20.309:Homeworks/Homework1 - Revision history

Steven Wasserman at 23:42, 9 March 2009

2009-03-09T23:42:09Z

←Older revision		Revision as of 23:42, 9 March 2009
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	<div style="padding: 10px; width: 774px; border: 3px solid #000000;">		<div style="padding: 10px; width: 774px; border: 3px solid #000000;">

-	<center><big>'''20.309 ~~Fall~~ Semester ~~2007~~'''</big></center>	+	<center><big>'''20.309 Spring Semester 2009'''</big></center>
-	<center><big>'''Homework Set 1'''</big></center>	+	<center><big>'''Homework Set 2'''</big></center>
-	<center>''Due by 12:00 ~~noon~~ on Tuesday ~~Sept. 16~~, ~~2008~~''</center>	+	<center>''Due by 5:00 PM on Tuesday March 17, 2009''</center>
	<br/>		<br/>

Scottm at 14:22, 3 September 2008

2008-09-03T14:22:51Z

←Older revision		Revision as of 14:22, 3 September 2008
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	<center><big>'''20.309 Fall Semester 2007'''</big></center>		<center><big>'''20.309 Fall Semester 2007'''</big></center>
	<center><big>'''Homework Set 1'''</big></center>		<center><big>'''Homework Set 1'''</big></center>
-	<center>''Due by 12:00 noon on ~~Friday~~ Sept. 21, ~~2007~~''</center>	+	<center>''Due by 12:00 noon on Tuesday Sept. 16, 2008''</center>
	<br/>		<br/>

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-	~~==Question 6:Transimpedence Amplifier==~~
-
-	~~Lab module 0 introduced the op-amp circuit shown in Figure 5.~~
-
-	~~[[Image:Hw1invopamp.JPG\|250px\|center]]<br/>~~
-	~~<center>'''Figure 5: Inverting Voltage Amplifier'''</center><br/>~~
-
-	~~(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/>~~
-	~~<center>'''Figure 6: Transimpedence Amplifier'''</center><br/>~~
-
-	(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.~~
-
-	~~(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 6 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.
-
-	~~[[Image:Hw1highgopamp.JPG\|250px\|center\|Figure 5: high gain transimpedance amplifier]]<br/>~~
-	~~<center>'''Figure 7: High gain transimpedance amplifier'''</center><br/>~~
-
-	(f) 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 7, where would you connect the capacitor and how would you choose its size?
-
-	~~(g) Now write down the expression for this new circuit's output with respect to the current input for AC signals.~~

	</div>		</div>

Rumi: /* Question 6:Transimpedence Amplifier */

2007-09-19T19:33:12Z

Question 6:Transimpedence Amplifier

←Older revision		Revision as of 19:33, 19 September 2007
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	<center>'''Figure 7: High gain transimpedance amplifier'''</center><br/>		<center>'''Figure 7: High gain transimpedance amplifier'''</center><br/>

-	(f) 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?	+	(f) 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 7, where would you connect the capacitor and how would you choose its size?

	(g) Now write down the expression for this new circuit's output with respect to the current input for AC signals.		(g) Now write down the expression for this new circuit's output with respect to the current input for AC signals.

	</div>		</div>

Rumi: /* Question 6:Transimpedence Amplifier */

2007-09-17T17:21:49Z

Question 6:Transimpedence Amplifier

←Older revision		Revision as of 17:21, 17 September 2007
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	<center>'''Figure 7: High gain transimpedance amplifier'''</center><br/>		<center>'''Figure 7: High gain transimpedance amplifier'''</center><br/>

-	(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?	+	(f) 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.	+	(g) Now write down the expression for this new circuit's output with respect to the current input for AC signals.

	</div>		</div>

Rumi: /* Question 6:Transimpedence Amplifier */

2007-09-17T17:21:31Z

Question 6:Transimpedence Amplifier

←Older revision		Revision as of 17:21, 17 September 2007
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	(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.

-	(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 6 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.		(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.
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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?		(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<sub>L</sub><math>\parallel</math>C'' for the resistor ''R<sub>x</sub>'' that you chose)~~.	+	(d) Now write down the expression for this new circuit's output with respect to the current input for AC signals.

	</div>		</div>

Steven Wasserman: /* Question 2: Measuring Physical Quantities with a Wheatstone Bridge */

2007-09-06T15:12:21Z

Question 2: Measuring Physical Quantities with a Wheatstone Bridge

←Older revision		Revision as of 15:12, 6 September 2007
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	<br/>[[Image:ThermistorMan.jpg\|150px\|center]]<br/>		<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/>	+	<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.		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.

Steven Wasserman: /* Question 2: Measuring Physical Quantities with a Wheatstone Bridge */

2007-09-06T14:51:44Z

Question 2: Measuring Physical Quantities with a Wheatstone Bridge

←Older revision		Revision as of 14:51, 6 September 2007
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	==Question 2: Measuring Physical Quantities with a Wheatstone Bridge==		==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>'', ~~(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>'', 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/>		<br/>[[Image:ThermistorMan.jpg\|150px\|center]]<br/>
-	<center>Figure 2: Mister Thermistor (With Apologies to Horowitz and Hill)</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.		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.

Steven Wasserman: /* Question 2: Measuring Physical Quantities with a Wheatstone Bridge */

2007-09-06T12:16:25Z

Question 2: Measuring Physical Quantities with a Wheatstone Bridge

←Older revision		Revision as of 12:16, 6 September 2007
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	<br/>[[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: Mister Thermistor (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>4</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>4</sub>'') is attached to an odd-looking blue apparatus that varies in temperature. The other three are maintained at a constant 20°C.

Steven Wasserman: /* Question 2: Measuring Physical Quantities with a Wheatstone Bridge */

2007-09-06T02:05:02Z

Question 2: Measuring Physical Quantities with a Wheatstone Bridge

←Older revision		Revision as of 02:05, 6 September 2007
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	<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>4</sub>'') is attached to an odd-looking blue apparatus that varies in temperature. The other three are maintained at a constant 20°C.

	<br/>[[Image:ThermistorBridge.jpg\|359px\|center]]<br/>		<br/>[[Image:ThermistorBridge.jpg\|359px\|center]]<br/>
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	(a) Derive an expression for ''V<sub>ab</sub>'' as a function of temperature.		(a) Derive an expression for ''V<sub>ab</sub>'' as a function of temperature.

-	(b) What if both ''R<sub>2</sub>'' and ''R<sub>3</sub>'' are attached to the apparatus? Which configuration is more sensitive to temperature variations?	+	(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==		==Question 3: Photodiode I-V Characteristics==

Steven Wasserman at 01:27, 6 September 2007

2007-09-06T01:27:17Z

←Older revision		Revision as of 01:27, 6 September 2007
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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>~~out~~</sub>/V<sub>in</sub>'' in terms of the input voltage and the two resistor values.	+	(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 ~~outut~~ current of a photodiode. Figure 6 shows a circuit that converts a current to a voltage ~~called a transimpedance 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.

-	~~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>''.	+	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 C. 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.	+	(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/>