# 20.309:Homeworks/Homework1

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- | <center><big>'''20.309 | + | <center><big>'''20.309 Spring Semester 2009'''</big></center> |

- | <center><big>'''Homework Set | + | <center><big>'''Homework Set 2'''</big></center> |

- | <center>''Due by | + | <center>''Due by 5:00 PM on Tuesday March 17, 2009''</center> |

<br/> | <br/> | ||

==Question 1:Wheatstone Bridge== | ==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<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). | |

- | + | ||

- | (a) The bridge is balanced when ''V<sub>ab</sub>'' is zero. Assuming ''R<sub>3</sub>'' is set | + | <br/>[[Image:Hw1wheatstone.JPG|250px|center]]<br/> |

+ | <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>''. | ||

(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. | ||

+ | ==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<sub>0</sub>'', 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=R<sub>0</sub>+αT'', where ''T'' is the temperature. | |

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

+ | <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/> | ||

+ | 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. | ||

- | ==Question 3:Photodiode I-V Characteristics== | + | <br/>[[Image:ThermistorBridge.jpg|359px|center]]<br/> |

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

+ | |||

+ | |||

+ | (a) Derive an expression for ''V<sub>ab</sub>'' as a function of temperature. | ||

+ | |||

+ | (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? | ||

+ | |||

+ | ==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. <br> | 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> | ||

+ | |||

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

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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. | 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 5:Power in a Voltage Divider== | ==Question 5:Power in a Voltage Divider== | ||

- | Referring to the circuit shown in Figure | + | 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? |

+ | |||

(''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>''. | (''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>''. | ||

[[Image:Hw1Divider.JPG|250px||center]]<br> | [[Image:Hw1Divider.JPG|250px||center]]<br> | ||

- | <center>Figure | + | <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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</div> | </div> |

## 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. *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}*, 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=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 _{4}*) 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 _{1}* and

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

_{4}## 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}