Maximum Power Transfer Theorem

[JJBladester]

Homework Statement

"If the load voltage at the output of a circuit keeps going up as the load resistance is increased, why doesn’t the load power keep going up as well?"

Homework Equations

The maximum power transfer theorem states:
A load will receive maximum power from a network when its resistance is exactly equal to the Thevenin resistance of the network applied to the load. That is,

R_L=R_th

R_L = Load resistance

P_L = Power dissipated by load

R_th = Thevenin resistance

$$I_L=\frac{E_{th}}{R_{th}+R_L}=\frac{E_{th}}{R_{th}+R_{th}}=\frac{E_{th}}{2R_{th}}$$

$$P_{L_{max}}=I_{L}^{2}R_L=\left (\frac{E_{th}}{2R_{th}} \right )^{2}\left (R_{th} \right )=\frac{E_{th}^{2}R_{th}}{4R_{th}^{2}}=\frac{E_{th}^{2}}{4R_{th}}$$

The Attempt at a Solution

Here's how I answered this question.

The power *will* keep going up until R_L = R_th. The graph of P_L versus R_L reaches a maximum (d/dR_L = 0) at that point. Thereafter, P_L will gradually decline as R approaches infinity.

R approaching infinity is analogous to an open circuit in which no current can flow. With no current flow, there can be no power.

That's how I answered the question, but I'm not fully satisfied with my answer. Maybe I'm actually not satisfied with the wording of the original question... A Thevenin voltage source wouldn't change; it would be fixed, right? Thus, this question is sort of mis-stated in my opinion.

Doesn't the maximum power transfer theorem assumes a fixed source voltage?

 

[berkeman]

I think your answer is fine. When the load resistance increases past the source resistance, less current flows throught the load, reducing the power dissipated.

BTW, have you learned yet that for circuits with reactive components (like inductors and capacitors), the maximum power transfer occurs when the load impedance is the complex conjugate of the source impedance? When the circuit only has resistors, there is no imaginary component to the impedances, so maximum power is transferred when the resistances are equal.

 

[JJBladester]

I think your answer is fine.

Thank you. I'm still curious whether the maximum power transfer theorem is applicable if the source voltage is allowed to increase. My understanding of a Thevenin circuit was that E_th was *fixed*.

BTW, have you learned yet that for circuits with reactive components (like inductors and capacitors), the maximum power transfer occurs when the load impedance is the complex conjugate of the source impedance?

Berkeman, I haven't learned this yet. We just got into capacitors. I'm excited to learn about the M.P.T.T. for cases other than simply resistors!

 

[berkeman]

Thank you. I'm still curious whether the maximum power transfer theorem is applicable if the source voltage is allowed to increase. My understanding of a Thevenin circuit was that E_th was *fixed*.

The source is usually a fixed value for these calculations. I'm not sure what it means for the source voltage to be "allowed" to increase. Sources are usually fixed, unless they have some strange behavior. They can be current limited, for example, as many real power supplies are. But that will result in a decrease in output voltage if the current limit is reached...

 

[gneill]

The initial statement of the problem said that the load voltage rises as the load resistance rises, which is perfectly true for even a fixed value of E_th; in the limit as R_L approaches infinite resistance the load voltage approaches E_th.