# [time-nuts] New Question on HP3048A Phase Noise Test Set

BriMDavis at aol.com BriMDavis at aol.com
Wed Jan 23 22:25:51 EST 2008

```Bruce Griffiths wrote:
>
> After low pass filtering
>
> Vo(t) = (A2/2)*sin((w1-w2)t)
>
>
> Thus the amplitude of the discrete spur related component in the low
> pass filtered phase detector output is 1/2 (6dB) the amplitude of the
> discrete spur itself.
>
>
> This result is independent of any additional calculations that may be
> done to derive L(f) from the observed spectrum.
>

I agree completely that the phase detector output amplitude
of your ideal multiplier is half the input spur amplitude.

But I believe that particular 0.5 drops out later along the
way in the computation of S_phi(f) and L(f), as converting
from voltage to phase requires division by the phase detector
constant Kd, which is also 1/2 for your ideal multiplier.

Here's my attempt at continuing on to calculate S_phi(f) and L(f)
for your example phase detector ( somewhat simpler than a real
world calculation requiring gain and calibration corrections ) :

S_phi is defined as the square of the RMS phase
:
:  S_phi = phase_rms ^ 2
:

RMS phase is the RMS voltage over the phase detector constant:
:
:  phase_rms =  Vrms / Kd
:

Substituting to get S_phi in terms of Vrms and Kd:
:
:  S_phi = [ Vrms / Kd ] ^ 2
:

:
:  Vphase_detector_out = 0.5 * Vspur
:

The rms voltage at your ideal phase detector output is therefore:
:
:  Vrms = 0.707 * Vphase_detector_out
:       = 0.707 * 0.5 * Vspur
:

The phase detector constant, Kd, is 0.5 for your ideal multiplier:
:
:  Kd = 0.5
:

Plugging in, we get:
:
:  S_phi = [ Vrms / Kd ] ^ 2
:        = [ 0.707 * 0.5 * Vspur / 0.5 ] ^ 2
:        = [ 0.707 * Vspur ] ^ 2
:        = 0.5  * Vspur^2
:
:      L = 0.5  * S_phi         ( small angle approximation L ~= S_phi/2 )
:        = 0.5  * 0.5 * Vspur^2
:        = 0.25 * Vspur^2
:

As these are powers, S_phi(f) is -3 dB, and L(f) is -6 dB

Which is exactly what Martyn's 3048A has been telling him !

Brian

p.s.

One interesting detail of the calculation of S_phi
is that the phase detector constant Kd depends upon
the _peak_ value of the phase detector output.

In an actual system with the phase detector constant
calibrated by a beat note measurement, if the system
instrumentation measuring the beat note amplitude
returns an RMS value, that tosses another 1.414 into
the calculation of Kd to convert from RMS to peak.

This is the source of the 'mystery' 3dB correction
John Miles mentioned elsewhere in the recent discussions:
>
> According to the notes written by the guy whose office
> door the 3048A authors would have knocked on for advice
> (see _www.ke5fx.com/Scherer_Art_of_PN_measurement.pdf_
(http://www.ke5fx.com/Scherer_Art_of_PN_measurement.pdf)  page 12),
> you need to subtract 6 dB from the noise trace for two reasons.
> 3 dB of it comes from a mysterious "Accounts for RMS value of
> beat signal (3 dB)" clause in Scherer's app note.
>

The top of page 35 ( original page numbering ) of the following
HP app note describes the beatnote correction :