Notes on Inductors - Voltage and Current Relationship

Understanding the Voltage-Current Relationship in an Inductor

  • Ohm's Law Overview

  • In resistors, the relationship is linear:

    • V = I * R

  • Capacitor Current-Voltage Relationship

  • Current (I) is related to capacitance (C) and voltage (V) by the formula:

    • I = C * dV/dt

  • Voltage can be found by integrating the current.

  • Inductive Behavior

  • An inductor opposes changes in current.

  • When current increases, an opposing voltage is induced; when current decreases, a voltage of opposite polarity is induced to maintain current flow.

  • Magnetic Flux

  • Flux (Φ) measures magnetic field strength across an area.

  • It can be determined by the formula:

    • Φ = μ * n * I
    • Where μ is permeability, n is the number of turns, and I is the current.

  • Induced Voltage Formula

  • Induced voltage (V) in an inductor is expressed as:

    • V = n * dΦ/dt

  • Since μ and n are constants, we derive the relation considering the change in current:

    • V = n * μ * (di/dt)
    • This leads to:
    • V = L * (di/dt), where L = n² * μ is the inductance.

  • Inductance (L)

  • Inductance quantifies an inductor's ability to induce voltage in response to current change.

  • Dependent on the number of turns and the permeability of the core material.

  • Relationship to Capacitors

  • Voltage-current relationships in inductors and capacitors exhibit similar forms with reversed roles.

  • This points to their opposing effects in circuits, especially in frequency-selective applications.

  • Current as a function of Voltage in an Inductor

  • The current as a function of voltage is expressed as:

    • I = (1/L) * ∫V(x)dx + I0, where I0 is the initial current and x is the integration variable.

  • Direction of Induced Voltage

  • If current (I) increases (dI/dt > 0),

    • Induced voltage (V) opposes this increase, effectively pushing back.

  • If current decreases (dI/dt < 0),

    • The induced voltage attempts to keep current flowing, creating a positive response.

  • Analogy of Rubber Bands

  • Visualize voltage as rubber bands resisting changes in current.

  • Increasing current stretches the bands (voltage induction), while decreasing current allows them to relax, attempting to maintain flow.

  • Constant vs. Changing Current

  • If dI/dt = 0 (constant current), then V = 0 across the inductor:

    • Under DC conditions, an inductor behaves like a wire with no voltage drop across it.

  • Key Concept: A constant current can exist without a voltage drop, but a voltage requires a change in current.