By definition, maximum power is transferred to a load when the resistance or complex conjugate of the load matches the source. The term resistance is generally reserved for DC loads, while impedance is used for AC to accommodate reactive components such as inductors and capacitors.
Keep in mind that we are talking about theorems. The Maximum Power Transfer Theorem is as solid and dependable as the Pythagorean Theorem. Both are mathematically derived based on a set of assumptions. The Pythagorean Theorem is limited to geometry within a flat plane – it fails in a curved surface. Likewise, the Maximum Power Transfer Theorem depends on the source resistance being fixed – one that does not change with load current.
You already knew this, so where’s the myth?
The myth is in the application. For example, the antique generator shown in Figure 1 has winding resistance as well as inductance. For the sake of discussion, let’s assume the generator winding is 1 Ohm resistive in series with 1 Ohm reactive (inductive) resulting in a complex impedance of (1 + j1) Ω.
What is the most appropriate load impedance for maximum efficiency?
Did you say (1 - j1) Ω?
Figure 1: Image of an antique generator.
Tech Tip: If you are new to electronics, you may not yet be familiar with the complex impedance such as (1 + j1) Ω. For now, just ignore the j (imaginary) component. For simplicity, you can assume the circuit is DC and has a 1 Ω resistance.
The Myth is in the Application
If you said yes, you fell into the trap—arguing about the tires on a spaceship.
Yes, you read that correctly, talking about the Maximum Power Transfer Theorem and efficiency of a generator has the same relationship as bias-ply tires and starships. They could be “related”, but they don’t naturally fall into the same conversation. The myth is an assumption that efficiency is somehow related to the Maximum Power Transfer Theorem.
Nothing could be further from the truth.
Understand that the Maximum Power Transfer Theorem assumes that half of the power will be burned (dissipated) in the generator and half of the power will make it to the load to do useful work. This atrocious 50% system efficiency will melt the generator’s windings! Instead, we want a high system efficiency which implies that the generator’s winding resistance should be insignificant relative to the load resistance. Technically, we aim to minimize the I^2R losses in the generator.
Tech Tip: This conversation involves system efficiency which implies minimizing power loss inside the generator. In an ideal setting, all mechanical shaft horsepower is converted to electrical energy. This would imply a generator efficiency (Greek letter η) of unity (100%). Real-world generators have an efficiency in the 95% range.
On a related note, don’t fall for over-unity snake oil. These fictional perpetual motion machines break the laws of thermodynamics.
Why do textbook authors like the Maximum Power Transfer Theorem?
Have you ever noticed that textbook authors like to overlay AC topics onto the DC topics. It’s like an upward spiral of learning. We complete a loop of DC topics only to end up where we started but on a higher level. As we complete the AC loop, we can look down and see the familiar DC topics below us. This is a layered approach, where we revisit and reinforce each DC topic as we explore the AC topic.
Personally, I feel the Maximum Power Transfer Theorem is somehow out of place in the DC world, yet perfectly placed to enhance the AC learning sequence. From an AC perspective it serves to reinforce the complex (imaginary) mathematic associated with capacitors and inductors with a touch of series resonance.
If you agree with my higher level of learning analogy, you may also agree with the implication that the Maximum Power Transfer Theorem is sometimes artificially placed into the DC learning sequence.
Also, you may have noticed that efficiency is an early topic in the DC sequence. IMO, it often gets lost in the fog of learning. Students don’t truly appreciate the concepts until they encounter amplifiers (A, AB, B, C) or electromechanical systems such as our featured generator. This leads to a misunderstanding of the theorem and the myth featured in this article.
When do we apply the Maximum Power Transfer Theorem?
Clearly, the Maximum Power Transfer Theorem has no business being applied to power systems such as generators and batteries. For these applications, we want an internal resistance that is very low relative to the load resistance. This keeps the system efficiency high – much higher than the paltry 50% associated with the Maximum Power Transfer Theorem.
Direct applications of the Maximum Power Transfer Theorem are rare and somewhat specialized. Communication systems immediately come to mind.
Perhaps the best examples involve low power systems. As an example, consider a radio antenna. Each antenna and associated transmission line is built to transfer the miniscule, received radio signal into a specific impedance. The 50 Ω termination is one of the most common. In this application, we optimize the system for maximum power transfer from the antenna into the receiver’s preamplifier section.
As a counterexample, consider the loudspeakers in a HiFi audio system. Occasionally, people will point to stereo system and state that the amplifier must be matched to a specific speaker impedance e.g., 8 Ω. While this is true from a power delivery perspective, it is emphatically not an application of the Maximum Power Transfer Theorem. In practice we see that the amplifier’s impedance is orders of magnitude less than the loudspeaker. To do otherwise is to have a “squishy sound” with a low (uncontrolled) damping factor where the speaker dynamics color the sound.
Parting Thoughts
The Maximum Power Transfer Theorem is an excellent learning tool, especially for AC analysis. It reinforces the complex (imaginary numbers) math and reinforces the properties of capacitors and inductors.
Unfortunately, the concept gets twisted in our minds and then misapplied to power systems. In the real world, efficiency dominates; period – full stop! Therefore, the Maximum Power Transfer Theorem is relegated to low power system. This allows us to extract the maximum power from these systems.
Best wishes,
APDahlen
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About This Author
Aaron Dahlen, LCDR USCG (Ret.), serves as an application engineer at DigiKey. He has a unique electronics and automation foundation built over a 27-year military career as a technician and engineer which was further enhanced by 12 years of teaching (interwoven). With an MSEE degree from Minnesota State University, Mankato, Dahlen has taught in an ABET-accredited EE program, served as the program coordinator for an EET program, and taught component-level repair to military electronics technicians. Dahlen has returned to his Northern Minnesota home and thoroughly enjoys researching and writing articles such as this.