A proof that the power factor of an arbitrary network of resistances and homogeneous reactances will have a predictable numeric sign

All linear electrical loads either consist of or can be modeled by a network of resistive and reactive circuit elements.  The circuit elements comprising the network exist in some overall composite of combinations of series and parallel relationships.

The simplest network of resistive and reactive elements is a single resistor in parallel with a single reactive component, either a capacitor or an inductor.  The impedance of such a network can be expressed as a rectangular phasor in the following manner:

Z = R + jX

If the impedance above is the only load connected to a source, then basic AC power analysis tells us that the load will have a positive power factor if X=XL is an inductive reactance.  Conversely, the load will have a negative power factor if X=XC is a capacitive reactance.  One reason we are able to make these pronouncements by simple inspection is because the impedance Z is given in a form which enables us to express the real and imaginary parts of the phasor with no additional analysis.  Re(Z) = R, and Im(Z) = X.  Since the formula for inductive reactance is XL = jωL and the formula for capacitive reactance in XC = -j/(ωC), the sign of X can be shown in a rather straightforward graphical manner to be the same as the sign of the power factor.

We noted earlier that a load network (or load network model) consists of some combination of resistive and reactive elements in series or parallel relationships or some combination of the two.  It is not difficult to see that if two impedances are placed in series, the imaginary part of the total resulting impedance Im(Z1 + Z2) = Im(R1 + jX1+ R2 + jX2) = X1 + X2 will have the same numeric sign as the imaginary parts of the individual series impedances if the reactances involved are homogeneous in the sense that they are either both inductive or both capacitive.

As an aside, it may be worth recalling that the formula for series equivalent capacitance is analogous to the formula for parallel equivalent inductance.  But when adding two rectangular phasors, these contrasting approaches are addressed by the fact that while the inductive reactance is directly proportional to the inductance, the capacitive reactance is inversely proportional to the capacitance (see formulae above).

From what we have discussed so far, it is convincing that any series combination of capacitive reactances will have a total reactance that is capacitive.  Likewise, any series combination of inductive reactances will have a total reactance that is inductive.  Furthermore, it may be intuitive that any parallel combination of homogeneous reactances will yield a total impedance with imaginary part having the same numeric sign as that of the imaginary parts of its component impedances.  Even though two impedances in parallel result in a total impedance magnitude less than that of either of the component impedances, we “just know” that the total impedance magnitude will never decrease so far that it crosses the zero point and hence changes sign, no matter how many parallel homogeneous impedances we introduce.  However, we cannot prove this by direct inspection of the real and imaginary parts of the expression for parallel impedances.  Consider the following expression for the equivalent impedance of two impedances in parallel:

We have no direct means by which we may explicitly write the real and imaginary parts of the above expression.  Hence, we can draw no conclusions about the numeric sign of the imaginary part.  Our only course of action is to carry out the implied calculations symbolically in the hopes that we may arrive at an expression with explicit real and imaginary parts.  Let us proceed.

We now perform the indicated division by multiplying both the numerator and denominator by the complex conjugate of the denominator as follows:

We now have an expression for which the real and imaginary parts may be directly written because the numerator is separated into explicit real and imaginary parts, and the denominator is a real number.  However, the imaginary part in the numerator appears to be of indeterminate numeric sign.  Therefore we must continue in hopes of obtaining an expression for the imaginary part whose sign can be determined assuredly.

The imaginary part of the above expression is

There are no negative signs in the expression.  R1 and R2 will always be positive.  If both X1 and X2 are positive, the value of the expression will be positive.  If both X1 and X2 are negative, the first two terms of the numerator will clearly be negative.  The third term of the numerator will also be negative because X1X2 will be positive, but X1 + X2 will be negative.  The denominator will always be positive because the two main terms are both squared.  We have shown that the sign of the imaginary part of the total impedance of a parallel arrangement of two impedances with homogeneous reactances will have the same sign as the individual impedances making up that parallel arrangement.  Adding further parallel impedances meeting the reactance homogeneity stipulation will also result in a total impedance with imaginary part of the same numeric sign.

Recall that we established early that impedances with homogeneous reactances placed in series transparently exhibit a property analogous to that in the above paragraph.  With this in mind, any combination of parallel and series impedances with homogeneous reactances will result in a total impedance whose power factor as a load will be lagging if the reactances were all inductive and will be leading if the reactances were all capacitive.