MATLAB amplifier in detail

The model of the SiPM amplifier is a system of 24 equations in 24 variables that has been linearized so that it can be solved by MATLAB.

Parameters and variables

The MATLAB model has a number of parameters and variables to describe the amplifier circuit, including the 24 unknowns, 4 inputs, and numerous constants.

Input parameters

There are four input parameters:

Input current: $I_{in}$ (A)
Bias voltage: $V_{b}$ (V)
Power voltage: $V_{c}$ (V)
Frequency: $f$ (Hz)

Unknown variables

There are twenty-four unknown variables. The locations (and directions in the case of currents) are labeled on the circuit diagram. All unknowns are assumed to be of the form

X(t)=X_{0}e^{i\omega t}+X_{1}

,

where $X_{0}$ gives the amplitude of oscillation, or the AC component, and $X_{1}$ gives the DC offset.

Node voltages: $V_{1}$ $V_{1}$ , $V_{2}$ $V_{2}$ , $V_{3}$ $V_{3}$ , $V_{4}$ $V_{4}$ , $V_{5}$ $V_{5}$ , $V_{7}$ $V_{7}$ , $V_{out}$ $V_{out}$
- Note: there is no $V_{6}$ on this circuit; it was a redundant variable with $V_{c}$ .
Resistor currents: $I_{1}$ , $I_{2}$ , $I_{3}$ , $I_{4}$ , $I_{5}$ , $I_{6}$ , $I_{7}$ , $I_{t}$
Transistor currents: $j_{b}$ , $j_{c}$ , $j_{e}$ , $k_{b}$ , $k_{c}$ , $k_{e}$
Capacitor currents: $h_{1}$ , $h_{2}$ , $h_{3}$

Constants

Resistors

The resistance values are mostly the same as those marked on the actual amplifier itself, however $R_{4}$ and $R_{6}$ were changed for better agreement of the model with the desired responses. See the article on the actual SiPM Amplifier for details on that circuit.

Component	Resistance
$R_{1}$	$100{\mbox{k}}\Omega$
$R_{2}$	$10{\mbox{k}}\Omega$
$R_{3}$	$5.6{\mbox{k}}\Omega$
$R_{4}$	$1.2{\mbox{k}}\Omega$
$R_{5}$	$1{\mbox{k}}\Omega$
$R_{6}$	$53\Omega$
$R_{7}$	$240\Omega$
$R_{t}$	$50\Omega$

Capacitors

The capacitors are not labeled on the amplifier itself or in the documentation supplied with the amplifier, so the following values are guesses as to the capacitances. Note that $C_{4}$ does not exist.

Component	Capacitance
$C_{1}$	$100{\mbox{nF}}$
$C_{2}$	$0.1{\mbox{nF}}$
$C_{3}$	$100{\mbox{nF}}$
$C_{5}$	$10{\mbox{nF}}$

Transistors

The transistor parameters used are selections from the Gummel-Poon SPICE model parameters for these two parts.

Parameter	Description	$T_{1}$ value	$T_{2}$ value
VT	temperature voltage	0.0259	0.0259
BF	ideal forward maximum $\beta$	93	34
NF	forward current emission coefficient	0.99	1.0
IS	transport saturation current	0.24 fA	0.44 fA
ISE	B-E leakage saturation current	2.4 fA	87 fA
NE	B-E leakage emission coefficient	1.46	1.94
RB	zero-bias base resistance	21 $\Omega$	5 $\Omega$
RE	emitter resistance	0.37 $\Omega$	1 $\Omega$

Equations

There are five categories of equations, which give a set of twenty-four equations in total. Two categories of equations are non-linear and need to be linearized to solve this system as a linear model using matrices.

Resistor voltage drop

The resistor voltage drop equations all take the form

\Delta V=IR

or alternately

V_{\alpha }-IR=V_{\beta }

.

They describe the voltage drop associated with current crossing a resistor, according to Ohm's Law. As such, there is one equation per resistor in the circuit.

$R_{1}$ : $V_{b}-I_{1}\!\cdot \!R_{1}=V_{1}$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_2} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_2 - I_2 \!\cdot\! R_2 = 0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_3} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_4 - I_3 \!\cdot\! R_3 = V_3}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_4} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_3 - I_4 \!\cdot\! R_4 = 0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_5} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_5 - I_5 \!\cdot\! R_5 = V_4}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_6} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_c - I_6 \!\cdot\! R_6 = V_5}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_7} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_c - I_7 \!\cdot\! R_7 = V_7}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_t} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{out} - I_t \!\cdot\! R_t = 0}

Node charge flow

Each node must maintain a dynamic equilibrium of charge during steady-state operation. That means that flow of charge (current) into a given node must equal flow of charge (current) out of that same node. Thus the node charge flow equations take the form of

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum I = 0}