RMS voltage between two arbitrarily phased AC lines

Question

As part of a thought experiment to determine the barriers to electrification of a street running railway using 25kVAC single phase overhead line, I found myself consulting the clearance rules in the US National Electrical Safety Code (ANSI/IEEE C2). From my reading, rule 233B1 applies to determine the horizontal clearance between live OCS (OHLE) parts and conductors and any overhead power or telecommunications conductors hanging from roadside poles:

B. Horizontal Clearance

1. Clearance requirements

The horizontal clearance between crossing or adjacent wires, conductors, or cables carried on different supporting structures shall not be less than 1.50 m (5 ft). For voltages between the wires, conductors, or cables exceeding 22 kV, additional clearance of 10 mm (0.4 in) per kV over 22 kV shall be provided.

Easy enough, right? Well, there's a catch: we know the phase-to-ground voltage of the live OHLE parts, and we know the phase-to-ground voltage of the parallel utility conductors in this hypothetical, but the phase relationship between the two conductors is essentially arbitrary (due to transpositions, if nothing else, although I'm certain there are other things that could introduce a phase difference between two different lines derived from different points on the MV distribution system).

So, given those two nominal RMS phase-to-ground voltages (railroad-OHLE and electric-utility), how can one find the maximum possible (worst case phasing) nominal line-to-line RMS voltage between the two live lines in my hypothetical? If you want numbers for the utility voltage in the hypothetical to work examples with, 13.8kV and 34.5kV systems can be used for the utility side, as those are the most common distribution voltages one would expect to run into in a situation like this.

Answer 1

Let A and B be the two line-to-ground voltages.

The voltage between the two lines is |A - B*(cos θ + j sin θ)|, where θ is the phase difference and |Z| is the complex magnitude. Using only real numbers, that's sqrt((A - B*cos θ)^2 + (B*sin θ)^2).

This takes a minimum value of A - B when θ = 0° and a maximum value of A + B when θ = 180°. It varies almost (but not quite) sinusoidally for intermediate values. Your worst case is simply A+B.

(This is valid for RMS and the result will be RMS when dealing with two sine waves of identical frequencies. This doesn't give you the RMS value of the not-a-sine-wave that results from adding two different frequencies, nor is it particularly useful if there is significant distortion. In both cases the ratio between peak and RMS voltage becomes something other than it was to start with.)