The technical capacitor model shows that the impedance real part is the sum of two contributions: the capacitor series resistance serie r_{s} and the parallel resistance r_{p}.
The contribution due to r_{s} becomes important at high frequency
while the contribution due to r_{p} has a dominating effect essentially at low frequency,
in particular at the industrial frequency.

In the practical measurements, it's not possible to differentiate the 2 contributions. To overcome this problem equivalent models have been developped.
At low frequency, the Equivalent Parallel Resistance (EPR) model is used; at high frequency the Equivalent Series Resistance (ESR) model is used.

Equivalent Series Resistance (ESR)

All the dissipative composants are collected in a new resistance which is called the Equivalent Series Resistance and is specified as "ESR".

The impedance in the technical capacitor model is given by:
Z = R + j X = (r_{s} + r_{p}/a) + jw(L - C r_{p}^{2}/a) where a = 1 +(w C r_{p})^{2}

The impedance in the Equivalent Series Resistance model is given by:

Z = ESR - j/wC_{s}

Comparing both the real and imaginary part, it gives:

ESR = r_{s} + r_{p}/a
and
C_{s}= 1/w^{2}/(C r_{p}^{2}/a-L)

The plot below shows a simulation of the ESR with:
C = 100 nF
L = 100 nH
r_{s} = 10 mΩ
r_{p} = 100 kΩ

At the resonnance the impedance module is equal to the real part of the impedance.

At a given frequency the power losses are given by: P_{L} = ESR i^{2}.