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: 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_b} (V)
• Power voltage: 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} (V)
• Frequency: 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 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.

• Node voltages: 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_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 V_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_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} , 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} , 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_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_{out}}
  • Note: there is no 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_6} on this circuit; it was a redundant variable with 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} .
• 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
  • $R_{1}=100{\mbox{k}}\Omega$
  • $R_{2}=10{\mbox{k}}\Omega$
  • $R_{3}=5.6{\mbox{k}}\Omega$
  • $R_{4}=1.2{\mbox{k}}\Omega$
  • $R_{5}=1{\mbox{k}}\Omega$
  • $R_{6}=53\Omega$
  • $R_{7}=240\Omega$
  • $R_{t}=50\Omega$
• Capacitors
  • $C_{1}=100{\mbox{nF}}$
  • $C_{2}=0.1{\mbox{nF}}$
  • $C_{3}=100{\mbox{nF}}$
  • $C_{5}=10{\mbox{nF}}$

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 $\mathrm {B}$	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$\Omega$