20.309:Homeworks/Homework1

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<div style="padding: 10px; width: 774px; border: 3px solid #000000;">
<div style="padding: 10px; width: 774px; border: 3px solid #000000;">
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<center><big>'''20.309 Fall Semester 2007'''</big></center>
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<center><big>'''20.309 Spring Semester 2009'''</big></center>
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<center><big>'''Homework Set 1'''</big></center>
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<center><big>'''Homework Set 2'''</big></center>
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<center>''Due by 12:00 noon on Friday Sept. 21, 2007''</center>
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<center>''Due by 5:00 PM on Tuesday March 17, 2009''</center>
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==Question 1:Wheatstone Bridge==
==Question 1:Wheatstone Bridge==
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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).
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[[Image:Hw1wheatstone.JPG|250px|center]]<br>
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<br/>[[Image:Hw1wheatstone.JPG|250px|center]]<br/>
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<center>Figure 1: A Wheatstone bridge circuit.</center>
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<center>Figure 1: Schematic Diagram of a Wheatstone Bridge</center><br/>
(a) The bridge is balanced when ''V<sub>ab</sub>'' is zero. Assuming ''R<sub>3</sub>'' is set so the bridge is balanced, derive an expression for ''R<sub>x</sub>'' in terms of ''R<sub>1</sub>'', ''R<sub>2</sub>'' and ''R<sub>3</sub>''.
(a) The bridge is balanced when ''V<sub>ab</sub>'' is zero. Assuming ''R<sub>3</sub>'' is set so the bridge is balanced, derive an expression for ''R<sub>x</sub>'' in terms of ''R<sub>1</sub>'', ''R<sub>2</sub>'' and ''R<sub>3</sub>''.
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(b) Now let ''R<sub>3</sub>'' also be a fixed resistor. Suppose that ''R<sub>x</sub>'' varies in a way that makes ''V<sub>ab</sub>'' nonzero. Derive an expression for the current that would flow if you connected an [http://en.wikipedia.org/wiki/Ammeter ammeter] from ''a'' to ''b''. Assume the ammeter has zero internal resistance.
(b) Now let ''R<sub>3</sub>'' also be a fixed resistor. Suppose that ''R<sub>x</sub>'' varies in a way that makes ''V<sub>ab</sub>'' nonzero. Derive an expression for the current that would flow if you connected an [http://en.wikipedia.org/wiki/Ammeter ammeter] from ''a'' to ''b''. Assume the ammeter has zero internal resistance.
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==Question 2:Current in a Wheatstone Bridge==
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==Question 2: Measuring Physical Quantities with a Wheatstone Bridge==
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Referring again to the Wheatstone Bridge in Figure 1, suppose that ''R<sub>x</sub>'' varies with temperature as: ''R<sub>x</sub>=100+3.85*T<sub>x</sub>'', where ''T<sub>x</sub>'' is the temperature of ''R<sub>x</sub>''. The rest of the resistors are kept at a constant temperatiure. Assume you will measure temperatures in the range of 20°C to 100°C.
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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>'', at a given temperature and a temperature coefficient, ''&alpha;''. As shown in Figure 2, a small person inside the thermistor observes the temperature on a thermometer and adjusts a variable resistor so that ''R=R<sub>0</sub>+&alpha;T'', where ''T'' is the temperature.
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What values of ''R<sub>1</sub>'', ''R<sub>2</sub>'' and ''R<sub>3</sub>'' should you choose to make a sensitive measurement? Explain your reasoning. (Hint: using <tt>matlab</tt> to plot the output as a function of the varying resistances is a very useful way to think about this problem).
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<br/>[[Image:ThermistorMan.jpg|150px|center]]<br/>
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<center>Figure 2: Mister Thermistor (with apologies to [http://books.google.com/books?id=bkOMDgwFA28C&pg=PA64&lpg=PA64&dq=horowitz+hill+transistor+man&source=web&ots=F1goPL6_Tt&sig=2BkT_t2YQRLSUheqws2BUE8z9k8 Horowitz and Hill])</center><br/>
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==Question 3:Photodiode I-V Characteristics==
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Now imagine a Wheatstone bridge made out of four identical thermistors, as shown in figure 3. One of the thermistors (''R<sub>4</sub>'') is attached to an odd-looking blue apparatus that varies in temperature. The other three are maintained at a constant 20°C.
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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. <br>
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<br/>[[Image:ThermistorBridge.jpg|359px|center]]<br/>
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Give a brief (qualitative) explanation for why photodiodes are best used in reverse bias?
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<center>Figure 3: Wheatstone Bridge Made of 4 Thermistors</center><br/>
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==Question 4:Unknown Transfer Functions==
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(a) Derive an expression for ''V<sub>ab</sub>'' as a function of temperature.
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'''''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.<br>
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(b) What if both ''R<sub>1</sub>'' and ''R<sub>4</sub>'' are attached to the apparatus? Which configuration is more sensitive to temperature variations?
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Of the four boxes, "D" is required, and you should choose one of either "A" or "C". You can fit "B" for bonus credit.
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==Question 3: Photodiode I-V Characteristics==
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==Question 5:Power in a Voltage Divider==
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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. <br>
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Referring to the circuit shown in Figure 2, what value of ''R<sub>L</sub>'' (in terms of ''R<sub>1</sub>'' and ''R<sub>2</sub>'') will result in the maximum power being dissipated in the load?<br>
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Give a brief (qualitative) explanation for why photodiodes are best used in reverse bias?
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(''Hint:'' this is much easier to do if you first remove the load, and calculate the equivalent Thevenin output resistance ''R<sub>T</sub>'' of the divider looking into the node labeled ''V<sub>out</sub>''. Then express ''R<sub>L</sub>'' for maximal power transfer in terms of ''R<sub>T</sub>''.
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[[Image:Hw1Divider.JPG|250px||center]]<br>
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==Question 4: Unknown Transfer Functions==
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<center>Figure 2: A voltage divider formed by ''R<sub>1</sub>'' and ''R<sub>2</sub>'' driving a resistive load ''R<sub>L</sub>''.</center>
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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.
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==Question 6:Transimpedence Amplifier==
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Of the four boxes, "D" is required, and you should choose one of either "A" or "C". You can fit "B" for bonus credit.
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Lab module 0 introduced the op-amp circuit shown in Fig. 3.
 
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[[Image:Hw1invopamp.JPG|250px|center]]<br/>
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==Question 5:Power in a Voltage Divider==
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<center>'''Figure 3: Inverting Voltage Amplifier'''</center><br/>
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(a) Calculate the gain of this circuit, ''V<sub>out</sub>/V<sub>in</sub>'' in terms of the input voltage and the two resistor values.
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Referring to the circuit shown in Figure 4, what value of ''R<sub>L</sub>'' (in terms of ''R<sub>1</sub>'' and ''R<sub>2</sub>'') will result in the maximum power being dissipated in the load?
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[[Image:TransimpedenceAmplifierSchematic.jpg|250px|center]]<br/>
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(''Hint:'' this is much easier to do if you first remove the load, and calculate the equivalent Thevenin output resistance ''R<sub>T</sub>'' of the divider looking into the node labeled ''V<sub>out</sub>''. Then express ''R<sub>L</sub>'' for maximal power transfer in terms of ''R<sub>T</sub>''.
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<center>'''Figure 4: Transimpedence Amplifier'''</center><br/>
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(b) In the DNA melting lab, fluorescence intensity will be determined by measuring the outut current of a photodiode. Figure 4 shows a circuit that converts a current to a voltage called a transimpedance amplifier. 
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[[Image:Hw1Divider.JPG|250px||center]]<br>
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<center>Figure 4: A voltage divider formed by ''R<sub>1</sub>'' and ''R<sub>2</sub>'' driving a resistive load ''R<sub>L</sub>''.</center>
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Determine 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 in your calculation.) Express your answer in the form of a transfer function, ''V<sub>out</sub>/I<sub>in</sub>''.
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(c) What is the high frequency gain of the circuit in Figure 4. 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?
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(e) It is undesirable to use the large value of resistor you computed in part C. The schematic diagram in Figure 5 shows another possible implementation of the transimpedence amplifier. Derive an expression for the output voltage of the circuit in figure 5 in terms of the input current and the three resistor values.
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[[Image:Hw1highgopamp.JPG|250px|center|Figure 5: high gain transimpedance amplifier]]<br/>
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<center>'''Figure 5: High gain transimpedance amplifier'''</center><br/>
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(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?
 
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(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<sub>L</sub><math>\parallel</math>C'' for the resistor ''R<sub>x</sub>'' that you chose).
 
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Current revision

20.309 Spring Semester 2009
Homework Set 2
Due by 5:00 PM on Tuesday March 17, 2009


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. Rx is the component being measured, and R3 is a variable resistor (often called a potentiometer or just a pot for no sensible reason).



Figure 1: Schematic Diagram of a Wheatstone Bridge

(a) The bridge is balanced when Vab is zero. Assuming R3 is set so the bridge is balanced, derive an expression for Rx in terms of R1, R2 and R3.

(b) Now let R3 also be a fixed resistor. Suppose that Rx varies in a way that makes Vab nonzero. Derive an expression for the current that would flow if you connected an ammeter from 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, R0, at a given temperature and a temperature coefficient, α. As shown in Figure 2, a small person inside the thermistor observes the temperature on a thermometer and adjusts a variable resistor so that R=R0+αT, where T is the temperature.



Figure 2: Mister Thermistor (with apologies to Horowitz and Hill)

Now imagine a Wheatstone bridge made out of four identical thermistors, as shown in figure 3. One of the thermistors (R4) is attached to an odd-looking blue apparatus that varies in temperature. The other three are maintained at a constant 20°C.



Figure 3: Wheatstone Bridge Made of 4 Thermistors


(a) Derive an expression for Vab as a function of temperature.

(b) What if both R1 and R4 are attached to the apparatus? Which configuration is more sensitive to temperature variations?

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 RL (in terms of R1 and R2) will result in the maximum power being dissipated in the load?

(Hint: this is much easier to do if you first remove the load, and calculate the equivalent Thevenin output resistance RT of the divider looking into the node labeled Vout. Then express RL for maximal power transfer in terms of RT.


Figure 4: A voltage divider formed by R1 and R2 driving a resistive load RL.


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