RL circuits, composed of resistors (R) and inductors (L), are essential components in electrical and electronic systems. These circuits exhibit unique behavior during transient processes, such as when a voltage source is connected or disconnected. RL circuits are used in various applications, including filters, oscillators, and damping systems.

Energizing Process:

When an RL circuit is connected to a DC voltage source, current starts to flow through the resistor and inductor. As the current increases, the inductor generates a back electromotive force (EMF) that opposes the change in current. The key equations for the energizing process are:

- Current through the inductor: I(t) = (V_source / R) * (1 – e^(-t/τ))
- Voltage across the inductor: V_L(t) = V_source * e^(-t/τ)
- Time constant (τ) for the RL circuit: τ = L / R

De-energizing Process:

When the voltage source is disconnected, and the inductor is allowed to discharge through the resistor, the stored energy in the inductor’s magnetic field is released, generating a back EMF that causes the current to decay over time. The key equations for the de-energizing process are:

- Current through the inductor: I(t) = I_initial * e^(-t/τ)
- Voltage across the inductor: V_L(t) = -V_initial * e^(-t/τ)
- Time constant (τ) for the RL circuit: τ = L / R

Applications:

- Filters: RL circuits can be used as low-pass or high-pass filters, attenuating specific frequencies while allowing others to pass through. In a low-pass filter configuration, the output is taken across the inductor, while in a high-pass filter configuration, the output is taken across the resistor.
- Oscillators: RL circuits can be combined with other circuit elements, such as capacitors, to create oscillators that generate continuous, periodic waveforms. These oscillators can be utilized in signal generation, frequency synthesis, and clock circuits.
- Damping Systems: RL circuits can be employed as damping elements in systems where oscillations or vibrations need to be controlled, such as in mechanical or electrical systems.
- Pulse Generators: RL circuits can be used to generate pulses or pulse trains with specific rise and fall times, which can be adjusted by choosing appropriate resistor and inductor values.

Understanding the behavior, equations, and applications of RL circuits is crucial for designing and analyzing various electrical and electronic systems. These fundamental circuits are widely used in signal processing, control, and power systems, making them an essential topic for engineers and technicians.

## Example of Calculation

An RL (Resistor-Inductor) circuit is a simple electrical circuit composed of a resistor and an inductor connected either in series or parallel. Here, we will discuss a series RL circuit, which is a common configuration. The current through the circuit and the voltage across the inductor change over time when a voltage is applied or removed from the circuit.

Let’s consider an example of an RL circuit calculation involving a circuit response to a voltage source:

Given values:

- Voltage source (V_source): 12 V
- Resistor (R): 200 Ω
- Inductor (L): 400 mH (0.4 H)

We will calculate the current through the circuit (I) and the voltage across the inductor (V_L) at a specific time (t) after the voltage source is applied.

- Calculate the time constant (τ) of the RL circuit: τ = L / R τ = 0.4 H / 200 Ω = 0.002 s (2 ms)

The time constant (τ) indicates the time required for the current through the circuit to reach approximately 63.2% of its final value or the time needed for the voltage across the inductor to decrease to 36.8% of its initial value.

- Choose a specific time (t) after the voltage source is applied: For this example, let’s pick t = 0.001 s (half of the time constant).
- Calculate the current through the circuit (I) at time t: I(t) = (V_source / R) × (1 – e^(-t/τ)) I(0.001) = (12 V / 200 Ω) × (1 – e^(-0.001/0.002)) I(0.001) ≈ (0.06 A) × (1 – e^(-0.5)) ≈ 0.06 A × (1 – 0.6065) ≈ 0.0236 A

At t = 0.001 s, the current through the circuit is approximately 0.0236 A (23.6 mA).

- Calculate the voltage across the inductor (V_L) at time t: V_L(t) = V_source × e^(-t/τ) V_L(0.001) = 12 V × e^(-0.001/0.002) V_L(0.001) ≈ 12 V × e^(-0.5) ≈ 12 V × 0.6065 ≈ 7.278 V

At t = 0.001 s, the voltage across the inductor is approximately 7.278 V.

This example demonstrates how to calculate the current through the circuit and the voltage across the inductor at a specific time after the RL circuit is connected to a voltage source. The same approach can be used for a circuit response when the voltage source is removed, with some modifications to the equations.