The impedance real part is due to the series resistance serie rs and to the parallel resistance rp. rs becomes important at high frequency when rp has a dominating effect essentially at low frequency, in particular at the industrial frequency.
In rs are collected the resistances of the conductors, connections, electrodes and electrolyte if necessary.In rp are collected the losses due to the dielectric polarisation, the dielectric resistance, the losses due to the leakage current and the charges redistribution phenomena if necessary.
In a theoretical or mathematical capacitor, the power is purely reactive. There are no losses.
Definition of tangent delta
δ = 90° - φ Delta is the angle between the impedance vector and the imaginary axis in the complex impedance plane. A mathematical capacitor (or perfect capacitor) has a tangent delta = 0
Quality factorIn the technical capacitor there is an active power PD which is due to the energy dissipation in the resistances. The capacitor quality factor Q is given by:
where PR is the capacitor reactive power, the one which is used. The loss factor is the inverse of the quality factor. It may be written with the complementary angle of φ , which is δ . Tangent delta may also be given by the ration of the losses and reactive power.
PD the mean losses power averaged during a cycle, in the case where rp is very large, is given by:
With U(t) = Uo sin ωt, the current is given by:
PD is finally given by:
PR the mean reactive power is given by:
Tan δ at “higher” frequency
At higher frequency where the effect of the series resistance is dominating, the electric circuit of a technical capacitor may be simplified to an equivalent series resistance ESR ~ Rs. The loss factor is given by:
Tan δ at “lower frequency”
At lower frequency where the effect of the parallel resistance is dominating, the electric circuit of a technical capacitor may be simplified to an equivalent parallel resistance EPR ~ Rp. The loss factor is given by:
Tan δ of the dielectric material beweeen the electrodeThe parallel part has two contributions which behavior differs as a function of the frequency: the « ohmic » losses contribution and the polarisation losses.
Total tan δ
To determine the total losses, a series resistance is added in series in front of the circuit Cp,rp.
Introducing the tangent delta expression found in the case of a equivalent parallel circuit, it comes out that:
Introducing the tangent delta expression found in the case of an equivalent series circuit, and defining the equivalent series capacitance Cs to simplify the equation:
By comparison with the impedance expression in the case of a series equivalent circuit (Rs ,Cs) of a technical capacitor (rp,rs, Cp):
Total tan δ frequency behavior
Dielectric materialComplex permittivity
The loss factor is given by
Interface polarization to be considered for « DC » applications.
Polarization by orientation to be considered at industrial frequency.
Ionic polarization to be considered in the microwave and infra-red frequency domain.
Electronic polarization to be considered in the optical frequency domain.