Generating DC signal using a phase detector

Question

I can generate a DC signal using a splitter and a phase detector. I do this by splitting a signal to two and then mixing them in a phase detector (I can also use a mixer as well for this purpose). By adjusting the length of the wires between the two split signals I can maximize the DC signal voltage.

I know that output voltage from the phase detector is $$V_{\mathrm {out}}=(2E_s/\pi)\sin(\Delta\omega)t$$ (This is from The Art of Electronics, third edition, page 576.)

I have done this empirically, but I wish to understand mathematically how the length of a wire are can increase or decrease the value of the DC signal.

Jens kindly reminded me to put a schematic of the setup:

Splitter model number: ZFRSC-2050+ Phase detector model number: ZRPD-1+ PD: Newport 1801 LPF:BLP-1.9+

Answer 1

As pointed out @Uwe, just calculate the phase shift along with the length of the line delay (generally 5 ns/m, classical coaxial cable speed = 200 000 km/s).

Remember that phase is a function of "delay" and the period (or frequency) of the signal used. EE&O.
(\$ phi= 2*pi* delay/period\$ or \$2*pi* delay*frequency \$)
Here is a Maple sheet to explain calculus (example plotted for a delay= ~ 180 us).
The picture is "animated", and the delay value is "evaluated".
NB: if del = 0, phase voltage is max and mpl_filtered = 0.5 V.

Answer 2

If you use a longer wire for a signal, you get a delay, the signal needs more time to pass the longer wire. You use a sin wave as signal, therefore the longer wire causes a phase shift. The phase detector with the low pass filter make a DC signal change out of the wire length change.