Full System Calibration of the BPMs at ATF2

Origin of the Calibration Constants

  • A clearer version of the text in this first section is attached here.

Electronics Processing

The monopole signal produced in the reference cavity has a frequency ωR, is independent of beam offset and is used to account for charge and bunch length. It is assumed that the signal is at a maximum at time t=0 and decays exponentially with a decay time τR.
 \begin{equation*} V_R(t)=R_a\exp\left[-\frac{t}{2\tau_R}\right]\cos(\omega_Rt) \end{equation*}(1)

The processing shall be split into three parts: transmission to the mixer (T1), the rest of the electronics (E) and transmission to the digitiser (T2). Each of the two transmission sections has a specific length Δx and an impedance that determines the wave number k. The wave number in T1 will be very different from in T2 because the frequency upstream of the mixer is much larger. The electronics are simplified to a single impedance and an effective length. Therefore, there is a phase advance between the cavity and the digitiser given by

 \begin{equation*} r_T=k_R^{T1}\Delta x_R^{T1}+k_R^{E}\Delta x_R^{E}+k_R^{T2}\Delta x_R^{T2} \end{equation*}(2)

where the superscript R refers to the reference signal and the superscripts denote the hardware section. The mixer uses a local oscillator of amplitude RLO and phase rLO relative to the reference signal to mix down to the intermediate frequency ωRLO. The total gain/attenuation between the cavity and the digitiser arises from the gain of the amplifier and mixer in the electronics, the attenuation arising from the cable losses and any added attenuation. These have been combined in a single factor Rg. The signal at the digitiser is therefore given by

 \begin{equation*} V_R^d(t)=R_aR_{LO}R_g\exp\left[-\frac{t}{2\tau^d_R}\right]\cos((\omega_R-\omega_{LO})t+r_T-r_{LO}) \end{equation*}(3)

The decay constant has also changed to τdR because of the filters in the electronics.

The dipole signal from the position cavity is linearly dependent on the beam offset x as well as the bunch tilt α and the beam angle θ. The signal also peaks at a time t+Δtbeam that accounts for the difference in the time at which the beam arrives at each cavity.

 \begin{eqnarray*} V_P(t\ge \Delta t_{beam})&=\exp\left[-\frac{t-\Delta t_{beam}}{2\tau_P}\right][A_x x\cos(\omega_P(t-\Delta t_{beam}))+(A_\alpha\alpha-A_\theta\theta)\sin(\omega_P(t-\Delta t_{beam}))]\\ V_P(t\ge \Delta t_{beam})&=\exp\left[-\frac{t-\Delta t_{beam}}{2\tau_P}\right]\sqrt{(A_x x)^2+(A_\alpha\alpha-A_\theta\theta)^2}\cos\left(\omega_P(t-\Delta t_{beam})-\tan^{-1}\left(\frac{A_\alpha\alpha-A_\theta\theta}{A_xx}\right)\right) \end{eqnarray*}(4)

This signal goes through similar processing but the constants are different because of the frequency differences of the two cavities, the length differences of the cables and manufacturing differences of the electronics. Furthermore, the local oscillator used for mixing down to the intermediate frequency will be at a different amplitude and phase after being transmitted between the two cavities. Amplitude factors and phase changes are therefore written as P and p instead of R and r respectively. The position cavity signal at the digitiser is given by Eq. (5) where the square root in Eq. (4) is replaced by Pa and the arctangent term, that is the phase change due to the tilt signal, by Θ.

 \begin{equation*} V_P^d(t)=P_aP_{LO}P_g\exp\left[-\frac{t-\Delta t_{beam}}{2\tau^d_P}\right]\cos((\omega_P-\omega_{LO})t-\omega_P \Delta t_{beam}+p_T-p_{LO}+\Theta) \end{equation*}(5)

Digital Processing

The digitiser sampling window opens at a time t0 before the arrival of the position or reference signal. The window start time is adjusted to take Δtbeam into account so that t0 is the same for both position and reference signals. This has the effect of boosting the position signal such that t→ t+Δtbeam which swaps the phase change of ωPΔtbeam to ωLOΔtbeam Both signals are then mixed down to zero frequency using a complex oscillator that starts at the beginning of the sampling window t-t0 at the same frequency as the digitised signal. The high frequency component from the mixing is then removed using a digital filter that again, changes the amplitude by a factor Rf and the decay constant to τ0.
 \begin{eqnarray*} V_R^0(t)&=\frac{R_aR_{LO}R_gR_f}{2}\exp\left[-\frac{t}{2\tau^0_R}\right]\exp\left[i((\omega_R-\omega_{LO})t_0+r_T-r_{LO}\right)]\\ V_P^0(t)&=\frac{P_aP_{LO}P_gP_f}{2}\exp\left[-\frac{t}{2\tau^0_P}\right]\exp\left[i((\omega_P-\omega_{LO})t_0-\omega_{LO} \Delta t_{beam}+p_T-p_{LO}+\Theta\right)]. \end{eqnarray*}(6)

Both signals are then sampled at a time ts and the position signal is divided by the reference signal and the real and imaginary parts are

 \begin{eqnarray*} I&=\frac{P_aP_{LO}P_gP_f}{R_aR_{LO}R_gR_f}\exp\left[\frac{t_s}{2\tau^0_R}-\frac{t_s}{2\tau^0_P}\right]\cos((\omega_P-\omega_R)t_0-\omega_{LO}\Delta t_{beam}+p_T-r_T+r_{LO}-p_{LO}+\Theta)\\ Q&=\frac{P_aP_{LO}P_gP_f}{R_aR_{LO}R_gR_f}\exp\left[\frac{t_s}{2\tau^0_R}-\frac{t_s}{2\tau^0_P}\right]\sin((\omega_P-\omega_R)t_0-\omega_{LO}\Delta t_{beam}+p_T-r_T+r_{LO}-p_{LO}+\Theta) \end{eqnarray*}(7)

The IQ rotation angle is given by

 \begin{equation*} \theta_{IQ}=(\omega_P-\omega_R)t_0-\omega_{LO}\Delta t_{beam}+p_T-r_T+r_{LO}-p_{LO}. \end{equation*}(8)

This rotates the real and imaginary signals to pure position and angle signals. The position signal I is

 \begin{eqnarray*} I^\prime&=I\cos(\theta_{IQ})+Q\sin(\theta_{IQ})\\ I^\prime&=\frac{P_aP_{LO}P_gP_f}{R_aR_{LO}R_gR_f}\exp\left[\frac{t_s}{2\tau^0_R}-\frac{t_s}{2\tau^0_P}\right](\cos(\theta_{IQ}+\Theta)\cos(\theta_{IQ})+\sin(\theta_{IQ}+\Theta)\sin(\theta_{IQ}))\\ I^\prime&=\frac{P_aP_{LO}P_gP_f}{R_aR_{LO}R_gR_f}\exp\left[\frac{t_s}{2\tau^0_R}-\frac{t_s}{2\tau^0_P}\right]\cos\Theta \end{eqnarray*}(9)

This is equivalent to substituting t=-Δtbeam into Eq. (4) and multiplying by a constant.

 \begin{eqnarray*}  I^\prime&=\frac{P_{LO}P_gP_f}{R_aR_{LO}R_gR_f}\exp\left[\frac{t_s}{2\tau^0_R}-\frac{t_s}{2\tau^0_P}-\frac{\Delta t_{beam}}{2\tau_P}\right]V_P(-\Delta t_{beam})\\ I^\prime&=\frac{P_{LO}P_gP_f}{R_aR_{LO}R_gR_f}\exp\left[\frac{t_s}{2\tau^0_R}-\frac{t_s}{2\tau^0_P}\right]A_xx. \end{eqnarray*}(10)

Using both forms in Eq. (4), the position scale is therefore given by

 \begin{equation*} S=\left[\frac{P_{LO}P_gP_f}{R_aR_{LO}R_gR_f}\exp\left[\frac{t_s}{2\tau^0_R}-\frac{t_s}{2\tau^0_P}\right]A_x\right]^{-1} \end{equation*}(11)


 \begin{equation*} x=SI^\prime. \end{equation*}(12)


From Eq. (8) and Eq. (11), it is possible to see all the sources of the calibration constants and how they might vary. Besides transmission of electronics, the IQ rotation angle will vary with timing changes. Any change in the trigger timing Δt0 will alter the IQ rotation angle by PR)Δt0. Any change in the frequency of the two cavities will also affect the IQ rotation angle through this term. Δbeam can be the same order of magnitude as t0 but should not cause changes in phase from pulse to pulse barring changes in the frequency of the local oscillator ωLO since it is almost completely accounted for in the digitiser window spacing. Δtbeam itself should be relatively stable since the beam is relativistic and is moving in longitudinally much faster than it is transversely so it's path length will not change by a great deal. It is possible to account for trigger timing variation in the analysis. Frequency changes are possible to measure by phase flattening but this is not possible on a pulse by pulse basis.

Phase changes in transmission may occur with changes in impedance. The effect would be smaller for the C-band cavities than for the S-band cavities since the electronics are much closer to the cavities and so there is less time before down-mixing when the wavenumber is highest and so the phase change is greatest. Changes in the phase of the local oscillator should show up in both rLO and pLO and the two effects should cancel. However, impedance changes in cables could affect the local oscillator at the position cavity only.

The electronic sources that affect the position scale are mostly measurable in a system of the size at ATF2. However, long term stability may be an issue. The coupling of the cavity to the beam is less easy to determine but may be equal between similar cavities so may only need to be measured once The effect of the filters on the decay constants is not so trivial. However, it is not expected to be a large effect and should be similar for each cavity and therefore, predictable.

Phase Determination using Beam Data

Two files were used to determine the effectiveness of determining the phase of a calibration from a single quadrupole/corrector scan. These were:

  • /home/sboogert/Physics/bpm/atf2/bpmBeamCalibration_20100217_024015.dat
  • /home/sboogert/Physics/bpm/atf2/bpmAllMoverCalibration_20110215_022335.dat
It was assumed that any changes in I/Q during the scans were due to changes in position. The results were compared with the IQ rotation angles recorded in the next archive file.

  • Phases determined using a single magnet position scan:

  • Phases determined using a single corrector scan:

Simulated Full System Calibration

A single, simulated corrector scan was used to calibrate the entire system. The same fits as above were used to determine the phases while the scales were determined by orbit reconstruction from 6 pre-calibrated BPMs. The results are better than with real data because of the unrealistic BPM simulation.

  • Simulated phases using a single corrector scan:

  • Simulated scale determination using orbit reconstruction and 6 calibrated BPMs:

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Topic revision: r6 - 29 Aug 2013 - FrankieCullinan

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