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): $\mu = \mu</em>0\mu_r,\quad B = \mu H$
• 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; $F</em>m = N I$ where N is turns and I is current
• Magnetic circuit law (Ohm’s law analog): $\Phi = \frac{F<em>m}{\mathcal{R}},\quad F</em>m = \Phi\mathcal{R}$
• 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 $B = \mu<em>0\mu</em>r\frac{N}{\ell} I$ 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: $u<em>B = \dfrac{B^2}{2\mu</em>0}$; 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: $\Phi<em>1 = M</em>{12} I<em>2, \quad \Phi</em>2 = M<em>{21} I</em>1$; for passive coils, $M<em>{12} = M</em>{21} = M$
• Inductors in networks: series/parallel combinations use standard impedance rules (see impedance section)
• Transformer basics: $\frac{V<em>p}{V</em>s} = \frac{N<em>p}{N</em>s},\quad \frac{I<em>p}{I</em>s} = \frac{N<em>s}{N</em>p}$
• 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: $Z<em>R = R,\quad Z</em>L = j\omega L,\quad Z_C = \frac{1}{j\omega C} = -\frac{j}{\omega C}$
• 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
  $I<em>1 = I</em>{tot}\frac{Z<em>2}{Z</em>1+Z<em>2},\quad I</em>2 = I<em>{tot}\frac{Z</em>1}{Z<em>1+Z</em>2}$

Resonance in RLC circuits

• Series resonance: occurs when $X<em>L = X</em>C\Rightarrow \omega L = \frac{1}{\omega C}$
• Resonant frequency: $\omega<em>0 = \frac{1}{\sqrt{L C}},\quad f</em>0 = \frac{\omega_0}{2\pi}$
• At resonance, Z = R (minimum impedance in the ideal case) and current is maximum
• Bandwidth and quality factor (Q):
• $Q = \frac{\omega<em>0 L}{R} = \frac{1}{\omega</em>0 R C}$
• 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: $\Delta\omega = \omega<em>2 - \omega</em>1$
• 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: $i(t) = I<em>{final}\left(1 - e^{-t/\tau</em>{RL}}\right)$
• 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: $P<em>{in} \approx P</em>{out}$
• 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: $\mathcal{R} = \frac{l}{\mu A},\quad F<em>m = N I,\quad \Phi = \frac{F</em>m}{\mathcal{R}}$
• Solenoid field: $B = \mu<em>0\mu</em>r\frac{N}{\ell}I$
• 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: $\frac{V<em>p}{V</em>s} = \frac{N<em>p}{N</em>s},\quad \frac{I<em>p}{I</em>s} = \frac{N<em>s}{N</em>p}$
• Energy density in magnetic field: $u_B = \dfrac{B^2}{2\mu}$ (material-dependent: use μ or μ0 depending on context)