Question

The equation governing the flow of current \(i\) in a series \(L R\) circuit with applied constant voltage \(E\) is $$ L \frac{\mathrm{d} i}{\mathrm{~d} t}+R i=E $$ (a) Solve this equation subject to the condition \(i(0)=0 .\) (b) State the final value of the current. (c) Find the time taken for the current to reach \(95 \%\) of its final value.

Short answer

Question: Determine the time taken for the current to reach 95% of its final value in an LR circuit given by the differential equation, \(L \frac{di}{dt} + Ri = E\) and initial condition \(i(0) = 0\). Answer: The time taken for the current to reach \(95\%\) of its final value is \(-\frac{L}{R}\ln{(0.05)}\).

Step-by-step solution

Separate the variables

Rewrite the equation as $$ \frac{\mathrm{d}i}{E - Ri} = \frac{\mathrm{d}t}{L}. $$ This form of the equation allows us to separate the variables, so we can integrate both sides.

Integrate both sides

Integrate both sides of the equation with respect to their respective variables: $$ \int \frac{\mathrm{d}i}{E - Ri} = \int \frac{\mathrm{d}t}{L}. $$

Apply integration techniques

Use integration techniques such as u-substitution or integration by parts to solve the integrals. In this case, let \(u=E-Ri\), so \(\mathrm{d}u = -R\mathrm{d}i\). Rewrite the integral as: $$ \int \frac{-1}{R}\frac{\mathrm{d}u}{u}=\int \frac{\mathrm{d}t}{L}. $$ Now integrate both sides: $$ -\frac{1}{R} \ln |u| = \frac{t}{L} + C, $$ where \(C\) is the constant of integration.

Solve for the current i(t)

Substitute \(u = E-Ri\) back into the equation and solve for \(i(t)\): $$ -\frac{1}{R} \ln |E - Ri| = \frac{t}{L} + C. $$ Rearrange the equation to isolate \(i\) on one side: $$ i(t) = \frac{E}{R} - \frac{E}{R}e^{-\frac{R}{L} (t - C)}. $$

Apply the initial condition

Use the initial condition \(i(0) = 0\) to find the value of the constant \(C\): $$ 0 = \frac{E}{R} - \frac{E}{R}e^{-\frac{R}{L} (0 - C)}. $$ This leads to \(C=-\frac{L}{R}\ln{(-1)}\). Insert the constant \(C\) into the equation for \(i(t)\): $$ i(t)=\frac{E}{R}\left(1-e^{-\frac{Rt}{L}}\right). $$ (a) The solution to the equation subject to the condition \(i(0)=0\) is: $$ i(t) = \frac{E}{R}\left(1-e^{-\frac{Rt}{L}}\right). $$

Determine the final value of current

As \(t\rightarrow\infty\), the exponential term \(e^{-\frac{Rt}{L}}\) goes to 0. The final value of the current is given by: $$ I_{final}=\lim_{t\to\infty} \frac{E}{R}\left(1-e^{-\frac{Rt}{L}}\right)= \frac{E}{R}. $$ (b) The final value of the current is \(\frac{E}{R}\).

Determine the time for current to reach \(95 \%\) of its final value

Set the current \(i(t)\) equal to \(0.95 I_{final}\) and solve for \(t\): $$ 0.95\frac{E}{R} = \frac{E}{R}\left(1-e^{-\frac{Rt}{L}}\right). $$ Simplify the equation and solve for \(t\): $$ 0.95 = 1 - e^{-\frac{Rt}{L}}. $$ $$ e^{-\frac{Rt}{L}} = 0.05, $$ $$ t = -\frac{L}{R}\ln{(0.05)}. $$ (c) The time taken for the current to reach \(95\%\) of its final value is \(-\frac{L}{R}\ln{(0.05)}\).

Key concepts

Differential Equations

Differential equations are mathematical tools we use to describe relationships involving rates of change. They are central in engineering, physics, economics, and other fields. For instance, in an LR circuit, the rate at which the current changes is directly influenced by the circuit's resistance (R) and inductance (L). The equation we use to model this behavior is \[L \frac{\mathrm{d} i}{\mathrm{d} t}+R i=E\], where \(i\) is the current, \(E\) is the constant voltage, \(t\) is time, and \(L\) and \(R\) represent inductance and resistance, respectively. This particular form of a differential equation is a 'first-order linear ordinary differential equation' and is powerful because it allows us to predict the behavior of the current over time. We solve it using separation of variables and integration, which gives us the current as a function of time — a crucial insight for designing and understanding electrical circuits.

Transient Response

The transient response in an electrical circuit refers to how the current and voltage behave between their initial and final steady states — essentially, it’s what happens right after a circuit is switched on or off. In our LR circuit, the transient response is described by the equation \[i(t) = \frac{E}{R}\left(1-e^{-\frac{Rt}{L}}\right)\]. This represents how the current builds up from zero to its final steady state value. We call this process 'transient' because it's temporary, diminishing over time until the system stabilizes and the response is 'steady'. Understanding the transient response is critical for predicting how quickly a circuit will reach its stable operating condition and for ensuring that the circuit can handle the initial surge in current.

Final Value Theorem

The Final Value Theorem is a concept used in control theory and signal processing to predict the steady-state value of a system's response as time approaches infinity, assuming the final value exists. In the context of our LR circuit example, the Final Value Theorem helps confirm that the current eventually settles at a final value, \(I_{final}\). Using the theorem, we have \[I_{final}=\lim_{t\to\infty} \frac{E}{R}\left(1-e^{-\frac{Rt}{L}}\right) = \frac{E}{R}\], indicating that, regardless of the transients, the final current in the circuit stabilizes at a value determined by the voltage and resistance. This theorem is crucial for ensuring that long-term system behavior is consistent with design specifications and safety requirements.

Exponential Decay

Exponential decay is a process that decreases at a rate proportional to its current value, characterized by a consistent halving time or 'half-life'. In electrical engineering, it describes how transient effects, like the changing current in an LR circuit, diminish over time. The current's exponential decay is captured by the term \(e^{-\frac{Rt}{L}}\) in our equation for \(i(t)\). This term shows that, as time increases, the exponent becomes more negative, causing \(e^{-\frac{Rt}{L}}\) to approach zero, and the system to move towards its stable state. Understanding exponential decay helps engineers determine how quickly things like currents in circuits will approach their steady-state values, which is essential for timing and safety considerations.