20.309:Homeworks/Homework1

20.309 Fall Semester 2007 Homework Set 1 Due by 12:00 noon on Friday Sept. 21, 2007

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 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 V_ab is zero. Assuming R₃ is set so the bridge is balanced, derive an expression for R_x in terms of R₁, R₂ and R₃.

(b) Now let R₃ also be a fixed resistor. Suppose that R_x varies in a way that makes V_ab 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

Figure 2: Thermistor Man (With Apologies to Horowitz and Hill)

A thermistor is a resistor whose value varies with temperature. Thermistors are specified by a zero power resistance, R₀, at a specified temperature and a temperature coefficient, α. A small person inside the thermistor adjusts a variable resistor to ensure that R=R₀+αT, where T is the temperature.

Figure 3: Wheatstone Bridge Made of 4 Thermistors

Now imagine a Wheatstone bridge made out of four identical thermistors, as shown in figure 3. One of the thermistors (R₃) is attached to an odd-looking blue lab 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 R₂ and R₃ 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

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 2, what value of R_L (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 R_T of the divider looking into the node labeled V_out. Then express R_L for maximal power transfer in terms of R_T.

Figure 2: A voltage divider formed by R₁ and R₂ driving a resistive load R_L.

Question 6:Transimpedence Amplifier

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 10⁸ 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

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[math]\displaystyle{ \parallel }[/math]C for the resistor R_x that you chose).