# 20.309:Homeworks/Homework1

**20.309 Fall Semester 2007**

**Homework Set 1**

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

## Question 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}(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 _{3}* also be a fixed value, and 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

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 1-10k . 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).

## 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

* 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

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}*R*and

_{1}*R*driving a resistive load

_{2}*R*.

_{L}

## Question 6

Lab module 0 introduced the op-amp circuit shown in Fig. 3.

**Figure 3: Inverting Voltage Amplifier**

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

**Figure 4: Transimpedence Amplifier**

(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.

Determine 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 in your calculation.) 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 4. 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 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.

**Figure 5: 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} C* for the resistor

*R*that you chose).

_{x}