Phase angle, complex impedance phase, in relation to power factor compensation. Exercise 13.

If the phase angle or phase shift of an impedance is not 0°, the impedance is not a pure resistance. The phase angle or phase shift is related to the complex number j in the frequency domain.
Inductive and capacitive reactance result from dividing the phase of the voltage by the phase of the current.
For inductance L, XL = + j.(W.L); for capacitance C, Xc = - j/(W.C).
The abscissa of the orthogonal reference system is used to store real numbers and can also be used as a reference to zero degrees of an angle, i.e. 0°.
The vertical axis is used to store the imaginary part j.
For example, the real impedance of a pure resistor without imaginary part j is placed on the real axis together with its real amplitude (1, 13, 100, etc.) and is on the 0° reference axis, so the phase angle is zero.
If the angle of the impedance with respect to the 0° reference axis is positive or negative, the real number is followed by +j.(W.L) if the angle is positive and -j/(W.C) if it is negative.
The reactive power Q of the circuit is caused by the presence of XL=+j.(W.L) or Xc=-j/(W.C).
In other words, without reactance XL or Xc, the sinusoidal voltage source would not provide reactive power Q. This is because it is L or C that draws reactive power Q.
The reactive power (Q) in the distribution line can be neutralised (or compensated) by cancelling the angle (or phase angle) of the impedance of the electrical installation.
Simply cancel the reactive part of +j.(W.L) or -j/(W.C) that produces the angle in the orthogonal reference frame. If the angle is positive, add a capacitor C; if it is negative, add an inductor L. In this practice, the RL circuit is rather inductive, as the impedance angle of the RL circuit is positive 89,909°.
Therefore, to counteract the 89,909°, a capacitor must be added that generates the opposite angle, i.e. -89,909°. Despite the presence of capacitor C, the terminal voltage of RL remains unchanged and the RL circuit continues to receive the same source voltage, and secondly, the capacitor must be placed in parallel with the RL circuit so that the charge on L corresponds to the discharge of C. If the capacitors are connected in series, the voltages across the RL will no longer be the same. After the capacitor is installed, the total impedance no longer has an angle, so that COS(0°) = 1. The voltage source considers/deems the circuit RL to be a pure resistance, as the total phase angle is zero in the presence of the capacitor C.
As a result, the voltage source no longer supplies reactive power and it is the capacitor that supplies reactive energy locally to the inductor L. As the voltage source only supplies active power to R, the supply line only carries active current, reactive current no longer flows into the supply line and heating of the supply line is reduced.
The purpose of reactive power compensation is to increase the power factor COS(phi) by canceling the phase angle of the impedance of the electrical installation; the closer COS(phi) is to 1, the better; if COS(phi) is equal to 1, the circuit receives no reactive power from the voltage source. As a result, no reactive power or reactive current flows into the distribution line (or transmission line). However, the active current or active power is not affected by the presence of a compensating capacitor. The active power (or active current) is always supplied by the power source.