RL circuit equation

Explore the RL circuit equation for series and parallel configurations, understand its applications, and learn how to calculate current.

Understanding the RL Circuit Equation

An RL circuit, also known as a resistor-inductor circuit, is an electrical circuit composed of a resistor (R) and an inductor (L) connected either in series or parallel. These circuits play a vital role in various applications, including power electronics, filters, and oscillators. To analyze and design RL circuits effectively, it is essential to understand the RL circuit equation.

The RL Circuit Equation: The Basics

In an RL circuit, when a voltage is applied, the current flowing through the circuit changes with time due to the inductor’s behavior. The inductor opposes the change in current by producing a voltage across its terminals, which is proportional to the rate of change of current. This property of inductors is defined by Faraday’s law of electromagnetic induction and can be mathematically represented using the following equation:

1. v_L = L * (di/dt)

Where:

• v_L represents the voltage across the inductor
• L is the inductance of the inductor
• di/dt is the rate of change of current with respect to time

RL Circuit Equation for Series Configuration

In a series RL circuit, the resistor and inductor are connected end-to-end, allowing the same current to flow through both components. By applying Kirchhoff’s voltage law, we can derive the equation governing the circuit’s behavior:

1. v_in = v_R + v_L

Where:

• v_in represents the input voltage
• v_R is the voltage across the resistor
• v_L is the voltage across the inductor

Since v_R = R * i and v_L = L * (di/dt), the equation can be rewritten as:

1. v_in = R * i + L * (di/dt)

RL Circuit Equation for Parallel Configuration

In a parallel RL circuit, the resistor and inductor share the same voltage across their terminals, but the current flowing through them can be different. By applying Kirchhoff’s current law, we can derive the equation governing the circuit’s behavior:

1. i_in = i_R + i_L

Where:

• i_in represents the input current
• i_R is the current through the resistor
• i_L is the current through the inductor

Since i_R = v / R and i_L = (1/L) * ∫v * dt, the equation can be rewritten as:

1. i_in = (v / R) + (1/L) * ∫v * dt

Example of RL Circuit Calculation

Let’s consider a series RL circuit with a 10V DC voltage source, a 20-ohm resistor, and a 0.1 H inductor. We will calculate the time constant and the current through the circuit after a certain time.

Step 1: Calculate the Time Constant

The time constant, τ, of an RL circuit is given by the following equation:

1. τ = L / R

Where:

• τ represents the time constant
• L is the inductance of the inductor
• R is the resistance of the resistor

Using the given values, we can calculate τ:

1. τ = 0.1 H / 20 Ω
2. τ = 0.005 s

Step 2: Calculate the Current at a Specific Time

For a series RL circuit, the current at any given time t can be calculated using the following equation:

1. i(t) = (V / R) * (1 – e^-t/τ)

Where:

• i(t) is the current at time t
• V is the voltage across the circuit
• e is Euler’s number, approximately equal to 2.718

Let’s calculate the current after 0.01 s:

1. i(0.01) = (10 V / 20 Ω) * (1 – e^{-(0.01 / 0.005)})
2. i(0.01) = 0.5 A * (1 – e^-2)
3. i(0.01) ≈ 0.432 A

After 0.01 s, the current through the series RL circuit is approximately 0.432 A.