74
2 – Methods and techniques
For the R //C circuit
Z
1 jR C
R
R //C
ω
= +
The expression for the impedance of a R //C circuit in the form a + jb is
Z
1 R C
R
j 1 R C
R C
R //C
2 2 2
2 2 2
2
ω
ω
ω
= +
−
+
This is the equation of a circle centered on the abscissa and passing through
the origin.
For the circuit R //CPE, the equation for associating dipole devices gives
Z
1 RA(j )
R
R //CPE
p
ω
= +
Because
j
cos 2
p
jsin 2
p
p
π
π
=
+
we obtain Z
1 RA cos 2
p
jRA sin 2
p
R
R //CPE
p
p
ω
π
ω
π
=
+
+
The expression for the impedance of a R //CPE circuit in the form a + jb is
Z
1 RA cos 2
p
RA sin 2
p
R 1 RA cos 2
p
j
1 RA cos 2
p
RA sin 2
p
R RA sin 2
p
R //CPE
p
2
p
2
p
p
2
p
2
p
ω
π
ω
π
ω
π
ω
π
ω
π
ω
π
=
+
+
+
−
+
+
`
`
`
`
`
`
j
j
j
j
j
j
This is the equation for a circle with its center under the abscissa and that
passes through the origin. For p = 1, we find the expression for the impedance of the a R //C circuit (ideal capacitor, zero decentering).
2. The complex impedance of a R //CPE circuit is represented in the complex
plane in figure 29. β 1 and β 2 are the decentering angles for p = 0.75 and
p = 0.5, respectively.