MATLAB amplifier in detail - Revision history

Senderovich at 13:48, 30 June 2008

2008-06-30T13:48:24Z

Senderovich: /* Circuit diagram */

2008-03-04T02:55:52Z

Circuit diagram

Krueger at 15:20, 10 July 2007

2007-07-10T15:20:39Z

Krueger at 15:14, 10 July 2007

2007-07-10T15:14:52Z

Krueger at 15:11, 10 July 2007

2007-07-10T15:11:28Z

Krueger at 15:06, 10 July 2007

2007-07-10T15:06:38Z

Krueger at 15:02, 10 July 2007

2007-07-10T15:02:45Z

Krueger at 14:57, 10 July 2007

2007-07-10T14:57:12Z

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Krueger at 14:30, 10 July 2007

2007-07-10T14:30:15Z

Krueger at 14:28, 10 July 2007

2007-07-10T14:28:59Z

@@ Line 1: / Line 1: @@
 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 ==
@@ Line 6: / Line 7: @@
 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 ==
 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 ===
@@ Line 18: / Line 21: @@
 * Power voltage: V<sub>c</sub> (V)
 * Frequency: f (Hz)
 === Unknown variables ===
@@ Line 29: / Line 33: @@
 * 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>
 === Constants ===
@@ Line 55: / Line 60: @@
 | R<sub>t</sub> || 50&#937;
 |}
 ==== Capacitors ====
@@ Line 71: / Line 77: @@
 | C<sub>5</sub> || 10nF
 |}
 ==== Transistors ====
@@ Line 95: / Line 102: @@
 | RE || emitter resistance || 0.37&#937; || 1&#937;
 |}
 === Transistor operating point ===
@@ Line 101: / Line 109: @@
 * V<sub>be1</sub> = V<sub>3</sub>
 * V<sub>be2</sub> = V<sub>7</sub> - V<sub>4</sub>.
 === Derived parameters ===
@@ Line 114: / Line 123: @@
 * <math>Q = \frac{IBF}{V_0}\,\!</math>
 * <math>Z = 1 + Q \!\cdot\! \left( RB + RE \!\cdot\! BF \right)\,\!</math>
 == 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 ===
@@ Line 135: / Line 146: @@
 * 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
 === Node charge flow ===
@@ Line 152: / Line 164: @@
 * 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>
 === Capacitors ===
@@ Line 167: / Line 180: @@
 * C<sub>3</sub> : h<sub>3</sub> = i&#969;C<sub>3</sub>V<sub>5</sub>
 * C<sub>5</sub> : I<sub>t</sub> = i&#969;C<sub>5</sub>(V<sub>7</sub> - V<sub>out</sub>)
 === Transistor current gain ===
@@ Line 176: / Line 190: @@
 * T<sub>1</sub>: j<sub>c</sub> = &#946;<sub>1</sub>j<sub>b</sub>
 * T<sub>2</sub>: k<sub>c</sub> = &#946;<sub>2</sub>k<sub>b</sub>
 === Transistor exponential response ===
@@ Line 187: / Line 202: @@
 * 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 ==
 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]].

@@ Line 3: / Line 3: @@
 == 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.

@@ Line 77: / Line 77: @@
 {| 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

@@ Line 180: / Line 180: @@
 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
-: <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
-* <math>T_1</math>: <math>Z_1 \!\cdot\! j_b = Q_1 \!\cdot\! (V_{01} + V_3 - U_1)</math>
+* 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>)
-* <math>T_2</math>: <math>Z_2 \!\cdot\! k_b = Q_2 \!\cdot\! (V_{02} + V_7 - V_4 - U_2)</math>
+* 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 ==
-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]].

@@ Line 161: / Line 161: @@
 so the capacitor equation can be linearized as
 : <math>I = i \omega C V\,\!</math>
-where &#969; = 2&#960;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 &3969; = 0 so the equation still holds.  There is one such equation for each capacitor.
+where &#969; = 2&#960;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 &#969; = 0 so the equation still holds.  There is one such equation for each capacitor.
-* <math>C_1</math>: <math>h_1 = i \omega C_1 V_1</math>
+* C<sub>1</sub> : h<sub>1</sub> = i&#969;C<sub>1</sub>V<sub>1</sub>
-* <math>C_2</math>: <math>h_2 = i \omega C_2 (V_2 - V_3)</math>
+* C<sub>2</sub> : h<sub>2</sub> = i&#969;C<sub>2</sub>(V<sub>2</sub> - V<sub>3</sub>)
-* <math>C_3</math>: <math>h_3 = i \omega C_3 V_5</math>
+* C<sub>3</sub> : h<sub>3</sub> = i&#969;C<sub>3</sub>V<sub>5</sub>
-* <math>C_5</math>: <math>I_t = i \omega C_5 (V_7 - V_{out})</math>
+* C<sub>5</sub> : I<sub>t</sub> = i&#969;C<sub>5</sub>(V<sub>7</sub> - V<sub>out</sub>)
 === Transistor current gain ===
 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.
-* <math>T_1</math>: <math>j_c = \beta_1 \!\cdot\! j_b</math>
+* T<sub>1</sub>: j<sub>c</sub> = &#946;<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> = &#946;<sub>2</sub>k<sub>b</sub>
 === Transistor exponential response ===

@@ Line 144: / Line 144: @@
 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 ===
@@ Line 158: / Line 158: @@
 : <math>I = C \frac{dV}{dt}</math>.
 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
-: <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 &#969; = 2&#960;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 &3969; = 0 so the equation still holds.  There is one such equation for each capacitor.
 * <math>C_1</math>: <math>h_1 = i \omega C_1 V_1</math>

@@ Line 122: / Line 122: @@
 The resistor voltage drop equations all take the form
-: <math>\Delta V = IR</math>
+: <math>\Delta V = IR\,\!</math>
 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.
-* <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 ===

@@ Line 106: / Line 106: @@
 The following parameters relate to the transistor constants.  They are also constants, but are derived from the more fundamental constants given above.  As the fundamental parameters are different for each transistor, there is a different set of derived parameters for each transistor.
-* <math>V_0 = NF \!\cdot\! Vt</math>
+* <math>V_0 = NF \!\cdot\! Vt\,\!</math>
-* <math>IBF = IS \!\cdot\! \exp \left( \frac{U}{V_0} \right)</math>
+* <math>IBF = IS \!\cdot\! \exp \left( \frac{U}{V_0} \right)\,\!</math>
-* <math>IBL = ISE \!\cdot\! \exp \left( \frac{U}{NE\!\cdot\! VT} \right)</math>
+* <math>IBL = ISE \!\cdot\! \exp \left( \frac{U}{NE\!\cdot\! VT} \right)\,\!</math>
-* <math>IB = IBF + IBL</math>
+* <math>IB = IBF + IBL\,\!</math>
-* <math>IC = BF \!\cdot\! IBF</math>
+* <math>IC = BF \!\cdot\! IBF\,\!</math>
-* <math>\beta = \frac{IC}{IB}</math>
+* <math>\beta = \frac{IC}{IB}\,\!</math>
-* <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 ==

@@ Line 22: / Line 22: @@
 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>,
+: <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.-->
 where <math>X_0</math> gives the amplitude of oscillation, or the AC component, and <math>X_1</math> 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>