Problem 31 - Advanced Engineering Mathematics

Question

An \(L R\)-series circuithas a variable inductor with the inductance defined by $$ L(t)= \begin{cases}1-\frac{1}{10} t, & 0 \leq t<10 \\ 0, & t \geq 10 .\end{cases} $$ Find the current \(i(t)\) if the resistance is \(0.2 \mathrm{ohm}\), the impressed voltage is \(E(t)=4\), and \(i(0)=0\). Graph \(i(t)\).

Step-by-Step Solution

Answer

The current \(i(t)\) can be expressed as a piecewise function, using solutions found over different intervals.

Step 1: Write Down the Differential Equation

In an \( L R \)-series circuit, the equation is \( E(t) = L(t) \frac{di(t)}{dt} + R i(t) \). Substitute the given values \( E(t) = 4 \), \( R = 0.2 \), and the piecewise function \( L(t) \). We have:\[4 = \left(1 - \frac{t}{10}\right) \frac{di}{dt} + 0.2 \cdot i(t) \text{ for } 0 \leq t < 10\]

Step 2: Solve the Differential Equation from 0 to 10

For \( 0 \leq t < 10 \), solve the equation \(4 = \left(1 - \frac{t}{10}\right) \frac{di}{dt} + 0.2 \cdot i(t)\).This is a first-order linear non-homogeneous differential equation. Rearrange as:\[\left(1 - \frac{t}{10}\right) \frac{di}{dt} = 4 - 0.2i(t)\]Divide through by \(1 - \frac{t}{10}\):\[\frac{di}{dt} = \frac{4 - 0.2i(t)}{1 - \frac{t}{10}}\]Solve this differential equation using an integrating factor or separation of variables.

Step 3: Apply Initial Condition

Using the general solution from Step 2, apply the initial condition \(i(0) = 0\) to find the specific solution for \(i(t)\). This will give you the form of \(i(t)\) for \(0 \leq t < 10\).

Step 4: Determine Behavior for t ≥ 10

For \(t \geq 10\), the inductance \(L(t) = 0\). The equation becomes simple:\[4 = 0.2i(t)\]Solve for \(i(t)\):\[i(t) = \frac{4}{0.2} = 20\]

Step 5: Compile the Complete Solution for i(t)

Use the solutions obtained in steps 3 and 4 to write a piecewise solution for the current \(i(t)\) as:\[i(t)= \begin{cases}function \, \text{from} \, Step \, 3, & 0 \leq t < 10 \20, & t \geq 10\end{cases}\]Substitute the calculated function from Step 3 into this piecewise format.

Step 6: Graph the Current i(t)

Plot the function \(i(t)\) over the interval \([0, 15]\) using the piecewise function obtained in Step 5. This graph will show the current increasing according to the found function until \(t = 10\) and then staying constant at 20 for \(t \geq 10\).

Key Concepts

LR-series circuit

An LR-series circuit is a type of electrical circuit that includes an inductor (L) and a resistor (R) connected in sequence. These circuits are commonly used to study transient responses, where the current or voltage takes time to reach its final value after a change, such as a sudden application of a voltage. This time-dependent behavior differentiates an LR-series circuit from static circuits.

**Key Components of an LR-series Circuit**:
- **Inductor (L):** This coil stores energy in a magnetic field when electric current flows through it, and opposes changes in the current.
- **Resistor (R):** This element opposes the flow of electric current, converting electrical energy into heat.
- **Voltage Source (E):** Supplies the power driving the current through the circuit.

In an LR-series circuit, the voltage drop across the inductor and the resistor must sum up to equal the imposed voltage from the source. This relationship helps us understand how current changes with time as the circuit reaches its steady state.

Variable Inductance

Variable inductance involves an inductor whose inductance value can change rather than remain constant. This means that the inductor can dynamically adjust its opposition to changing currents over time, providing a more flexible control over the circuit behavior.

In the given problem, the inductance is defined as a piecewise function, where:\[L(t)= \begin{cases}1-\frac{1}{10} t, & 0 \leq t<10 \ 0, & t \geq 10\end{cases}\]**Implications of Variable Inductance**:- By having a variable inductance, the circuit's response to an applied voltage is time-dependent, exactly following the given inductance formula.
- As the inductor varies over time, its impact on the circuit's impedance changes, influencing how quickly the circuit reaches steady state.
- Understanding this time-variance is crucial for modeling real-world applications, where inductors may not always maintain a constant inductance due to factors like core saturation.

Initial Condition

The initial condition specifies the state of a system, such as an LR-series circuit, at the start of an analysis or simulation. For differential equations, initial conditions are essential because they help determine a unique solution.

For instance, in our problem, the initial condition is given as \( i(0) = 0 \). This tells us that the current through the circuit is zero at time zero. Initial conditions are utilized for:

- **Solving Differential Equations:** They provide the specific values needed to solve for constants present in the solution.
- **Setting System Start Points:** By defining what happens at the very start, we influence how the solution behaves over time.
- **Ensuring System Stability:** Knowing the initial condition aids in predicting whether a system will stabilize or escalate under certain conditions.

Without these initial conditions, we might get a general solution but not one tailored to the specific moment or most applicable scenario.

Piecewise Function

A piecewise function is a math function defined by multiple sub-functions, each corresponding to a certain interval in its domain. In real-world applications, like circuits, piecewise functions are essential for modeling behaviors that change over time or under specific conditions.

**Key Aspects of Piecewise Functions**:
- They allow us to describe complex behaviors in a simplified way. Each segment of the function caters to a different operational region or condition.
- In our problem, the inductance function \( L(t) \) is a piecewise function that changes its value based on time \( t \).
- Piecewise functions are crucial when certain properties of a system (like inductance) suddenly change at determined points, impacting the system's overall behavior.

In the exercise, transitioning from one piece of the function to another (at \( t = 10 \)) signifies a change in the circuit dynamics, influencing the solution and the graphical representation of the current \( i(t) \). This highlights their role in accurately depicting scenarios for different time periods or conditions.