Source Modeling

4.10.1 Source Modeling

Source modeling provides an example of the usefulness of the Thevenin or the Norton equivalent. An active source such as a battery

is often characterized by its Thevenin or Norton equivalent circuit. An ideal voltage source provides a constant voltage irrespective of the cur- rent drawn by the load, while an ideal current source supplies a con- stant current regardless of the load voltage. As Fig. 4.58 shows,

(b)

practical voltage and current sources are not ideal, due to their inter- Figure 4.58

nal resistances or source resistances R s and R p . They become ideal as (a) Practical voltage source, (b) practical R s S 0 and R p S ⬁ . To show that this is the case, consider the effect current source.

Chapter 4

Circuit Theorems

of the load on voltage sources, as shown in Fig. 4.59(a). By the volt- age division principle, the load voltage is

R s ⫹R s L

v (4.25)

As R L increases, the load voltage approaches a source voltage v s , as illustrated in Fig. 4.59(b). From Eq. (4.25), we should note that:

1. The load voltage will be constant if the internal resistance R s of the source is zero or, at least, R s VR L . In other words, the smaller R s is compared with R L , the closer the voltage source is to being ideal.

Ideal source

Practical source

(a) Practical voltage source connected to a load R L , (b) load volt- age decreases as R L decreases.

2. When the load is disconnected (i.e., the source is open-circuited so that R L S ⬁ ), v oc ⫽v s . Thus, v s may be regarded as the unloaded source voltage. The connection of the load causes the terminal volt- age to drop in magnitude; this is known as the loading effect.

The same argument can be made for a practical current source when connected to a load as shown in Fig. 4.60(a). By the current division principle,

i s (4.26)

R p ⫹R L

(a)

Figure 4.60(b) shows the variation in the load current as the load resist- ance increases. Again, we notice a drop in current due to the load (load-

ing effect), and load current is constant (ideal current source) when the

Ideal source i

p WR L ). Sometimes, we need to know the unloaded source voltage v s and the internal resistance R s of a voltage source. To find v s and R s , we fol-

internal resistance is very large (i.e., R p S ⬁ or, at least, R

Practical source

low the procedure illustrated in Fig. 4.61. First, we measure the open- circuit voltage v oc as in Fig. 4.61(a) and set

Then, we connect a variable load R L across the terminals as in (a) Practical current source connected to a

Fig. 4.61(b). We adjust the resistance R L until we measure a load volt- load R L , (b) load current decreases as R L

age of exactly one-half of the open-circuit voltage, v L ⫽v oc 兾2, increases.

because now R L ⫽R Th ⫽R s . At that point, we disconnect R L and measure it. We set

R s ⫽R L

(a) Measuring v oc , (b) measuring v L .

The terminal voltage of a voltage source is 12 V when connected to a

Example 4.16

2-W load. When the load is disconnected, the terminal voltage rises to

12.4 V. (a) Calculate the source voltage v s and internal resistance . R s (b) Determine the voltage when an 8-⍀ load is connected to the source.