3/3/2015 Day 4: Nodal Analysis and Temperature Measurement Systems

Today in class we first took a quiz regarding finding the voltage across a resistor as a function of the other two resistors and two currents provided by current sources. After much of a disappointment, we did the "Temperature Measurement System" lab, where we before found the temperature and voltage resolutions of our circuit in the lab. In the lab we designed a circuit that would use output voltage to calculate the temperature of the environment. After all of the design and trial and error of the lab we then learned about nodal analysis, where we later solved for the voltage across resistor 1 in the circuit used in our quiz. It was found that the nodal analysis method was much simpler algebraically and physically than KCL and KVL for this circuit. Lastly, we reviewed some blogs and we picked out what was necessary for a good blog.

LECTURE:

We started the day with a quiz involving the circuit seen above. The objective was to find the voltage across resistor 1 (v1) as a function of resistors 2 and 3 and currents 1 and 2. This quiz was done before learning about nodal analysis, so the algebra involving this problem done via Kirchhoff's laws was disturbing and found to be difficult to complete. As seen above, once we learned about nodal analysis this problem became much easier to solve for. First Kirchhoff's current laws were used to obtain two equations. Then the three unknown currents placed in the circuit were solved for as a function of the voltage difference divided by the resistance faced by the currents. Then, these values were substituted into the KCL equations and v1 was solved for using much simpler algebra.

Knowing that the system used in the lab uses an 8 bit architecture, the computer records 2 to the 8th power or 256 bits of data when supplying 5 V to the circuit. Dividing the voltage supplied to the system by the number of domains between bits (255), the voltage resolution was found to be 0.0196 V/degC. Then, the number of bits required to read a 0.5 V difference was found using the voltage resolution, which was calculated to be 25 bits. Dividing that number of bits by the change in the temperature it was found that the temperature resolution was a "bit" shy of half a degree Celsius per bit.

LAB (Temperature Measurement System):

Purpose:

The purpose of this experiment was to create a circuit that would use the output voltage to provide a temperature measurement. This was done using a thermistor, a resistor that changes its resistance with changes in temperature of the environment. This circuit displayed the output voltage, which was then used to convert to the temperature of the thermistor using the necessary temperature to voltage ratio. 

Apparatus:

The apparatus of this experiment consisted of an analog discovery voltage source, a breadboard, a digital multimeter, a thermistor, a resistor, a laptop with "Waveforms" software, wires , alligator clips and a temperature probe. The breadboard connected the whole circuit together, and the digital multimeter was used as an ohmmeter and a voltmeter. The analog discovery provided a constant 5 V to the circuit and the Waveforms software powered the analog. The temperature probe, which was connected to the multimeter, measured the ambient temperature of the lab. The thermistor used has a negative temperature coefficient; its resistance decreases as temperature increases.

Prelab:

Shown above is the prelab to this experiment (I lost the picture of the one we did in lab so I redid it on this paper). In this activity the output voltage as a function of the input voltage and the resistances of the resistor and thermistor was obtained. In addition, the theoretical resistances of the resistor that would result in a 0.5 V change in output voltage over a 12 degree Celsius domain was calculated. Then, the resistances were used to pick resistors whose values were close to the theoretical and that are cheap and easy to obtain. Then, the output voltage at room temperature was obtained using one of the values for a chosen resistor.

Procedure/Data:

The circuit designed in this experiment had to meet some criteria. First, it had to have an input voltage of 5 V. Also, the output voltage had to vary by at least 0.5 V over a temperature range of 12 degrees Celsius. Lastly, the output voltage had to increase as the temperature increased.

The schematic of the circuit shown in the design process was built using the provided materials. The circuit was fairly simple; the 5 V analog source was connected to the thermistor, which was then connected to the resistor. The ground was then connected to the resistor. The voltmeter was connected in parallel to the ends of the 3600 ohm resistor. 

The design in the prelab had to be greatly tweaked and modified using the experimental values for temperature and resistance of the thermistor at those temperatures. The initial design from the prelab did  meet at all the requirements per se, but the change in output voltage was large (0.61 V) and the staring voltage was large (1.82 V). This is due to the thermistor starting at a lower temperature (23 instead of 25 degrees Celsius) and the thermistor not being able to reach 37 degrees Celsius (only up to 34 degrees). This resulted in a smaller temperature range (11 degrees). In addition, the resistance of the thermistor at those temperatures were different than the theoretical seen in the prelab; they were close to the ones in the prelab (2.3% and 4.7% percent differences at lower and higher temperatures, respectively) but at different temperatures. This resulted in the modified use of a 3500 ohm resistor, which theoretically resulted in a 0.527 V change in output voltage over an eleven degree Celsius range. In truth, the change in output voltage was experimentally determined to be 0.54 V, which is nearly equal to the theoretical (2.5% percent error). Our final circuit, which met all requirements (albeit voltage-to-temperature ratio was 0.075 V higher than needed) is shown in action in the video below:

Post-Lab:

The design now had to be changed again in order for the output sensitivity to be at least 0.1 V/degC. As seen above though, an imaginary number was obtained for the needed resistance value, which shows that the output sensitivity was not obtainable using the design we had. Therefore, to find the smallest output sensitivity obtained, the derivative of the change in voltage with respect to the resistance was taken and set to zero (i.e. the design function was optimized), as shown below:

The smallest obtainable output sensitivity using our design was found to be 0.0457 V/degC. It may seem strange at first that the lowest output sensitivity is less than the needed (0.1 V/degC), but this result shows us that the circuit design can be used to reach the wanted output sensitivity. The only thing we would need to do is increase the resistance of the resistor (to 10290 ohms) and/or increase the voltage supply to higher than 5 V.

Overall, as seen in the results and the video, the design of our system turned out to be great with the resources that were available to us. The design requirements were all met and the circuit worked as expected; the voltage increased with increasing temperature.