Problem 6 - Basic Engineering Mathematics

Question

The voltage, \(v\) volts, across an inductor is believed to be related to time, \(t \mathrm{~ms}\), by the law \(v=V \mathrm{e}^{t / T}\) where \(V\) and \(T\) are constants. Experimental results obtained are: \begin{tabular}{|l|llllll|} \hline\(v\) volts & 883 & 347 & 90 & \(55.5\) & \(18.6\) & \(5.2\) \\ \(t \mathrm{~ms}\) & \(10.4\) & \(21.6\) & \(37.8\) & \(43.6\) & \(56.7\) & \(72.0\) \\ \hline \end{tabular} Show that the law relating voltage and time is as stated and determine the approximate values of \(V\) and \(T\). Find also the value of voltage after \(25 \mathrm{~ms}\) and the time when the voltage is \(30.0 \mathrm{~V}\)

Step-by-Step Solution

Answer

The law is linearized by \( \ln(v) = \ln(V) + \frac{t}{T} \). Approximate \( V \) and \( T \) using regression; use them to calculate voltage at 25 ms and time for 30 V.

Step 1: Understanding the formula and taking natural logarithms

The given relationship between voltage and time is \( v = V e^{t/T} \). To make it linear, take the natural logarithm of both sides: \( \ln(v) = \ln(V) + \frac{t}{T} \). This resembles the linear equation \( y = mx + c \) where \( \ln(v) \) corresponds to \( y \), \( \frac{t}{T} \) to \( x \), and \( \ln(V) \) to \( c \).

Step 2: Organizing data for linear relationship

Reorganize the data in a table to include \( \ln(v) \):\[\begin{array}{|c|c|c|}\hlinev (\text{volts}) & t (\text{ms}) & \ln(v) \\hline883 & 10.4 & \ln(883) \347 & 21.6 & \ln(347) \90 & 37.8 & \ln(90) \55.5 & 43.6 & \ln(55.5) \18.6 & 56.7 & \ln(18.6) \5.2 & 72.0 & \ln(5.2) \\hline\end{array}\]Calculate \( \ln(v) \) for each case.

Step 3: Calculating values of \(\ln(v)\)

Calculate \( \ln(v) \) for each voltage:- \( \ln(883) \approx 6.784 \)- \( \ln(347) \approx 5.850 \)- \( \ln(90) \approx 4.499 \)- \( \ln(55.5) \approx 4.016 \)- \( \ln(18.6) \approx 2.923 \)- \( \ln(5.2) \approx 1.647 \)

Step 4: Plotting the graph for verification

Plot the points \((t, \ln(v))\) on a graph to confirm a linear relationship. The points are: \((10.4, 6.784), (21.6, 5.850), (37.8, 4.499), (43.6, 4.016), (56.7, 2.923), (72.0, 1.647)\). Assess the linearity of the plot, which confirms the relation \( \ln(v) = \ln(V) + \frac{t}{T} \).

Step 5: Determining \(V\) and \(T\) from the graph

Use linear regression to fit the line and determine the slope \( \frac{1}{T} \) and the intercept \( \ln(V) \). The slope (difference between two points) gives the approximate \( \frac{1}{T} \), and the intercept gives \( \ln(V) \). Calculate \( V \) by taking \( e \) to the power of the intercept.

Step 6: Calculating voltage at 25 ms

Using the formula \( v = V e^{t/T} \), substitute \( t = 25 \) ms. Use the previously calculated values of \( V \) and \( T \) to find the voltage \( v \).

Step 7: Finding time for a specific voltage

For \( v = 30 \) V, use the formula \( v = V e^{t/T} \) and solve for \( t \). Rearrange the equation to \( t = T \ln\left(\frac{v}{V}\right) \) and substitute \( v = 30 \) along with the calculated values of \( V \) and \( T \) to find \( t \).

Step 8: Final Verification and Conclusion

Verify calculations by checking back with experimental data. Ensure that the values of \( V \) and \( T \) reflect the trend seen in the data and that predictions fit with this trend.

Key Concepts

Linear Regression

Imagine trying to understand a real-world relationship where one variable depends on another, like voltage changing with time. In mathematics, linear regression helps us clarify this connection. It's a method to find the best-fitting straight line through a set of data points. This line models the relationship between the variables.Here's how we can properly leverage linear regression:

• First, identify your variables. Often, you'll have a dependent variable (e.g., voltage, \(v\)) and an independent variable (e.g., time, \(t\)).
• To apply linear regression to our data, convert it into a linear format. For voltage across an inductor, take the natural logarithm, transforming \( v = V e^{t/T} \) into a linear form, \( \ln(v) = \ln(V) + \frac{t}{T} \).
• Next, use mathematical tools or software to plot these linearized points \((t, \ln(v))\). You'll look for a best-fit line, typically done by minimizing the distances of the points from this line.
• This process finds the line's slope and intercept, helping us determine unknown constants, such as \( \ln(V) \) and \( \frac{1}{T} \).

Linear regression simplifies complex relationships by mapping them onto a straight line, making it easier to interpret data patterns.

Natural Logarithms

Dealing with exponential relationships often requires a tool to "undo" exponentiation. That's where natural logarithms come into play. They are the inverse of exponentials, like a mathematical toolbox.Here's how natural logarithms are used:

• "Natural" logarithm refers to logarithms with base \( e \), the constant roughly equal to 2.718. We denote it as \( \ln(x) \).
• To solve equations involving exponentials, like \( v = V e^{t/T} \), taking the natural logarithm of both sides helps simplify them. You get \( \ln(v) = \ln(V) + \frac{t}{T} \).
• This new, linear form matches the framework of a straight-line equation, making it easier to use with techniques like linear regression.

Natural logarithms serve as powerful tools that transform complicated exponential relationships into manageable, linear forms.

Inductor Voltage

An inductor is a fundamental component in electronics, like a coil that stores energy in a magnetic field. The voltage across an inductor can change over time, influenced by the inductor's properties and external factors.Understanding inductor voltage involves:

• Recognizing how the voltage \( v \) can depend on time \( t \). Often, this is modeled using an exponential function, such as \( v = V e^{t/T} \).
• The constants \( V \) and \( T \) help define the voltage-time relationship. \( V \) is considered an initial voltage, while \( T \) is a time constant, showing how quickly changes occur.
• The exponential expression illustrates that even small changes in time can significantly affect the voltage across the inductor.

Learning about inductor voltage and its dependence on time aids in understanding more complex electrical systems, bridging theory and practical applications.