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Norton Equivalent Circuits

Module by: Don Johnson. E-mail the author

Summary: Introduction of Norton equivalent circuits.

As you might expect, equivalent circuits come in two forms: the voltage-source oriented Thévenin equivalent and the current-source oriented Norton equivalent (see figure).

Figure 1: All circuits containing sources and resistors can be described by simpler equivalent circuits. Choosing the one to use depends on the application, not on what is actually inside the circuit.
(equivalents.png)

To derive the latter, the v-i relation for the Thévenin equivalent can be written as

v= R eq ⁢i+ v eq v R eq i v eq
(1)
or
i=v R eq − i eq i v R eq i eq
(2)
where i eq = v eq R eq i eq v eq R eq is the Norton equivalent source. The Norton equivalent shown in the above figure be easily shown to have this v-i relation. Note that both variations have the same equivalent resistance. The short-circuit current equals the negative of the Norton equivalent source.

Exercise 1

Find the Norton equivalent circuit for the circuit below.

Figure 2
(circuit13.png)

Solution

i eq = R 1 R 1 + R 2 ⁢ i in i eq R 1 R 1 R 2 i in and R eq = R 3 ∥ R 1 + R 2 R eq ∥ R 3 R 1 R 2 .

Equivalent circuits can be used in two basic ways. The first is to simplify the analysis of a complicated circuit by realizing the any portion of a circuit can be described by either a Thévenin or Norton equivalent. Which one is used depends on whether what is attached to the terminals is a series configuration (making the Thévenin equivalent the best) or a parallel one (making Norton the best).

Another application is modeling. When we buy a flashlight battery, either equivalent circuit can accurately describe it. These models help us understand the limitations of a battery. Since batteries are labeled with a voltage specification, they should serve as voltage sources and the Thévenin equivalent serves as the natural choice. If a load resistance R L R L is placed across its terminals, the voltage output can be found using voltage divider: v= v eq ⁢ R L R L + R eq v v eq R L R L R eq . If we have a load resistance much larger than the battery's equivalent resistance, then, to a good approximation, the battery does serve as a voltage source. If the load resistance is much smaller, we certainly don't have a voltage source (the output voltage depends directly on the load resistance). Consider now the Norton equivalent; the current through the load resistance is given by current divider, and equals i=− i eq ⁢ R eq R L + R eq i i eq R eq R L R eq . For a current that does not vary with the load resistance, this resistance should be much smaller than the equivalent resistance. If the load resistance is comparable to the equivalent resistance, the battery serves neither as a voltage source or a current course. Thus, when you buy a battery, you get a voltage source if its equivalent resistance is much smaller than the equivalent resistance of the circuit you attach it to. On the other hand, if you attach it to a circuit having a small equivalent resistance, you bought a current source.

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