Question

For an RL circuit with \(R=21.84 \Omega\) and \(L=55.93 \mathrm{mH}\), how long does it take the current to reach \(75 \%\) of its maximum value?

Short answer

Answer: Approximately 3.56 x 10^-3 seconds.

Step-by-step solution

Finding the time constant

The time constant of an RL circuit is given by \(\tau = \frac{L}{R}\). We can calculate the time constant with the given values for \(R\) and \(L\): \(\tau = \frac{L}{R} = \frac{55.93 \mathrm{mH}}{21.84 \Omega} = \frac{55.93 \times 10^{-3} \mathrm{H}}{21.84 \Omega} = 2.56 \times 10^{-3} \mathrm{s}\)

Finding the time for the current to reach 75% of its maximum value

Using the formula for the current, \(i(t)=I_0 (1 - e^{-\frac{t}{\tau}})\), set \(i(t)\) to be \(75 \%\). Since we're looking for the time when i(t) is \(75 \%\) of its maximum value, we may represent this as \(0.75I_0 = I_0 (1 - e^{-\frac{t}{\tau}})\). Now, we will find the value of \(t\). Divide both sides by \(I_0\): \(0.75 = 1 - e^{-\frac{t}{\tau}}\) Subtract \(0.75\) from both sides: \(0.25 = e^{-\frac{t}{\tau}}\) Take the natural logarithm of both sides to get: \(ln(0.25) = -\frac{t}{\tau}\) Multiply by \(-\tau\) to solve for \(t\): \(t = -\tau\cdot ln(0.25) = -2.56 \times 10^{-3} \mathrm{s}\cdot ln(0.25) \approx 3.56 \times 10^{-3} \mathrm{s}\) So, it takes approximately \(3.56 \times 10^{-3} \mathrm{s}\) for the current to reach \(75\%\) of its maximum value.

Key concepts

Time Constant Calculation

In an RL circuit, the time constant is a crucial value that provides an insight into how quickly the circuit responds to changes in voltage or current. The time constant, represented by the symbol \(\tau\), is calculated by dividing the inductance \(L\) by the resistance \(R\) of the circuit. Formally, the equation is \(\tau = \frac{L}{R}\).

For example, if an RL circuit has an inductance of \(L = 55.93 \mathrm{mH}\) and a resistance of \(R = 21.84 \Omega\), the time constant is:\[\tau = \frac{55.93 \times 10^{-3} \mathrm{H}}{21.84 \Omega} = 2.56 \times 10^{-3} \mathrm{s}\]
This value indicates the time in seconds taken for the current to reach about 63.2\% of its final value after the voltage is applied. The smaller the time constant, the quicker the circuit will respond. Understanding the time constant is essential for designing circuits with desired electrical response speeds.

Exponential Growth and Decay

Exponential growth and decay describe processes that change proportionally to their current value, which is characteristic in many natural phenomena like population growth, radioactive decay, and in electrical circuits, including the charging and discharging of currents in RL circuits.

In our RL circuit example, once the switch is closed to allow current to flow, the current doesn't immediately jump to its maximum value. Instead, it 'grows' exponentially according to a special formula:\[i(t)=I_0 (1 - e^{-\frac{t}{\tau}})\]
Here, \(i(t)\) is the current at time \(t\), \(I_0\) is the maximum current value, \(\tau\) is the time constant, and \(e\) is the base of the natural logarithms. This formula reflects how the current 'decays' towards the remaining percentage to reach the maximum value over time. As time goes on, the changing rate becomes slower until the current steadies at \(I_0\).

Natural Logarithm Applications

The natural logarithm, written as \(\ln\), is a mathematical function that undoes the exponential function, much like subtraction undoes addition. It's incredibly useful in solving equations where the variable is an exponent, particularly in fields like physics, engineering, and economics. One common application is finding the time it takes for an exponential process to reach a certain level.

In our RL circuit scenario, to find the time it takes for the current to reach 75\% of its maximum value, we rearrange the exponential equation and take the natural logarithm on both sides. The key steps are to isolate the exponential term and then apply \(\ln\) to both sides, leading to:\[ t = -\tau\cdot \ln(0.25) \]
By applying the value of the time constant \(\tau\), we calculate the specific time for the circuit. In practical situations, natural logarithms are used to analyze circuit behavior over time, calculate the half-life of radioactive elements, and determine the time for investments to grow, showcasing its versatility and importance in various fields.