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Line 171: − + − + − + + + + + + − * <math>C_1</math>: <math>h_1 = i \omega C_1 V_1</math> − * <math>C_2</math>: <math>h_2 = i \omega C_2 (V_2 - V_3)</math> − * <math>C_3</math>: <math>h_3 = i \omega C_3 V_5</math> − * <math>C_5</math>: <math>I_t = i \omega C_5 (V_7 - V_{out})</math> − + − + − + + − + − + − + + + + − * <math>T_1</math>: <math>Z_1 \!\cdot\! j_b = Q_1 \!\cdot\! (V_{01} + V_3 - U_1)</math> − * <math>T_2</math>: <math>Z_2 \!\cdot\! k_b = Q_2 \!\cdot\! (V_{02} + V_7 - V_4 - U_2)</math> − +
no edit summary
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 ==
[[Image:SSPM Amplifier Circuit Diagram.jpg|thumb|Circuit diagram]]
[[Image:SSPM Amplifier Circuit Diagram.jpg|thumb|500px|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 ====
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:left"
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:left"
| ''Parameter'' || ''Description'' || T<sub>1</sub> ''value'' || T<sub>2</sub> ''value''
| ''Parameter'' || ''Description'' || ''T<sub>1</sub> value'' || ''T<sub>2</sub> value''
|-
|-
| VT || temperature voltage || 0.0259 || 0.0259
| VT || temperature voltage || 0.0259 || 0.0259
| 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 ===
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.
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.
* <math>V_1</math>: <math>I_1 = I_{in} + h_1</math>
* V<sub>1</sub>: I<sub>1</sub> = I<sub>in</sub> + h<sub>1</sub>
* <math>V_2</math>: <math>I_{in} = I_2 + h_2</math>
* V<sub>2</sub>: I<sub>in</sub> = I<sub>2</sub> + h<sub>2</sub>
* <math>V_3</math>: <math>I_3 + h_2 = I_4 + j_b</math>
* V<sub>3</sub>: I<sub>3</sub> + h<sub>2</sub> = I<sub>4</sub> + j<sub>b</sub>
* <math>V_4</math>: <math>I_5 + k_b = I_3 + j_c</math>
* V<sub>4</sub>: I<sub>5</sub> + k<sub>b</sub> = I<sub>3</sub> + j<sub>c</sub>
* <math>V_5</math>: <math>I_6 = I_5 + h_3</math>
* V<sub>5</sub>: I<sub>6</sub> = I<sub>5</sub> + h<sub>3</sub>
* <math>V_7</math>: <math>I_7 = I_t + k_e</math>
* V<sub>7</sub>: I<sub>7</sub> = I<sub>t</sub> + k<sub>e</sub>
* <math>T_1</math>: <math>j_b + j_c = j_e</math>
* T<sub>1</sub>: j<sub>b</sub> + j<sub>c</sub> = j<sub>e</sub>
* <math>T_2</math>: <math>k_e = k_b + k_c</math>
* T<sub>2</sub>: k<sub>e</sub> = k<sub>b</sub> + k<sub>c</sub>
=== Capacitors ===
=== Capacitors ===
: <math>I = C \frac{dV}{dt}</math>.
: <math>I = C \frac{dV}{dt}</math>.
As stated above, the unknown voltages and currents are assumed to be of the form
As stated above, the unknown voltages and currents are assumed to be of the form
: <math>X(t) = X_0 e^{i \omega t} + X_1</math>
: <math>X(t) = X_0 e^{i \omega t} + X_1\,\!</math>
so the capacitor equation can be linearized as
so the capacitor equation can be linearized as
: <math>I = i \omega C V</math>
: <math>I = i \omega C V\,\!</math>
where <math>\omega = 2 \pi f</math>. 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 <math>\omega = 0</math> so the equation still holds. There is one such equation for each capacitor.
where ω = 2πf. 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 ω = 0 so the equation still holds. There is one such equation for each capacitor.
* C<sub>1</sub> : h<sub>1</sub> = iωC<sub>1</sub>V<sub>1</sub>
* C<sub>2</sub> : h<sub>2</sub> = iωC<sub>2</sub>(V<sub>2</sub> - V<sub>3</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>)
=== Transistor current gain ===
=== Transistor current gain ===
One of the characteristic equations of a transistor is
One of the characteristic equations of a transistor is
: <math>I_c = \beta I_b</math>.
: <math>I_c = \beta I_b\,\!</math>.
There is one such equation associated with each transistor.
There is one such equation associated with each transistor.
* <math>T_1</math>: <math>j_c = \beta_1 \!\cdot\! j_b</math>
* T<sub>1</sub>: j<sub>c</sub> = β<sub>1</sub>j<sub>b</sub>
* <math>T_2</math>: <math>k_c = \beta_2 \!\cdot\! k_b</math>
* T<sub>2</sub>: k<sub>c</sub> = β<sub>2</sub>k<sub>b</sub>
=== Transistor exponential response ===
=== Transistor exponential response ===
Another characteristic equation of transistors is
Another characteristic equation of transistors is
: <math>Z \!\cdot\! I_b = IS \!\cdot\! \exp \left( \frac{V_{be}}{V_0} \right)</math>.
: <math>Z \!\cdot\! I_b = IS \!\cdot\! \exp \left( \frac{V_{be}}{V_0} \right)\,\!</math>.
This equation is linearized by performing a Taylor expansion up to the first degree, which gives
This equation is linearized by performing a Taylor expansion up to the first degree, which gives
: <math>Z \!\cdot\! I_b = Q \!\cdot\! (V_0 + V_{be} - U)</math>.
: <math>Z \!\cdot\! I_b = Q \!\cdot\! (V_0 + V_{be} - U)\,\!</math>.
Under AC conditions this equation is modified by defining <math>V_0 = U</math>. There is one such equation for each transistor
Under AC conditions this equation is modified by defining V<sub>0</sub> = U. There is one such equation for each transistor
* 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>)
== Solution ==
== Solution ==
The solution (that is, <math>V_{out}</math>) 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 <math>V_{out}</math>. "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 <math>V_{out}</math> 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]].