Understanding Inductor, Inductive Reactance and Impedance

Understanding Inductor

1. What is an inductor?

An inductor is a passive, two-terminal electrical component that stores energy in a magnetic field when electric current flows through it.

By the principle of electromagnetic induction, it is characterized by its inductance, which is the property of opposing any change in the electric current passing through it by generating a voltage across its terminals.

2. Characteristics of an Indctor

An inductor opposes change in current. This is the defining property of an inductor.

An inductor generates a "back" voltage (or electromotive force, EMF) that fights any attempt to change the current flowing through it.

• Characteristic: The voltage across an ideal inductor is zero if the current is constant (DC). It is non-zero only when the current is changing.
• Equation (Voltage-Current Relationship): The voltage (v) across the inductor is directly proportional to the rate of change of the current (i) through it.

${\displaystyle V(t)=L\frac{dI(t)}{dt}}$

Where,

• L is the inductance in Henrys.
• V(t) is the instantaneous voltage in Volts.
• ${\displaystyle \frac{dI(t)}{dt}}$ is the rate of change of current in Amperes per second.

Interpreting the equation:

• This equation shows that to change the current quickly (a large ${\displaystyle \frac{dI(t)}{dt}}$ ), you must apply a very large voltage.
• Conversely, a sudden change in voltage (like in a square wave) will cause the current to change as a ramp (a steady increase or decrease).

3. Stores Energy in a Magnetic Field

An ideal inductor does not "use up" energy like a resistor; it temporarily stores it within a magnetic field created by the current flowing through its coils.

When current increases, the inductor absorbs energy from the circuit to build its magnetic field. When the current decreases, it returns this stored energy to the circuit.

Equation (Energy Stored): The energy (W) stored in an inductor is proportional to its inductance (𝐿) and the square of the current (I) flowing through it.

${\displaystyle W=\frac{1}{2}L{I}^2}$

Where,

• W is the energy in Joules.
• L is the inductance in Henrys.
• I is the current in Amperes.

4. Behavior in DC Circuits

In a DC Circuit:

• Characteristic: When a DC voltage is first applied, the inductor opposes the rise in current. The current builds up gradually (not instantly).
• At Steady State: Once the current is stable and no longer changing (${\displaystyle \frac{dI}{dt}=0}$), the voltage across the ideal inductor becomes zero (v = L × 0 = 0).
• Implication: A fully energized ideal inductor in a DC circuit acts like a short circuit (a piece of wire with zero resistance).

5. Behavior in AC Circuits

In an AC Circuit:

• Characteristic: In an AC circuit, the current is constantly changing, so the inductor continuously opposes it. This opposition is called inductive reactance (X_L).
• Inductive Reactance: The reactance is proportional to the frequency (f) of the AC signal and the inductance (L). Continute reading the next section below.

6. Inductive Reactance

The magnitude of the opposition an inductor offers to AC due to its inductance.

Meaning: How strongly the inductor resists AC, without considering phase information.

The inductive reactance is given by:

${\displaystyle X_L=2\mathrm{\pi <!--\; \pi \; -->}fL=\mathrm{\omega <!--\; \omega \; -->}L}$

Where,

• X_L is the inductive reactance in Ohms (Ω).
• f is the frequency in Hertz (Hz).
• ω is the angular frequency in radians/sec (2 π f).

6.1 Low frequencies vs. High frequencies:

• At low frequencies (like DC, where f=0), the reactance is zero (X_L = 0). This matches its "short circuit" behavior for DC.
• At very high frequencies, the reactance becomes very high (${\displaystyle X_L\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}$). An ideal inductor acts like an "open circuit" to high-frequency signals.

6.2 Phase Shift:

In an AC circuit, the changing voltage and current are not in sync.

For an ideal inductor, the current lags the voltage by exactly 90 degrees. This means the current reaches its peak value one-quarter of a cycle after the voltage does.

7. Impedance of an Inductor

The full complex opposition to AC, including magnitude and phase.

${\displaystyle Z_L=j{X}_L=j\mathrm{\omega <!--\; \omega \; -->}L}$

The impedance tell us the following:

• The magnitude (which is the reactance 𝑋_𝐿)
• The phase shift (for an ideal inductor, +90°)