Introduction / Context:
Dielectric materials in alternating electric fields are described by a complex relative permittivity ε_r = ε_r' − j ε_r'' . The real part ε_r' represents the electric energy stored elastically in polarization, while the imaginary part ε_r'' represents energy dissipated as heat during each cycle. The dielectric loss angle δ quantifies this dissipation and is widely used in capacitor design, RF engineering, insulation diagnostics, and materials characterization.
Given Data / Assumptions:
- Complex relative permittivity: ε_r = ε_r' − j ε_r'' with ε_r', ε_r'' > 0 for passive materials.
- Loss angle δ is defined from the phase lag between the electric field and the displacement (or polarization) in linear dielectrics under sinusoidal steady state.
- Small-signal, linear, isotropic medium; angular frequency ω is fixed.
Concept / Approach:
The phasor form of the electric displacement is D = ε_0 ε_r E = ε_0 (ε_r' − j ε_r'') E. In the complex plane, the in-phase component (with E) is ε_0 ε_r' E and the quadrature (loss) component is ε_0 ε_r'' E. The loss tangent tan δ is defined as the ratio of the magnitudes of these two components. Therefore tan δ equals ε_r'' / ε_r'. This dimensionless ratio is sometimes written as tan δ = σ_eff / (ω ε_0 ε_r') when losses are represented by an equivalent conductivity σ_eff.
Step-by-Step Solution:
Write ε_r = ε_r' − j ε_r'' and D = ε_0 ε_r E.Identify stored (in-phase) component: D_s = ε_0 ε_r' E.Identify loss (quadrature) component: D_l = ε_0 ε_r'' E.Define loss tangent: tan δ = |D_l| / |D_s| = (ε_0 ε_r'' |E|) / (ε_0 ε_r' |E|).Cancel common terms to obtain: tan δ = ε_r'' / ε_r'.
Verification / Alternative check:
If ε_r'' = 0 (perfect lossless dielectric), tan δ = 0 as expected.For small losses (ε_r'' ≪ ε_r'), δ ≈ tan δ, so δ directly indicates the small phase lag; this matches standard capacitor dissipation factor measurements.
Why Other Options Are Wrong:
tan δ = ε_r' / ε_r'' reverses the ratio and would imply infinite loss when ε_r'' is small.tan δ = ε_r' * ε_r'' has incorrect dimensions and no physical basis.tan δ = 1 − ε_r' mixes a dimensionless ratio with an absolute permittivity value; meaningless physically.tan δ = (ε_r'^2 + ε_r''^2)^(1/2) gives the magnitude of ε_r, not the loss tangent.
Common Pitfalls:
Confusing the sign convention (ε_r = ε_r' − j ε_r'')—regardless of convention, the loss tangent remains the ratio of imaginary to real parts.Mixing absolute permittivity ε with relative permittivity ε_r; the ratio tan δ is the same in either case because ε_0 cancels.
Final Answer:
tan δ = ε_r'' / ε_r'
Discussion & Comments