Circuit Theorems: Source Transformation

 

Source Transformation

We have noticed that series-parallel combination and wye-delta transformation help simplify circuits. Source transformation is another tool for simplifying circuits. Basic to these tools is the concept of equivalence. We recall that an equivalent circuit is one whose v-i characteristics are identical with the original circuit. In Section 3.6, we saw that node-voltage (or mesh-current) equations can be obtained by mere inspection of a circuit when the sources are all independent current (or all independent voltage) sources. It is therefore expedient in circuit analysis to be able to 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. 4.15. Either substitution is known as a source transformation.

Figure 4.15 Transformation of independent sources.

A source transformation is the process of replacing a voltage source vs in series with a resistor R by a current source is in parallel with a resistor R, or vice versa

The two circuits in Fig. 4.15 are equivalent provided the y have the same voltage-current relation at terminals a-b. It is easy to show that they are indeed equivalent. If the sources are turned off, the equivalent resistance at terminals a-b in both circuits is R. Also, when terminals a-b are short-circuited, the short-circuit current flowing from a to b is i_sc = v_s ∕ R in the circuit on the left-hand side and i_sc = i_s for the circuit on the right-hand side. Thus, v_s ∕ R = i_s in order for the two circuits to be equivalent. Hence, source transformation requires that

Source transformation also applies to dependent sources, pro vided we carefully handle the dependent variable. As shown in Fig. 4.16, 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 where we make sure that Eq. (4.5) is satisfied.

Figure 4.16 Transformation of dependent sources.

Like the wye-delta transformation we studied , a source transformation does not affect the remaining part of the circuit. When applicable, source transformation is a powerful tool that allows circuit manipulations to ease circuit analysis. However, we should keep the following points in mind when dealing with source transformation.

1. Note from Fig. 4.15 (or Fig. 4.16) that the arrow of the current source is directed toward the positive terminal of the voltage source.
2. Note from Eq. (4.5) that source transformation is not possible when R = 0, which is the case with an ideal voltage source. However, for a practical, non-ideal voltage source, R ≠ 0. Similarly, an ideal current source with R = ∞ cannot be replaced by a finite voltage source

Example:

Use source transformation to find v_o in the circuit : 

Solution:

We first transform the current and voltage sources to obtain the circuit in Figure:

Combining the 4-Ω and 2-Ω resistors in series and transforming the 12-V voltage source gives us Figure:

We now combine the 3-Ω and 6-Ω resistors in parallel to get 2-Ω. We also combine the 2-A and 4-A current sources to get a 2-A source. Thus, by repeatedly applying source transformations, we obtain the circuit in Figure:

We use current division in Figure to get:

Alternatively, since the 8-Ω and 2-Ω resistors are in parallel, they have the same voltage v_o across them. Hence,