Thevenin and Norton Equivalent Circuits

February 6, 2020 Cadence PCB Solutions

Emperor Augustus statue in front of deteriorating stone wall

 

The Latin phrase “primus inter pares” roughly translates to “first among equals.” While someone standing first among equals seems contradictory, the Emperor Augustus used the phrase to separate himself from the dictatorial ambitions of Julius Caesar—who wanted everyone to recognize him as the First Man in Rome or Princeps. After all, the senator’s disapproval of Julius Caeser and other influencers in Rome led to his murder.

In a more modern context, the concept of “primus inter pares” becomes more recognizable in the duties of officials such as prime ministers in many countries who have higher status and more power but still serve as an equal to their peers.

Thevenin and Norton Knew “Primus Inter Parest”

If we think about Kirchoff’s Laws, Ohm’s Law, and the Superposition theorem, we draw closer and closer to solving linear circuits. Those foundational theories and laws tell us that we can designate all the sources in a circuit and then use linear equations to find the values of any voltage or current in the circuit.

Certainly, Georg Simon Ohm, Gustav Robert Kirchoff, Hermann von Helmholtz, and Emil du Bois-Reymond have their places in the electronic theory hall-of-fame. Our exploration of linear circuit theory drives us towards equivalent circuits and two additional giants of electronic theory. Leon Charles Thevenin tells us that:

Any two-terminal network containing voltage or current sources can be replaced by an equivalent circuit consisting of a voltage equal to the open-circuit voltage of the original circuit in series with the resistance measured back into the original circuit.

Let’s think about this for a moment. Equivalent circuits show that a simple, functional form of a complex circuit may exist. Two circuits that differ physically have the same electrical characteristics.

While Thevenin’s theory focuses on voltage sources, Edward Norton took a slightly different path by focusing on current sources. As a result, the two theories complement one another. Norton tells us that:

Any two-terminal network containing voltage or current sources can be replaced by an equivalent circuit consisting of a current source equal to the short -circuit current from the original network in parallel with the resistance measured back into the original circuit.

We can see how the two theories complement one another with the simple circuit diagrams shown in figure one.

Figure One - Finding Equivalent Circuits

 Pair of Thevenin and Norton’s theorem circuits

Thevenin’s Theorem                    Norton’s Theorem

 

When looking at Thevenin’s Theorem, the output voltage measured under open circuit conditions or EOC is the Thevenin voltage. The circuit shown in figure 1B represents the electrical equivalent of the two-terminal network shown in figure 1A. Thevenin’s Theorem treats one branch of the network as a load while the remainder of the network functions as a two-terminal network that supplies the load. We calculate the Thevenin resistance or RTH by looking back into the network. Removing the sources allows the internal resistances to remain.

Now, let’s compare the network and equivalent circuit for Norton’s Theorem. Remember that we are considering a current source rather than a voltage source. The circuit shown in figure 1D is the electrical equivalent of the two-terminal network shown in figure 1C. With this arrangement, the Norton current or IN equals the current under short-circuit conditions or ISC. Here’s an interesting twist. The parallel resistance obtained as the Norton resistance or RN equals the Thevenin resistance or RTH. Applying Norton’s Theorem follows the same approach used for applying Thevenin’s Theorem with one exception. Removing the load from the two terminals of the network causes the terminals to electrically short and produce ISC.

Whether we work with a Thevenin or Norton equivalent network, we always determine the Thevenin resistance since that resistance equals the Norton resistance. If we decide to use a Thevenin equivalent, we can solve for the Thevenin voltage. Or...we can first determine the Norton current and then determine the Thevenin voltage.

Bilateral Agreements Work in Thevenin and Norton Equivalent Circuits

We can only apply Thevenin’s Theorem and Norton’s Theorem to linear bilateral networks. A linear network contains capacitances, resistances, and inductances that remain constant even if the voltage changes. The relationship between voltage and current is a straight line. Bilateral networks retain the same characteristics regardless of the direction of the current through individual elements of the network. When we combine the two types of network into a linear bilateral network, we have components that have a linear relationship while the magnitude of current remains independent of the polarity of the voltage.

To this point, we have only worked with dc circuits when applying the two theorems. However, both Thevenin’s Theorem and Norton’s Theorem allow us to also simplify AC networks. Rather than looking back for the Thevenin resistance, we look for the internal impedance or Thevenin impedance. 

Any two-terminal network containing voltage or current sources can be replaced by an equivalent circuit consisting of a voltage equal to the open-circuit voltage of the original circuit in series with the impedance measured back into the original circuit.

AC transformer working within an electronic system

Ensuring a proper AC network Thevenin equivalent circuit will add security to your design.

 

Any two-terminal network containing voltage and/or current sources can be replaced by an equivalent circuit consisting of a current source equal to the short-circuit current from the original network in parallel with the impedance measured back into the original circuit.

When we attempt to analyze circuits, we look for the quickest method for solving the circuit. Thevenin’s Theorem allows us to analyze power circuits that have a changing load. Applying Thevenin’s Theorem to a power circuit calculates the voltage and current flowing through the load. Because of the complementary nature of Norton’s Theorem, we can calculate the load voltage, load current, and load power by reducing the circuit to its equivalent value. All this allows us to select a load resistance that transfers the maximum power to the load.

With the suite of design and analysis tools from Cadence, you’ll be sure to be able to work through any Thevenin or Norton circuits with ease. PSpice simulator, too, will be able to accurately model the impact your component and circuit choices make on the overall integrity of your design, and help you prepare your circuit for the projected functionality you desire. 

If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts

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