Download PSD Chip Calculations

PSD Chip Calculations
Energy Conversions
Erad
Energy of incident radiation (MeV)
evis
Energy of visible photon radiation (eV)
εcon
Conversion efficiency
εcoll
Visible light collection efficiency
εq
Photocathode quantum efficiency
εsplit
Split signal efficiency
evis = 3
LY = 10
εvis ⋅ εcon
6
kcn = LY ⋅ εcoll ⋅ εq
Ne = Erad ⋅ kcn
⎧ 0.17 for CsI(Tl)
εcon = ⎨
⎩0.045 for Liquid Scintillator
εcoll = 0.8
εq = 0.25
εsplit = 0.5
Transresistive Gain Calculation
ArGAIN =
VIN ,max
Emax ⋅ kcn ⋅ q ⋅ max( f (t ))
Argain is the Transresistive Gain following the energy
conversion of an incoming charge pulse.
VIN,max is the maximum voltage allowed at the input of the
chip.
Emax is the maximum energy value that will produce the
maximum voltage.
Pulse Model
Multi-Exponential (with rise and fall times)
(Normalized)
Pulse Creation Equations
t
t
n ⎛
−
−
⎛
τ
τ
f (t ) = n 1 ⋅ ∑ ⎜ τ F ,iA−1τ R ,i ⋅ ⎜ e F ,i − e R ,i
⎜
∑ Ai i =1 ⎜
i =1
⎝
⎝
f (t )
fV (t ) =
max( f (t ))
Pulse Integration Equations
F (t1 , t 2 ) = ∫
t2
t1
FV (t1 , t 2 ) =
⎞⎞
⎟ ⎟ for n exponentials
⎟⎟
⎠⎠
t
⎛
⎛
− 2
⎛ −τ t1
τ
A
⎜
f (t )dt = n 1 ⋅ ∑ ⎜ τ F ,i −iτ R ,i ⋅ ⎜τ F ,i ⋅ ⎜ e F ,i − e F ,i
⎜
⎜
∑ Ai i =1 ⎜⎝
⎝
⎝
i =1
n
1
τ INT
⋅
F (t1 , t 2 )
max( f (t ))
∞
∫ f (t ) = 1
0
max( fV (t )) = 1
t
− 2
⎞
⎛ −τ t1
τ
⎟ − τ ⋅ ⎜ e R ,i − e R ,i
⎟ R ,i ⎜
⎠
⎝
⎞ ⎞ ⎞⎟
⎟⎟
⎟ ⎟ ⎟⎟
⎠⎠⎠
Noise Sources
Poisson – noise due to random arrival of discrete electrons
Electronics Noise
– Jitter – noise created by an uncertainty in the integration start time
and in the width of integration period
– RI – thermal noise from the integrating resistor sampled onto the
integrating capacitor
– OTA – thermal noise of the op amp sampled onto the integrating
capacitor
– OTA (+) – continuous additive input-referred thermal noise of the op
amp
– 1/f – 1/f noise of the op amp sampled onto the integrating capacitor
– 1/f (+) – continuous additive input-referred 1/f noise of the op amp
ADC – quantization noise of a 12-bit converter
Poisson Noise
kOUT =
q ⋅ ArGAIN
τ INT
σ p 2 = kOUT ⋅ VOUT
kOUT represents the gain from incoming charge
packet to voltage output
σp2 is the variance of the Poisson noise at the
output of the integrator.
Jitter Noise
VOFi =
A
τ Fi −τ Ri
VORi = −
⋅τ Fi ⋅ e
A
τ Fi −τ Ri
− τTi
Fi
⋅τ Ri ⋅ e
(
⋅ 1− e
− τTi
Ri
(
− τT
F
⋅ 1− e
)
− τT
Ri
ci ,Ti = −
)
(
VOFi
τ Fi
i
ci ,T = VOF
τ Fi
i
+ VOR
τ Ri
e − T τ Fi
1− e − T τ Fi
)
i
+ VOR
τ Ri
e − T τ Ri
1− e − T τ Ri
where i = 1, 2, …, n
for n exponentials
2
σ j2
2
⎛ n
⎞
⎛ n
⎞
2
2
= ⎜ ∑ ci ,Ti ⎟ σ Ti + ⎜ ∑ ci ,T ⎟ σ T
⎝ i =1
⎠
⎝ i =1
⎠
VOF and VOR are the separate voltages at the output for the falling
and rising exponentials.
Ci,Ti and Ci,T are the constants for n exponentials involved in the
calculation of variance at the output.
σj2 is the variance at the output due to jitter in the starting
integration, Ti, and integration period, T at the input.
Integrating Resistor
Thermal Noise (Sampled)
σ RI ,t 2 = 4 CkT
INT
T
τ INT
σRI,t2 is the variance sampled onto the integrating
capacitor due to thermal noise in the integrating resistor.
OTA Thermal Noise (Sampled)
σ OTA,t 2 = σ RI ,t 2 ⋅ RRN
INT
RN is the equivalent thermal resistance of the OTA.
σOTA,t2 is the variance sampled onto the integrating
capacitor due to thermal noise in the OTA.
OTA Thermal Noise (Continuous)
σ OTA+ ,t 2 = 4 ⋅ kT ⋅ RN ⋅ BW
RN is the equivalent thermal resistance of the OTA.
BW is the close-loop bandwidth of the OTA.
σOTA,t2 is the continuous-time variance at the output due
to thermal noise in the OTA.
OTA 1/f Noise (Sampled)
[
σ OTA, f = Kf ⋅ ln (2 ⋅ Tcal ⋅ Fs )
2
]
0.62 2
Tcal is the time span between calibrations of the output voltage.
Fs is the sampling frequency, or twice the bandwidth of the voltage
at the output.
Kf is the fitted 1/f constant that models the 1/f noise in the OTA.
σOTA,f2 is the variance sampled onto the integrating capacitor due to
1/f noise in the OTA.
OTA 1/f Noise (Continuous)
σ OTA + , f 2 = σ OTA, f 2 ⋅ τ T
INT
σOTA+,f2 is the continuous-time variance at the
output due to 1/f noise in the OTA.
ADC Quantization Noise
Qbin =
σ ADC =
2
VOmax
2 ADCbits
Qbin 2
12
Qbin is the quantization bin size of an ADC with
ADCbits of resolution.
σADC2 is the variance of the ADC at the output.
Variance and SNR at the output
σ TOTAL 2 = σ p 2 + σ j 2 + σ RI ,t 2 + σ OTA,t 2 + σ OTA+ ,t 2 + σ OTA, f 2 + σ OTA+ , f 2
SNR = σVOUT
TOTAL
Since each noise variance at the output is independent
of each other, the total variance at the output is simply
the sum of the variances.
SNR = Signal to Noise Ratio
Analytical Predictions of Variance
of Angular PSD Plots
sin 2 2θ
var(θ ) =
4
FOM =
⎡ 1
1 ⎤
⋅⎢
+
2
2 ⎥
SNR B ⎦
⎣ SNR A
θ1 − θ 0
var(θ1 ) + var(θ 0 )
Variance of angular PSD plot depends on the signal-to-noise ratio
of the A and B integrators.
Small signal-to-noise ratios, which correspond to low-energy
particles, results in a larger variance in angle which is consistent
with simulation.
Figure of merit (FOM) is computed as the difference between the
means divided by the square root of the sum of the variances.