Max power in in sinusoidal circuits

[dexterbla]

in a sinusoidal circuit, only the resistive part gives the average power. so why is the load impedance the conjugate of thevenin resistance, consisting of both the resistive part and inductive part, required for maximum average power?

 

[marcusl]

Part of the power never enters the network if the impedances are not conjugate matched, so the dissipation in the resistance is lower than it could be.

 

[tiny-tim]

impedance matching

hi dexterbla!

in a sinusoidal circuit, only the resistive part gives the average power.

he he … it depends how you write it!

as a function of current, yes …

P_av = (I_r.m.s.)²Re(Z_load). ​

but as a function of voltage, it's …

P_av = (I_r.m.s.)²Re(Z_load)

= (V_{source,r.m.s.}/|Z_total|)²Re(Z_load) ​

so if the input current is fixed, yes the reactance has no effect on the maximum power ; unfortunately, however, life isn't that simple , and it's always the input voltage that's fixed ,

and then the average power in the load is maximised (for a given Re(Z_load)) by minimising |Z_total| (= |Z_source + Z_load|), ie by reducing the total reactance to zero.

(ie by making Im(Z_load) = -Im(Z_source), and since you can prove that power in the load in a purely resistive circuit is maximised when load resistance equals source resistance, that means that the impedances must be conjugates … see http://en.wikipedia.org/wiki/Maximum_power_theorem#Proof" for details )

 

[Naty1]

Decent discussion here:

http://en.wikipedia.org/wiki/AC_power

 

[PJC01]

This is because in a sinusoidal circuit, the inductive and capacitive components cause the voltage and current to be out of phase, resulting in a reactive power component. This reactive power does not contribute to the average power dissipated in the circuit. Therefore, in order to achieve maximum average power, the load impedance must be matched to the Thevenin resistance, which takes into account both the resistive and reactive components, in order to minimize the reactive power and maximize the average power. This is known as the maximum power transfer theorem.