Inductance & Transformers

Electromagnetic induction, self and mutual inductance, and transformer operation.

Definition

Inductance is the property of an electrical conductor by which a change in current induces an electromotive force (EMF). An inductor (also called a choke) consists of a conductor wound into a coil. When current flows, energy is stored temporarily in the coil's magnetic field; when current changes, the induced voltage opposes that change (Faraday's law). Typical inductance values range from $1\ \mu\text{H}$ to $1\ \text{H}$. Transformers use mutual inductance to transfer electrical energy between circuits via a magnetic field, typically using a laminated soft-iron core to maximize flux linkage and minimize eddy-current losses.

Magnetic Flux

Magnetic flux $\Phi_B$ through a loop of area $A$ in a magnetic field $B$:

$$\Phi_B = BA\cos\theta = B_\perp A$$

  • $\theta$ = angle between $\vec{B}$ and the normal to the loop area
  • Unit: weber (Wb)
  • Only the perpendicular component of area contributes to flux
  • $\theta = 0^\circ$ → $\Phi_B$ is maximum; $\theta = 90^\circ$ → $\Phi_B = 0$

Key Concepts

  • Faraday's Law — induced EMF proportional to rate of change of flux: $$\mathcal{E} = -\frac{d\Phi_B}{dt}$$
    • Discrete form (lecture): $\mathcal{E} = N\left(\frac{\Delta\Phi_B}{\Delta t}\right)$ where $N$ = number of loops
    • Magnitude from Faraday's Law; direction from Lenz's Law
  • Lenz's Law — induced current opposes the change causing it
    • Two fields to consider: external changing field and field produced by induced current
    • Increasing flux: induced field points in opposite direction to oppose the increase
    • Decreasing flux: induced field points in same direction to oppose the decrease
  • Self-Inductance — $L = \frac{N\Phi}{I}$, unit: Henry (H)
  • Back EMF — the self-induced emf $\mathcal{E} = -L\frac{dI}{dt}$ opposes changes in current
    • When $I$ is increasing, induced emf is in the opposite direction to $I$
    • When $I$ is decreasing, induced emf is in the same direction as $I$
  • Inductor Energy Storage — $U = \frac{1}{2}LI^2$
  • Mutual Inductance — $M = \frac{N_2\Phi_{21}}{I_1} = \frac{N_1\Phi_{12}}{I_2}$; for coaxial solenoids $M = \frac{\mu_0 N_p N_s A}{l}$
  • Mutually Induced EMF — $\mathcal{E}_1 = M\frac{dI_2}{dt}$ and $\mathcal{E}_2 = M\frac{dI_1}{dt}$
  • Transformers — AC voltage transformation using a soft-iron core; primary ($N_p$) driven by AC source, secondary ($N_s$) has induced voltage
  • Turns Ratio — $\frac{V_s}{V_p} = \frac{N_s}{N_p}$; rate of change of flux is same for both coils
  • Step-Up Transformer — $N_s > N_p$ → increases voltage, decreases current
  • Step-Down Transformer — $N_s < N_p$ → decreases voltage, increases current
  • Power Conservation — $V_p I_p = V_s I_s$ (ideal); real transformers have losses
  • Transformer Losses — copper loss ($I^2R$), hysteresis loss, flux leakage, eddy currents
  • Power Transmission — high-voltage transmission reduces $I^2R$ losses; voltage stepped up at source and stepped down at destination
  • RL Circuits — transient current growth/decay
  • Time Constant — $\tau = \frac{L}{R}$

Transformer Operation

flowchart LR
    A[AC Source] -->|V_p, I_p| P[Primary Coil<br/>(N_p turns)]
    P -->|Magnetic flux Φ| C[Soft-Iron Core]
    C -->|Induced flux Φ| S[Secondary Coil<br/>(N_s turns)]
    S -->|V_s, I_s| L[Load]

    C -.->|N_s > N_p| U[Step-Up<br/>V_s ↑ I_s ↓]
    C -.->|N_s < N_p| D[Step-Down<br/>V_s ↓ I_s ↑]

    style U fill:#e1f5e1
    style D fill:#ffe1e1

Key Formulas

Formula Description
$\mathcal{E} = -L\frac{dI}{dt}$ Self-induced EMF (back emf)
$L = \mu_0 n^2 A l = \frac{\mu_0 N^2 A}{\ell}$ Solenoid inductance
$U = \frac{1}{2}LI^2$ Stored energy in inductor
$\mu_0 = 1.2567 \times 10^{-6}\ \text{H/m}$ Permeability of free space
$M = \frac{N_2\Phi_{21}}{I_1} = \frac{N_1\Phi_{12}}{I_2}$ Mutual inductance
$\mathcal{E}_2 = -M\frac{dI_1}{dt}$ Mutually induced EMF
$M = \frac{\mu_0 N_p N_s A}{l}$ Mutual inductance (coaxial solenoids)
$\frac{V_s}{V_p} = \frac{N_s}{N_p}$ Transformer voltage ratio
$\frac{I_s}{I_p} = \frac{N_p}{N_s}$ Transformer current ratio (ideal)
$P_{\text{loss}} = I^2R$ Power loss in transmission
$\tau = \frac{L}{R}$ RL time constant
$i(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau})$ RL current growth

Capacitor–Inductor Analogy

Both components depend on geometric factors and store energy, but with complementary roles:

Aspect Capacitor Inductor
Geometry $C = \frac{\varepsilon_0 A}{d}$ $L = \frac{\mu_0 N^2 A}{\ell}$
Energy stored $U = \frac{1}{2} C V^2$ $U = \frac{1}{2} L I^2$
Defining relation $C = \frac{Q}{V}$ $L = \frac{N\Phi}{I}$

Self Induction vs Mutual Induction

flowchart TB
    subgraph Self ["Self-Inductance (L)"]
        direction TB
        S1[Current change in Coil 1] --> S2[Magnetic flux change in Coil 1]
        S2 --> S3[Back EMF induced in Coil 1]
        S3 --> S4[Opposes change in current]
        S4 --> S5[Energy stored in magnetic field]
    end

    subgraph Mutual ["Mutual Inductance (M)"]
        direction TB
        M1[Current change in Coil 1] --> M2[Magnetic flux links Coil 2]
        M2 --> M3[EMF induced in Coil 2]
        M3 --> M4[Energy transferred between coils]
        M4 --> M5[Applications: transformers, wireless charging]
    end

    style Self fill:#e3f2fd
    style Mutual fill:#fff3e0
Aspect Self Induction ($L$) Mutual Induction ($M$)
Definition Opposes change in current in the same coil Induces EMF in one coil due to current change in another
Dependence Geometry of coil and core material Geometry of both coils, distance, and orientation
Energy Stores energy in magnetic field Transfers energy between coils via magnetic field
Cause Change in current in same coil Change in current in neighboring coil
Interaction Single coil Two or more coils
Applications Inductors, chokes, tuning circuits Transformers, wireless charging, inductive coupling

Related Concepts

Quick Quiz 2026 — Key Insights

From FAD1022 Quick Quiz 2026 — Inductance & Transformers:

Mutual Inductance Independence

  • Mutual inductance $M$ depends only on geometry (turns, area, length, separation, core material) and not on the current flowing through either coil. Changing current changes the induced emf, but $M$ itself is invariant.

Reciprocal EMF Relationship

  • For two coupled coils, the ratio of induced emf to current change is reciprocal: $$\frac{\varepsilon_2}{\Delta i_1} = \frac{\varepsilon_1}{\Delta i_2}$$ This follows directly from $M_{12} = M_{21} = M$.

Inductor Transient Behaviour

  • In an RL circuit immediately after closing a switch, an inductor opposes current flow via back emf ($\varepsilon = -L,dI/dt$).
  • At steady state ($t \to \infty$), $dI/dt \to 0$ and the inductor behaves like a connecting wire (short circuit) with only its internal wire resistance remaining.
stateDiagram-v2
    [*] --> TransientGrowth : Close switch at t=0
    state TransientGrowth {
        [*] --> CurrentRising
        CurrentRising --> CurrentRising : di/dt > 0, Back EMF opposes
        CurrentRising --> SteadyState : t → ∞
    }

    TransientGrowth --> SteadyState

    state SteadyState {
        [*] --> MaxCurrent
        MaxCurrent --> MaxCurrent : I = ℰ/R, acts as wire
    }

    SteadyState --> TransientDecay : Open switch

    state TransientDecay {
        [*] --> CurrentFalling
        CurrentFalling --> CurrentFalling : di/dt < 0, Back EMF sustains
        CurrentFalling --> [*] : t → ∞
    }

Maximum Self-Induced EMF

  • $|\varepsilon| = L|dI/dt|$; the magnitude depends on the rate of change of current, not the current magnitude itself. Rapidly increasing current produces the maximum emf.

Transformer Losses Detail

  • Iron Loss (Core Loss) = Hysteresis loss + Eddy current loss, both occurring in the magnetic core.
  • Copper Loss = $I^2R$ heating in the primary and secondary windings — distinct from core loss.
  • Transformer efficiency: $$\eta = \frac{P_{\text{out}}}{P_{\text{in}}} \quad \Rightarrow \quad P_{\text{loss}} = P_{\text{in}} - P_{\text{out}}$$

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