# SC Cavities Feedback Controls

![](media/image1.wmf)

Cavity represented by resonance circuit with L,C, and R, and two current
sources: the klystron and the beam

Differential equation:

> $$\overset{¨}{U} + \frac{\overset{˙}{U}}{\text{RC}} + \frac{U}{\text{LC}} = \frac{\overset{˙}{I}}{C}$$

The voltage V is made up by a "slowly" varying amplitude and a fast
oscillation

$U = V(t)e^{i\omega t}$

The amplitude approximately follows a first order equation

$\overset{˙}{V} = - (\omega_{\frac{1}{2}} - i\Delta\omega)V + \frac{1}{2}\left( \frac{r}{Q} \right)\omega_{0}I$

The detuning is approximated by

$\tau\Delta\overset{˙}{\omega} = - \Delta\omega - 2\pi K\left( \frac{V}{l} \right)^{2}$

The simulation is done such, that

a)  If there is beam, the beam is propagated through the linac based on
    transit time functions and drifts between cavities

b)  The corresponding beam phases are calculated

c)  The induced field is subtracted from the cavity field, assuming that
    all the charge comes in one δ-peak at the beginning of the time step

d)  The refill of the cavity is simulated assuming constant klystron
    power

e)  The klystron power is then adjusted by a vector sum feedback loop
    and the phase shifter loop

The transit time factor for a standing wave structure is

$$T = \frac{1}{\pi} \cdot \frac{\text{cos}\left( \frac{\pi}{2} \cdot \frac{\beta_{0}}{\beta} \right)}{1 - \left( \frac{\beta_{0}}{\beta} \right)^{2}} \cdot \frac{\text{sin}\left( \frac{N\pi}{2}\left( 1 - \frac{\beta_{0}}{\beta} \right) \right)}{N\text{sin}\left( \frac{\pi}{2}\left( 1 - \frac{\beta_{0}}{\beta} \right) \right)}$$

Here I deviate from the usual definition by normalizing T to 1.

## The Phase Shifter

The heart of the phase shifter (electrically) is the solenoid coil
creating the bias field. This acts as an integrator. Additionally there
are filters and screening effects acting as low-pass filters

![](media/image2.wmf)

The elements:

1)  "black box" for the loop controls

2)  ideal amplifier

3)  limiter, say +/-150V

4)  low-pass τ=10μs

5)  low-pass τ=20μs

6)  solenoid L=1mH

![](media/image3.wmf)

I take the phase shifter as approximately linear in the interesting
regime

![](media/image4.png){width="5.999305555555556in"
height="7.874305555555556in"}

Open loop Bode-plot with proportional gain 1

![](media/image5.png){width="5.999305555555556in"
height="7.874305555555556in"}

Open loop Bode-plot with differential gain 1

![](media/image6.png){width="5.622916666666667in"
height="4.222222222222222in"}

energy along macro pulse slew rate 45deg/300us

![](media/image7.png){width="5.622916666666667in"
height="4.227777777777778in"}

voltage

![](media/image8.png){width="5.622916666666667in"
height="4.222222222222222in"}

cavity phases

![](media/image9.png){width="5.622916666666667in"
height="4.222222222222222in"}

forward power before phase shifter

![](media/image10.png){width="5.999305555555556in"
height="6.750694444444444in"}

phase shifter attenuation beta 0.47-0.81

![](media/image11.png){width="5.375in" height="4.036111111111111in"}

phase shift of the two stubs

![](media/image12.png){width="5.375in" height="4.036111111111111in"}

first 16 elements (1^st^ module)

![](media/image13.png){width="5.999305555555556in"
height="4.504166666666666in"}

gradient of the phase shift, due to the differential gain it oscillates
between the two maximal positions

![](media/image14.png){width="5.999305555555556in"
height="7.874305555555556in"}

include condition to reduce attenuation

(i.e. from \<0dB towards 0dB)

![](media/image15.png){width="5.999305555555556in"
height="4.504166666666666in"}

energy with attenuation reduction

![](media/image16.png){width="5.999305555555556in"
height="4.504166666666666in"}

slew rate 45deg/ 500us

![](media/image17.png){width="5.999305555555556in"
height="7.874305555555556in"}

corresponding phase shifts

![](media/image18.png){width="5.622916666666667in"
height="4.222222222222222in"}

beam energy without klystron linearizer (arctan saturation)

![](media/image19.png){width="5.622916666666667in"
height="4.222222222222222in"}

corresponding forward power

![](media/image20.png){width="5.503472222222222in"
height="4.131944444444445in"}

cavity voltages, saturated case

![](media/image21.png){width="5.503472222222222in"
height="4.131944444444445in"}

cavity phases in that case