PHY-212 Magnetic Circuits and Induction — Quick Reference

Magnetic concepts

• Magnetic flux (Φ): total ‘lines of force’ crossing a surface
• Flux density (B): amount of flux per unit area, perpendicular to surface
• Flux relation: \Phi = B\,A when B ⟂ surface with area A
• Magnetic field sources: bar magnets produce a field from North to South and back via the magnetic circuit
• Permeability (μ) and relative permeability (μr): \mu = \mu0\mu_r,\quad B = \mu H
• Permeability of vacuum: \mu_0 = 4\pi \times 10^{-7}\ \text{H/m}
• Reluctance (𝓡) and magnetic circuit analogy: \mathcal{R} = \frac{l}{\mu A}
• Magnetomotive force (MMF, Fm): the cause of the magnetic field; Fm = N I where N is turns and I is current
• Magnetic circuit law (Ohm’s law analog): \Phi = \frac{Fm}{\mathcal{R}},\quad Fm = \Phi\mathcal{R}
• Flux linkage and energy relation: flux linkage in a coil relates to current via inductance (see inductors)

Inductors, inductance and energy storage

• Inductance (L): L = \frac{N\Phi}{I}\quad\Rightarrow\quad \Phi = \frac{L I}{N}\quad\text{(for a coil with N turns)}
• Solenoid (straight solenoid): inside B-field B = \mu0\mur\frac{N}{\ell} I where \ell is length, N turns, area A
• Energy stored in an inductor: U = \tfrac{1}{2} L I^2
• Magnetic energy density in vacuum: uB = \dfrac{B^2}{2\mu0}; in material: u_B = \dfrac{B^2}{2\mu}

Induction and Faraday’s law

• Induced EMF in a coil: \mathcal{E} = -N\frac{d\Phi}{dt}
• Flux linkage change drives current; direction given by Lenz’s law: induced current opposes the change in flux
• Flux definition for a uniform B and area: \Phi = B A\cos\theta; maximum when θ = 0 (B ⟂ surface)

Right-Hand Rule and solenoid/toque concepts

• Right-Hand/Corkscrew rule: direction of magnetic field lines around a current: if you grip the conductor with the right hand, the thumb points in current direction and fingers show the magnetic field direction
• Direction of rotation and magnetic field for solenoids: a coil carrying current behaves like a magnet; a loop in a uniform field B experiences torque
• Torque on a current loop: \tau = N I A B \sin\phi where φ is the angle between the loop’s normal and B; for maximum torque φ = 90°
• Inside a solenoid, a loop experiences a magnetic torque: if B is along the loop’s normal, torque magnitude is as above

Electric vs magnetic circuits (key differences)

• Electric circuit: current is the flow; resistance largely constant (with fixed T) for fixed materials
• Magnetic circuit: reluctance depends on flux density (nonlinear with B); energy is required to create flux but not to maintain it unless B changes
• Permeability and reluctance can vary with flux, so reluctance is not strictly constant
• Flux in magnetic circuits does not “flow” like current in electric circuits; the analogy is with flux playing the role of current and MMF as the driving force

Inductance, mutual inductance and transformers

• Mutual inductance (two coils): when current in one coil changes, it induces a voltage in the other
• Definitions: \Phi1 = M{12} I2, \quad \Phi2 = M{21} I1; for passive coils, M{12} = M{21} = M
• Inductors in networks: series/parallel combinations use standard impedance rules (see impedance section)
• Transformer basics: \frac{Vp}{Vs} = \frac{Np}{Ns},\quad \frac{Ip}{Is} = \frac{Ns}{Np}
• Equivalent inductance for coupled coils relates to L1, L2 and M; high coupling gives large M; ideal transformer assumes perfect coupling

Alternating current, impedance and admittance

• Impedance of basic elements: ZR = R,\quad ZL = j\omega L,\quad Z_C = \frac{1}{j\omega C} = -\frac{j}{\omega C}
• Total impedance in a series RLC: Z = R + j\omega L - \frac{j}{\omega C}
• Admittance: Y = \frac{1}{Z} = G + jB where G = \Re(Y) = \frac{1}{R},\quad B = \Im(Y) = \text{susceptance}
• Parallel/series impedance combinations follow standard circuit rules
• Current divider rule (parallel impedances): for two branches in parallel
  I1 = I{tot}\frac{Z2}{Z1+Z2},\quad I2 = I{tot}\frac{Z1}{Z1+Z2}

Resonance in RLC circuits

• Series resonance: occurs when XL = XC\Rightarrow \omega L = \frac{1}{\omega C}
• Resonant frequency: \omega0 = \frac{1}{\sqrt{L C}},\quad f0 = \frac{\omega_0}{2\pi}
• At resonance, Z = R (minimum impedance in the ideal case) and current is maximum
• Bandwidth and quality factor (Q):
• Q = \frac{\omega0 L}{R} = \frac{1}{\omega0 R C}
• Half-power frequencies (approximately, for series RLC):
  \omega_{1,2} = \frac{R}{2L} \pm \sqrt{\left(\frac{R}{2L}\right)^2 - \frac{1}{LC}}
• Bandwidth: \Delta\omega = \omega2 - \omega1
• Parallel resonance: occurs when Im(Y) = 0, for a parallel LC network loaded with resistance
• Bandwidth and Q apply similarly in parallel configurations

Nonlinear elements (overview)

• Nonlinear elements do not obey Ohm’s law (V ≠ IR in general, or R not constant)
• Examples: diodes, transistors (BJT, JFET, MOSFET), certain nonlinear resistors (thermistors, varistors)
• Key features: non-linear I–V characteristics, superposition and linearity do not apply
• Uses: rectification, regulation (Zener diodes), amplification, switching, modulation, sensing

Energy and inductive time constants

• Time constant for RL circuit: \tau_{RL} = \frac{L}{R}
• Transient response for current in an RL circuit after a step input follows: i(t) = I{final}\left(1 - e^{-t/\tau{RL}}\right)
• Inductive energy decay/charging follows the same exponential form with the RL time constant

Induction and energy through a coil (example use-cases)

• Energy stored when current changes: detailed derivations use U = \tfrac{1}{2}LI^2 and related integrals
• Mutual inductance and energy transfer between coils depend on M and coupling coefficient k (0 ≤ k ≤ 1)

Practical transformer and power concepts

• Power transformation: input power approximates output power with losses neglected: P{in} \approx P{out}
• Transformer sizing: given primary voltage Vp, secondary voltage Vs, turns ratio Np:Ns, and load, compute currents and power using the above relations

Brief notes on waveforms and signal context

• Periodic waveforms repeat patterns; key characteristics: amplitude, frequency, period, phase, and cycle definition
• Common waveforms: sine, triangular, square, complex waveforms

Summary formulas to memorize

• Flux and MMF relation: \mathcal{R} = \frac{l}{\mu A},\quad Fm = N I,\quad \Phi = \frac{Fm}{\mathcal{R}}
• Solenoid field: B = \mu0\mur\frac{N}{\ell}I
• Induced emf: \mathcal{E} = -N\frac{d\Phi}{dt}
• Energy in inductor: U = \tfrac{1}{2}LI^2
• Impedance: Z = R + j\omega L + \frac{1}{j\omega C} (with the sign convention shown above)
• Admittance: Y = \frac{1}{Z} = G + jB
• Resonant frequency: \omega_0 = \frac{1}{\sqrt{LC}}
• Time constant: \tau = \frac{L}{R} for RL; for RC, \tau = RC
• Transformer ratios: \frac{Vp}{Vs} = \frac{Np}{Ns},\quad \frac{Ip}{Is} = \frac{Ns}{Np}
• Energy density in magnetic field: u_B = \dfrac{B^2}{2\mu} (material-dependent: use μ or μ0 depending on context)