Inductance in DC Circuits — Study Notes

Inductance in DC Circuits — Study Notes

• Introduction to inductors

  • An inductor is a passive device that stores energy in its magnetic field and can return energy to the circuit when required.
  • Structure: a cylindrical core with many turns of conducting wire; basic structure and schematic symbol are shown.
  • Types mentioned: Ferromagnetic-cored inductor; Variable inductor.
  • Inductance definition: The unit is the henry (H). A circuit has an inductance of 1 H if an emf of 1 volt is induced in the circuit when the current varies uniformly at the rate of 1 ampere per second.
  • Key concepts: inductance is related to how current changes induce a voltage via magnetic flux linkage; self-inductance is the inductance of a circuit due to its own magnetic field.

• Fundamental principle and physical picture

  • A straight piece of wire produces a weak magnetic field; winding it into a coil around a core strengthens and concentrates the magnetic field, enabling useful work (opening/closing valves, relays, etc.).
  • Inductance is a consequence of a conductor linking a magnetic field.

• Inductance from flux linkage (core relations)

  • Faraday-like relation for an inductor: the emf induced is proportional to the rate of change of flux linkage Λ:
    $e = - \frac{d\Lambda}{dt}$
    where the flux linkage is $\Lambda = N \Phi$ with N = number of turns and Φ = magnetic flux.
  • If flux changes from 0 to Φ in time t, the average emf is $\bar{e} = \frac{\Delta \Lambda}{\Delta t} = \frac{N \Delta \Phi}{\Delta t}.$
  • Reluctance and inductance connection: in a magnetic circuit, the inductance is related to core properties as
    $L = \frac{\mu N^2 A}{\ell}$
    where μ is the permeability of the core material, A is cross-sectional area, and ℓ is the mean length through the core.
  • In SI units, μ = μ0 μr, with μ0 ≈ 4π×10^{-7} H/m and μr the relative permeability of the core material.
  • The physical interpretation: inductance is proportional to how effectively the coil links magnetic flux with the current; a higher μ, larger A, more turns N, or shorter l increases L.

• Core geometry and the inductance formula

  • Explicit formula:
    $L = \frac{\mu N^2 A}{\ell} = \frac{\mu<em>0 \mu</em>r N^2 A}{\ell}.$
  • Increasing inductance can be achieved by:
  • increasing the cross-sectional area A of the core,
  • increasing the number of turns N,
  • decreasing the mean length ℓ of the magnetic path, and/or
  • using a core with higher permeability μ (larger μr).
  • Permittivity term μ (the permeability) injects the core’s physical characteristics into the computation of inductance.

• Voltage-current relation and DC behavior

  • The terminal voltage of an inductor is proportional to the time rate of change of current:
    $v = L \frac{di}{dt}.$
  • If the current is constant (di/dt = 0), the voltage across an ideal inductor is zero; the inductor behaves as a short circuit for DC.
  • A key implication: current cannot change instantaneously in an inductor; attempting to change it instantaneously would require an infinite voltage (not physically possible).
  • If either the inductance or the rate of change of current is doubled, the induced emf is doubled (since emf scales with L and di/dt).

• Polarity of the induced emf

  • The induced emf follows Lenz’s law: it opposes changes in current.
  • There are two common sign conventions:
  • The induced emf e obeys the relation $e = - \frac{d\Lambda}{dt}$ (Lenz’s law).
  • Under the passive sign convention, the voltage and current are related by $v = L \frac{di}{dt}$, which is often displayed in circuit diagrams to reflect volt drop.
  • An inductor is described as an energy-storing element; there is discussion in the material about classifying it as active vs passive depending on sign convention.

• Properties of inductors (summarized)

  • If current is constant, voltage across the inductor is zero in the ideal case (DC short).
  • The current in an inductor cannot change instantaneously; abrupt interruption can generate a high voltage across the winding.
  • Ideal inductors do not dissipate energy (energy stored is returned later); real inductors include winding resistance in series and a small winding capacitance between windings.
  • At high frequencies, the winding capacitance may become significant and affect behavior.

• Series combination of inductors (KVL and equivalent inductance)

  • For a series connection of inductors with voltages v1, v2, …, vN and currents (same current) i:
    $v = v<em>1 + v</em>2 + \cdots + v_N.$
  • Each inductor relates its voltage to the current by $v<em>k = L</em>k \frac{di}{dt}.$
  • Therefore, the total voltage is
    $v = \left(\sum<em>{k=1}^{N} L</em>k\right) \frac{di}{dt}.$
  • The equivalent inductance in series is
    $L<em>{eq} = \sum</em>{k=1}^{N} L_k.$
  • Series inductors behave similarly to series resistors in terms of the combination rule for L.

• Parallel combination of inductors (KCL and equivalent inductance)

  • For a parallel connection, currents add:$i = i<em>1 + i</em>2 + \cdots + i_N.$
  • The same voltage is across all inductors: $v<em>1 = v</em>2 = \cdots = v.$
  • Using the relation $vk = Lk \frac{dik}{dt}$, one can derive that $\frac{1}{L</em>{eq}} = \sum<em>{k=1}^{N} \frac{1}{L</em>k},$
    so that
    $L<em>{eq} = \left( \sum</em>{k=1}^{N} \frac{1}{L_k} \right)^{-1}.$
  • Parallel inductors follow the same “resistance-like” rule as parallel resistors; Delta-Wye transformations can be applied to inductors/capacitors when all elements are the same type.

• Worked examples and problems (selected from the transcript)

  • Problem 2: Coil of 500 turns, non-magnetic core, L = 15 mH. Find:

  • (i) Flux produced by I = 5 A:
    $\Phi = \frac{L I}{N} = \frac{(15\times 10^{-3}) \times 5}{500} = 1.5\times 10^{-4} \text{ Wb}.$

  • (ii) Average emf when the current is reversed from +5 A to -5 A in Δt = 10 ms:

    • Change in current ΔI = -10 A, so di/dt magnitude is 1000 A/s.
    • Induced emf magnitude: $|v| = L \frac{di}{dt} = (15\times 10^{-3}) \times 1000 = 15 \text{ V}.$
    • The average emf during reversal is 15 V (sign depends on the direction).

  • Example 3: Inductance of a square-shaped coil (500 turns, mean length 0.850 m) wound on a steel core (A = 100 mm^2, μr = 100), later on ferrite core (μr = 650).

  • Geometry: mean length ℓ = 0.850 m; cross-sectional area A = 100 mm^2 = 1.0×10^{-4} m^2; μ0 ≈ 4π×10^{-7} H/m.

  • (1) Steel core inductance:
    $L<em>{steel} = \frac{\mu</em>0 μ_r N^2 A}{\ell} = \frac{(4\pi\times10^{-7})(100) (500)^2 (1.0\times10^{-4})}{0.850} \approx 3.69\times10^{-3} \text{ H} = 3.69\text{ mH}.$

  • (2) Ferrite core inductance (μr = 650):
    $L<em>{ferrite} = L</em>{steel} \times \frac{μ<em>{r, ferrite}}{μ</em>{r, steel}} = 3.69\text{ mH} \times \frac{650}{100} \approx 24.0\text{ mH}.$
    Alternatively, directly compute: ≈ 0.0240 H.

  • (3) Core flux at I = 5 A:

    • For steel: $\Phi<em>{steel} = \frac{L</em>{steel} I}{N} = \frac{(3.69\times10^{-3}) \times 5}{500} \approx 3.69\times10^{-5} \text{ Wb}.$
    • For ferrite: $\Phi<em>{ferrite} = \frac{L</em>{ferrite} I}{N} = \frac{(24.0\times10^{-3}) \times 5}{500} \approx 2.40\times10^{-4} \text{ Wb}.$

  • (4) Which core is better magnetic material? Ferrite (higher μ_r) yields a larger flux for the same current, hence is the better magnetic material in this context.

• Properties recap (concise)

  • Constant current → zero voltage across ideal inductor (DC short).
  • Inductor current cannot change instantaneously; changing it quickly requires high voltage.
  • Real inductors have winding resistance and inter-winding capacitance; high-frequency behavior is affected by these parasitics.
  • The energy stored in an ideal inductor is recovered when the current decreases; energy is not dissipated in the ideal model.

• Practical sign conventions (summary)

  • Faraday/Lenz viewpoint: emf opposes the change in flux: $e = - \frac{d\Lambda}{dt}.$
  • Circuit-oriented viewpoint (passive sign convention): $v = L \frac{di}{dt}.$

• Quick physical intuition

  • Increase in current strengthens the magnetic field; inductance captures how effectively a coil links magnetic flux with current.
  • Higher μ, larger cross-section, more turns, or shorter magnetic path length all raise inductance.

• Key constants and units

  • Permeability: $\mu = \mu<em>0 μ</em>r, \quad \mu_0 \approx 4\pi\times10^{-7}\ \text{H/m}.$
  • Inductance unit: $[L] = \text{H}.$
  • Flux in webers: $[Φ] = \text{Wb}.$
  • Flux linkage: $Λ = N Φ.$