Application note #8
Impedance, admittance, Nyquist, Bode, Black, etc…
I- Impedance or admittance Nyquist’s
diagrams
Impedance Z and admittance Y are two
inverse transfer functions linked by the
following very simple relation:
Z
1
Y
(1)
Let us consider the electrical circuit shown in
Fig. 1 corresponding to circuit #1 of the Test
Box-3 [1].
To highlight the high frequency part of the
diagram, it is better to plot the admittance
diagram instead of the impedance diagram as
it is shown in Fig. 3.
The admittance diagram in Fig. 3 shows the
high frequency semi-circle better. Does the
graph of the admittance contain more
information than the graph of the impedance?
No, the admittance diagram only presents
information differently.
PEIS_10loops.m pr
Im(Y) vs. Re(Y), loop 1
Fig. 1: Voigt circuit made of three Rs and two
Cs.
The experimental Nyquist diagram of the
impedance Z is show in Fig. 2 [1]. Since
frequency values are lost in the Nyquist
diagram, it is useful to indicate the frequency
of some characteristic points (top of the semicircles).
-Im(Z) vs. Re(Z)
21.6 Hz
-Im(Z)/kOhm
0.6
0.4
68.2 Hz
0.2
0
-0.2
1
Re(Y)/kOhm-1
II- Impedance
diagrams
1.5
1
46.4 kHz
Fig. 3: Nyquist admittance diagram of the
electrical circuit shown in Fig. 1.
PEIS_10loops.m pr
2
Im(Y)/kOhm-1
0.8
or
admittance
Bode
To be convinced of that, we can plot the
impedance and admittance Bode diagrams as
shown in Fig. 4. Let us recall that plotting the
Bode diagram of a transfer function H consists
of plotting the decimal logarithm of the
magnitude of H given by:
14.7 kHz
0.5
0
-0.5
2
4
Re(Z)/kOhm
(Re H) 2
H
(Im H) 2
and the phase of H given by:
Fig. 2: Nyquist impedance diagram of the
electrical circuit shown in Fig. 1. Arrow always
indicates increasing frequencies.
Obviously the high frequency semi-circle is
smaller than the low frequency semi-circle.
φH
arctan
Im H
Re H
versus the decimal logarithm of frequency or
radial frequency.
Bio-Logic Science Instruments, 1 rue de l'Europe, F-38640 Claix - tel: +33 476 98 68 31 – Fax: +33 476 98 69 09
Web: www.bio-logic.info
1
According to Eq. (1), it is obvious that
log Y
Electricians use other representations, such as
Black diagrams, for example, where the
decimal logarithm of the magnitude is plotted
versus the phase (Fig. 5).
log Z
and
φY
φZ
PEIS_10loops.m pr
log ( |Z| ) vs. Phase(Z), loop 1
II- Impedance or admittance Black
diagrams
0.6
log ( |Z|/kOhm )
The graphs showing magnitude and phases
on Fig. 4 are symmetrical with respect to the
horizontal axis. There is no more information
in an admittance diagram than in an
impedance diagram.
31.7 Hz
0.4
0.2
0
31.6 kHz
-0.2
PEIS_10loops.m pr
log ( |Z| ) vs. log ( freq ), loop 1
-30
Phase(Z) vs. log ( freq ), loop 1 #
-20
-10
Phase(Z)/deg
log ( |Y| ) vs. Phase(Y), loop 1
-15
0.2
-20
0
-25
-0.2
-30
0
2
4
0.2
log ( |Y|/kOhm-1 )
-10
0.4
Phase(Z)/deg
log ( |Z|/kOhm )
PEIS_10loops.m pr
-5
0.6
31.6 kHz
0
-0.2
-0.4
31.7 Hz
-0.6
log ( freq/Hz )
PEIS_10loops.m pr
log ( |Y| ) vs. log ( freq ), loop 1 #
10
Phase(Y) vs. log ( freq ), loop 1
20
30
Phase(Y)/deg
30
25
0
20
-0.2
15
-0.4
10
-0.6
Phase(Y)/deg
log ( |Y|/kOhm-1 )
0.2
5
0
2
Fig. 5: Black impedance and admittance
diagrams of the electrical circuit shown in Fig.
1.
As with the Nyquist diagram, frequency values
are lost in Black Adiagram. Therefore, it is
useful to indicate the frequency of some
characteristic points.
4
log ( freq/Hz )
Fig. 4: Bode impedance and admittance
diagrams of the electrical circuit shown in Fig.
1.
Reference:
[1]
Bio-Logic
Application
(http://www.bio-logic.info)
Note
Bio-Logic Science Instruments, 1 rue de l'Europe, F-38640 Claix - tel: +33 476 98 68 31 – Fax: +33 476 98 69 09
Web: www.bio-logic.info
#9
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