Simulation of Probe Motion Effects on Harmonics

I. Transverse Vibrations

H. Glass

11/11/1998

This note is the first in a series investigating the effects of various
probe motions and probe manufacturing defects on the harmonics measured
by the rotating coil which will be built for use with the LHC high
gradient quadrupoles. Since this probe doesn't exist yet, the approach
used in this study was a simulation \[ J. DiMarco, Probesim,
http://tsmtf.fnal.gov/\~dimarco/probesim/probesim.html \]. The program
uses 3 data files for input: 1) a file containing the 'true' harmonics
of the field being measured, 2) a file describing the probe parameters,
and 3) a file that describes errors in probe motion. For this note I
varied the parameters in file 3 and kept the other 2 files unchanged,
with the exception of comparing effects on a pure quadrupole field vs a
field having lots of harmonics.

*Magnetic field*: For the gradient strength, I used the nominal value of
205 T/m (3.485 T @ 17mm). The field quality specs for the MQXB magnets
were prescribed in \[ W. Chou et al., Fermilab Pub-97/378, Nov 1997 \].
In that report, the harmonics of the lead end, return end, and straight
section were specified separately. In the table below, I calculate
harmonics for the whole magnet using a total effective length of 5.5 m.
The table lists the nominal value for the mean, the uncertainty in the
actual value of the mean, and the standard deviation (magnet-to-magnet
variation). The reference radius is 17 mm.

Table I. MQXB nominal harmonics

  ---------- ---------- ---------- ---------- ---------- ---------- ----------
  *n*        *\<bn\>*   *d(bn)*    *σ(bn)*    *\<an\>*   *d(an)*    *σ(an)*

  2          10000                            2.833                 

  3          0          0.34       0.85       0          0.34       0.85

  4          0          0.26       0.87       0          0.26       0.87

  5          0          0.20       0.34       0          0.20       0.34

  6          0.482      0.17       0.25       0.015      0.17       0.25

  7          0          0.14       0.11       0          0.14       0.11

  8          0          0.10       0.07       0          0.10       0.07

  9          0          0.08       0.07       0          0.08       0.07

  10         -0.033     0.06       0.03       -0.007     0.06       0.03
  ---------- ---------- ---------- ---------- ---------- ---------- ----------

If we wish to measure the mean and standard deviation through 20-pole,
it seems reasonable to demand a measurement accuracy approaching 0.01
units.

Probe motion has generally worse effects on the measurements of magnetic
fields with high harmonic content (see below). For this study, I
constructed a field having reasonably bad harmonics by using

$b_{n}\text{=<}b_{n} > \pm \left\lbrack d(b_{n})^{2} + \sigma(b_{n})^{2} \right\rbrack^{\frac{1}{2}}$
(1)

where the minus sign was used for b~10~ and a~10~ to get a larger
absolute value for the harmonic. This resulted in the following set of
harmonics, and these are the ones used in this study:

Table II. MQXB with rich harmonic content.

  ----------------------- ----------------------- -----------------------
  *n*                     *bn*                    *an*

  3                       0.915                   0.915

  4                       0.908                   0.908

  5                       0.394                   0.394

  6                       0.784                   0.317

  7                       0.178                   0.178

  8                       0.122                   0.122

  9                       0.106                   0.106

  10                      -0.100                  0.060
  ----------------------- ----------------------- -----------------------

*Probe parameters*: For simplicity I considered a probe fully contained
in the magnet body, with probe length = 4.0 m. I used the nominal coil
parameters for the probe \[ ref? \], which is a tangential probe having
both dipole and quadrupole bucking coils:

radius 18.33 m

tangential coil: opening angle 13º\
number of turns 15

> D1 vertices (degrees) 31.89, 211.89
>
> D2 vertices (degrees) 328.11, 148.11
>
> Q1 vertices (degrees) 28.76, 118.76, 208.76, 298.76
>
> Q2 vertices (degrees) 61.24, 151.24, 241.24, 331.24

####### Transverse probe vibrations

The simulation program allows one to insert, among other things, modes
of vibration. I studied horizontal vibrations (parameter VIB_X), which
displaces the probe axis (as a function of rotation angle) by

$\text{vib}_{x}(\theta) = A\text{cos}(p\theta + \phi)$ (2)

The adjustable parameters are the amplitude A, the phase φ, and the
frequency p. I studied the effect on the harmonics of each of these
parameters.

An analytical treatment is given by \[ A. Jain, Harmonic coils, CERN
98-05 \]. Considering a displacement function D(, and keeping only terms
linear in D, the spurious harmonics in a 2n-pole magnet are given by

$C^{'}(m)e^{{- \text{im \{}\alpha}_{m}^{'}} \approx (n - 1)\left\lbrack \left( \frac{K_{\text{n-1}}}{K_{m}} \right)\left( \frac{D_{m - (n - 1)}}{R_{\text{ref}}} \right)C(n)e^{- \text{in}\alpha_{n}} + \left( \frac{K_{n - 1}^{}}{K_{m}} \right)\left( \frac{D_{- m - (n - 1)}^{}}{R_{\text{ref}}} \right)C(n)e^{\text{in}\alpha_{n}} \right\rbrack$
(3)

where the Kn are the probe sensitivities. In the case of a quadrupole
magnet (n=2), the spurious harmonics depend on the probe sensitivity to
dipole (K1). For single-mode vibrations, D(θ) = A sin(pθ), the only
spurious harmonics are for m = p±1.

As an example calculation, suppose the probe has a octupole mode
vibration (p=4) with amplitude 1.0 mm. If we use the tangential coil in
unbucked mode, we expect spurious harmonics at m = 3 and 5. Calculations
using Eq. 3 gives C'(3) = 515 units and C'(5) = 459 units. Using the
simulation program, we get 385 and 765 units, respectively, which is
reasonably close but tells us that the non-linear effects that were
neglected may be important. In any case, these harmonics are intolerably
large.

*Bucking and magnet harmonic content*: We expect bucking to improve the
situation dramatically. In fact, the bucking coil geometry has been
chosen so that K1 = 0 ideally, and thus according to Eq. 3 we should
have no spurious harmonics. When the simulation is run on a pure
quadrupole field (no higher-order harmonics at all), we indeed see that
the probe measures cn = 0 for all n\>2. In this ideal case, bucking
allows us to completely recover from vibrations. Of course, once we
start to measure real magnets, the situation becomes more complicated.
Measuring the magnet described by Table II (the rich-harmonic magnet),
we find spurious amplitudes c3 = 0.85 and c5 = 0.60 units. While these
are far smaller than those obtained without bucking, they are large
enough to corrupt the measurement.

Thus, the effect of vibrations becomes worse as the harmonic content of
the magnet grows. For the purpose of doing a worst-case analysis, it
makes sense to use the Table II harmonics in the simulation.

###### Simulation studies

*Control run*: This used no probe motion, but with a probe offset of ∆x
= 0.1 mm. The program has difficulty calculating the correct bucking
coefficients when the probe is exactly centered, so I used this small
(reasonable) offset. This does not introduce any error, since the
program uses the measured b1 and a1 to find the harmonic center, and the
probe measures the harmonics perfectly (at least to an accuracy of 0.001
units). In the runs having probe motion, the difference in harmonics
between these test runs and the control run are reported in the
discussion below. All test runs also used ∆x = 0.1 mm.

*Frequency dependence*: I kept the amplitude fixed at 1.0 mm, the phase
at 0., and varied the frequency, n, between 1 and 10. Of course, we
should be able to control vibration amplitudes to better than 1 mm, but
I chose it this large to be guaranteed of seeing some effect. The
results on the normal harmonics are shown in Figures 1 and 2.

![](media/image1.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 1. Normal harmonics vs vibration frequency

![](media/image2.wmf){width="5.995138888888889in"
height="4.0993055555555555in"} Figure 2. Skew harmonics vs vibration
frequency

From these figures we see that the largest effects are for harmonics
m=p±1, as expected from the analytical discussion above. However, all
the other harmonics are also affected. The magnitude of these effects do
not appear to have an obvious dependence on frequency, although the p=4
and p=6 modes seem to more troublesome than the others.

*Phase dependence*: In this series of runs I kept the amplitude fixed at
1.0 mm and varied the phase of the p=4 mode. The results for sextupole
and decapole, which are affected the most by this mode are plotted in
Figure 3, along with a more benign spurious harmonic, octupole. All of
the results have a 180 degree period (probably because we're measuring a
quadrupole magnet). The sextupole varies between about --1.5 and +0.5
units as a function of phase. The phases at which extrema occur vary for
each harmonic in a way that does not appear simple to predict.

![](media/image3.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 3. Spurious harmonics (sextupole through decapole only) vs phase
of vibration.

*Amplitude dependence*: This is the crucial part of the study, because
the vibration amplitude is something we should have a reasonable ability
to control, and the question we really need to answer is, how much
amplitude can we tolerate? In a study made on the VMTF probe \[ J.
DiMarco \], analysis of the raw flux signals showed that transverse
vibrations were in the range 5 -- 100 μm for p≤3, and were \< 2 μm for
higher order modes.

For this simulation, I examined the amplitude dependence of the p=4 mode
at a phase of 120 degrees, where the harmonics were generally large (see
Fig. 3), and varied the amplitude from 1.0 mm down to 50 μm. I didn't go
to lower amplitudes because the effects were very small by that point
and the trend was clear. I also examined modes p=5 and 6, but at 0
degrees, since I didn't know where their worst phase was. The results,
showing spurious amplitudes vs amplitude, are in Figures 4, 5, and 6.

![](media/image4.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 4. Spurious harmonics vs vibration amplitude (freq=4).

The plots are shown with log-log scales so as to highlight the power law
dependence. Also, the absolute value of the change in the harmonics are
shown; the change can be either positive or negative, but that isn't as
important as understanding the growth of the size of the error with
vibration amplitude. Two obvious groups are seen: the harmonics at
n=p±1, which vary approximately as A^2^, and nearly everything else,
which depends linearly on amplitude A. There is also a third group
having a higher-than-linear dependence, but with generally small values
of ∆bn or ∆an.

![](media/image6.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 5. Spurious harmonics vs vibration amplitude (freq=5).

![](media/image7.wmf){width="5.995138888888889in"
height="4.0993055555555555in"}

Figure 6. Spurious harmonics vs vibration amplitude (freq=6).

Although the p±1 group causes the most trouble at large vibration
amplitude (typically \> 0.5 mm), their A^2^ dependence means they dampen
quickly as one works to limit vibrations to smaller amplitudes. It's the
linearly dependent ones that become somewhat annoying in the 100 μm
range. Fortunately, between 50 and 100 μm all the harmonics cn fall
below 0.01 units. This is achievable, if this probe works as well as the
VMTF probe, and if we work to limit transverse vibrations to \< 50 μm
then we'll be in good shape.

*What's next*: The obvious path for the friendly neighborhood simulator
is to investigate torsional motion, which will be the subject of the
next note in this series. This will be followed by studying other probe
motions such as sagging and bowing. After that, one can explore the
effects of coil defects (incorrect radii and vertices). This should keep
me busy for another few hours.