CIRCUIT THEOREMS

CHAPTER FOUR
CIRCUIT THEOREMS
4.1 INTRODUCTION
The growth in areas of application of electric circuits has led to an evolution from simple to complex circuits. To handle the complexity, engineers over the years have developed some theorems to simplify circuit analysis. Such theorems include Thevenin’s and Norton’s theorems. Since these theorems are applicable to linear circuits, we first discuss the concept of circuit linearity. In addition to circuit theorems, we discuss the concepts of superposition, maximum power transfer, Millman’s theorem, Substitution theorem, and Reciprocity theorem in this chapter.
4.2 LINEARITY PROPERTY
Linearity is the property of an element describing a linear relationship between cause and effect. The property is a combination of both the homogeneity property and the additivity property.
The homogeneity property requires that if the input (also called the excitation) is multiplied by a constant, then the output (also called the response) is multiplied by the same constant. For a resistor, for example, Ohm’s law relates the input i to the output v,
v = iR (4.1)
If the current is increased by a constant k, then the voltage increases correspondingly by k, that is,
kiR = kv (4.2)
The additivity property requires that the response to a sum of inputs is the sum of the responses to each input applied separately. Using the relationship of a resistor, if
v1 = i1R and v2 = i2R (4.3)
then applying (i1 + i2) gives
v = (i1 + i2) R = i1R + i2R = v1 + v2 (4.4)
A linear circuit is one whose output is linearly related (or directly proportional) to its input.
4.3 SUPERPOSITION
The idea of superposition rests on the linearity property.
The superposition principle states that the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone.
However, to apply the superposition principle, we must keep two things in mind:
1. We consider one independent source at a time while all other independent sources are turned off. This implies that we replace every voltage source by 0 V (or a short circuit), and every current source by 0 A (or an open circuit).
2. Dependent sources are left intact because they are controlled by circuit variables. With these in mind, we apply the superposition principle in three steps:

Analyzing a circuit using superposition has one major disadvantage: it may very likely involve more work. Keep in mind that superposition is based on linearity.
4.4 SOURCE TRANSFORMATION
We have noticed that series-parallel combination and wye-delta transformation help simplify circuits. Source transformation is another tool for simplifying circuits. We can substitute a voltage source in series with a resistor for a current source in parallel with a resistor, or vice versa, as shown in Fig. 4.4. Either substitution is known as a source transformation.

Figure 4.4 Transformation of independent sources.


We need to find the relationship between vs and is that guarantees the two configurations in Fig. 4.4 are equivalent with respect to nodes a, b.
Suppose RL, is connected between nodes a, b in Fig. 4.4(a). Using Ohms law, the current in RL is.
i_L=v_s/((R+R_L ) ) R and RL in series (4.5)
If it is to be replaced by a current source then load current must be V/((R+R_L ) )
Now suppose the same resistor RL, is connected between nodes a, b in Fig. 4.4 (b). Using current division, the current in RL, is
i_L=i_s R/((R+R_L ) ) (4.6)
If the two circuits in Fig. 4.4 are equivalent, these resistor currents must be the same. Equating the right-hand sides of Eqs.4.5 and 4.6 and simplifying
i_s=v_s/R or v_s= i_s R (4.7)
Source transformation also applies to dependent sources, provided we carefully handle the dependent variable. As shown in Fig. 4.5, a dependent voltage source in series with a resistor can be transformed to a dependent current source in parallel with the resistor or vice versa.

Figure 4.5 Transformation of dependent sources.
However, we should keep the following points in mind when dealing with source transformation.
1. Note from Fig. 4.4 (or Fig. 4.5) that the arrow of the current source is directed toward the positive terminal of the voltage source.
2. Note from Eq. (4.7) that source transformation is not possible when R = 0, which is the case with an ideal voltage source. However, for a practical, nonideal voltage source, R ? 0. Similarly, an ideal current source with R =?cannot be replaced by a finite voltage source.