Consider a realistic sinusoidal voltage source with output impedance Rth driving
ID: 1919619 • Letter: C
Question
Consider a realistic sinusoidal voltage source with output impedance Rth driving a load impedance ZL (see figure below). Show that the power dissipated in the load impedance ZL is maximized if the load impedance is purely resistive and has a magnitude equal to t he output impedance (Rth) of t he source. (In other words, if ZL = RL + j XL , then the power dissipated is maximized when RL = Rth and XL = 0.) Hint: Determine the circulating current i, and voltage across the load impedance v, and then calculate t he power using P = 1 /2Re[vi*]. The maximum of the power occurs when t he "gradient" with respect to the variables RL and XL is zero (i.e. p / RL = 0 and P / XL = 0).Explanation / Answer
P = power dissipated across load
p = Vth2*zl/(R+Zl)2
p will be maximum when dp/dz = 0
1/(R+Zl)2 -2zl/(R+Zl)3 =0
R -zl = 0
or zl =R
hence Zl is real because R is real .
maximum power is p =vth2/4R , when zl =R