J. Kissel, R. McCarthy At my request, after seeing that the EY BSC 10 vacuum pressure has dropped below 1e-5 [Torr] (see attached trend), Richard has turned on the H1 SUS ETMY ESD at ~2pm PST. I'm continuing to commission the chain, and will post functionality results shortly. Also -- I've found the ESD linearization force coefficient (H1:SUS-ETMX_L3_ESDOUTF_LIN_FORCE_COEFF) to be -180000 [ct]. I don't understand from where this number came, and I couldn't find any aLOGs explaining it. I've logged into to LLO, their coefficient is -512000 [ct]. There's no aLOG describing their number either, but I know from conversations with Joe Betz in early December 2014 that he installed this number when the LLO linearization was switched from before the EUL2ESD matrix to after. When before the EUL2ESD matrix the coefficient was -128000 = - 512000/4 so we was accounting for the factor of 0.25 in EUL2ESD matrix. I suspect that -128000 [ct] came from the following simple model of longitudinal force, F_{tot} on the optic as a result of the quadrant's signal voltage, V_{S} and the bias voltage V_{B}, (which we know is incomplete now -- see LLO aLOG 14853): F_{tot} = a ( V_{s} - V_{B} )^2 F_{tot} = a ( V_{s}^2 - 2 V_{s} V_{B} + V_{B}^2) F_{lin} = 2 a V_{s} V_{B} where F_{lin} is the linear term in the force model, and a< is the force coefficient that turns whatever units V_{S} and V_{B} are in ([ct^2] or [V_{DAC}^2] or [V_{ESD}^2]) into longitudinal force on the test mass in [N]. I *think* the quantity (2 a V_{B}) was mistakenly treated as simply (V_{B}) which has always been held at 128000 [ct] (or the equivalent of 390 [V] on the ESD bias pattern) and the scale factor (2 a) was ignored. Or something. But I don't know. So I try to make sense of these numbers below. Looking at what was intended (see T1400321) and what was eventually analytically shown (see T1400490), we want the quantity F_{ctrl} ------- 2 k V_{B}^2 to be dimensionless, where F_{ctrl} is the force on the optic caused by the ESD. Note that comparing John / Matt / Den's notation against Brett / Joe / my notation, k = a. As written in T1400321, F_{ctrl} was assumed to have units of [N], and V_{B} to have units of [V_{esd}], such that k has units of [ N / (V_{esd}^2) ], and it's the number we all know from John's thesis, k = a = 4.2e-10 [N/V^2]. We now know the number is smaller than that because of the effects of (we think) charge (see, e.g. LHO aLOG 12220, and again LLO aLOG 14853). In the way that the "force coefficient" has been implemented in the front end code -- as an epics variable that comes into the linearization blockas "Gain_Constant_In," (see attached) -- I think the number magically works out to be ... close. As implemented, the linearized quadrant's signal voltage is as shown in Eq. 13 of T1400490, except that the EPICs record, we'll call it G, is actually multiplied in V_{S} = V_{C} + V_{B}(1 - sqrt{ 2 [ (F_{ctrl} / V_{B}^2) * G + 1 + (V_{C}/V_{B}) + (V_{C}/V_{B})^{2} * 1/4 ] )} Note, that we currently have all of the effective charge voltages set to 0 [ct], so the equation just boils down to the expected V_{S} = V_{B}(1 - sqrt{ 2 [ (F_{ctrl} / V_{B}^2) * G + 1] )} which means that G == 1 / (2 k) or k = 1 / (2 G) and has fundamental dimensions of [V_{esd}^2 / N]. So let's take this "force coefficient," G = -512000 [ct], and turn into fundamental units: G = 512000 [ct] {{LLO}} * (20 / 2^18) [V_{dac} / ct] * 40 [V_{esd} / V_{dac}] * 1 / (V_{B} * a) [(1 / V_{esd}) . (V_{esd}^{2} / N)] G = 9.5391e9 [V_{ESD}^2 / N] ==> k = 5.37e-11 [N/V_{ESD}^2] {{LLO}} where I've used V_{B} = 400 [V_{esd}] and the canonical a = 4.2e-10 [N/V_{esd}^2] originally from pg 7 of G0900956. That makes LLO's coefficient assume the actuation strength is a factor of 8 lower from the canonical number. For the LHO number, G = 180000 [ct] {{LHO}} * (20 / 2^18) [V_{dac} / ct] * 40 [V_{esd} / V_{dac}] * 1 / (V_{B} * a) [(1 / V_{esd}) . (V_{esd}^{2} / N)] G = 3.2697e9 [V_{ESD}^2 / N] ==> k = 1.53e-10 [N/V_{ESD}^2] {{LHO}} Which is within a factor of 3 lower, and if the ESD's as weak as we've measured it to be it may be dead on. So maybe whomever stuck in 180000 is much smarter than I. For now I leave in 180000 [ct], which corresponds to a force coefficient of a = 1.53e-10 [N/V_{ESD}^2].
B. Shapiro, J. Kissel As usual, two heads are better than one when it comes to these nasty dealings with factors of two (go figure). Brett has caught a subtlety in the front-end implementation that further makes it different from the analytical approach used in T1400321 and T1400490. In summary, we now agree that the LLO and LHO EPICs force coefficients that have been installed are closer to the measured values by a factor of 4, i.e. G = 512000 [ct] ==> k = 2.0966e-10 [N/V^2] {{LLO}} and G = 180000 [ct] ==> k = 6.1168e-10 [N/V^2] {{LHO}} which means, though they still differ from the canonical value (from pg 7 of G0900956) k = 4.2e-10 [N/V^2] {{Canonical Model}} and what we've measured (including charge) (see LHO aLOG 12220, and LLO aLOGs 14853 and 15657) k = 2e-10 +/- 1.5e-10** [N/V^2] {{Measured}} they're much closer. **I've quickly guesstimated the uncertainty based on the above mentioned measurement aLOGs. IMHO, we still don't have a systematic estimate of the uncertainty because we've measured it so view times, in so many different ways, infrequently, and with the ion pumps still valved in, and each test mass has a different charge mean, charge location, and charge variance. Here's how the aLOG 15809 math should be corrected: The F_{ctrl} and k = a in the analytic equations is assumed to be for full longitudinal force. However, as implemented in the front end, the longitudinal force F_{ctrl} has already been passed through the EUL2ESD matrix, which splits transforms into quadrant basis force F_{ii}, dividing F_{ctrl} by 4. The EPICs force coefficient, G, should therefore *also* be divided by 4, to preserve the ratio F_{ctrl} F_{ii} ------- = ------------ 2 k V_{B}^2 2 k_{ii} V_{B}^2 inside the analytical linearization algorithm. In other words, as we've physically implemented the ESD, on a quadrant-by-quadrant basis, F_{ctrl} = F_{UL} + F_{LL} + F_{UR} + F_{LR} where F_{ii} = k_{ii} (V_{ii} - V_{B})^2 and k_{ii} = k / 4 = a / 4. As such, the implemented front-end equation V_{ii} = V_{B}(1 - sqrt{ 2 [ (F_{ii} / V_{B}^2) * G + 1] )} means that G == 1 / 2 k_{ii} = 2 / k = 2 / a and still has the fundamental units of [V_{esd}^2 / N]. So nothing changes about the above conversation from G in [ct] to G in [V_{esd}^2 / N], its simply that the conversion from G to the more well-known analytical quantity k was off by a factor of 4, G = 512000 [ct] {{LLO}} * (20 / 2^18) [V_{dac} / ct] * 40 [V_{esd} / V_{dac}] * 1 / (V_{B} * a) [(1 / V_{esd}) . (V_{esd}^{2} / N)] G = 9.5391e9 [V_{ESD}^2 / N] ==> k = 2.0966e-10 [N/V_{ESD}^2] {{LLO}} where I've used V_{B} = 400 [V_{esd}] and the canonical a = 4.2e-10 [N/V_{esd}^2] originally from pg 7 of G0900956. That makes LLO's coefficient assume the actuation strength is a factor of 2 lower from the canonical number, pretty darn close to the measured value and definitely within the uncertainty. For the LHO number, G = 180000 [ct] {{LHO}} * (20 / 2^18) [V_{dac} / ct] * 40 [V_{esd} / V_{dac}] * 1 / (V_{B} * a) [(1 / V_{esd}) . (V_{esd}^{2} / N)] G = 3.2697e9 [V_{ESD}^2 / N] ==> k = 6.1168e-10 [N/V_{ESD}^2] {{LHO}} both of which are closer to the measured value as described above.
N. Smith, (transcribed by J. Kissel) Nic called and fessed up to being the one who installed the -180000 [ct] force coefficient at LHO. Note -- this coefficient only is installed in ETMX, the ETMY coefficient is still the original dummy coefficient of 1.0 [ct]. He informs me that this number was determined *empirically* -- he drove a line at some frequency, and made sure that the requested input amplitude (driven before the linearization algorithm) was the same as the requested output amplitude (the MASTER_OUT channels) at the that frequency, with the linearization both ON and BYPASSED. He recalls measuring this with a DTT session, not just looking at the MEDM screen (good!). Why does this work out to be roughly the right number? Take a look at the front-end equation again: V_{ii} = V_{B}(1 - sqrt{ 2 [ (F_{ii} / V_{B}^2) * G + 1] ) } ) and let's assume Nic was driving V_{ii} at a strength equal and opposite sign to the bias voltage V_{B}. With the linearization OFF / BYPASSED, V_{ii} = - V_{B} Duh. With the linearization in place, V_{ii} = - V_{B} = - V_{B} (1 - sqrt{ 2 [ (F_{ii} / V_{B}^2) * G + 1] ) } ) so we want the quantity (1 - sqrt{ 2 [ (F_{ii} / V_{B}^2) * G + 1] )} = 1 which only happens if (F_{ii} / V_{B}^2) * G = 1. If Nic wants to create a force close to the maximum, it needs to be close to the maximum of F_{ii,max} = 2 k_{ii} V_{B}^2, which makes 2 k_{ii} * G = 1 or G = 1 / (2 k_{ii}) = 2 / k which is the same result as in LHO aLOG 15873. Granted, it's late and I've waved my hands a bit, but this is me trying to justify why it feels like it makes sense, at least within the "factor of two-ish" discrepancy between the canonical value and the accepted measurements of the right number.
I've summarized this exploration of Linearization Science in G1500036.