Bias-induced impedance effect of the current-carrying conductors


ACTA IMEKO 
ISSN: 2221-870X 
June 2021, Volume 10, Number 2, 88 - 97 

 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 88 

Bias-induced impedance effect of the current-carrying 
conductors 

Sioma Baltianski 1 

1 Wolfson Dept. of Chemical Engineering, Technion – Israel Institute of Technology, Haifa, Israel 

 

 

Section: RESEARCH PAPER  

Keywords: bias-induced impedance; impedance spectroscopy; ZBI-effect 

Citation: Sioma Baltianski, Bias-induced impedance effect of the current-carrying conductors, Acta IMEKO, vol. 10, no. 2, article 13, June 2021, identifier: 
IMEKO-ACTA-10 (2021)-02-13 

Section Editor: Giuseppe Caravello, Università degli Studi di Palermo, Italy 

Received January 18, 2021; In final form April 15, 2021; Published June 2021 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution, and reproduction in any medium, provided the original author and source are credited. 

Corresponding author: Sioma Baltianski, e-mail: cesema@technion.ac.il  

 

1. INTRODUCTION 

The study of various physical objects using impedance 
spectroscopy under applied dc bias is widespread. These objects 
may be of different physical nature: semiconductor structures [1], 
electroceramic structures [2], [3], electrochemical objects [4], etc. 
In all cases, the external offset sets the operating point in the 
vicinity of which the impedance measurements are made. The set 
offset (bias) makes it possible to tie the parameters obtained from 
impedance measurements to the physical state of the object 
under study. 

The idea to investigate the current-carrying conductors under 
the influence of bias arose because of a misunderstanding of the 
behavior of a fairly simple object as a load in the process of 
testing the different impedance meters. The first research results 
were published in [5]. Possible measurement errors were checked 
in various ways, and literature sources describing the detected 
effects were not found. 

This work relates to impedance spectroscopy for several 
reasons. On the one hand, based on the described phenomena, 

sensitive elements can be created that require the use of 
impedance spectroscopy as a method for extracting informative 
parameters. On the other hand, it is a challenge to build more 
sensitive impedance meters (with an appropriate offset function) 
that allow obtaining reliable data under relatively difficult 
measurement conditions: low frequency and high tangent of the 
loss angle. The goal of this work is to reveal the discovered 
properties, which are important in the context of impedance 
research. 

Low-frequency impedance spectroscopy was used as a 
method. The complex conductivity function and its 
determination by approximating experimental data utilising 
different models serve as theoretical basis [6]-[8]. 

The impedance of current-carrying conductors is well known 
and described in the literature [9]. As an example, we give the 
behaviour of the impedance of a silver conductor 0.5 m long and 
0.25 mm in diameter. The initial experiment in Figure 1 without 
bias (at Vdc = 0 V) represents a typical behaviour of the real and 
imaginary parts of the impedance in the frequency range 1 MHz 
… 100 mHz. First, we use the full frequency range to verify the 
processes. Later, we will be interested in events only in the low 

ABSTRACT 
The paper presents the previously unstudied properties of current-carrying conductors utilising impedance spectroscopy. The purpose 
of the article is to present discovered properties that are the significant context of impedance research. The methodology is based on 
the superposition of test signals and bias affecting the objects under study. These are the main results obtained in this work: the studied 
objects have an additional low-frequency impedance during the passage of an electric current; the bias-induced impedance effect (ZBI-
effect) is noticeably manifested in the range of 0.01 Hz … 100 Hz and it has either capacitive or inductive nature or both types, depending 
on the bias level (current density) and material types. The experiments in this work were done using open and covered wires made of 
pure metals, alloys, and non-metal conductors, such as graphite rods. These objects showed the ZBI-effect that distinguishes them from 
other objects, such as standard resistors of the same rating, in which this phenomenon does not occur. The ZBI-effect was modeled by 
equivalent circuits. Particular attention is paid to assessing the consistency of experimental data. Understanding the nature of this effect 
can give impetus to the development of a new type of instrument in various fields. 

mailto:cesema@technion.ac.il


 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 89 

frequency part of the spectrum. The initial and all subsequent 
potentiostatic experiments were carried out at the amplitude of 
the test signal Vac = 10 mV. This signal corresponds to a small 
signal approach. The view of graph results is quite trivial at zero 
bias. Three approximate frequency domains can be distinguished 
in this graph: the high frequency region (HF) of the spectrum 
f = 1 MHz … 100 kHz; the mid frequency region (MF) 
f = 100 kHz … 100 Hz and the low frequency region (LF) 
f = 100 Hz … 0.1 Hz.  

In the HF region there is an increase of a real Re(Z) and 
imaginary Im(Z) part of impedance with increasing frequency. 
This part is well described by a parallel connection of resistance 
Rp and inductance Lp (Figure 1). 

In the MF region, a constant value of Re(Z) is observed. 
Series resistance Rs must be added to the model. A linear 
decrease in the imaginary component occurs with a decrease in 
frequency (log-log scale). The relative noise level increases at the 
same time. This noise is natural and associated with the 
measuring system capabilities. The LF part of the spectrum at 
Vdc = 0 V demonstrates the constancy of Re(Z) and the strong 
noise of Im(Z). This area is not informative for interpretation 
using the imaginary part of the impedance. Measurements were 
made using a Faraday cage to improve the signal-to-noise ratio 
mainly for the imaginary impedance component. 

In this simple model, the specific resistance of the conductor 
determines the series resistance Rs. Mainly the length of the 
conductor determines the inductance Lp. The parallel resistance 
Rp that is connected to the inductance characterises the active 
loss in the conductor due to the skin effect at high frequencies. 
The experimental values at zero bias correspond to the expected 
values and are quite common.  

The situation changes significantly when measurements are 
carried out under bias. The experimental characteristics are 
shown in the same Figure 1 at biases Vdc = 0.09 V … 0.9 V. The 
increment of bias was 0.09 V, and the measuring test signal was 
the same, namely Vac = 10 mV. The HF and the MF imaginary 
part of the impedance do not change with bias. However, the LF 
part changes considerably. This response can be reflected by 

including an additional non-linear impedance ZBI (Figure 1). A 
corresponding increase of the real part of the impedance occurs 
in the considered region, which meets the Kramers–Kronig 
relationships [6]. Besides, we observe a monotonic change in the 
real component of the impedance in the entire frequency range 
(the model element Rs). This change is caused by a shift in the 
temperature of the conductor due to biases. The model indicated 
in Figure 1 is intuitive, but it well describes and fits the object 
under study in the specified frequency range.  

The bias phenomenon is sometimes difficult to detect 
because experimental data are often on the limit of the sensitivity 
of measuring instruments, namely, limitations on phase 
measurement with a high value of the loss tangent. 
Potentiostat/Galvanostat Biologic SP-240 (BioLogic Science 
Instruments) was used as a measuring instrument. In doubtful 
cases, data were verified using the Potentiostat/Galvanostat 
Gamry Reference 3000 (Gamry Instruments). A Biologic’s 
contour plot defines an error not more than 0.3% and 0.3° in the 
desired measuring range [10]. Thus, the experimental data in 
Figure 1 is reliable. Moreover, in this case, we are interested in 
relative changes in the impedance components.  

A homemade four-wire sample holder was used to connect 
the samples under test (SUT), as shown in Figure 2. Electrode 
designation is taken from the manual [10]. This sample holder 
gives accurate measurement of low-resistance objects as well as 
negligible influence of contact phenomena. 

In addition to the devices used in this work (Biologic and 
Gamry), which are based on Frequency Response Analyzer, also 
the devices of the Lock-in Amplifier type can be used (see for 
example [11]). We used standard not wire wound resistors as 
references to verify measuring results and estimate artifacts. This 
phenomenon was not observed when dealt with standard 
resistors of the same rating as SUT.  

The impedance change in the LF region can be caused not 
only by bias using direct current but also by alternating current – 
by a large amplitude test signal at zero bias. It should be 
emphasised that in this work, we use a small signal approach in 
which a change in the LF impedance is not observed at zero bias. 
Thus, the occurrence of the additional impedance in the LF 
region will be determined solely by the level of bias. This physical 
phenomenon is named here as a bias-induced impedance (ZBI-
effect).  

In Section 2 we will systematise the experimental results. 
Section 3 is devoted to the interpretation of experimental results 

lo g  ( f r e q /Hz )

50

lo
g

 (
|R

e
(
Z

)
/O

h
m

|)

0

-1

-2

-3

-4

-5

lo
g

 (
|-

Im
(
Z

)
/O

h
m

|)

 0

 -1

 -2

 -3

 -4

 -5

Re(Z) at Vdc = 0 V

Im(Z) at Vdc = 0 V

Rs

Rp

Lp
Vdc t

ZBI

lo
g
(|
R

e
(Z

)/
O

h
m

|) lo
g
(|-Im

(Z
)/O

h
m

|)

log(freq/Hz)

 

Figure 1. Re(Z) and Im(Z) of the silver wire at Vdc = 0 V … 0.9 V with bias step 
0.09 V; length 500 mm and diameter 0.25 mm; frequency range 1 MHz …  
0.1 Hz. 

 

Figure 2. Four-Wire Sample Holder. 1 - High force-and-sense Kelvin clip; 2 – 
Sample Under Test (wire); 3 – Working Electrode; 4 – Counter Electrode; 5 - 
Low force-and-sense Kelvin clip; 6 – Reference Electrode; 7- Working Sence 
Electrode.  



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 90 

using electrical models. Section 4 discusses significant differences 
in impedance behavior of open and covered objects. Special 
attention is paid to checking the consistency of experimental 
data: this is outlined in the Section 5. Section 6 discusses and 
proves the main hypotheses that explain the revealed effect. 
Finally, the main results are presented in the concluding section.  

2. SYSTEMATISATION OF THE EXPERIMENTAL RESULTS  

We studied pure metals: nickel, copper, silver, tungsten, 
platinum, gold; alloys: constantan, nichrome, manganin; non-
metals - graphite rods. Although the frequency scan started from 
1 MHz toward low frequencies, the analysis of the results was 
carried out only for the LF part of the spectrum, where the ZBI-
effect manifested.  

According to the type of ZBI-effect, all studied materials were 
grouped into three categories: (i) the ZBI-effect has a capacitive 
nature, (ii) - an inductive nature, and (iii) - mixed when both types 
of reactance occur. Table 1 summarises the properties of the 
investigated materials. Below are the experimental characteristics 
of one of the representatives of each group.  

2.1. Pure metals 

We find significant changes in the behaviour of the imaginary 
part of the impedance after a critical frequency of about 30 Hz 
(resonance point) when applying bias (Figure 1). The inductive 
nature of reactance sharply changes to a capacitive nature from 
this point to the direction of LF. We observe a monotonic 
change in the imaginary and real component of the impedance, 
depending on the applied bias. 

Changes in the real part of impedance in the mid frequency 
region also occur; however, this is due to a change in the 
temperature of the conductor upon bias. For example, with a 
maximum bias of 0.9 V for this experiment and a conductor 
resistance of about 0.23 Ω, the current flowing through the 
conductor will be approximately 3.9 A. The power dissipation 
will be approximately 3.5 W, which will lead to certain heating of 
the conductor and consequential increase in its resistance. Figure 
3 shows a Nyquist graph of the LF part of the same experiment, 
which is shown in Figure 1. 

At mid and high frequencies, there is no change in the 
behaviour of the imaginary component of the impedance under 
the influence of bias. Henceforward, we will limit visualisation 
within the LF part of experimental data in the form of Nyquist 
plots. Similar in appearance, but numerically different in values, 
the characteristics were obtained in studies of other pure metals: 
nickel, copper, tungsten, platinum, and gold.  

2.2. Alloys 

The impedance characteristics of alloys as a function of bias 
differ from pure metals. Manganin demonstrated the inductive 
nature of reactance at moderate bias. In our nichrome and 
constantan samples, the ZBI-effect had both capacitive and 
inductive reactance. The nature of the reactance depends on the 
level of bias.  

As an example, studies of a nichrome sample with a diameter 
of 0.1 mm and a length of 57 mm are presented in Figure 4. 

The experiment was carried out using a bias in the range of 
2.8 V… 8.5 V in increments of 0.1 V. The data were taken in the 
frequency spectrum 1 MHz … 0.1 Hz, but only the range of 
interest is presented here: 100 Hz … 0.1 Hz.  

Three areas of bias were identified. In the bias of 2.8 V … 
4.6 V, the capacitive nature of the reactance was observed. In the 
range of 4.6 V … 6.7 V, the increasing portion of the inductive 
nature of the reactance added to the decreasing portion of the 
capacitive nature of the reactance. With a subsequent increase in 
bias, the reverse process occurs. Also, in the bias range of 6.7 V 
… 8.5 V, the capacitive nature of the reactance was again 

Re ( Z ) / Oh m

0.30.250.20.15

-
Im

(
Z

)
/O

h
m

0.08

0.06

0.04

0.02

0

-0.02

-0.04

-0.06

-0.08

Z at
Vdc = 0 V

Z at
Vdc = 0.9 V

F
 <

 3
0

 H
z

F
 >

 3
0

 H
z

-I
m

(Z
)/
O

h
m

Re(Z)/Ohm

 

Figure 3. Nyquist plot of silver wire at Vdc = 0 V … 0.9 V with bias step 0.09 V; 
length 500 mm and diameter 0.25 mm. 

Re ( Z ) /Oh m

9.59

-
Im

(
Z

)
/O

h
m

0.3

0.2

0.1

0

-0.1

-0.2

-0.3

-0.4

-0.5

Z at Vdc = 6.7 V Z at Vdc = 4.6 V

Re(Z)/Ohm

-I
m

(Z
)/
O

h
m

 

Figure 4. Nyquist plot of nichrome wire at Vdc=4.6 V … 6.7 V with bias step         
0.1 V; length 57 mm and diameter 0.1 mm. 

Table 1. Systematisation of the investigated materials by the nature of ZBI-
effect. 

Type of 
conductors 

Pure  
metals 

Alloys 
Non-metals 
(Graphite) 

 
Nature of 
ZBI-effect 

 
Capacitive 

mixture: 
Capacitive and 

Inductive 

 
Inductive 



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 91 

observed, the same as with a small bias. Figure 4 shows a 
transient state when both types of reactance are present. 

2.3. Non-metals 

Measurements were carried out on graphite rods. Samples of 
various diameters were investigated.  

Figure 5 shows Nyquist plots of the impedance of the 
graphite rod 0.5 mm in diameter and 57 mm in length. The 
inductive nature of reactance was demonstrated over the entire 
range of biases.  

3. INTERPRETATION USING ELECTRICAL MODELS  

First, we consider a simple case of interpreting experimental 
data related to pure metals in which the ZBI-effect of capacitive 
nature is manifested. As an example, Figure 6 presents a fitting 
result of the LF part of one of the experiments shown in Figure 
3, specifically at bias Vdc = 0.72 V. 

The fitting was carried out using the impedance model which 
consists of serial resistor Rs connected to parallel C1 and R1. The 
resistor Rs reflects specific resistance of the sample under test 
and its geometry. The resistance varies with the applied bias, 

which affects the temperature of the sample (see a right shift of 
characteristics in Figure 3 with increasing bias). The parallel 
circuit C1-R1 exactly describes the ZBI-effect. Figure 6 shows a 
good fitting quality. 

A similar approach for fitting can be used for materials in 
which the ZBI-effect is purely inductive (Figure 5) by using the LR 
- circuit. The situation becomes more complicated in the case of 
a complex ZBI-effect (Figure 4). One of the possible electrical 
models that satisfactorily approximate the experimental data is 
embedded in Figure 7.  

A system function as a rational fraction [12] that corresponds 
to this model has the following form: 

𝑍(𝑠) =
𝐴0 + 𝐴1𝑠 + 𝐴2𝑠

2

1 + 𝐵1𝑠 + 𝐵2 𝑠
2

 , (1) 

where 𝑠 = 𝑗 2 π 𝑓 and 𝐴𝑖, 𝐵𝑖  are unknown coefficients. 
Although the system function uniquely approximates the 

experimental data, its coefficients are difficult to fill with a 
physical meaning. It is easier to do this using circuit functions 
which reflect the topology of the corresponding equivalent 
circuits [12]. The circuit function corresponding to the model in 
Figure 7 is described by the following equation: 

𝑍(𝑠) = 𝑅𝑠 +
𝑅1

1 + 𝜏𝐶 ⋅ 𝑠
+

𝐿1 ⋅ 𝑠

1 + 𝜏𝐿 ⋅ 𝑠
 , (2) 

where: 𝜏𝐶 = 𝑅1 ⋅ 𝐶1; 𝜏𝐿 = 𝐿1/𝑅2 and Rs, R1, R2, C1, L1 requested 
parameters from fitting.  

The system function (1) covers several equivalent circuits. 
Figure 7 represents one of the possible implementations. The 
results of fitting utilising this circuit for one of the characteristics 
represented in Figure 4, specifically at bias Vdc = 5.3 V, are given 
in Figure 7. The selection of a suitable electrical model can be 
made empirically by iterating through the available set of models 
determined by the system function (1). To implement this 
process, experimental data can be approximated using an 
acceptable set of equivalent circuits and utilising available fitting 
programs, such as LEVM [6] or method described in [13]. 

Figure 8 represents the dependencies of the model parameters 
versus bias of the silver wire corresponding to the data shown in 
Figure 3.  

Re ( Z ) /Oh m

1.31.2

-
Im

(
Z

)
/O

h
m

0.05

0

-0.05

-0.1

-0.15

Z at 
Vdc = 0V

Z at 
Vdc = 1V

F
 >

 1
0
0
 H

z

-I
m

(Z
)/
O

h
m

Re(Z)/Ohm

 

Figure 5. Nyquist plot of graphite rod at Vdc=0 V … 1 V with bias step 0.1 V; 
length 57 mm and diameter 0.5 mm. 

R1

Rs
C1

ZBI

-I
m

(Z
)/

O
h
m

Re(Z)/Ohm  

Figure 6. Fitting result of LF part of data (f = 10 Hz … 0.1 Hz). Silver wire at 
Vdc=0.72 V: Rs=0.214 Ω; R1=0.07 Ω; C1=21.34 F. 

R2

Rs
C2

ZBI

R1

L1

-I
m

(Z
)/

O
h
m

Re(Z)/Ohm  

Figure 7. Fitting result of LF part of data (f = 10 Hz … 0.1 Hz). Nichrome wire 
at Vdc = 5.3 V: Rs = 8.261 Ω; R1 = 0.622 Ω; C1 = 0.373 F; R2 = 1.018 Ω; 
L1 = 1.091 H. 



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 92 

The capacitance С1 increases exponentially with decreasing 
bias. This leads to a decrease in the contribution of the reactive 
component into the ZBI-effect. At the same time, a monotonic 
decrease in resistance R1 is observed. As a result, at zero bias, the 
ZBI-effect demonstrated by the vanishingly small magnitude.  

The resistance Rs reflects a change in resistivity as a function 
of temperature, which in turn depends on the flowing bias 
current. The temperature value (it got by utilising resistivity) and 
power dissipation for this experiment shown in Figure 9. Similar 
results, but differing in values, were obtained for other pure 
metals. 

The graphite rod model behaves quite differently as shown in 
Figure 10. First, the series resistance Rs decrease with bias due to 
a negative temperature coefficient (NTC). It distinguishes them 
from metal in which there is a positive coefficient of resistance 
(PTC), see Figure 8. Secondly, the inductance L1, together with 
the parallel resistance R1, decreases with decreasing bias. This 
nullifies the ZBI-effect at zero bias.  

The behaviour of model parameters for alloys is more 
complex and is beyond the scope of this article.  

4. THE DIFFERENCE BETWEEN OPEN AND COVERED 
OBJECTS 

Some of the previously investigated objects were studied 
using various types of cover. This is necessary to validate the 
hypotheses put forward to explain the occurrence of the effect 
as described in Section 6. The results of the study of current-
carrying conductors covered with dielectric materials are quite 
informative for this purpose. Here, we present the results 
employing a shell in the form of thin Teflon and ceramic 
(alumina) tubes that is quite tight adjacent to the object under 
study. It is necessary to expand the frequency range towards 
lower frequencies down to 1 mHz to detect the effect of covering 
onto the impedance results. This requirement significantly 
increased the time of each experiment. 

Figure 11 shows the real and imaginary parts of the 
impedance without covering. We used the galvanostatic mode of 
the measurement device (Biologic SP-240) and the same sample 
holder as previously. The full range of bias current was Idc = 

R1

Rs
C1

ZBI

0.72 V

Vdc/V

R
/O

h
m C

/F

 

Figure 8. Model parameters of silver wire as a function of bias - refer to the 
experiment data in Figure 3. 

Vdc/V

T
  
 

P
/W

 

Figure 9. Power dissipation and temperature of silver wire - refer to the 
experiment data in Figure 3. 

Rs

ZBI

R1

L1

Vdc/V

L
/H

R
/O

h
m

 

Figure 10. Model parameters of graphite rod as a function of bias - refer to 
the experiment data in Figure 5. 

l o g  ( f r e q / Hz )

50

lo
g

 (
|R

e
(
Z

)
/O

h
m

|)

0

-1

-2

-3

-4

-5

lo
g

 (
|-

Im
(
Z

)
/O

h
m

|)

 0

 -1

 -2

 -3

 -4

 -5

lo
g
(|
R

e
(Z

)/
O

h
m

|) lo
g
(-|Im

(Z
)/O

h
m

|)

log(freq/Hz)

 

Figure 11. Re(Z) and Im(Z) of the open silver wire; galvanostatic mode at 
amplitude Iac = 100 mA and bias Idc = 0 A … 4A with step 0.4 A; length 500 
mm and diameter 0.25 mm; frequency range 1 MHz … 1 mHz. 



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 93 

0 A … 4 A with steps 0.4 A, and the test signal was Iac = 100 mA 
(it satisfies the low signal approach). 

Figure 12 and Figure 13 show the results of the silver wire 
inside alumina and Teflon tubes, respectively. 

A comparison between these experiments in the form of a 
Nyquist plot at one of the biases (at Idc = 2 A) is shown in Figure 
14.  

It can be seen with the naked eye that an open wire has one 
time constant while the covered conductors have two time 
constants. Yet Teflon covering shows more overlapping and 
more distributed impedance spectra. 

The fitting of the low-frequency part of the data (100 Hz … 
1 mHz) using the equivalent RC-circuits is also presented in 
Figure 14. The fitting results are summarised in Table 2. The 
indexes in the equivalent circuits in the table have the following 
meaning: p - parallel connection and s - serial connection. 

Evaluating the fitting results, we can say that these objects are 
quite satisfactorily approximated by models utilising lumped 

elements. A better result can be obtained for the case of a 
conductor surrounded by Teflon using the Gaussian distribution 
function, convolved into impedance [14]. But in this case, to 
demonstrate the ZBI-effect when exposed by the covering, this is 
not essential.  

A simple calculation of the ratios of the time constants (τ = 
RC) taken from Table 2 gives the following values: τ2 / τ1 = 247 
in the case of alumina covering and τ2 / τ1 = 40 in the case of 
Teflon covering. 

Let us cite as an example the behaviour of an object exhibiting 
an inductive nature of ZBI-effect also in a free and covered state. 
Figure 15 shows resulting Nyquist graphs of a graphite rod at 
bias Idc = 0.9 A and the test signal Iac = 100 mA in the open state 
and covered with alumina and Teflon tubes. 

It is important to emphasise that the imaginary part has the 
opposite sign compared to the previous graphs of the silver wire. 

The fitting results are summarised in Table 3. The ratio of the 
two time constants (τ =L/R) for the graphite rod covered by 
alumina is τ2 / τ1 = 284. In the case of Teflon covering, the ratio 
is τ2 / τ1 = 22. The difference in the ratios of the time constants 
is similar to that found earlier in experiments with silver wire 
covered with the same materials. 

5. CHECK OF THE DATA CONSISTENCY 

Current-voltage characteristics were acquired on the same 
samples to check a data set for internal consistency. A sweep rate 
of 1 mV/s which is reasonable to our low frequency 0.1 Hz (in 
the first studies) was selected. This speed allows getting quasi-
static characteristics. Static and differential parameters, namely 
resistances, were calculated and compared with parameters 
obtained from impedance measurements. 

l o g  ( f r e q / Hz )

50

lo
g

 (
|R

e
(
Z

)
/O

h
m

|)

0

-1

-2

-3

-4

-5

lo
g

 (
|-

Im
(
Z

)
/O

h
m

|)

 0

 -1

 -2

 -3

 -4

 -5

lo
g
(|
R

e
(Z

)/
O

h
m

|)

lo
g
(-|Im

(Z
)/O

h
m

|)

log(freq/Hz)

 

Figure 12. Re(Z) and Im(Z) of the silver wire inside the alumina tube. The same 
experimental conditions as pointed in Figure 11. 

l o g  ( f r e q / Hz )

50

lo
g

 (
|R

e
(
Z

)
/O

h
m

|)

0

-1

-2

-3

-4

-5

lo
g

 (
|-

Im
(
Z

)
/O

h
m

|)

 0

 -1

 -2

 -3

 -4

 -5

lo
g
(|
R

e
(Z

)/
O

h
m

|) lo
g
(-|Im

(Z
)/O

h
m

|)

log(freq/Hz)

 

Figure 13. Re(Z) and Im(Z) of the silver wire inside the Teflon tube. The same 
experimental conditions as pointed in Figure 11. 

Re(Z)/Ohm

-I
m

(Z
)/

O
h
m

 

Figure 14. Nyquist plot of experimental and fitted data of non-covered silver 
wire, the wire inside alumina, and the wire inside Teflon. Low frequency part 
of data: 100 Hz … 1mHz at the bias Idc = 2 A. 

Table 2. Fitting results of non-covered silver wire, the wire inside alumina, 
and the wire inside Teflon (according to Figure 14). 

SUT Equivalent circuit Fit parameters: 
R/Ohm; C/F 

Open 
wire 

(R1p + C1p)s + 
Rs 

R1p = 0.032; C1p = 64.27 
Rs = 0.185 

Wire 
inside alumina 

(R1p + C1p)s + 
(R2p + C2p)s + 
Rs 

R1p = 6.0e-3; C1p = 42.3 
R2p = 0.025; C2p = 2.48e3 
Rs = 0.214 

Wire 
inside Teflon 

(R1p + C1p)s + 
(R2p + C2p)s + 
Rs 

R1p = 6.2e-3; C1p = 91.8 
R2p = 0.013; C2p = 1.86e3 
Rs = 0.179 



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 94 

The I-V characteristics of a silver sample are shown in Figure 
16. The parameters of the sample under test correspond to the 
parameters indicated in Figure 3. The setup for I-V 
measurements was identical to the setup for impedance 
measurements. 

Figure 16 represents I-V curve, static resistance Rstat = 
Vdc/Idc and differential resistance Rdiff = d(Vdc)/d(Idc). 
Parabolic spline interpolation was used for analytical 
differentiation. A fairly good accordance was obtained between 
the model parameters extracted from impedance measurements 
and the parameters calculated from the current-voltage 
characteristics. The parameter Rs extracted from the impedance 
ZBI and the Rstat extracted from the I-V well fit each other in an 
error of not more than 0.3%. The total resistance Rsum=Rs+R1 
found from the impedance measurements corresponds to the 
resistance Rdiff calculated from the I-V (Figure 16). As an 
example, the bias point Vdc = 0.72 V was taken in Figure 16 for 
indicating these correlations. It is corresponding to this bias 
point in Figure 6, Figure 8 and Figure 9. 

The dependence of power dissipation Pi +ΔP on the influence 
of bias and the test signal at an operating point i will have the 
form: 

𝑃𝑖 + 𝛥𝑃 = (𝑉𝑖 + 𝛥𝑉) ⋅ (𝐼𝑖 + 𝛥𝐼), (3) 

therefore, changes in power due to only the test signal is 
determined as 

𝛥𝑃 = 𝑉𝑖 ⋅ 𝛥𝐼 + 𝐼𝑖 ⋅ 𝛥𝑉 + 𝛥𝑉 ⋅ 𝛥𝐼 , (4) 

where: Vi, Ii - voltage and current at a working point and ΔV, ΔI 
– amplitudes of voltage and current of the test signal.  

From (4), it can be noticed that with increasing bias the 
dissipated power caused by the test signal will increase. Hence, 
the temperature variation will increase, which will lead to an 
increase in a change of the resistivity by an influence of the test 
signal. This is an explanation of the magnification of the ZBI-effect 
at all the experiments with increasing a bias. 

The most mysterious case is the presence of both types of 
reactivity in experiments with alloys (Figure 4 and Figure 7). It 
required an additional consistency check through an independent 
method. For these purposes, current-voltage characteristics with 
different sweep rates were used. The sweep rates were chosen to 
match the transient point around 0.5 Hz (Figure 7). The 
corresponding graphs of the I-V characteristics are shown in 
Figure 17. The setup was the same as for impedance 
measurements. 

Figure 18 shows the graphs of the calculated values of the 
static resistances Rstat obtained from the I-V characteristics with  
different sweep rates (Figure 17). 

Now we can see that, in the vicinity of 5.3 V bias voltage, the 
static resistance changes the trend from decreasing its value with 
increasing a bias at the sweep 0.1 mV/s to increasing its value 
with increasing a bias at the sweep 100 mV/s. The sign of the 

-I
m

(Z
)/

O
h
m

Re(Z)/Ohm

 

Figure 15. Nyquist plot of experimental and fitted data of non-covered 
graphite rod, the rod inside alumina, and the rod inside Teflon. Low 
frequency part of data 100 Hz … 1 mHz at the bias Idc = 0.9 A. 

Table 3. Fitting results of the non-covered graphite rod, the rod inside 
alumina, and the rod inside Teflon (according to Figure 15) 

SUT Equivalent circuit Fit parameters: 
R/Ohm; L/H 

Open 
rod 

(R1p + L1p)s + 
Rs 

R1p = 0.160; L1p = 0.343 
Rs = 1.09 

Rod 
inside alumina 

(R1p + L1p)s + 
(R2p + L2p)s + 
Rs 

R1p = 0.01; L1p = 1.33e-3 
R2p = 0.121; L2p = 4.572 
Rs = 1.128 

Rod 
inside Teflon 

(R1p + L1p)s + 
(R2p + L2p)s + 
Rs 

R1p = 0.041; L1p = 0.027 
R2p = 0.095; L2p = 1.38 
Rs = 1.041 

R
e
s
is

ta
n
c
e
/O

h
m

Voltage/V

C
u
rre

n
t/A

 
Figure 16. I-V curve; static and differential resistance curves of silver wire 
with the same dimensions as in Figure 3. 

C
u

rr
e

n
t/

A

Voltage/V

C
u
rr

e
n
t/
A

Voltage/V
   

Figure 17. I-V curves with voltage sweeps: 0.1; 1; 10 and 100 mV/s. Nichrome 
wire, the same dimensions as pointed in Figure 4.  



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 95 

differential resistance, which is the essence of impedance 
measurements, will change accordingly. Thus, there is a 
consistency between measurements in the frequency domain 
(Figure 7) and time domain (Figure 18). 

Several temperature studies were carried out to check the data 
consistency. The platinum conductor was heated by a separate 
heating element at various biases. 

Figure 19 shows the results of the LF part of data 100 Hz – 
0.1 Hz at different values of temperature controller: at room 
temperature, at 165 C°, and 265 C°. The actual temperature of 
the wire can be calculated considering the resistivity of the wire 
– via real part of impedance at frequency 100 Hz or using DC – 
measurement. For qualitative analysis, knowing the actual 
temperature of the wire is not essential in our case. 

The main results of this experiment are as follows. We see 
that the ZBI-effect does not occur in the absence of bias at any 
temperature. At the same time, this effect takes place in the 
presence of a bias at any temperature. It is also noticeable that 
the external temperature additively shifts the values of the real 
impedance component, almost without affecting the imaginary 
one. 

A good illustration in this graph is the occasion when, in the 
case of room temperature and bias Idc = 2 A, the actual 
conductor temperature is practically identical with the case when  
the temperature controller value shows 265 C° without bias 
applied to the conductor (Idc = 0 A). 

6. DISCUSSIONS 

The impedance of quite ordinary current conductors under 
bias in the low-frequency region demonstrates remarkable 
properties - named as ZBI-effect. The term "current-carrying 
conductor" refers to extended conductors designed to carry 
electric current. The research that is outlined in this article is also 
relevant to other types of objects. These can be conductive or 
semiconductive materials of various compositions and shapes. 
An example relates to experiments carried out with thermistors 
(they are out of the scope of the paper).  

The ZBI-effect is counter-intuitive. It was natural to assume that 
the measurement of the impedance of objects at the lowest 
frequency brings it closer to the measurement result at direct 
current. This is what happens in the absence of bias. Yet, if a bias 
is applied to an object the feeling deceives. It looks paradoxical, 
but the infra-low frequency is not an asymptotic approximation 
to DC when measuring the impedance of a conductor under bias.  

Now, the question of how to explain the occurrence of such 
significant reactive elements in the impedance models of the 
studied objects. In particular, the capacitance of pure metals 
reaches the order of farads (Figure 8). The inductance of graphite 
rods reaches about hundreds of millihenry (Figure 10). In reality, 
of course, such reactance does not exist in the studied objects. 
This phenomenon may be called as a “phantom” reactance.  

This effect can be explained considering two necessary 
properties of the studied objects. The first is nonlinearity and the 
second one is inertia. The nonlinearity of current conductors is 
the second kind of nonlinearity (indirect). This property 
distinguishes them from objects of the first kind of nonlinearity 
(direct), such as p-n junctions or Schottky diodes. 

In the case of the first kind of nonlinearity, the nonlinearity 
reveals itself directly, without any delay. For the second kind of 
nonlinearity, it manifests itself due to a resistivity dependence on 
temperature. This is a factor of the studied material. The 
nonlinearity of the second kind has a significant delay. The bias 
sets a specific operating point. The test signal acts in the vicinity 
of this point. No matter how small the test signal is, it will change 
the temperature of the investigated object in the locality of the 
operating point with a certain delay. Consequently, the resistance 
of the material under investigation will cyclically change with the 
test signal.  

Eventually, the resistance is modulated by the test signal. The 
difference between the phase of this modulation and the phase 
of the acting test signal determines the occurrence of phantom 
reactance. If the investigated object has a PTC property, a 
capacitive reactance arises. This behavior is typical for pure 
metals or PTC thermistors. If the object under study has an NTC 
feature, then an inductive reactance appears. It is specific, for 
example, to graphite and NTC thermistors. 

In terms of electrical measurements, impedance is properly 
defined only for systems satisfying stationarity [6]. In our case, 
we have a dynamic structure with one exception - the system 
changes cyclically and synchronously with the influence of the 
test signal. The amplitude and phase response depends on the 
frequency of the test signal. A purely active resistance, which 
changes are synchronously according to the test signal, but with 

R
e

si
st

a
n

ce
/O

h
m

Voltage/VVoltage/V

R
es

is
ta

n
ce

/O
h
m

 

Figure 18. Static resistances calculated from the I-V curves represented in 
Figure 17. 

-I
m

(Z
)/

O
h
m

Re(Z)/Ohm

  

Figure 19. Nyquist plot of platinum wire at different temperatures and biases; 
length 83 mm and diameter 0.2 mm; frequency range 100 Hz … 0.1 Hz. 



 

ACTA IMEKO | www.imeko.org June 2021 | Volume 10 | Number 2 | 96 

a different phase relative to the test signal, generates a reaction 
that looks like a complex resistance. As a result, a complex value 
will be estimated during the measurements as an impedance of 
the studied object.  

Successive experiments revealed a significant feature. It 
turned out that the time constant following from the ZBI-effect’s 
model (τ = RC for PTC objects and τ = L / R for NTC objects) 
weakly depends on the applied bias. It is reasonable to assume 
that these time constants related to the time constant of the heat 
exchange between the object under study and the environment 
(air in our initial study case). Studies using a various covering of 
current-carrying conductors support this hypothesis. There are 
two time constants (Figure 12 and Figure 13). The first of which 
is apparently determined by the thermal properties of the 
covering and the thermal interaction between the conductor and 
the covering. The second time constant is determined by the 
thermal interaction between the covering and the environment.  

An accurate description of thermal processes requires special 
knowledge and is beyond the scope of this article. However, it 
seems possible that the discovered effect could allow to develop 
specialised sensors to assess the thermal conductivity of various 
materials. This approach may be an alternative to the methods 
described in [15], [16].  

The experimental results obtained on the nichrome alloy 
motivates the appearance of additional ideas. In particular, both 
capacitive and inductive reactive components are observed at a 
bias Vdc = 5.3 V (Figure 7) and in the areas close to it (Figure 4) 
depending on the frequency of the test signal. Specifically, for 
Figure 7 capacitive nature takes place in the frequency range of 
10 Hz to 0.5 Hz, and inductive nature takes place in the range of 
0.5 Hz to 0.1 Hz. Such effects are possible if we assume that the 
temperature coefficient of resistance (TCR) has dynamic 
properties. In other words, the TCR changes its character 
depending on the rate of temperature change. In turn, the rate of 
temperature variations at the selected operating point will be 
related to the frequency of the test signal. Therefore, in the 
higher frequency range will be observed a PTC-feature, in the 
lower frequency range will be observed an NTC-feature (Figure 
7). This assumption was confirmed using I-V experiments with 
different sweep rates (Figure 17 and Figure 18). It looks like 
impedance spectroscopy may provide a more sensitive tool for 
assessing the dynamic properties of the TCR. 

Since the dynamic properties of the TCR will depend on the 
composition of the objects under study, there is a potential 
possibility of indirect composition estimation by checking the 
moment of the TCR sign change utilising impedance 
spectroscopy.  

The revealed new properties (i.e., the possibility of evaluating 
the thermal conductivity and estimating the composition of the 
material by dynamic TCR, arising from the discovered ZBI-effect) 
may represent a significant contribution to the scientific and 
technical community, in particular, in the development of the 
theory and practice of impedance spectroscopy of objects that 
cyclically change their parameters synchronously with the test 
signal. Understanding the nature of this effect can foster the 
development of a new type of instruments in various fields and 
various scientific institutions. 

7. CONCLUSIONS 

In this work, the phenomenon of bias-induced impedance 
was described. This effect was most evident in the low frequency 
spectra of the reactive part of the impedance. The manifestation 

of the different nature of this issue was shown experimentally. 
The ZBI-effect may be capacitive, inductive, or complex, which 
includes both types of reactance. The nature of the reactance 
depends on the type of test material. Pure metals showed 
capacitive reactance. Graphite rods showed inductive reactance. 
The alloys showed reactance of both types depending on the 
level of bias. The investigated objects can be attributed to the 
inertial nonlinear resistances.  

The ZBI-effect is caused by the thermal interaction between the 
conductor and the environment under the superposition of the 
bias combined with a test signal. 

Relatively simple equivalent circuits were found to describe 
the experimental data. Additional studies should be undertaken 
to better understand the behaviour of alloys and other 
composites under the bias, especially unexpected dynamic TCR 
properties.  

New possibilities arise for assessing the thermal conductivity 
of various materials. This requires the synthesis of knowledge in 
the fields of electrical and thermal measurements and the 
construction of specialised sensor devices. 

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