20.309:Homeworks/Homework1

20.309 Spring Semester 2009 Homework Set 2 Due by 5:00 PM on Tuesday March 17, 2009

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

A thermistor is a resistor whose value varies with temperature. Thermistors are specified by a zero power resistance, R₀, 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=R₀+αT, where T is the temperature.

Figure 2: Mister Thermistor (with apologies to Horowitz and Hill)

Now imagine a Wheatstone bridge made out of four identical thermistors, as shown in figure 3. One of the thermistors (R₄) is attached to an odd-looking blue apparatus that varies in temperature. The other three are maintained at a constant 20°C.

Figure 3: Wheatstone Bridge Made of 4 Thermistors

(a) Derive an expression for V_ab as a function of temperature.

(b) What if both R₁ and R₄ 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

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