Equivalent resistance models
The technical capacitor model shows that the impedance real part is the sum of two contributions: the capacitor series resistance serie rs and the parallel resistance rp.
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 = (rs + rp/a) + jw(L - C rp2/a) where a = 1 +(w C rp)2
The impedance in the Equivalent Series Resistance model is given by:
Z = ESR - j/wCs
Comparing both the real and imaginary part, it gives:
ESR = rs + rp/a
Cs= 1/w2/(C rp2/a-L)
The plot below shows a simulation of the ESR with:
C = 100 nF
L = 100 nH
rs = 10 mΩ
rp = 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: PL = ESR i2.