๐Ÿ  / circuit / ๐Ÿงฒ Electrical Materials

Electrical Materials

Crystal structures, magnetic materials, dielectric properties, and insulating materials.

a) Crystal Structure

Atomic arrangements, unit cells, and packing efficiency

1. Crystal Structure: Fundamentals

Space Lattice

$$ \text{Lattice} + \text{Basis} \rightarrow \text{Crystal structure} $$

Unit Cell Volume: $ V = \bar{a} \cdot \bar{b} \cdot \bar{c} $

2. Atom Counting in 3D Unit Cells

APF

$$ N = N_c(\frac{1}{8}) + N_e(\frac{1}{4}) + N_f(\frac{1}{2}) + N_b(1) $$

$$ APF = \frac{N_{atoms} \times \frac{4}{3}\pi r^3}{a^3} \quad;\quad \text{Void} = 1 - APF $$

3. Simple Cubic (SC) Structure

Polonium

$$ N = 1 \quad;\quad CN = 6 \quad;\quad r = \frac{a}{2} $$

$$ APF = \frac{\pi}{6} \approx 0.52 \quad (52\%) $$

Touch points: 2r = a

4. Body Centred Cubic (BCC)

Na, Li, Cr

$$ N = 2 \quad;\quad CN = 8 \quad;\quad r = \frac{\sqrt{3}}{4}a $$

$$ APF = \frac{\sqrt{3}\pi}{8} \approx 0.68 \quad (68\%) $$

Body diagonal: 4r = aโˆš3

5. Face Centred Cubic (FCC)

Al, Cu, Ag

$$ N = 4 \quad;\quad CN = 12 \quad;\quad r = \frac{\sqrt{2}}{4}a $$

$$ APF = \frac{\sqrt{2}\pi}{6} \approx 0.74 \quad (74\%) $$

Face diagonal: 4r = aโˆš2 (Most efficient)

6. Unit Cell Properties Comparison

Diamond: 34% APF

$$ DC(0.34) < SC(0.52) < BCC(0.68) < FCC(0.74) $$

Diamond Cubic (Ge, Si, C): $N=8$, $CN=4$, $r = \frac{\sqrt{3}}{8}a$

7. Miller Indices & Bragg's Law

Crystallography

d-spacing: $$ d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}} $$

Bragg's Law: $$ n\lambda = 2d\sin\theta $$

b) Magnetic Materials

Dipoles, domains, hysteresis, and magnetic classification

8. Magnetic Materials: Fundamentals

Dipoles

Biot-Savart: $ dB = \frac{\mu_0}{4\pi}\cdot\frac{Id\ell\sin\theta}{r^2} $

Ampere's Law: $ \oint \vec{H}\cdot d\vec{l} = I_{enc} $

Intensity: $ B = \mu_0(H + M) \quad;\quad M = \chi_m H $

$\mu_B = 9.27\times10^{-24} \text{ Am}^2$

9. Anisotropy & Magnetostriction

Physical Changes

Magnetostriction: $\Delta L/L$ vs applied $H$.

Villari Effect: Converse of magnetostriction (Stress $\rightarrow$ $\Delta B$).

10. Curie Temperature & Magnetic Laws

$T_c$ Phase Transition

Paramagnetic (Curie's Law):

$$ \chi_m = \frac{C}{T} $$

Ferromagnetic ($T > T_C$):

$$ \chi_m = \frac{C}{T - \theta} $$

Anti-ferro ($T > T_N$):

$$ \chi_m = \frac{C}{T + \theta} $$

11. Types of Magnetic Materials

Classification

Dia ($\chi < 0$) | Para ($\chi > 0$) | Ferro ($\chi \gg 0$) | Anti-ferro ($\chi \approx 0$) | Ferri (large $\chi$)

12. Magnetization: Paramagnetic Materials

Langevin Function

$$ M = N\mu_0\left(\coth\alpha - \frac{1}{\alpha}\right) = N\mu_0 L(\alpha) \quad;\quad \alpha = \frac{\mu_0 H}{k_B T} $$

Weak fields ($\alpha \ll 1$): $M = \frac{N\mu_0^2 H}{k_B T}$ | Strong fields ($\alpha \gg 1$): $M \rightarrow N\mu_0 = M_{sat}$

13. B-H Hysteresis Loop Terms

Energy Loss

$$ B_r = \text{Retentivity} \quad;\quad H_C = \text{Coercivity} \quad;\quad W_{hyst} = \oint H\,dB $$

Steinmetz: $P_h = k_h f B_{max}^n$

c) Dielectric Materials

Polarization, permittivity, breakdown mechanisms, and dielectric loss

14. Internal Field in Solids & Liquids

Local Field

Lorentz Field: $$ E_{int} = E + \frac{P}{3\varepsilon_0} $$

Clausius-Mossotti: $$ \frac{\varepsilon_r - 1}{\varepsilon_r + 2} = \frac{N\alpha}{3\varepsilon_0} \quad;\quad \varepsilon_r = n^2 $$

15. Types of Dielectric Materials

Classification

Total Polarizability: $$ \alpha_{total} = \alpha_e + \alpha_i + \alpha_d + \alpha_{space} $$

$\alpha_e$: Optical ($10^{15}$ Hz) $\alpha_i$: Infrared ($10^{13}$ Hz) $\alpha_d$: Microwave ($10^{10}$ Hz) $\alpha_{space}$: Audio ($10^3$ Hz)

16. Piezoelectric, Ferroelectric & Pyroelectric

Active Dielectrics

Piezoelectric:

$$ P = dT \text{ (Direct)}, S = dE \text{ (Inverse)} $$

Ferroelectric:

$$ \varepsilon_r = \frac{C}{T - T_C} \text{ (Curie-Weiss)} $$

Pyroelectric: $$ \Delta P = \lambda \Delta T $$

17. Dielectric Properties Summary

Overview

$$ \varepsilon_r = \frac{\varepsilon}{\varepsilon_0} \quad;\quad \chi_e = \varepsilon_r - 1 \quad;\quad P = \varepsilon_0\chi_e E $$

$$ D = \varepsilon_0 E + P = \varepsilon_0\varepsilon_r E $$

Loss Tangent: $$ \tan\delta = \frac{\varepsilon''}{\varepsilon'} \quad \rightarrow \quad \text{Loss} = \omega\varepsilon_0\varepsilon''\tan\delta\,E^2 $$

18. Anti-Ferroelectric & Breakdown in Gases

Gases

Antiparallel alignment: $$ P_{net} = 0 $$ Examples: Lead zirconate, Sodium nitrate

Gas Mobility: $$ \bar{V} = \mu E $$ Townsend Criterion: $$ \gamma(e^{\alpha d}-1) = 1 $$

Paschen's Law: $$ V_{breakdown} = f(p \cdot d) $$

19. Dielectric Breakdown in Liquids

Liquids

Bubble Theory: $$ E_b = \frac{3\varepsilon_L}{2\varepsilon_L+1}E_0 $$

Liquid Globule stability: $$ E = 487.7\sqrt{\frac{\sigma}{R\varepsilon_L}} \text{ V/cm} $$

20. Dielectric Breakdown in Solids

Solids

Von Hippel (Intrinsic): $$ E_c = \frac{2\pi\nu em}{h}\left(\frac{1}{n_c^2} - \frac{1}{\varepsilon}\right) $$

Thermal Power loss: $$ W = E^2 f \cdot \frac{\varepsilon_r\tan\delta}{1.8\times10^{10}} \text{ W/cm}^3 $$

21. Dielectric Loss & Complex Permittivity

Losses

Complex Permittivity: $$ \varepsilon^* = \varepsilon' - j\varepsilon'' \quad;\quad \tan\delta = \frac{\varepsilon''}{\varepsilon'} $$

Power Loss: $$ P_{loss} = V^2\omega C\tan\delta \quad Q = \frac{1}{\tan\delta} $$

Debye: $$ \varepsilon' = \varepsilon_\infty + \frac{\varepsilon_s - \varepsilon_\infty}{1+\omega^2\tau^2} \quad \varepsilon'' = \frac{(\varepsilon_s - \varepsilon_\infty)\omega\tau}{1+\omega^2\tau^2} $$

22. Dielectric Strength & Energy Absorbed

Strength

Energy Lost: $$ W(t) = \frac{\omega}{2}\varepsilon_0\varepsilon_r\tan\delta\,E_0^2 \text{ W/m}^3 $$

Dielectric Strength: $$ E_{bd} = \text{Max Field before Breakdown} $$ Mica: 200 kV/cm (Highest)

d) Insulating Materials

Properties, thermal classification, and material selection

23. Insulating Materials: Basics & Properties

Insulators

NTC behavior: $$ dR/dT < 0 $$ (Resistance decreases as Temp increases)

Max Voltage: $$ V_{max} = E_{bd} \times d $$

24. Thermal Classification of Insulators

Thermal

Y (90ยฐ) < A,E (105ยฐ) < B (130ยฐ) < F (150ยฐ) < H (180ยฐ) < C (>180ยฐ)

Montsinger's Rule: $$ L_2 = L_1 \times 2^{-(T_2-T_1)/10} $$ (Life halves every 10ยฐC rise)

25. Classification of Insulating Materials

Categories

SFโ‚† (Sulphur Hexafluoride): $$ E_{bd} \approx 3 \times \text{Air} $$ (GIS Switchgear)

Organic/Inorganic | Natural/Synthetic | Solid, Liquid, Gas States

โ† Previous: ๐Ÿ”„ Two Port Networks