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{\mu0 \mur 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 = v1 + v2 + \cdots + v_N.
- Each inductor relates its voltage to the current by vk = Lk \frac{di}{dt}.
- Therefore, the total voltage is
  v = \left(\sum{k=1}^{N} Lk\right) \frac{di}{dt}.
- The equivalent inductance in series is
  L{eq} = \sum{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 = i1 + i2 + \cdots + i_N.
- The same voltage is across all inductors: v1 = v2 = \cdots = v.
- Using the relation $vk = Lk \frac{dik}{dt}$, one can derive that \frac{1}{L{eq}} = \sum{k=1}^{N} \frac{1}{Lk},
  so that
  L{eq} = \left( \sum{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{steel} = \frac{\mu0 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{ferrite} = L{steel} \times \frac{μ{r, ferrite}}{μ{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{steel} = \frac{L{steel} I}{N} = \frac{(3.69\times10^{-3}) \times 5}{500} \approx 3.69\times10^{-5} \text{ Wb}.
  - For ferrite: \Phi{ferrite} = \frac{L{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 = \mu0 μr, \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.