FAD1022 L31-L33 — Inductance & Transformers — Formula Sheet
A comprehensive collection of all formulas, equations, and key relationships from Lectures 31–33 on Self-Inductance, Mutual Inductance, and Transformers.
1. Fundamental Constants
| Symbol | Value | Description |
|---|---|---|
| $\mu_0$ | $1.2567 \times 10^{-6}\ \text{H/m}$ (or $4\pi \times 10^{-7}\ \text{H/m}$) | Permeability of free space |
2. Self-Inductance
2.1 Magnetic Field of a Solenoid
$$B = \mu_0 n I = \mu_0 \left(\frac{N}{\ell}\right) I$$
| Variable | Meaning | Unit |
|---|---|---|
| $B$ | Magnetic field inside the solenoid | T (Tesla) |
| $\mu_0$ | Permeability of free space | H/m |
| $n$ | Number of turns per unit length | turns/m |
| $N$ | Total number of turns | — |
| $\ell$ | Length of the solenoid | m |
| $I$ | Current through the solenoid | A |
2.2 Self-Induced EMF (Back EMF)
$$\mathcal{E} = -L \frac{dI}{dt}$$
| Variable | Meaning | Unit |
|---|---|---|
| $\mathcal{E}$ | Self-induced emf (back emf) | V |
| $L$ | Self-inductance | H (Henry) |
| $\frac{dI}{dt}$ | Rate of change of current | A/s |
Lenz's Law: The negative sign indicates that the induced emf opposes the change in current.
- $I$ increasing → induced emf is in the opposite direction.
- $I$ decreasing → induced emf is in the same direction.
2.3 Inductance of a Solenoid
$$L = \frac{\mu_0 N^2 A}{\ell}$$
| Variable | Meaning | Unit |
|---|---|---|
| $L$ | Self-inductance | H |
| $\mu_0$ | Permeability of free space | H/m |
| $N$ | Total number of turns | — |
| $A$ | Cross-sectional area | m² |
| $\ell$ | Length of the solenoid | m |
2.4 General Definition of Inductance
$$L = \frac{N\Phi_B}{I}$$
| Variable | Meaning | Unit |
|---|---|---|
| $L$ | Self-inductance | H |
| $N$ | Number of turns | — |
| $\Phi_B$ | Magnetic flux through one turn | Wb (Weber) |
| $I$ | Current | A |
3. Energy Stored in an Inductor
3.1 Energy in the Magnetic Field
$$U = \frac{1}{2} L I^2$$
| Variable | Meaning | Unit |
|---|---|---|
| $U$ (or $E$) | Energy stored in the inductor | J (Joules) |
| $L$ | Self-inductance | H |
| $I$ | Current | A |
3.2 Capacitor–Inductor Analogies
| Property | Capacitor | Inductor |
|---|---|---|
| Depends on geometry | $C = \dfrac{\varepsilon_0 A}{d}$ | $L = \dfrac{\mu_0 N^2 A}{\ell}$ |
| Energy stored | $U = \dfrac{1}{2} C V^2$ | $U = \dfrac{1}{2} L I^2$ |
| Defining relation | $C = \dfrac{Q}{V}$ | $L = \dfrac{N\Phi}{I}$ |
| Variable | Meaning | Unit |
|---|---|---|
| $C$ | Capacitance | F (Farad) |
| $\varepsilon_0$ | Permittivity of free space | F/m |
| $A$ | Plate / cross-sectional area | m² |
| $d$ | Plate separation | m |
| $Q$ | Charge | C (Coulomb) |
| $V$ | Voltage | V |
4. Mutual Inductance
4.1 Definition of Mutual Inductance
$$M_{21} = \frac{N_2 \Phi_{21}}{i_1}$$
| Variable | Meaning | Unit |
|---|---|---|
| $M_{21}$ | Mutual inductance of coil 2 with respect to coil 1 | H |
| $N_2$ | Number of turns in coil 2 | — |
| $\Phi_{21}$ | Magnetic flux through one turn of coil 2 due to current in coil 1 | Wb |
| $i_1$ | Current in coil 1 | A |
4.2 EMF Induced in Coupled Coils
EMF induced in coil 2 due to changing current in coil 1:
$$\varepsilon_2 = \frac{N_2 , d\Phi_{21}}{dt} = M_{21} \frac{di_1}{dt}$$
EMF induced in coil 1 due to changing current in coil 2:
$$\varepsilon_1 = \frac{N_1 , d\Phi_{12}}{dt} = M_{12} \frac{di_2}{dt}$$
| Variable | Meaning | Unit |
|---|---|---|
| $\varepsilon_2$ | Induced emf in coil 2 | V |
| $\varepsilon_1$ | Induced emf in coil 1 | V |
| $M_{21}, M_{12}$ | Mutual inductance | H |
| $\frac{di_1}{dt}, \frac{di_2}{dt}$ | Rate of change of current | A/s |
4.3 Reciprocity Theorem
$$M_{12} = M_{21} = M$$
Therefore:
$$\varepsilon_1 = M \frac{di_2}{dt} \quad \text{and} \quad \varepsilon_2 = M \frac{di_1}{dt}$$
4.4 Alternative Definition of Mutual Inductance
$$M = \frac{N_2 \Phi_2}{I_1} = \frac{N_1 \Phi_1}{I_2}$$
4.5 Mutual Inductance for Coaxial Solenoids
For two coaxial solenoids with common cross-sectional area $A$ and length $l$:
Magnetic field produced by primary coil:
$$B = \frac{\mu_0 N_p I}{l}$$
Rate of change of magnetic field:
$$\frac{dB}{dt} = \frac{\mu_0 N_p}{l_p} \frac{dI}{dt}$$
Induced emf in secondary coil:
$$\varepsilon_s = N_s A_s \frac{dB}{dt} = \frac{\mu_0 N_p N_s A_s}{l_p} \frac{dI_p}{dt}$$
Mutual inductance for coaxial solenoids:
$$M = \frac{\mu_0 N_p N_s A}{l}$$
| Variable | Meaning | Unit |
|---|---|---|
| $N_p$ | Number of turns in primary coil | — |
| $N_s$ | Number of turns in secondary coil | — |
| $A$ (or $A_s$) | Cross-sectional area | m² |
| $l$ (or $l_p$) | Length of the solenoid | m |
| $I$ (or $I_p$) | Current in primary coil | A |
| $\frac{dI_p}{dt}$ | Rate of change of primary current | A/s |
Key Point: As the separation distance between circuits increases, mutual inductance decreases because the magnetic flux linking the circuits decreases.
5. Transformers
5.1 Fundamental Principle
Rate of change of magnetic flux is the same for both coils:
$$\frac{d\Phi_1}{dt} = \frac{d\Phi_2}{dt}$$
5.2 Faraday's Law Applied to Transformer Coils
Secondary coil:
$$V_s = -N_s \frac{d\Phi}{dt}$$
Primary coil:
$$V_p = -N_p \frac{d\Phi}{dt}$$
| Variable | Meaning | Unit |
|---|---|---|
| $V_s$ | Voltage across secondary coil | V |
| $V_p$ | Voltage across primary coil | V |
| $N_s$ | Number of turns in secondary coil | — |
| $N_p$ | Number of turns in primary coil | — |
| $\frac{d\Phi}{dt}$ | Rate of change of magnetic flux | Wb/s |
5.3 Transformer Equation (Voltage–Turns Ratio)
$$\frac{V_s}{V_p} = \frac{N_s}{N_p}$$
- Step-up transformer: $N_s > N_p$ → increases voltage
- Step-down transformer: $N_s < N_p$ → decreases voltage
5.4 Current Ratio (Ideal Transformer)
For an ideal transformer (energy losses = zero), power conservation gives:
$$\frac{I_s}{I_p} = \frac{V_p}{V_s} = \frac{N_p}{N_s}$$
| Variable | Meaning | Unit |
|---|---|---|
| $I_s$ | Current in secondary coil | A |
| $I_p$ | Current in primary coil | A |
5.5 Power Conservation (Ideal Transformer)
$$P_p = P_s \implies V_p I_p = V_s I_s$$
| Variable | Meaning | Unit |
|---|---|---|
| $P_p$ | Power in primary coil | W |
| $P_s$ | Power in secondary coil | W |
Rule: A transformer that steps up the voltage simultaneously steps down the current, and vice versa.
6. Transformer Energy Losses (Real Transformers)
6.1 Copper Loss
$$P_{\text{copper}} = I^2 R$$
| Variable | Meaning | Unit |
|---|---|---|
| $P$ | Power lost as heat | W |
| $I$ | Current through the coil | A |
| $R$ | Resistance of the coil | Ω |
Fix: Use thick, low-resistance copper wire.
6.2 Eddy Current Loss
Time-varying magnetic field produces eddy currents in the iron core. Heat produced:
$$P_{\text{eddy}} \propto B^2 f^2 t^2$$
Fix: Use laminated iron core to minimize energy loss.
6.3 Hysteresis Loss
Energy wasted as heat during reversal of magnetization.
Fix: Use "soft" magnetic materials like Silicon Steel.
6.4 Flux Leakage
Not all magnetic field lines reach the secondary coil.
Fix: Wrap coils on top of each other.
7. Power Transmission
7.1 Power Loss in Transmission Lines
$$P_{\text{loss}} = I^2 R_{\text{line}}$$
| Variable | Meaning | Unit |
|---|---|---|
| $P_{\text{loss}}$ | Power lost in transmission | W |
| $I$ | Current in the transmission line | A |
| $R_{\text{line}}$ | Resistance of the transmission line | Ω |
7.2 Percentage Power Loss
$$% \text{ Power Loss} = \frac{P_{\text{loss}}}{P_{\text{total}}} \times 100% = \frac{I^2 R_{\text{line}}}{P_{\text{total}}} \times 100%$$
| Variable | Meaning | Unit |
|---|---|---|
| $P_{\text{total}}$ | Total power generated / transmitted | W |
7.3 RMS Voltage for Sinusoidal AC
$$V_{\text{rms}} = \frac{V_{\text{max}}}{\sqrt{2}}$$
| Variable | Meaning | Unit |
|---|---|---|
| $V_{\text{rms}}$ | Root-mean-square voltage | V |
| $V_{\text{max}}$ | Peak voltage | V |
High-voltage transmission dramatically reduces $I^2R$ losses by lowering the current for the same power.
8. Summary Table: Self vs Mutual Inductance
| Aspect | Self Inductance, $L$ | Mutual Inductance, $M$ |
|---|---|---|
| Definition | Property of a coil to oppose change in current flowing through itself | Property of two coils to induce EMF in one coil due to change in current in the other |
| Dependence | Geometry of the coil and core material | Geometry of both coils, their distance, and orientation |
| Unit | Henry (H) | Henry (H) |
| 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 |
| Application | Inductors, chokes, tuning circuits | Transformers, wireless charging, inductive coupling |
9. Key Relationships & Process Chains
Self-Induction Chain:
$$\Delta I \rightarrow \Delta B \rightarrow \Delta\Phi \rightarrow \mathcal{E}_{\text{ind}}$$
Mutual Induction Chain:
- Current in 1st coil changes
- Magnetic field in 1st coil changes
- Magnetic flux in 2nd coil changes
- Induced EMF in 2nd coil
- Induced current in 2nd coil
Transformer Power Transmission Chain:
$$\text{Power Plant} \rightarrow \text{Step-Up Transformer} \rightarrow \text{High Voltage Transmission} \rightarrow \text{Step-Down Transformer} \rightarrow \text{Consumer}$$
Compiled from FAD1022 L31-L33 — Inductance & Transformers