3.4 Node Voltage Analysis
Node voltage analysis is a systematic method for analyzing circuits based on determining the voltages at all circuit nodes relative to a reference, or ground node, in the circuit. To motivate this analysis method, consider the circuit shown in Figure 3.26. The circuit has 6 elements, including a known voltage source, , and five resistors having values . (Pretend we had actual numbers in place of the variables.) Consider the effort involved in determining all the voltages (five unknowns) and all the currents (six unknowns) in this circuit. Or, consider the effort involved in simply obtaining the current through resistor . There are no series or parallel-connected resistors in this circuit, so circuit simplification based on equivalent resistance does not help to simplify the work. We could solve this circuit by repeated application of KCL, KVL, and Ohm’s Law to solve for all the circuit unknowns. The circuit has 11 unknowns, so we would need to develop a system of 11 simultaneous linear equations by applying KVL, KCL, and Ohm’s law to the various loops, nodes, and resistors in the circuit. Solving this system of equations would, in principle, complete the circuit analysis problem. We develop these equations below to illustrate, by way of this example circuit, the effort involved in such a “brute force” application of circuit analysis laws. Then, we introduce node voltage analysis, which often leads to results with significantly less effort.
This circuit is re-drawn in figure 2.27 with annotations for: the voltage across and current through each circuit element, which are the sought-after variables; the four circuit nodes labeled A, B, C, D; several loops, namely, ABDA, BCDB, ACBDA, and DACD (these are not all of the possible loops). Writing Ohm’s law for the five resistors, applying KCL at the four nodes, and applying KVL around several circuit loops gives us a set of simultaneous equations.
This set of equations includes five Ohm’s law equations:
(1)
(2)
(3)
(4)
(5)
in addition to four KCL equations, including KCL at node A,
(6)
KCL at node B,
(7)
KCL at node C,
(8)
and KCL at node D,
(9)
and five KVL equations, including KVL around loop ABDA,
(10)
KVL around loop BCDB,
(11)
KVL around loop ACBA,
(12)
KCL around loop ACDA,
(13)
and (finally) KCL around loop ACBD,
(14)
Equations (1) – (14) represent a set of 14 simultaneous linear equations for this system. The system has only 11 unknowns, and we could have, in fact, written additional equations by including additional circuit loops (such as ABCDA). Algebraic manipulation of eq (1) – (14) (which we will not do here!) however would reveal they these equations are not all linearly independent. Among the four KCL equations, only three are independent, and the fourth can be derived from a linear combination of any three. The number of independent KCL equations is actually equal to the number of nodes in the circuit minus one. This circuit has four nodes, therefore, there are