Application of Simple Negative Feedback model for
Avalanche Photo Detectors investigation
Kushpil Vasily
(1),(2)
Avalanche multiplication coefficient
1
M o=
n
V
1−[
]
V br
Miller's formula
Mo
M=
1B⋅M o 
can be expressed using Mo:
n
where
B=
n
V −[V −V fb ] 
V
1
M=
n
V −i⋅R fb 
1−[
]
V br
and
n
br
V - bias voltage;
Vbr – breakdown voltage;
n - parameter;
Rfb – feedback resistance;
Cfb – feedback capacitance;
(1)
V fb =i⋅R fb
Formula (1) can be seen as an expression for response of the system with negative feedback:
Ao
Output
=
Input 1B⋅Ao 
Feedback is a process whereby some
proportion of the output signal of a
system is passed (feedback) to the
input. This is often used to control
the dynamic behavior of the system .
Ao- transfer function for system without feedback
B- feedback coefficient
We can differentiate (1) obtaining:
K ef
dM
=
2
dV 1B⋅M o 
where
dM o
dB
2
K ef =
− M o ⋅

dV
dV
Using Kef We can distinguish 4 types of APD's behavior (Figure 1):
B = 0 no negative feedback;
K > 0 feedback rising slowly then gain (unstable operation mode);
K = 0 feedback rising equal to gain rise (stable operation mode);
K < 0 feedback rising faster then gain (stable operation mode);
Figure 1. Gain vs. biasing voltage for different Kef
From this we can see that the presented model reproduces features of different produced APD exhibiting different gain characteristics.
Feedback coefficient B can be found experimentally as a difference between experimental
dependences of 1/Me(V) and simulated 1/Ms(V) assuming realistic value of Rfb(Figure 2). This
way we can easily characterize different APDs. For example for HAMAMATSU (Figure 3) and
JINR/Dubna (Figure 4) APD's.
Figure 2. Calculation of negative feedback coefficient B(V) for APD
from experimental data.
Figure 3. Calculation of negative feedback coefficient B(V) for
HAMAMATSU S8664 ( http://sales.hamamatsu.com/assets/pdf/parts_S/S8664_series.pdf).
The feedback can describe the dynamics
of APD. For example for
t
B t =B 0 t ⋅1 −exp−[
]
T fb
Figure 5. Investigation of response APD for step pulse from
red LED (650nm). Simulation and experimental data (1997 for
JINR/Dubna APD).
(1) NPI of ASCR in Řěž near Prague, (2) JINR/Dubna
where Tfb=Cfb·Rfb which describes well the
experimental data (Figure 5).
Figure 4. Calculation of negative feedback coefficient B(V) for
JINR/Dubna APD produced and measured in 1996 by author .
(http://sunhe.jinr.ru/struct/neeo/apd/index.html, http://hp.ujf.cas.cz/~kushpil/FTP/Diser/A/ch5.ps (in Russian)).
Conclusion:
1 APD is described as a system with
negative feedback.
2 Kef and B were found as sensitive
variables useful for description of APD
gain behavior.
3 The model describes also dynamics
features of the avalanche detector
response.
June 2008 for NDIP-08 (kushpil@ujf.cas.cz)