What is the difference between the formule Pmax = (|vth|^2)/(4*Rth) and Pmax = (|vth|^2)/(8*Rth) in the maximum power transfer theorem?

 What is the difference between the formule Pmax = (|vth|^2)/(4*Rth) and Pmax = (|vth|^2)/(8*Rth) in the maximum power transfer theorem?

Jose Soares Augusto

, developed simulators of dynamic systems; atomic collisions

Updated January 24, 2018

Unless your data is flawed, probably you have seen a formula for the case when Vth$V_{th}$ is a DC voltage, and the other formula is when Vth$V_{th}$ is the amplitude of an AC voltage.

A) - For the DC voltage case, the Thévenin equivalent is VthDC$V_{thDC}$ in series with a resistor R. You extract maximum power (see Maximum power transfer theorem - Wikipedia) when you load the Thévenin equivalent with RL=R$R_L=R$. In this case, the voltage in the load resistor is

VL=VthDC/2$V_L=V_{thDC}/2$

and the power in the load P=V2L/RL$P=V_L^2/R_L$ is given by

max power at load DC: Pmax=V2thDC/(4RL)$P_{max}=V_{thDC}^2/(4R_L)$

B) - For the AC case, with a sinusoidal voltage of amplitude VthAC$V_{thAC}$ you can apply the same formulas of the DC case, if you use RMS (“root-mean-square”) or effective values; in a sinusoidal, the RMS value is

VRMSAC=VthAC/2–√$V_{RMSAC}=V_{thAC}/\sqrt{2}$

and replacing in the power formula

max power at load DC: Pmax=V2RMSAC/(4RL)=V2thAC/(8RL)