L-R circuits
L-R Circuits
L-R circuits are electrical circuits that consist of an inductor (L) and a resistor (R) connected in series with a power supply. These circuits are fundamental in understanding how inductors behave in a circuit when direct current (DC) is applied or removed. The behavior of L-R circuits is characterized by the time-dependent change of current, which is governed by the inductor's ability to resist changes in current flow through it.
Key Concepts
Inductance (L)
Inductance is a property of an electrical conductor by which a change in current flowing through it induces an electromotive force (EMF) in both the conductor itself (self-inductance) and in any nearby conductors (mutual inductance). This is often referred to as the "inertia" of the circuit, as it tends to oppose changes in current.
Resistance (R)
Resistance is a measure of the opposition to the flow of electric current through a conductor. It is the property that dissipates energy in the form of heat.
Time Constant (τ)
The time constant, denoted by τ (tau), is a measure of the time it takes for the current to rise to approximately 63.2% of its final value after the voltage is applied, or to fall to 36.8% of its initial value after the voltage is removed. It is given by the formula:
$$ \tau = \frac{L}{R} $$
where L is the inductance and R is the resistance.
Charging and Discharging of an L-R Circuit
Charging Phase
When a DC voltage is applied to an L-R circuit, the current does not immediately reach its maximum value due to the inductor's opposition to the change in current. Instead, the current increases gradually and can be described by the equation:
$$ i(t) = I_{max} \left(1 - e^{-\frac{t}{\tau}}\right) $$
where:
- $i(t)$ is the current at time t,
- $I_{max}$ is the maximum current (which would be $V/R$ if the inductor were not present),
- $e$ is the base of the natural logarithm,
- $t$ is the time after the voltage is applied,
- $\tau$ is the time constant.
Discharging Phase
When the voltage source is removed, the inductor releases its stored energy and the current decreases exponentially according to the equation:
$$ i(t) = I_{0} e^{-\frac{t}{\tau}} $$
where:
- $i(t)$ is the current at time t,
- $I_{0}$ is the initial current at the time the voltage source is removed,
- $e$ is the base of the natural logarithm,
- $t$ is the time after the voltage is removed,
- $\tau$ is the time constant.
Differences and Important Points
| Property | Charging Phase | Discharging Phase |
|---|---|---|
| Current Behavior | Increases exponentially from 0 to $I_{max}$ | Decreases exponentially from $I_{0}$ to 0 |
| Time Constant | $\tau = \frac{L}{R}$ (same for both phases) | $\tau = \frac{L}{R}$ (same for both phases) |
| Voltage Across Inductor | Starts at maximum and decreases to 0 | Starts at 0 and increases to a maximum negative value |
| Voltage Across Resistor | Increases from 0 to $V$ (supply voltage) | Decreases from $V$ to 0 |
Examples
Example 1: Charging Phase
Suppose we have an L-R circuit with an inductor of 2 H and a resistor of 4 Ω. The time constant for this circuit would be:
$$ \tau = \frac{L}{R} = \frac{2 \text{ H}}{4 \text{ Ω}} = 0.5 \text{ s} $$
If a 12 V battery is connected to the circuit, the maximum current ($I_{max}$) is:
$$ I_{max} = \frac{V}{R} = \frac{12 \text{ V}}{4 \text{ Ω}} = 3 \text{ A} $$
The current at any time t can be calculated using:
$$ i(t) = 3 \text{ A} \left(1 - e^{-\frac{t}{0.5 \text{ s}}}\right) $$
Example 2: Discharging Phase
Using the same circuit, if the battery is disconnected when the current is at its maximum (3 A), the current decay can be described by:
$$ i(t) = 3 \text{ A} e^{-\frac{t}{0.5 \text{ s}}} $$
After 0.5 seconds (one time constant), the current will be:
$$ i(0.5 \text{ s}) = 3 \text{ A} e^{-1} \approx 1.1 \text{ A} $$
This shows that the current has decreased to about 36.8% of its initial value.
Understanding L-R circuits is crucial for analyzing transient responses in electrical systems, designing filters, and understanding the behavior of circuits in various applications such as power electronics, automotive systems, and more.