A single-phase load is connected to a single-phase source via a transmission line with impedance 3 + j10 $$Omega$$. The load operates at a voltage of 13,800 V and consumes 2 MVA at 0.8 lagging power factor. There are real and reactive power losses in the transmission line. 

The real power loss (kW) is: 

A.  40
B.  63
C.  134
D.  210

In our problem, we are given that the load operates at a voltage of 13,800V and consumes 2MVA at 0.8 lagging power factor. 2MVA represents our apparent, or complex power, S. We know that in order to calculate real power from apparent power:

$P = S\cos \left(\theta \right) = S*(power factor)$

Therefore:

$P = 2MVA * (.8) = 1.6M$

Now, let's use our equation for real AC power loss found on pg. 362 of our FE Reference Handbook to find total current through the circuit:

$P = V_{RMS}{I}_{RMS}\cos \theta$

$$\begin{gathered} 1.6M = \left(13800\right)\left(I_{RMS}\right)(.8) \\ 144.93 = I_{RMS} \end{gathered}$$

Now that we know the total current travelling through the system, and hence the current travelling through the line, we can find the power loss that occurs across the line according the following equation:

$$\begin{gathered} P = I^{2}R \\ P = {(144.93\angle 0°)}^2(3+10j) \end{gathered}$$

Let's first convert our line impedance to polar form to make multiplication simpler:

$$\begin{gathered} Rec\tan gular Form:a+jb \\ Polar Form: c\angle \theta where c = \sqrt{a^2+b^2}, \theta = {\tan }^{-1}\left(\frac{b}{a}\right) \\ c = \sqrt{3^2+{10}^2}=10.44 \\ \theta = {\tan }^{-1}\left(\frac{10}{3}\right) = 73.30° \\ 3+10j = 10.44\angle 73.30° \end{gathered}$$

Plugging the polar form into our equation and knowing we multiply constants and add angles in complex form:

$$\begin{gathered} P = {(144.93\angle 0°)}^2(10.44\angle 73.30°) \\ P = \left(219289.12\angle 73.30°\right) \end{gathered}$$

Converting this number back to rectangular form:

$$\begin{gathered} \left(219289.12\angle 73.30°\right) = 219289.12\left(\cos \right(73.30°)+j(\sin (73.30°)) \\ 219289.12((.287)+j(.958)) \\ 62935+210079j \end{gathered}$$

The real part of the number, 62935, represents the real power loss across our transmission line. This corresponds most nearly to 63kW, answer B.