# 20.309:Homeworks/Homework1

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1. Figure 1 shows a resistor network known as a Wheatstone bridge. Part of the apparatus which you'll build shortly for the DNA melting curves module involves half of a "bridge". This is a common circuit used to measure the resistance of an unknown value, ''R<sub>x</sub>''. For now, we will look at it analytically. ''R<sub>x</sub>'' is a resistance you are trying to measure, and ''R<sub>3</sub>'' is a variable resistor. | 1. Figure 1 shows a resistor network known as a Wheatstone bridge. Part of the apparatus which you'll build shortly for the DNA melting curves module involves half of a "bridge". This is a common circuit used to measure the resistance of an unknown value, ''R<sub>x</sub>''. For now, we will look at it analytically. ''R<sub>x</sub>'' is a resistance you are trying to measure, and ''R<sub>3</sub>'' is a variable resistor. | ||

- | [[Image:Hw1wheatstone.JPG|250px]] | + | [[Image:Hw1wheatstone.JPG|250px]]<br> |

Figure 1: A Wheatstone bridge circuit. | Figure 1: A Wheatstone bridge circuit. | ||

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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>''. | (''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]] | + | [[Image:Hw1Divider.JPG|250px]]<br> |

Figure 2: A voltage divider formed by ''R<sub>1</sub>'' and ''R<sub>2</sub>'' driving a resistive load ''R<sub>L</sub>''. | Figure 2: A voltage divider formed by ''R<sub>1</sub>'' and ''R<sub>2</sub>'' driving a resistive load ''R<sub>L</sub>''. | ||

+ | |||

6. Consider the op-amp circuit shown in Fig. 3 which was first introduced in Lab Module 0. | 6. Consider the op-amp circuit shown in Fig. 3 which was first introduced in Lab Module 0. | ||

- | [[Image:Hw1invopamp.JPG|250px]] | + | [[Image:Hw1invopamp.JPG|250px]]<br> |

+ | Figure 3: An inverting op-amp circuit. | ||

(a) Calculate the gain of this circuit, ''V<sub>out</sub>/V<sub>in</sub>''. | (a) Calculate the gain of this circuit, ''V<sub>out</sub>/V<sub>in</sub>''. | ||

- | (b) At times, the signal you may be interested in measuring is in the form of a current, such as in the upcoming DNA melting curve lab. An example circuit for measuring a current is a | + | (b) At times, the signal you may be interested in measuring is in the form of a current, such as in the upcoming DNA melting curve lab. An example circuit for measuring a current is a transimpedance amplifier shown in figure 3. Determine an expression for the output voltage of the circuit with respect to a DC current input. Express your answer in ''V<sub>out</sub>/I<sub>in</sub>''. <br> |

- | [[Image:Hw1highgopamp.JPG|250px]] | + | [[Image:Hw1highgopamp.JPG|250px]]<br> |

+ | Figure 4: A high gain transimpedance amplifier. | ||

(c) Since this is such a high-gain circuit, it can be quite noisy, if the input current ''I<sub>in</sub>'' experiences high-frequency fluctuations. You can insert a capacitor to reduce the noise (i.e. make a low-pass filter to eliminate high-frequency content). Where would you insert it, and how would you choose its size? <br> | (c) Since this is such a high-gain circuit, it can be quite noisy, if the input current ''I<sub>in</sub>'' experiences high-frequency fluctuations. You can insert a capacitor to reduce the noise (i.e. make a low-pass filter to eliminate high-frequency content). Where would you insert it, and how would you choose its size? <br> | ||

(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 (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). |

## Revision as of 18:07, 4 September 2007

**Homework Set 1**

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

1. Figure 1 shows a resistor network known as a Wheatstone bridge. Part of the apparatus which you'll build shortly for the DNA melting curves module involves half of a "bridge". This is a common circuit used to measure the resistance of an unknown value, *R _{x}*. For now, we will look at it analytically.

*R*is a resistance you are trying to measure, and

_{x}*R*is a variable resistor.

_{3}

Figure 1: A Wheatstone bridge circuit.

(a) Assuming *R _{3}* is set such that the bridge is balanced (i.e.

*V*), derive an analytical expression for

_{ab}=0*R*in terms of

_{x}*R*,

_{1}*R*and

_{2}*R*.

_{3}(b) Now let

*R*also be a fixed value, and suppose that

_{3}*R*varies in a way that makes

_{x}*V*nonzero. Derive an expression for the dependence of

_{ab}*V*on

_{ab}*R*.

_{x}

2. Referring again to the Wheatstone Bridge in Figure 1, suppose that *R _{x}* varies with some physical parameter (strain, temperature, etc.) in the range of 1-10kΩ. You want to use the circuit to measure the physical variable by observing

*V*and correlating it to the resistance changes. In what range should the values of

_{ab}=0*R*,

_{1}*R*and

_{2}*R*be to make a sensitive measurement? Explain your reasoning. (Hint: using

_{3}`matlab`to plot the output as a function of the varying resistances is a very useful way to think about this problem).

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?

4. * 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.

5. Referring to the circuit shown in Figure 2, 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*of the divider looking into the node labeled

_{T}*V*. Then express

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

_{L}*R*.

_{T}

Figure 2: A voltage divider formed by *R _{1}* and

*R*driving a resistive load

_{2}*R*.

_{L}

6. Consider the op-amp circuit shown in Fig. 3 which was first introduced in Lab Module 0.

Figure 3: An inverting op-amp circuit.

(a) Calculate the gain of this circuit, *V _{out}/V_{in}*.

(b) At times, the signal you may be interested in measuring is in the form of a current, such as in the upcoming DNA melting curve lab. An example circuit for measuring a current is a transimpedance amplifier shown in figure 3. Determine an expression for the output voltage of the circuit with respect to a DC current input. Express your answer in *V _{out}/I_{in}*.

Figure 4: A high gain transimpedance amplifier.

(c) Since this is such a high-gain circuit, it can be quite noisy, if the input current *I _{in}* experiences high-frequency fluctuations. You can insert a capacitor to reduce the noise (i.e. make a low-pass filter to eliminate high-frequency content). Where would you insert it, 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*for the resistor

_{L}C*R*that you chose).

_{x}