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: $V_{1}$, $V_{2}$, $V_{3}$, $V_{4}$, $V_{5}$, $V_{7}$, $V_{out}$
  • 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.

• $R_{1}$: $V_{b}-I_{1}\!\cdot \!R_{1}=V_{1}$
• $R_{2}$: $V_{2}-I_{2}\!\cdot \!R_{2}=0$
• $R_{3}$: $V_{4}-I_{3}\!\cdot \!R_{3}=V_{3}$
• $R_{4}$: $V_{3}-I_{4}\!\cdot \!R_{4}=0$
• $R_{5}$: $V_{5}-I_{5}\!\cdot \!R_{5}=V_{4}$
• $R_{6}$: $V_{c}-I_{6}\!\cdot \!R_{6}=V_{5}$
• $R_{7}$: $V_{c}-I_{7}\!\cdot \!R_{7}=V_{7}$
• $R_{t}$: $V_{out}-I_{t}\!\cdot \!R_{t}=0$

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

$\sum I=0$

or alternately

$\sum I_{into}=\sum I_{out}$.

There is one such equation per node, and each node already is labeled on the above diagram by the voltage at that point; thus there is one equation per voltage. Additionally, each transistor acts as a node.

• $V_{1}$: $I_{1}=I_{in}+h_{1}$
• $V_{2}$: $I_{in}=I_{2}+h_{2}$
• $V_{3}$: $I_{3}+h_{2}=I_{4}+j_{b}$
• $V_{4}$: $I_{5}+k_{b}=I_{3}+j_{c}$
• $V_{5}$: $I_{6}=I_{5}+h_{3}$
• $V_{7}$: $I_{7}=I_{t}+k_{e}$
• $T_{1}$: $j_{b}+j_{c}=j_{e}$
• $T_{2}$: $k_{e}=k_{b}+k_{c}$

Capacitors

Capacitors relate current and voltage according to the equation

$I=C{\frac {dV}{dt}}$.

As stated above, the unknown voltages and currents are assumed to be of the form

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

so the capacitor equation can be linearized as

$I=i\omega CV$

where $\omega =2\pi f$. This equation works for both AC and DC cases, because in the DC case the derivative on the voltage eliminates any DC bias for the current, but $\omega =0$ so the equation still holds. There is one such equation for each capacitor.

• $C_{1}$: $h_{1}=i\omega C_{1}V_{1}$
• $C_{2}$: Failed to parse (syntax error): {\displaystyle h_2 = i \omega C_2 {V_2 - V_3)}
• $C_{3}$: $h_{3}=i\omega C_{3}V_{5}$
• $C_{5}$: $I_{t}=i\omega C_{5}(V_{7}-V_{out})$