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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 ===
There are four input parameters:
There are four input parameters:
* Input current: <math>I_{in}</math> (A)
* Input current: I<sub>in</sub> (A)
* Bias voltage: <math>V_b</math> (V)
* Bias voltage: V<sub>b</sub> (V)
* Power voltage: <math>V_c</math> (V)
* Power voltage: V<sub>c</sub> (V)
* Frequency: <math>f</math> (Hz)
* Frequency: f (Hz)
=== Unknown variables ===
=== 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
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
: <math>X(t) = X_0 e^{i \omega t} + X_1\,\!</math>,<!-- The \,\! is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
: <math>X(t) = X_0 e^{i \omega t} + X_1\,\!</math>,
where <math>X_0</math> gives the amplitude of oscillation, or the AC component, and <math>X_1</math> gives the DC offset.
where X<sub>0</sub> gives the amplitude of oscillation, or the AC component, and X<sub>1</sub> gives the DC offset.
* Node voltages: <math>V_1</math>, <math>V_2</math>, <math>V_3</math>, <math>V_4</math>, <math>V_5</math>, <math>V_7</math>, <math>V_{out}</math>
* Node voltages: V<sub>1</sub>, V<sub>2</sub>, V<sub>3</sub>, V<sub>4</sub>, V<sub>5</sub>, V<sub>7</sub>, V<sub>out</sub>
** Note: there is no <math>V_6</math> on this circuit; it was a redundant variable with <math>V_c</math>.
** Note: there is no V<sub>6</sub> on this circuit; it was a redundant variable with V<sub>c</sub>.
* Resistor currents: <math>I_1</math>, <math>I_2</math>, <math>I_3</math>, <math>I_4</math>, <math>I_5</math>, <math>I_6</math>, <math>I_7</math>, <math>I_t</math>
* Resistor currents: I<sub>1</sub>, I<sub>2</sub>, I<sub>3</sub>, I<sub>4</sub>, I<sub>5</sub>, I<sub>6</sub>, I<sub>7</sub>, I<sub>t</sub>
* Transistor currents: <math>j_b</math>, <math>j_c</math>, <math>j_e</math>, <math>k_b</math>, <math>k_c</math>, <math>k_e</math>
* 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: <math>h_1</math>, <math>h_2</math>, <math>h_3</math>
* Capacitor currents: h<sub>1</sub>, h<sub>2</sub>, h<sub>3</sub>
=== Constants ===
=== Constants ===
| ''Component'' || ''Resistance''
| ''Component'' || ''Resistance''
|-
|-
| <math>R_1</math> || <math>100\mbox{k}\Omega</math>
| R<sub>1</sub> || 100kΩ
|-
|-
| <math>R_2</math> || <math>10\mbox{k}\Omega</math>
| R<sub>2</sub> || 10kΩ
|-
|-
| <math>R_3</math> || <math>5.6\mbox{k}\Omega</math>
| R<sub>3</sub> || 5.6kΩ
|-
|-
| <math>R_4</math> || <math>1.39\mbox{k}\Omega</math>
| R<sub>4</sub> || 1.39kΩ
|-
|-
| <math>R_5</math> || <math>1\mbox{k}\Omega</math>
| R<sub>5</sub> || 1kΩ
|-
|-
| <math>R_6</math> || <math>53\Omega</math>
| R<sub>6</sub> || 53Ω
|-
|-
| <math>R_7</math> || <math>1.55\mbox{k}\Omega</math>
| R<sub>7</sub> || 1.55kΩ
|-
|-
| <math>R_t</math> || <math>50\Omega</math>
| R<sub>t</sub> || 50Ω
|}
|}
==== Capacitors ====
==== 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 <math>C_4</math> does not exist.
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<sub>4</sub> does not exist.
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:left"
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:left"
| ''Component'' || ''Capacitance''
| ''Component'' || ''Capacitance''
|-
|-
| <math>C_1</math> || <math>100\mbox{nF}</math>
| C<sub>1</sub> || 100nF
|-
|-
| <math>C_2</math> || <math>0.1\mbox{nF}</math>
| C<sub>2</sub> || 0.1nF
|-
|-
| <math>C_3</math> || <math>100\mbox{nF}</math>
| C<sub>3</sub> || 100nF
|-
|-
| <math>C_5</math> || <math>10\mbox{nF}</math>
| 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'' || <math>T_1</math> ''value'' || <math>T_2</math> ''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
|-
|-
| BF || ideal forward maximum <math>\beta</math> || 93 || 34
| BF || ideal forward maximum β || 93 || 34
|-
|-
| NF || forward current emission coefficient || 0.99 || 1.0
| NF || forward current emission coefficient || 0.99 || 1.0
| NE || B-E leakage emission coefficient || 1.46 || 1.94
| NE || B-E leakage emission coefficient || 1.46 || 1.94
|-
|-
| RB || zero-bias base resistance || 21<math>\Omega</math> || 5<math>\Omega</math>
| RB || zero-bias base resistance || 21Ω || 5Ω
|-
|-
| RE || emitter resistance || 0.37<math>\Omega</math> || 1<math>\Omega</math>
| RE || emitter resistance || 0.37Ω || 1Ω
|}
|}
=== Transistor operating point ===
=== Transistor operating point ===
Each transistor has an operating point, <math>U_1</math> and <math>U_2</math>. Both are initially assumed to be 0.7V. Under DC conditions (<math>I_{in} = 0</math>A, <math>f = 0</math>Hz) we iterate on U to refine these two parameters. Each iteration refines the operating point by averaging the current operating point with the associated <math>V_{be}</math>. <math>V_{be}</math> is the base-to-emitter voltage of the transistor and is given by
Each transistor has an operating point, U<sub>1</sub> and U<sub>2</sub>. Both are initially assumed to be 0.7V. Under DC conditions (I<sub>in</sub> = 0A, f = 0Hz) we iterate on U to refine these two parameters. Each iteration refines the operating point by averaging the current operating point with the associated V<sub>be</sub>. V<sub>be</sub> is the base-to-emitter voltage of the transistor and is given by
* <math>V_{be1} = V_3</math>
* V<sub>be1</sub> = V<sub>3</sub>
* <math>V_{be2} = V_7 - V_4</math>.
* 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 ===
The resistor voltage drop equations all take the form
The resistor voltage drop equations all take the form
: <math>\Delta V = IR</math>
: <math>\Delta V = IR\,\!</math>
or alternately
or alternately
: <math>V_\alpha - IR = V_\beta</math>.
: <math>V_\alpha - IR = V_\beta\,\!</math>.
They describe the voltage drop associated with current crossing a resistor, according to [http://en.wikipedia.org/wiki/Ohm's_law Ohm's Law]. As such, there is one equation per resistor in the circuit.
They describe the voltage drop associated with current crossing a resistor, according to [http://en.wikipedia.org/wiki/Ohm's_law Ohm's Law]. As such, there is one equation per resistor in the circuit.
* <math>R_1</math>: <math>V_b - I_1 \!\cdot\! R_1 = V_1</math>
* R<sub>1</sub>: V<sub>b</sub> - I<sub>1</sub>R<sub>1</sub> = V<sub>1</sub>
* <math>R_2</math>: <math>V_2 - I_2 \!\cdot\! R_2 = 0</math>
* R<sub>2</sub>: V<sub>2</sub> - I<sub>2</sub>R<sub>2</sub> = 0
* <math>R_3</math>: <math>V_4 - I_3 \!\cdot\! R_3 = V_3</math>
* R<sub>3</sub>: V<sub>4</sub> - I<sub>3</sub>R<sub>3</sub> = V<sub>3</sub>
* <math>R_4</math>: <math>V_3 - I_4 \!\cdot\! R_4 = 0</math>
* R<sub>4</sub>: V<sub>3</sub> - I<sub>4</sub>R<sub>4</sub> = 0
* <math>R_5</math>: <math>V_5 - I_5 \!\cdot\! R_5 = V_4</math>
* R<sub>5</sub>: V<sub>5</sub> - I<sub>5</sub>R<sub>5</sub> = V<sub>4</sub>
* <math>R_6</math>: <math>V_c - I_6 \!\cdot\! R_6 = V_5</math>
* R<sub>6</sub>: V<sub>c</sub> - I<sub>6</sub>R<sub>6</sub> = V<sub>5</sub>
* <math>R_7</math>: <math>V_c - I_7 \!\cdot\! R_7 = V_7</math>
* R<sub>7</sub>: V<sub>c</sub> - I<sub>7</sub>R<sub>7</sub> = V<sub>7</sub>
* <math>R_t</math>: <math>V_{out} - I_t \!\cdot\! R_t = 0</math>
* 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]].