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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.
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.
== Circuit diagram ==
== Circuit diagram ==
The schematic for the amplifier circuit is shown to the right. Click the thumbnail for a larger image. Node voltages and branch currents are marked on the diagram.
The schematic for the amplifier circuit is shown to the right. Click the thumbnail for a larger image. Node voltages and branch currents are marked on the diagram.
== Parameters and variables ==
== 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.
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 ===
=== Input parameters ===
* Power voltage: V<sub>c</sub> (V)
* Power voltage: V<sub>c</sub> (V)
* Frequency: f (Hz)
* Frequency: f (Hz)
=== Unknown variables ===
=== Unknown variables ===
* Transistor currents: j<sub>b</sub>, j<sub>c</sub>, j<sub>e</sub>, k<sub>b</sub>, k<sub>c</sub>, k<sub>e</sub>
* Transistor currents: j<sub>b</sub>, j<sub>c</sub>, j<sub>e</sub>, k<sub>b</sub>, k<sub>c</sub>, k<sub>e</sub>
* Capacitor currents: h<sub>1</sub>, h<sub>2</sub>, h<sub>3</sub>
* Capacitor currents: h<sub>1</sub>, h<sub>2</sub>, h<sub>3</sub>
=== Constants ===
=== Constants ===
| R<sub>t</sub> || 50Ω
| R<sub>t</sub> || 50Ω
|}
|}
==== Capacitors ====
==== Capacitors ====
| C<sub>5</sub> || 10nF
| C<sub>5</sub> || 10nF
|}
|}
==== Transistors ====
==== Transistors ====
| RE || emitter resistance || 0.37Ω || 1Ω
| RE || emitter resistance || 0.37Ω || 1Ω
|}
|}
=== Transistor operating point ===
=== Transistor operating point ===
* V<sub>be1</sub> = V<sub>3</sub>
* V<sub>be1</sub> = V<sub>3</sub>
* V<sub>be2</sub> = V<sub>7</sub> - V<sub>4</sub>.
* V<sub>be2</sub> = V<sub>7</sub> - V<sub>4</sub>.
=== Derived parameters ===
=== Derived parameters ===
* <math>Q = \frac{IBF}{V_0}\,\!</math>
* <math>Q = \frac{IBF}{V_0}\,\!</math>
* <math>Z = 1 + Q \!\cdot\! \left( RB + RE \!\cdot\! BF \right)\,\!</math>
* <math>Z = 1 + Q \!\cdot\! \left( RB + RE \!\cdot\! BF \right)\,\!</math>
== Equations ==
== 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.
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 ===
=== Resistor voltage drop ===
* R<sub>7</sub>: V<sub>c</sub> - I<sub>7</sub>R<sub>7</sub> = V<sub>7</sub>
* R<sub>7</sub>: V<sub>c</sub> - I<sub>7</sub>R<sub>7</sub> = V<sub>7</sub>
* R<sub>t</sub>: V<sub>out</sub> - I<sub>t</sub>R<sub>t</sub> = 0
* R<sub>t</sub>: V<sub>out</sub> - I<sub>t</sub>R<sub>t</sub> = 0
=== Node charge flow ===
=== Node charge flow ===
* T<sub>1</sub>: j<sub>b</sub> + j<sub>c</sub> = j<sub>e</sub>
* T<sub>1</sub>: j<sub>b</sub> + j<sub>c</sub> = j<sub>e</sub>
* T<sub>2</sub>: k<sub>e</sub> = k<sub>b</sub> + k<sub>c</sub>
* T<sub>2</sub>: k<sub>e</sub> = k<sub>b</sub> + k<sub>c</sub>
=== Capacitors ===
=== Capacitors ===
* C<sub>3</sub> : h<sub>3</sub> = iωC<sub>3</sub>V<sub>5</sub>
* C<sub>3</sub> : h<sub>3</sub> = iωC<sub>3</sub>V<sub>5</sub>
* C<sub>5</sub> : I<sub>t</sub> = iωC<sub>5</sub>(V<sub>7</sub> - V<sub>out</sub>)
* C<sub>5</sub> : I<sub>t</sub> = iωC<sub>5</sub>(V<sub>7</sub> - V<sub>out</sub>)
=== Transistor current gain ===
=== Transistor current gain ===
* T<sub>1</sub>: j<sub>c</sub> = β<sub>1</sub>j<sub>b</sub>
* T<sub>1</sub>: j<sub>c</sub> = β<sub>1</sub>j<sub>b</sub>
* T<sub>2</sub>: k<sub>c</sub> = β<sub>2</sub>k<sub>b</sub>
* T<sub>2</sub>: k<sub>c</sub> = β<sub>2</sub>k<sub>b</sub>
=== Transistor exponential response ===
=== Transistor exponential response ===
* T<sub>1</sub>: Z<sub>1</sub>j<sub>b</sub> = Q<sub>1</sub>(V<sub>01</sub> + V<sub>3</sub> - U<sub>1</sub>)
* T<sub>1</sub>: Z<sub>1</sub>j<sub>b</sub> = Q<sub>1</sub>(V<sub>01</sub> + V<sub>3</sub> - U<sub>1</sub>)
* T<sub>2</sub>: Z<sub>2</sub>k<sub>b</sub> = Q<sub>2</sub>(V<sub>02</sub> + V<sub>7</sub> - V<sub>4</sub> - U<sub>2</sub>)
* T<sub>2</sub>: Z<sub>2</sub>k<sub>b</sub> = Q<sub>2</sub>(V<sub>02</sub> + V<sub>7</sub> - V<sub>4</sub> - U<sub>2</sub>)
== Solution ==
== Solution ==
The solution (that is, V<sub>out</sub>) is found by first iterating as described above to find the transistor operating points to the desired precision, then solving under AC conditions to find the correct V<sub>out</sub>. "Solving" (both during iteration and for the final answer) involves running the 24-equation matrix through MATLAB and selecting out the solution generated for the V<sub>out</sub> variable. For responses, see the article on the [[SiPM Amplifier]].
The solution (that is, V<sub>out</sub>) is found by first iterating as described above to find the transistor operating points to the desired precision, then solving under AC conditions to find the correct V<sub>out</sub>. "Solving" (both during iteration and for the final answer) involves running the 24-equation matrix through MATLAB and selecting out the solution generated for the V<sub>out</sub> variable. For responses, see the article on the [[SiPM Amplifier]].