Overview The goal of this two-week laboratory is to develop a procedure to accurately measure a capacitance. In the first lab session, you will explore methods to measure capacitance, and their uncertainties. Instead of handing in your lab notebook after the first session, you will take it home, reflect on your results, and write a procedure to do an accurate capacitance measurement. In the next lab session, you will follow your own procedure (with amendments if necessary, and improvements if you can make them) to accurately measure a capacitor. The fundamental equation for a capacitor is Q = CV, where Q is charge in Coulombs, C is capacitance in Farads, and V is voltage in Volts. Voltage is relatively easy to measure. Charge is not so easy to measure directly, it is more common to measure current I = dq / dt. But capacitors do not pass DC current, so a time-dependent current measurement is necessary. Ohm s Law is V = IR, where V and I are voltage and current, and R is resistance in Ohms. This serves largely as a definition of resistance R, but it is fairly straightforward to fabricate components (resistors) that obey Ohm s Law to great accuracy with good stability. Since DC voltage and current are readily measurable, resistance is also readily measurable. If a capacitor charged to voltage V 0 is discharged through a resistor R, the capacitor voltage V follows V (t) = V 0 exp ( t/τ) where τ = R C is known as the time-constant. If the voltage is measured as a function of time, the time-constant τ can be determined. If the resistance R is known, then the capacitance is C = τ /R. So measuring capacitance can be reduced to measuring the time-constant with a known resistance. Measuring the time-constant requires both charging the capacitor, and discharging it, while observing the voltage. You have already seen one approach to that problem in the LC circuit lab: use a square-wave function generator and an oscilloscope. For an RC circuit, the basic scheme is shown below. 1
If the square wave is between 0 Volts and V 0, the capacitor charges to V 0 on the positive part of the square wave, then discharges (back into the function generator) on the zero Volt part of the square wave. If there is a resistor between the function generator and the capacitor, the discharge will not be instantaneous. If the period of the square wave is much larger than the time constant, the voltage will obey V (t) = V 0 exp ( t /τ), where t = 0 refers to the falling edge of the square wave. The charging waveform in the same case will be V (t) = V 0 [1 exp ( t /τ) ] where t = 0 refers to the rising edge of the square wave. Apparatus Micsig model TO1072 tbook Mini touch-screen tablet oscilloscope Instek SFG-2110 Synthesized Function Generator (use TTL output) or Interstate Electronics F-31 Function Generator (use analog OUTPUT and OFFSET to zero) Three coaxial cables with BNC connectors BNC Tee connector OWON B41T+ Digital Multimeter Breadboard with 3 BNC connectors Capacitor, approximately 10 nf Various resistors The meter has a capacitance mode, and it s OK to use it to measure your capacitor. But you must develop (and next week execute) a procedure to measure the capacitance without using the capacitance mode of the meter. It is definitely OK, and expected, for you to use the meter to measure resistance values. It is suggested that you construct a circuit resembling this on your breadboard: It is suggested you make a diagram of your physical breadboard setup. 2
Exploration Questions and Suggestions Below are questions and suggestions to think about. We don t expect a written answer to each one. We do expect you to write your thoughts about some of these issues, as you develop your capacitance measurement method. We do expect you to take some data and analyze it far enough to get a preliminary measurement of the capacitor value, and your uncertainty on the value, in the lab today. You may continue to think and write about the questions and methods when you take your lab notebook home. What would be a reasonable resistor value to use with your capacitor? Would some values not work? What would be a reasonable period for the square wave? Would some values not work? How would you connect the function generator and oscilloscope to the breadboard? Hint: draw a diagram. How should you set the trigger on the oscilloscope? Does it matter? Why? What should be the horizontal scale (microseconds/division) on the oscilloscope? Why? Does your answer depend on the resistor value? On the period? Where should you locate the horizontal position of the signal on the oscilloscope? Why? Does your answer depend on the resistor value? On the period? What should be the vertical scale (volts/division) on the oscilloscope? Why? Does your answer depend on the resistor value? On the period? Where should you locate the vertical position of the signal on the oscilloscope? Why? Does your answer depend on the resistor value? On the period? Make a quick sketch of the signal with what you think are good choices for the resistor value, the function generator frequency or period, and the oscilloscope scales and offsets. Be sure to specify the conditions (so you could reproduce it). Make a spreadsheet graph of the voltage vs time signal with about 10 points. Be sure to specify the conditions (so you could reproduce it). You can and should use the oscilloscope cursors. Be aware of uncertainties. 3
What happens when you change the vertical scale of the voltage vs time graph to log scale? What happens to the error bars? Why? Is there a formula you could apply to your data points to linearize them, so you would get a straight line on a graph with linear scales? What is it? Try it and see if it works. How could you transform the error bars so you could plot them too? Make the linearized plot with error bars. Compare it to the un-linearized voltage vs time log-scale graph. Is there a way to turn a single (voltage, time) pair into a measurement of the time-constant? Hint: yes, if you make a few extra measurements first to plug into a formula. Write the formula, defining your terms. How do you calculate the uncertainty on such a single-point time-constant measurement? How does the uncertainty depend on the inputs? Write the formula or algorithm, defining your terms. From your spreadsheet data, make a graph of single-point time-constant, with error bar, vs time. The calculation of the time-constant, and the error bar on it, isn t trivial. But spreadsheets are a good way of doing such repetitive things. Since there is only one time-constant, this graph should of course be flat. But it will have fluctuations due to measurement uncertainties, and the uncertainties will be different for different points. What can you conclude about the validity of your time-constant value uncertainty calculations from this graph? Ask an instructor if unsure what to conclude. Which point in time gives the single best measurement of the time-constant (smallest error bar)? How is the time of that measurement related to the time-constant itself? Is there some way to use more than one voltage-time data-point to get a more accurate time-constant value than the single best measurement? Hint: yes, although some ways give worse accuracy than the best single value. What is the way? What do you get for the value and uncertainty of the time-constant using all of your data? How does the value and uncertainty compare to the single best measurement? 4
How do you turn your time-constant value (and uncertainty) into a capacitor value (and uncertainty)? Do you need other measurements? What are their uncertainties? How do they contribute to the uncertainty on the capacitor measurement? Would it be possible to improve the accuracy of your capacitance measurement by getting data with a different resistor value? What would you do with the data? Would it be possible to check the accuracy of your capacitance measurement by getting data with a different resistor value? What would you do with the data? Would it be possible to improve and check the accuracy of your capacitance measurement and its uncertainty by getting data with several resistor values? As you learned in the LC circuit lab, the function generator has an output impedance. How does that affect the time-constant? The oscilloscope input has an input impedance in MegaOhms to ground, and also an input capacitance in picofarads to ground. The values are printed near the connectors. How do these affect the time-constant? Type RG-58 coaxial cable has a capacitance of 100 pf/meter. How does this affect the time-constant? At Home, Write a Procedure for Capacitance Measurement Write it as if you are an engineer that is supervising a co-op student, and you want the student to measure some capacitors for you using the procedure. It should have an overview briefly describing the goal (measuring capacitance) and the general method (RC circuit, square wave, time-constant, ) This will include some equations and sketches. You don t need to explain the theory to the co-op student, just give enough background that s/he understands the context. It should list the required apparatus, and how it should be configured. This will include some diagrams. You don t need to write knob-by-knob instructions, you can assume the co-op student is familiar with function generators and oscilloscopes and cables and breadboards. It should list relevant settings of the equipment. It should describe the measurements to be made and the conditions for making them. 5
It should describe what to do with the measurements, e.g., graphs of raw data, formulas for transforming data, graphs of transformed data, formulas for combining data, etc. Following the procedure should result in a capacitance value, and an uncertainty of the capacitance value. The amount of data to be taken, and the processing of the data into the answer, should be chosen such that the result for one capacitor can be obtained comfortably in 2 hours. The procedure should have some internal consistency checks, so when you read the report from the co-op student who executes the procedure, you can see if they did it properly, and estimated uncertainties reasonably. Next Week, Execute Your Procedure Be sure to bring your lab notebook back to class! Note that it s OK to bring a prepared spreadsheet to class, including space for data table(s), and the graphs and calculations required to execute your procedure. You can tune it up at home with invented data. In fact, a smart supervisor would give his co-op student such a spreadsheet. Also note that for the lab exam at the end of the course, you will need to be able to do (simple) data analysis without any spreadsheet, let alone a prepared spreadsheet. (You will be allowed a formula sheet). So make sure that you understand what your analysis steps are, and why. 6