# Difference between revisions of "20.309:Homeworks/Homework1"

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Figure 1 shows a resistor network known as a Wheatstone bridge. This is a common circuit used to measure an unknown resistance. ''R<sub>x</sub>'' is the component being measured, and ''R<sub>3</sub>'' is a variable resistor (often called a [http://en.wikipedia.org/wiki/Potentiometer potentiometer] or just a [http://en.wikipedia.org/wiki/Pot pot] for no sensible reason). | Figure 1 shows a resistor network known as a Wheatstone bridge. This is a common circuit used to measure an unknown resistance. ''R<sub>x</sub>'' is the component being measured, and ''R<sub>3</sub>'' is a variable resistor (often called a [http://en.wikipedia.org/wiki/Potentiometer potentiometer] or just a [http://en.wikipedia.org/wiki/Pot pot] for no sensible reason). | ||

− | [[Image:Hw1wheatstone.JPG|250px|center]]<br> | + | <br/>[[Image:Hw1wheatstone.JPG|250px|center]]<br/> |

<center>Figure 1: Schematic Diagram of a Wheatstone Bridge</center><br/> | <center>Figure 1: Schematic Diagram of a Wheatstone Bridge</center><br/> | ||

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A thermistor is a resistor whose value varies with temperature. Thermistors are specified by a zero power resistance, ''R<sub>0</sub>'', (given at a specified temperature) and a temperature coefficient, ''α''. As shown in Figure 2, a small person inside the thermistor adjusts a variable resistor to ensure that ''R=R<sub>0</sub>+αT'', where ''T'' is the temperature. | A thermistor is a resistor whose value varies with temperature. Thermistors are specified by a zero power resistance, ''R<sub>0</sub>'', (given at a specified temperature) and a temperature coefficient, ''α''. As shown in Figure 2, a small person inside the thermistor adjusts a variable resistor to ensure that ''R=R<sub>0</sub>+αT'', where ''T'' is the temperature. | ||

− | [[Image:ThermistorMan.jpg|150px|center]]<br> | + | <br/>[[Image:ThermistorMan.jpg|150px|center]]<br/> |

<center>Figure 2: Thermistor Man (With Apologies to Horowitz and Hill)</center><br/> | <center>Figure 2: Thermistor Man (With Apologies to Horowitz and Hill)</center><br/> | ||

Now imagine a Wheatstone bridge made out of four identical thermistors, as shown in figure 3. One of the thermistors (''R<sub>3</sub>'') is attached to an odd-looking blue apparatus that varies in temperature. The other three are maintained at a constant 20°C. | Now imagine a Wheatstone bridge made out of four identical thermistors, as shown in figure 3. One of the thermistors (''R<sub>3</sub>'') is attached to an odd-looking blue apparatus that varies in temperature. The other three are maintained at a constant 20°C. | ||

− | [[Image:ThermistorBridge.jpg|359px|center]]<br> | + | <br/>[[Image:ThermistorBridge.jpg|359px|center]]<br/> |

<center>Figure 3: Wheatstone Bridge Made of 4 Thermistors</center><br/> | <center>Figure 3: Wheatstone Bridge Made of 4 Thermistors</center><br/> | ||

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<center>'''Figure 5: Inverting Voltage Amplifier'''</center><br/> | <center>'''Figure 5: Inverting Voltage Amplifier'''</center><br/> | ||

− | (a) Calculate the gain of this circuit, ''V<sub> | + | (a) Calculate the gain of this circuit, ''V<sub>o</sub>/V<sub>i</sub>'' in terms of the input voltage and the two resistor values. |

[[Image:TransimpedenceAmplifierSchematic.jpg|250px|center]]<br/> | [[Image:TransimpedenceAmplifierSchematic.jpg|250px|center]]<br/> | ||

<center>'''Figure 6: Transimpedence Amplifier'''</center><br/> | <center>'''Figure 6: Transimpedence Amplifier'''</center><br/> | ||

− | (b) In the DNA melting lab, fluorescence intensity will be determined by measuring the | + | (b) In the DNA melting lab, fluorescence intensity will be determined by measuring the output current of a photodiode. Figure 6 shows a circuit called a transimpedance amplifier that converts a current to a voltage. |

− | + | Derive an expression for the output voltage of the circuit produced by a DC current input at ''i<sub>in</sub>''. (At DC, you can ignore the affect of the capacitor.) Express your answer in the form of a transfer function, ''V<sub>out</sub>/I<sub>in</sub>''. | |

(c) What is the high frequency gain of the circuit in Figure 6. Remember that a capacitor acts like an open circuit at low frequencies and a short circuit at high frequencies. | (c) What is the high frequency gain of the circuit in Figure 6. Remember that a capacitor acts like an open circuit at low frequencies and a short circuit at high frequencies. | ||

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(d) A transimpedence amplifier with a gain of approximately 10<sup>8</sup> V/A will be required for the DNA lab. What value of resistor in the circuit of Figure 4 would achieve this gain? | (d) A transimpedence amplifier with a gain of approximately 10<sup>8</sup> V/A will be required for the DNA lab. What value of resistor in the circuit of Figure 4 would achieve this gain? | ||

− | (e) It is undesirable to use the large value of resistor you computed in part | + | (e) It is undesirable to use the large value of resistor you computed in part (d). Figure 7 shows another possible implementation of the transimpedence amplifier. Derive an expression for the output voltage of the circuit in figure 7 in terms of the input current and the three resistor values. |

[[Image:Hw1highgopamp.JPG|250px|center|Figure 5: high gain transimpedance amplifier]]<br/> | [[Image:Hw1highgopamp.JPG|250px|center|Figure 5: high gain transimpedance amplifier]]<br/> |

## Revision as of 17:27, 5 September 2007

**20.309 Fall Semester 2007**

**Homework Set 1**

*Due by 12:00 noon on Friday Sept. 21, 2007*

## Contents

## Question 1:Wheatstone Bridge

Figure 1 shows a resistor network known as a Wheatstone bridge. This is a common circuit used to measure an unknown resistance. *R _{x}* is the component being measured, and

*R*is a variable resistor (often called a potentiometer or just a pot for no sensible reason).

_{3}(a) The bridge is balanced when *V _{ab}* is zero. Assuming

*R*is set so the bridge is balanced, derive an expression for

_{3}*R*in terms of

_{x}*R*,

_{1}*R*and

_{2}*R*.

_{3}(b) Now let *R _{3}* also be a fixed resistor. Suppose that

*R*varies in a way that makes

_{x}*V*nonzero. Derive an expression for the current that would flow if you connected an ammeter from

_{ab}*a*to

*b*. Assume the ammeter has zero internal resistance.

## Question 2: Measuring Physical Quantities with a Wheatstone Bridge

A thermistor is a resistor whose value varies with temperature. Thermistors are specified by a zero power resistance, *R _{0}*, (given at a specified temperature) and a temperature coefficient,

*α*. As shown in Figure 2, a small person inside the thermistor adjusts a variable resistor to ensure that

*R=R*, where

_{0}+αT*T*is the temperature.

Now imagine a Wheatstone bridge made out of four identical thermistors, as shown in figure 3. One of the thermistors (*R _{3}*) is attached to an odd-looking blue apparatus that varies in temperature. The other three are maintained at a constant 20°C.

(a) Derive an expression for *V _{ab}* as a function of temperature.

(b) What if both *R _{2}* and

*R*are attached to the apparatus? Which configuration is more sensitive to temperature variations?

_{3}## Question 3: Photodiode I-V Characteristics

Using the data that you collected in the lab for the photodiode, generate 3-4 *i-v* curves for a photodiode at different light levels (including in darkness). Plot these on the same graph to see how incident light affects diode *i-v* characteristics.

Give a brief (qualitative) explanation for why photodiodes are best used in reverse bias?

## Question 4: Unknown Transfer Functions

For the black boxes that you measured in the lab, determine what kind of circuit/filter each one is (two of them will look similar, but have an important difference - what is it?). Determine a transfer function that can model the circuit, and fit the model to the data to see whether the model makes sense.

Of the four boxes, "D" is required, and you should choose one of either "A" or "C". You can fit "B" for bonus credit.

## Question 5:Power in a Voltage Divider

Referring to the circuit shown in Figure 4, what value of *R _{L}* (in terms of

*R*and

_{1}*R*) will result in the maximum power being dissipated in the load?

_{2}(*Hint:* this is much easier to do if you first remove the load, and calculate the equivalent Thevenin output resistance *R _{T}* of the divider looking into the node labeled

*V*. Then express

_{out}*R*for maximal power transfer in terms of

_{L}*R*.

_{T}*R*and

_{1}*R*driving a resistive load

_{2}*R*.

_{L}

## Question 6:Transimpedence Amplifier

Lab module 0 introduced the op-amp circuit shown in Figure 5.

**Figure 5: Inverting Voltage Amplifier**

(a) Calculate the gain of this circuit, *V _{o}/V_{i}* in terms of the input voltage and the two resistor values.

**Figure 6: Transimpedence Amplifier**

(b) In the DNA melting lab, fluorescence intensity will be determined by measuring the output current of a photodiode. Figure 6 shows a circuit called a transimpedance amplifier that converts a current to a voltage.

Derive an expression for the output voltage of the circuit produced by a DC current input at *i _{in}*. (At DC, you can ignore the affect of the capacitor.) Express your answer in the form of a transfer function,

*V*.

_{out}/I_{in}(c) What is the high frequency gain of the circuit in Figure 6. Remember that a capacitor acts like an open circuit at low frequencies and a short circuit at high frequencies.

(d) A transimpedence amplifier with a gain of approximately 10^{8} V/A will be required for the DNA lab. What value of resistor in the circuit of Figure 4 would achieve this gain?

(e) It is undesirable to use the large value of resistor you computed in part (d). Figure 7 shows another possible implementation of the transimpedence amplifier. Derive an expression for the output voltage of the circuit in figure 7 in terms of the input current and the three resistor values.

**Figure 7: High gain transimpedance amplifier**

(c) In part C, you determined the effect of putting a capacitor across the feedback resistor in a transimpedence amplifier. High gain amplifiers are succeptible to noise couplig from a variety of sources. Since high frequences are not of interest in the DNA melting lab, it is beneficial to insert a capacitor to reduce the noise. In the circuit of Figure 5, where would you connect the capacitor and how would you choose its size?

(d) Now write down the expression for this new circuit's output with respect to the current input for AC signals (Hint: in the expression from part (a), substitute the parallel combination *R _{L}Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \parallel}
C* for the resistor

*R*that you chose).

_{x}