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. Rx 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 Vab is zero. Assuming R3 is set so the bridge is balanced, derive an expression for Rx in terms of R1, R2 and R3.
(b) Now let R3 also be a fixed resistor. Suppose that Rx varies in a way that makes Vab 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 Vab 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 RL (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 RT of the divider looking into the node labeled Vout. Then express RL for maximal power transfer in terms of RT.
Figure 4: A voltage divider formed by R1 and R2 driving a resistive load RL.
Question 6:Transimpedence Amplifier
Lab module 0 introduced the op-amp circuit shown in Figure 5.
Figure 5: Inverting Voltage Amplifier
(a) Calculate the gain of this circuit, Vo/Vi in terms of the input voltage and the two resistor values.
Figure 6: Transimpedence 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.
Derive an expression for the output voltage of the circuit produced by a DC current input at iin. (At DC, you can ignore the affect of the capacitor.) Express your answer in the form of a transfer function, Vout/Iin.
(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 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 (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.
Figure 7: 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 RLC for the resistor Rx that you chose).