Norton's Theorem.

Fundamental Series FS05 - May, 2006

Nortons's Theorem

Any linear circuit can be arranged in the form of two networks, A and B connected by two resistance-less conductors. Define a current Isc as the short-circuit current which would appear at the terminals of A if B were short-circuited so that no voltage is provided by B.

Then all the voltages and currents in B will remain unchanged if the network of A is replaced by ideal current source Isc connected in parallel with a resistance Rp equivalent to to the resistance looking into network A with all independent voltage and current sources replaced by open and short circuits respectively.

Notes:

Biography and Backgound

Edward Lawry Norton (1898 - 1983)

Not much is known about Edward Norton outside of what he wrote about his theories and observations internally while at Bell Labs.

From his job application, we know that he was born in Rockland Maine on July 28, 1898. He attended the University of Maine for one year before wartime service in the Navy (1917-1919.) where he served as a radio operator After his return from the service, he transferred to M.I.T. in 1920, receiving his Bachelor of Science (electrical engineering) in 1922.

His first job was at Western Electric Corporation in New York City. This company became Bell Laboratories in 1925. While working at Western Electric, he earned a M.A. degree in electrical engineering from Colombia University in 1925.

According to AT&T Archives, he had 19 patents. He published three papers during his lifetime---none of which mention the equivalent circuit that’s associated with his name. He wrote 92 technical memoranda while at Bell Labs. It is in one of these reports that he first describes Norton theorem. It was not a published paper.

Norton's theorem, an extension of Thevenin’s theorem, was also introduced the same month in 1926, by another researcher: Hans Ferdinan Mayer. Mayer was the only one of the two to actually publish it, but Norton made known his findings through an internal technical report at Bell Labs. In Europe, Norton’s theorem is known as the “Mayer-Norton theorem”.

He retired in 1961 and died on January 28 1983 at the King James Nursing Home in Chatham, New Jersey.

Edward Lawry Norton (1898 - 1983)

Norton's Theorem

Last time we discussed Thevenin's Theorem. Norton's Theorem can be thought as a corollary of Thevenin's Theorem with voltage replaced by current, and series resistance replaced by parallel resistance. In this article, we will see in-fact how closely related they are.

Any linear circuit an be arranged in the form of two networks, A and B connected by two resistance-less conductors. The entire network A can be replaced by a single generator of:

• current Ig equal to the current which flows between the short-circuited terminals of A if B were disconnected so that no current is drawn from A,
  
• and parallel resistance of Rg, equal to the resistance seen looking back into the terminals of A with all voltage and current sources in A replaced by short circuits and open circuits respectively.

Looking at the circuit of figure 1 from last time, once again suppose we wanted to determine the output current Vo through different RL load values

We can write a series of mesh equations for the circuit, substitute different values of RL into the circuit and use mesh analysis to solve for each variation.

Using Kirchoff's Voltage Law, let's write the five mesh equations for
figure 1:

[1] 10(i1) + 12(i1) + 8(i1 - i2) - 15 = 0 (mesh1)

[2] 8(i2 - i1) + 15(i2 - i3) + 24 = 0 (mesh2)

[3] 15(i3 - i2) + 8 (i3 - i4) - 24 = 0 (mesh3)

[4] 8(i4 - i3) + 9(i4 - i5) + 12 = 0 (mesh4)

[5] 9(i5 - i4) + 5(i5) + 10(i5) - 12 = 0 (mesh5)

There are five equations and five unknowns, so we can solve for i5. Working the problem through and solving for i5, after rounding errors we get approximately

i5 = 0.45A.

V0 is thus:

10(i5) = 10(0.45) = 4.5V

Again, depending upon the complexity of the circuit and how many values we wanted to test, this could become very tedious fast. This time we will apply Norton's Theorem to simplify the circuit to make the analysis easier

As we did last time. first step is to break the circuit into parts A and B. The A and B division point is completely arbitrary. Since we are interested of the effect of varying only the load resistance RL, like last time we will only place RL into the B network, and everything else is placed into the A network as shown in figure 2.

The Norton equivalent for circuit A as shown in figure 3.

In figure 4, The short-circuit current Ig is calculated by looking into the circuit to the left of RL. Unlike last time, R7 is in the circuit and cannot be ignored.

Using Kirchoff's Voltage Law, let's write the five mesh equations for figure 4:

[1] 10(i1) + 12(i1) + 8(i1 - i2) - 15 = 0 (mesh 1)

[2] 8(i2 - i1) + 15(i2 - i3) + 24 = 0 (mesh 2)

[3] 15(i3 - i2) + 8 (i3 - i4) - 24 = 0 (mesh 3)

[4] 8(i4 - i3) + 9(i4) + 12 = 0 (mesh 4)

[5] 9(i5 - i4) + 5(i5) - 12 = 0 (mesh 5)

There are five equations, and four unknowns. Working the problem through and solving for i5 = which is the Ig current, after rounding errors we get approximately

Ig = 1.08

Luckily you only have to do this once!

Now to calculate the Norton parallel resistance, short all independent voltage sources and open circuit all independent current sources. The reduced circuit is shown is shown in figure 5. The resistance of this circuit looking into the terminals of the A network is easily calculated using Ohm's law as:

Rg = 7.1 Ohms.

The equivalent Norton circuit to our original figure 3 circuit is is shown in figure 6.

Now replace the circuit of figure.1 with the Norton Equivalent circuit of figure 6. We can now easily find Vo for any RL value without resorting to complex analysis.

For example, if RL = 10-Ohms, we can calculate the voltage Vo across our load resistor,

Vo = Ig * (Rg || Rl) = 1.08 * (7.1 || 10) = 4.48V

As expected.

The relationship between Thevenin and Norton equivalents.

This circuit looks kind-of familiar to the Thevenin equivalent we derived last time ( figure 7) doesn't it?

This shows an important relationship between - the Norton equivalent can be derived from the Thevenin equivalent or vice-versa by applying the corresponding source transformation. This is a powerful tool - the more easily derived equivalent can be transformed into the other equivalent by simple a simple source transformation.

Using example of the sample network in figure 3, to solve for the Norton current, it required five equations. The same network required only four equations for the Thevenin voltage.

Finally, depending upon the circuit - especially if there are a number of dependent voltage and dependent current sources which we cannot simply replace with short- circuits and open-circuits to find Rg, it is often more convenient to find the Thevenin equivalent voltage Vg and the Norton equivalent current Ig, then calculate Rg using Ohm's Law:

Rg="Vg/Ig"

In conclusion

Norton's Theorem states that any linear combination of voltage and current sources and resistances can be replaced by a single ideal current source Ig and a single parallel resistor Rg.

Norton equivalent circuits are transformations of Thevenin equivalent circuit - if you know either equivalent, the other is easily derived.