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

hi dexterbla!
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..._theorem#Proof for details )

Naty1

Decent discussion here:

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