Norton's Theorem
Norton's Theorem states that it is possible
to simplify any linear circuit, no matter how complex, to an
equivalent circuit with just a single current source and
parallel resistance connected to a load. Just as with
Thevenin's Theorem, the qualification of "linear" is
identical to that found in the Superposition Theorem: all
underlying equations must be linear (no exponents or roots).
Contrasting our original example circuit
against the Norton equivalent: it looks something like this:
. . . after Norton conversion . . .
Remember that a current source is a
component whose job is to provide a constant amount of
current, outputting as much or as little voltage necessary
to maintain that constant current.
As with Thevenin's Theorem, everything in
the original circuit except the load resistance has been
reduced to an equivalent circuit that is simpler to analyze.
Also similar to Thevenin's Theorem are the steps used in
Norton's Theorem to calculate the Norton source current (INorton)
and Norton resistance (RNorton).
As before, the first step is to identify the
load resistance and remove it from the original circuit:
Then, to find the Norton current (for the
current source in the Norton equivalent circuit), place a
direct wire (short) connection between the load points and
determine the resultant current. Note that this step is
exactly opposite the respective step in Thevenin's Theorem,
where we replaced the load resistor with a break (open
circuit):
With zero voltage dropped between the load
resistor connection points, the current through R1
is strictly a function of B1's voltage and R1's
resistance: 7 amps (I=E/R). Likewise, the current through R3
is now strictly a function of B2's voltage and R3's
resistance: 7 amps (I=E/R). The total current through the
short between the load connection points is the sum of these
two currents: 7 amps + 7 amps = 14 amps. This figure of 14
amps becomes the Norton source current (INorton)
in our equivalent circuit:
Remember, the arrow notation for a current
source points in the direction opposite that of
electron flow. Again, apologies for the confusion. For
better or for worse, this is standard electronic symbol
notation. Blame Mr. Franklin again!
To calculate the Norton resistance (RNorton),
we do the exact same thing as we did for calculating
Thevenin resistance (RThevenin): take the
original circuit (with the load resistor still removed),
remove the power sources (in the same style as we did with
the Superposition Theorem: voltage sources replaced with
wires and current sources replaced with breaks), and figure
total resistance from one load connection point to the
other:
Now our Norton equivalent circuit looks like
this:
If we re-connect our original load
resistance of 2 Ω, we can analyze the Norton circuit as a
simple parallel arrangement:
As with the Thevenin equivalent circuit, the
only useful information from this analysis is the voltage
and current values for R2; the rest of the
information is irrelevant to the original circuit. However,
the same advantages seen with Thevenin's Theorem apply to
Norton's as well: if we wish to analyze load resistor
voltage and current over several different values of load
resistance, we can use the Norton equivalent circuit again
and again, applying nothing more complex than simple
parallel circuit analysis to determine what's happening with
each trial load.
-
REVIEW:
-
Norton's Theorem is a way to reduce a
network to an equivalent circuit composed of a single
current source, parallel resistance, and parallel load.
-
Steps to follow for Norton's Theorem:
-
(1) Find the Norton source current by
removing the load resistor from the original circuit and
calculating current through a short (wire) jumping across
the open connection points where the load resistor used to
be.
-
(2) Find the Norton resistance by removing
all power sources in the original circuit (voltage sources
shorted and current sources open) and calculating total
resistance between the open connection points.
-
(3) Draw the Norton equivalent circuit,
with the Norton current source in parallel with the Norton
resistance. The load resistor re-attaches between the two
open points of the equivalent circuit.
-
(4) Analyze voltage and current for the
load resistor following the rules for parallel circuits.
|