18-12-2012, 06:03 PM
NODAL ANALYSIS
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A branch of an electric circuit is a connection between two points in the circuit. In general a simple wire
connection, i.e., a 'short-circuit', is not considered a branch since it is known directly that there is no
voltage drop across a short-circuit and the current in the short-circuit is whatever is required to satisfy
KCL. Although it is neither required, nor always desirable, ordinarily for simplicity each branch
contains a single circuit element.
A node is a point of connection of two or more branches. In general 'dangling' branches, i.e., branches
each of which is connected only to a single node are assumed to have been removed from the circuit
insofar as analysis of the circuit is concerned. Dangling branches are known directly to have at most a
constant voltage drop (e.g., a voltage source) and carry no current. Finally it is assumed that the circuit
does not have 'separate parts', i.e., consist of two or more electrically disconnected parts. It must be
possible to trace a path along circuit branches between any two nodes. For circuits with separate parts
each part can be analyzed separately. In practice these conditions are rarely violated.
A circuit is analyzed by application of KVL, KCL, and the volt-ampere relations for the circuit branch
elements. Nothing else is needed nor used. The three requirements are applied until a sufficient
number of independent equations are obtained to solve for all branch voltages and all branch currents.
This is far more subtle a procedure in practice than it sounds. An electric circuit usually involves many
branches and many nodes, and a haphazard search for a sufficient number of independent equations can
be quite enervating. Therefore we consider various ways of undertaking a circuit analysis with the
general aim of assuring that a solution will be found with a minimum effort to do so.
Supplementary Example Involving a Controlled Source
The voltage-controlled current source in the circuit below introduces only a modest adjustment in the
nodal analysis. The dependent current source is treated as is any node current, except that the source
strength depends on a voltage in another part of the circuit. Simply observe that the control voltage Vx
is a branch voltage and it can be expressed in terms of node voltages as e2 – e3. This replacement can
be done in the course of writing the equation.