Spring 2019

ECE 110

Course Notes

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In lab you use equipment that displays readings to several digits, but the measurements are often so noisy that not all the digits hold still. You should take care to record only the significant digits (the ones that are meaningful). Specifically, you should record all the non-fluctuating digits and the average value of the first fluctuating digit you encounter. All other digits should be ignored.

Figure 1

Instrument measurement example
Fig. 1: Instrument measurement example. This 4-digit reading is stable in the first two digits. The third digit fluctuates between $0$ and $1$, but spends more time as $0$. The fourth digit has no meaningful value and so should be ignored. Therefore, the measurement should be recorded as $2.70 \times 10^{3}$ or equivalently as $2.70\text{E}3$. Note that recording the measurement as $2700$ is ambiguous because it is unclear whether the zeroes are significant or not.

In mathematical operations, the significant digits (also known as significant figures) are any digits except:

  • All leading zeros.
  • All trailing zeros that are used as placeholders to indicate the order of magnitude of the number. If the trailing zeros on your measurement are significant, use scientific notation to indicate which digits are significant.
  • Digits that were introduced by calculation. The results of multiplications and divisions should have the same number of significant digits as the least precise measurement.

Number Number of Significant Digits Reason
$0.0004$ 1 Leading zeros are not significant.
$0.00040$ 2 Leading zeros are not significant.
$2700$ 2, 3 or 4 The trailing zeros may or may not be significant. This recorded value is ambiguous.
$2.70 \times 10^{3}$ 3 Trailing zeros after a decimal point are not placeholders and are significant.
$(0.52)(0.10)=0.052$ 2 Both $0.52$ and $0.10$ have 2 significant digits.
$(0.52)(0.1)=0.05$ 1 The least precise number $0.1$ has 1 significant digit.
Table 1: Significant digit examples.

When recording a measurement, you must always include the corresponding units. The International System of Units (SI) defines standard units for all physical quantities, which are derived from 7 mutually independent base units. For example, the unit for force is defined as newton with symbol N, so that $1 \text{ N} = 1 \text{ kg}\cdot\text{m}\cdot\text{s}^{-2}$.

Quantity Unit Name Unit Symbol
energy joule J
electric charge coulomb C
voltage volt V
electric current ampere (amp) A
capacitance farad F
resistance ohm $\Omega$
power watt W
time second s
frequency hertz Hz
Table 2: Common SI units in ECE 110.

Any unit can be scaled by various factors of 10 by applying prefixes to the unit name and its symbol. For example, a millimeter (symbol mm) is one-thousandth of a meter.

Factor Name Prefix Symbol Prefix
$10^{-15}$ femto- f-
$10^{-12}$ pico- p-
$10^{-9}$ nano- n-
$10^{-6}$ micro- $\mu$-
$10^{-3}$ milli- m-
$10^{3}$ kilo- k-
$10^{6}$ mega- M-
$10^{9}$ giga- G-
$10^{12}$ tera- T-
$10^{15}$ peta- P-
Table 3: Common metric prefixes.

Note that when it comes to bits and bytes (as in data storage size), the prefix k- sometimes means a factor $2^{10}=1024$ and the prefix M- sometimes means a factor of $2^{20}=1048576$.

As an engineer, you will often need to characterize the behavior of a device utilized in larger systems by varying one parameter of the device and measuring some other parameter. This process typically requires filling tables with measurements, performing calculations and generating graphs with the collected data. For example, you might characterize a motor by varying the voltage applied across its terminals and measuring the current flowing through it. Table 4 shows a good example of how to record data for this type of experiment. Observe that each measurement has an appropriate number of significant digits and that the units are labeled in the column headers.

Voltage (V) Current (A) Comments
$0.00$ $0.000$ Power supply is off.
$0.11$ $0.001$ I knocked my motor off the table, but it still seems to work.
$0.20$ $0.011$
$0.31$ $0.021$ The wheel just started turning.
$0.40$ $0.034$ Something smells burnt!
$0.50$ $0.001$ Motor seems to have died.
Table 4: Example of a table with comments.

Notice that it is just as important to make descriptive notes. In this course, the tables provided in your lab procedures will have a column for comments. The notes you make while recording measurements can describe the behavior of the device you are testing (e.g. "the wheel just started turning!"), events in the surrounding environment (e.g. "something smells burnt"), or actions taken by the experimenter that might have had an effect on the measurement (e.g. "I knocked my motor off the table, but it still seems to work").

Graphs are visual presentations of measurements that make the data easy to read and understand. To avoid ambiguity and erroneous conclusions, you should describe a graph with a title or caption, label its axes completely and choose an appropriate resolution to show sufficient detail. Most of your graphs will be computer generated, but Fig. 2 shows a hand-drawn example.

Figure 2

Graph example
Fig. 2: Graph example. This graph plots the data found in Table 4. The title above the graph gives the reader an idea of what is plotted and why it matters. Notice that the axes are labeled with physical quantities, current and voltage, and their corresponding units, amperes and volts. The scale of the graph is chosen to show maximum detail. Each axis is also marked off with at least 3 divisions with numerical values. The linear fit is clearly labeled; you can use a legend if you have multiple curves on a graph.

In ECE 110 lab, you must follow all of the guidelines described in Fig. 2 to obtain full credit. Additionally, the titles of your graphs should also indicate which steps (or questions) in the procedure are being addressed.

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