Question from inbox:

In problem 34 of Vol II, I am having trouble understanding the answer given. I understand that if the sending angle is greater than the receiving angle then real power flows from the sending end to the receiving end. I am having trouble understanding in regards to max power and load admittance. I thought that max sending power occurs when the difference between the sending and receiving angles is 90 degrees. Do you have any additional information that help understand this problem?

Yes it is true what you said, but this question is not giving angles between Es and Vr. It is asking something very different. Hopefully the below explanation will clarify:

The current is given by:

Where θ is 80° and Φ is 10°

Now if we let K be the following via trig identities:

Than we have the following:

The power transmitted is max when the voltage drop in the line is equal in magnitude to Vr. This is another way of saying Z_Line/ Z_load=1. In the graph I have in the answer, I show that the Max power delivered (Pr/Pmax=1) occurs exactly at this point.

Now look at the formulas above for Vr and I. You can see that as Z_load  decreases, I increases and Vr decreases. At high values of Z_load  , the increase in I dominates over the decrease in Vr. Therefore Pr increases  with a decrease in Z_load . When Z_load approaches Z_line  the effect of the decrease in I is only slightly greater than that of the decrease in Vr.  When Z_load  is less than Z_line  the decrease in Vr dominates over the increase in I, and the net effect is a decrease in Pr.