Introductory Electrical Engineering With Math Explained in Accessible Language

24Millman’s Theorem: Circuit Analysis

24.1 Introduction

In this chapter, we examine Millman’s theorem, another very useful set of techniques for circuit analysis, specifically for circuits containing multiple voltage sources.

24.2 Millman’s Theorem

Millman’s theorem states that any circuit containing multiple voltage sources, each one in series with its own resistance, can be replaced by one voltage source (V_EQ) in series with a resistance (R_EQ) (see Figure 24.1).

Figure 24.1 Millman’s theorem.

24.2.1 The Theory

According to Millman’s theorem, each of the voltage sources and their respective resistors produce a current, and the sum of all currents is equal to the total current produced by the circuit, as shown in Figure 24.2.

Figure 24.2 Sum of individual currents.

Mathematically, this can be expressed as

or generically as the following postulate.

MILLMAN’S POSTULATE

• i represents the circuit’s total current, in Amperes.
• i_x represents each of the parallel currents, in Amperes.

The individual element being considered by the theorem is a voltage source in ...