FAD1022 — Inductance & Transformers — Formula Sheet
A comprehensive formula sheet extracted from the Quick Quiz on mutual inductance, self-inductance, and transformer principles.
1. Mutual Inductance
1.1 Definition from Flux Linkage
$$M_{21} = \frac{N_2 \phi_2}{i_1}$$
| Symbol | Meaning | Unit |
|---|---|---|
| $M_{21}$ | Mutual inductance of coil 2 with respect to coil 1 | Henry (H) |
| $N_2$ | Number of turns in coil 2 | dimensionless |
| $\phi_2$ | Magnetic flux through each turn of coil 2 due to current in coil 1 | Weber (Wb) |
| $i_1$ | Current flowing in coil 1 | Ampere (A) |
1.2 Induced EMF (Faraday's Law)
$$\varepsilon_2 = -M \frac{dI_1}{dt}$$
For average (finite change): $$|\varepsilon| = M \left|\frac{\Delta I}{\Delta t}\right|$$
| Symbol | Meaning | Unit |
|---|---|---|
| $\varepsilon$ | Induced electromotive force (emf) | Volt (V) |
| $M$ | Mutual inductance | Henry (H) |
| $\Delta I$ | Change in current | Ampere (A) |
| $\Delta t$ | Time interval | Second (s) |
1.3 Reciprocity Theorem
$$\frac{\varepsilon_2}{\Delta i_1} = \frac{\varepsilon_1}{\Delta i_2}$$
Or equivalently: $$\varepsilon_1 = \varepsilon_2 \frac{\Delta i_2}{\Delta i_1}$$
Note: $M_{12} = M_{21} = M$. Mutual inductance is symmetric between the two coils.
1.4 Factors Affecting Mutual Inductance
$M$ depends only on:
- Number of turns ($N_1$, $N_2$)
- Cross-sectional area ($A$)
- Length of solenoids ($l$)
- Separation distance ($d$)
- Core material (permeability $\mu$)
$M$ does NOT depend on:
- The magnitude of current flowing through either coil
Decreasing separation distance $d$ → increases $M$ (greater flux linkage).
2. Self-Inductance
2.1 Self-Induced EMF
$$|\varepsilon| = L \left|\frac{dI}{dt}\right|$$
| Symbol | Meaning | Unit |
|---|---|---|
| $L$ | Self-inductance | Henry (H) |
| $dI/dt$ | Rate of change of current | A/s |
| $\varepsilon$ | Self-induced (back) emf | Volt (V) |
The induced emf magnitude depends on the rate of change of current, not on the current itself. A rapidly increasing current produces the maximum emf.
2.2 Inductor Behaviour in DC RL Circuits
| Time | Behaviour | Equivalent |
|---|---|---|
| $t = 0$ | Opposes current flow with back emf equal to applied voltage | Open circuit |
| $t \to \infty$ | $dI/dt \to 0$, back emf vanishes | Short circuit / connecting wire |
3. Transformers (Ideal)
3.1 Turns Ratio and Voltage
$$\frac{V_s}{V_p} = \frac{N_s}{N_p}$$
3.2 Turns Ratio and Current
$$\frac{I_s}{I_p} = \frac{N_p}{N_s}$$
| Symbol | Meaning | Unit |
|---|---|---|
| $V_p$, $V_s$ | Primary and secondary voltage | Volt (V) |
| $I_p$, $I_s$ | Primary and secondary current | Ampere (A) |
| $N_p$, $N_s$ | Number of turns in primary and secondary coils | dimensionless |
Step-up transformer: $N_s > N_p$ → $V_s > V_p$ and $I_s < I_p$ Step-down transformer: $N_s < N_p$ → $V_s < V_p$ and $I_s > I_p$
3.3 Transformer Efficiency
$$\eta = \frac{P_{\text{out}}}{P_{\text{in}}}$$
$$P_{\text{loss}} = P_{\text{in}} - P_{\text{out}}$$
| Symbol | Meaning | Unit |
|---|---|---|
| $\eta$ | Efficiency | dimensionless (or %) |
| $P_{\text{out}}$ | Output power | Watt (W) |
| $P_{\text{in}}$ | Input power | Watt (W) |
| $P_{\text{loss}}$ | Power lost as heat | Watt (W) |
4. Transformer Losses
4.1 Iron Loss (Core Loss)
$$P_{\text{iron}} = P_{\text{hysteresis}} + P_{\text{eddy}}$$
Hysteresis Loss: Energy dissipated during cyclic magnetization reversal of the core.
Eddy Current Loss: $I^2R$ heating from induced circulating currents within the conducting iron core due to the time-varying magnetic field.
Iron losses occur in the iron core, not in the windings.
4.2 Copper Loss
$$P_{\text{copper}} = I_p^2 R_p + I_s^2 R_s$$
Copper losses occur in the primary and secondary windings due to their electrical resistance.
5. Key Conceptual Relationships
| Concept | Relationship |
|---|---|
| Mutual inductance vs current | $M$ is independent of current magnitude |
| Mutual inductance vs distance | $M$ increases as separation distance decreases |
| Self-induced emf maximization | Maximum when $|dI/dt|$ is largest (rapidly changing current) |
| Energy conservation in ideal transformer | $P_{\text{in}} = P_{\text{out}}$ |
6. Summary Table of All Symbols
| Symbol | Quantity | SI Unit |
|---|---|---|
| $M$ | Mutual inductance | H |
| $L$ | Self-inductance | H |
| $N$ | Number of turns | — |
| $\phi$ | Magnetic flux | Wb |
| $\varepsilon$ / $\mathcal{E}$ | Induced emf | V |
| $I$ / $i$ | Current | A |
| $V$ / $v$ | Voltage | V |
| $P$ | Power | W |
| $\eta$ | Efficiency | — / % |
| $R$ | Resistance | $\Omega$ |
| $t$ / $\Delta t$ | Time / time interval | s |
| $\mu$ | Permeability of core material | H/m |