On Thevenin’s theorem

Seeing another long running discussion on QRZ about transmitters and antennas, and maximum power transfer and Thevenin equivalent circuits prompts some discussion.

Thevenin published his theorem in 1883, the following is an English translation:

“Theorem: Assuming any system of linear interconnected conductors, and containing some electromotive forces E1, E2,… En distributed in any way, one considers two points A and A’ belonging to the system and actually having the potentials V and V’. If the points A and A’ are connected with wire ABA’ having resistance r, and not having an electromotive force, the potentials at the points A and A’ take on different values of V and V’, but the current flowing in the wire is given by the formula i=(V-V’)/(r+R) in which R represents the resistance of the primitive system, measured between the points A and A’ considered as electrodes.”

(The Principles of Superposition on which this partly depends were published some 30 years earlier, Kirchoff’s Laws prior a little earlier again, and Ohms law a little earlier.)

The meaning of linear here is that the relationship of V and I for the ‘resistances’ is a linear relationship, ie V=IR (Ohms law).

In time, classic theorems were adapted for AC and the notion of reactance and impedance, and importantly application to networks that might not be linear in an absolute sense, but are sufficiently linear over a range of V and I of interest to apply the formula i=(V-V’)/(r+R) with good accuracy and utility.

In ham speak, a linear amplifier’s is not the same thing as a linear circuit (as used above), its V,I characteristic is not necessarily a straight line in an absolute sense, or even over a usefully small part of its range.

Wikipedia gives a definition which is quite common:

“As originally stated in terms of DC resistive circuits only, Thévenin’s theorem holds that:

  • Any linear electrical network with voltage and current sources and resistances only can be replaced at terminals A-B by an equivalent voltage source Vth in series connection with an equivalent resistance Rth.
  • The equivalent voltage Vth is the voltage obtained at terminals A-B of the network with terminals A-B open circuited.
  • The equivalent resistance Rth is the resistance that the circuit between terminals A and B would have if all ideal voltage sources in the circuit were replaced by a short circuit and all ideal current sources were replaced by an open circuit.
  • If terminals A and B are connected to one another, the current flowing from A to B will be Vth/Rth. This means that Rth could alternatively be calculated as Vth divided by the short-circuit current between A and B when they are connected together.”

Well, it is not as originally stated, it is an interpretation and restatement, but it is broadly compatible, but potentially impractical in that it is specific about how to find Vth and Rth and it may be unsafe or impractical to operate the source with open circuit or short circuit load. It also does not permit application over a sub range of V,I where a source may be sufficiently linear to usefully apply the model over that range.

If your learning is from Wikipedia, you have seriously restricted your knowledge.

A simple case

Lets consider a source that has an ideal 10V battery in series with a 10Ω resistor and another 10Ω resistor. All these components are internal to the source, the output terminals AB are connected to one of the 10Ω resistors.

It is a trivial exercise to calculate the Thevenin equivalent of the source, Vth=5V and Rth=5Ω.

There are an infinite number of linear circuits that have the same Thevenin equivalent circuit as this. You cannot derive the original network from its Thevenin equivalent circuit, and if in any way you infer the original network from its Thevenin equivalent circuit, you demonstrate a lack of understanding of the concepts.

For this source, it can be shown that maximum power is developed in the load when Rload=Rth=5Ω. We can calculate the power in the load as E^2/R=(5/2)^2/5=5/4=1.25W.

Now in the ham world, it is commonly held that this condition (maximum power transfer) implies 50% system efficiency, but in this case the battery supplies 0.75A, it supplies 10*0.75=7.5W, and since power in the load is 1.25W, efficiency is 1.25/7.5=16.7%. Again, this fails because you cannot derive the original network from its Thevenin equivalent circuit, and if in any way you infer the original network from its Thevenin equivalent circuit, you demonstrate a lack of understanding of the concepts.

A follow up article will look at a network that is not linear for all V,I, but is sufficiently linear over a sub range to be usefully modelled using Thevenin’s equivalent circuit.

Conclusions

  • Thevenin’s theorem dates from 1883, and is founded on Ohms Law (1827) and the Principle of Superposition (1853), the stuff is well know and quite mature… albeit apparently less well known in the realm of ham radio.
  • A Thevenin equivalent circuit is only valid over the range where Vout=Vth-Iout*Rth, and that range may be less than the full V/I range of the source.
  • A Thevenin equivalent circuit of a source does not imply the internal implementation of the source.
  • Calculation of system efficiency based on a Thevenin equivalent circuit is invalid as it implies / assumes the internal implementation of the source.
  • Calculation of  Thevenin equivalent circuit Rth based on perceived efficiency is invalid as it implies / assumes the internal implementation of the source.
  • Assertions to the contrary, or arguments based on contrary assertions are simply wrongly based and are testament to the maker’s lack of understanding, and when stated as fact, question their credibility.