “-1”
for the destination. An example incidence matrix for a sample circuit graph
is shown in Figure 2.1. The reference directions chosen for the branches are
shown by arrows in the circuit graph.
It may be observed that by definition‚ the sum of all entries in each column
of is 0; as a consequence‚ is not of full rank
2
. It can be shown that
for a connected graph the rank of is and that any submatrix
of is nonsingular. In other words‚ eliminating one row of (typically
the row corresponding to the ground node) would convert it to a matrix of full
rank that we will denote as
A.
Let denote the vector of branch currents. With the aid of the incidence
matrix‚ the
KCL equations may be written as follows:
