Page 210 - The Mechatronics Handbook

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FIGURE 11.22 Branch current formulation in nodal
analysis.

FIGURE 11.23 Use of KCL in nodal analysis.
Once each branch current is defined in terms of the node voltages, Kirchhoff’s current law is applied
at each node. The particular form of KCL employed in the nodal analysis equates the sum of the currents
into the node to the sum of the currents leaving the node:

∑ i in ∑ i out (11.24)
=

Figure 11.23 illustrates this procedure.
The systematic application of this method to a circuit with n nodes would lead to writing n linear
equations. However, one of the node voltages is the reference voltage and is therefore already known,
since it is usually assumed to be zero. Thus, we can write n – 1 independent linear equations in the n – 1
independent variables (the node voltages). Nodal analysis provides the minimum number of equations
required to solve the circuit, since any branch voltage or current may be determined from knowledge of
nodal voltages.
The nodal analysis method may also be defined as a sequence of steps, as outlined below.

Node Voltage Analysis Method
1. Select a reference node (usually ground). All other node voltages will be referenced to this node.
2. Define the remaining n – 1 node voltages as the independent variables.
3. Apply KCL at each of the n – 1 nodes, expressing each current in terms of the adjacent node
voltages.
4. Solve the linear system of n – 1 equations in n – 1 unknowns.

In a circuit containing n nodes we can write at most n – 1 independent equations.

The Mesh Current Method
In the mesh current method, we observe that a current flowing through a resistor in a specified direction
defines the polarity of the voltage across the resistor, as illustrated in Fig. 11.24, and that the sum of
the voltages around a closed circuit must equal zero, by KVL. Once a convention is established regarding
the direction of current flow around a mesh, simple application of KVL provides the desired equation.
Figure 11.25 illustrates this point.

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