Use of Bioelectrical Impedance: General Principles and Overview
3.2 Impedance, Phase Angle, Resistance, and Reactance
In spite of the growing accessibility and use of BIA, there are still a lot of misconceptions concerning the appropriate analysis of BIA results. The basic concept of in-vivo bioimpedance monitoring is the injection of an alternating current between two specifi c points of the human body (usually the wrist and ipsilateral ankle for body composition assessments) and the measurement of the resulting voltage
Fig. 3.1 Number of annual peer-reviewed publications on BIA since 1980. Results based on PubMed search using the following citation:
(bioelectrical impedance analysis [Title/Abstract]) AND (“Year/MM/
DD”[PDAT]: “ Year/MM/DD
“[PDAT])
52 A. Stahn et al.
drop between these points. The data obtained from this measurement are impedance, phase angle, resistance, and reactance. To effectively interpret BIA results, a basic physical understanding of these concepts and their implications for physiologic processes is mandatory. To provide a simple and clear introduction to these concepts, they are fi rst addressed in a general context from a physical perspec- tive. In the following subsection they will then be linked to biological cells and tissues. The key con- cepts underlying these two subsections are summarized in Table 3.1 .
Table 3.1 Key concepts of bioelectrical impedance
Concept Defi nition Equation Unit
Impedance ( Z ) Electrical impedance is a measure of the total opposition of a conductor against an alternating current. This opposition is directly proportional to voltage across the two points and inversely proportional to the current between them. Given that the human body can be electrically modeled as an array of resistors and capacitors, the electrical impedance of the human body comprises both a real part (resistance R ) and an imaginary part (reactance X ).
Since in an alternating-current circuit voltage and current change as a function of frequency, impedance comprises both information about both the direction (phase) as well as the magnitude of impedance. Consequently, it is expressed as a complex number which can be graphically depicted as a vector.
Z= +R jX [ W ]
Magnitude of impedance ( Z )
The magnitude of impedance is a scalar quantity as it only refers to the distance of the impedance vector. It can be calculated as the vector sum of resistance and reactance.
Z= R2+X2c [ W ]
Phase ( j ) Phase refl ects the delay between voltage and current. Hence, it indicates to the direction of complex impedance. Graphically, it is represented as the angle between impedance and the x-axis.
1 Xc tan R j= − ⎛⎜⎝ ⎞⎟⎠ [°]
Resistance ( R ) Resistance refers to the real part of complex impedance. It is the opposition offered by a resistor to the fl ow of electrons due to friction. It is limited by the area and length of the conductor as well as its specifi c resistivity, the temperature-dependent intrinsic property of material to conduct a current.
l
R=rA [ W ]
Reactance ( X c ) Reactance refers to the imaginary part of complex impedance. It is the opposition generated by the storage and release of charges by a capacitor in an alternating current-circuit. Given that reactance is inversely proportional to frequency it decreases with increasing frequency.
1
c 2
X = pfC [ W ]
Extracellular resistance ( R e )
At very low frequencies or rather zero frequency obtained from extrapolation by bioimpedance spectroscopy, an alternating current will be able to penetrate the cell membrane. Accordingly, a measurement of resistance is only a refl ection of the extracellular space and therefore termed extracellular resistance.
R0=Re [ W ]
Intracellular resistance ( R i )
Intracellular resistance refers to the resistance of the intracellular space only. It cannot be measured, but computed from extracellular and infi nite resistance using Kichhoff’s laws.
0
0 i
R R R
R R
∞
∞
= − [ W ]
Infi nite resistance ( R ∞ )
At very high frequencies or rather infi nite frequency, cell membranes are continuously charged and recharged, so that virtually a current is also “conducted” in the intracellular space. Accordingly, resistance is a refl ection of the extracellular and intracellular resistance. Infi nite resistance can be extrapolated by bioimpedance spectroscopy and is related to intra- and extracellular resistance by Kichhoff’s laws.
∞
= 1 + 1[Ω]
1
e e
R R R [ W ]
The table above summarizes important key concepts underlying bioimpedance measurements, starting with the most important key point, impedance. Phase-sensitive analyzers additionally allow the differentiation between reactance and resistance. Extracellular resistance and intracellular resistance require performance of a bioimpedance spectros- copy, where measurements are performed between a range of frequencies (15–500) between 5 kHz and 1 MHz
53 3 Use of Bioelectrical Impedance: General Principles and Overview
3.2.1 Ohm’s Law
Electrical impedance is a measure of the total opposition of a conductor against an alternating current. It is defi ned by Ohm’s law, stating that the opposition of a conductor against an alternating current is directly proportional to voltage across the two points and inversely proportional to the cur- rent between them. Hence, if a known current is applied to a subject and voltage is measured, the underlying impedance can be calculated. Since in an alternating-current circuit voltage and current change as a function of frequency, impedance comprises information about both the direction (phase) as well as the magnitude of impedance. Whereas for direct currents resistance is simply defi ned as the ratio of voltage to current, the mathematical expression for impedance takes into account the dependence of voltage and current on frequency. This is summarized in the following equation:
max
max
sin( )
( ) [ ].
( ) sin( )
v i
V t
Z v t
i t I t
w j w j
= = + Ω
+ (3.1)
where Z is impedance, v ( t ), i ( t ), V max , and I max denote instantaneous voltage, instantaneous current, maximum voltage, and maximum current, respectively. t is the time since the waveform started, w is angular frequency, and j v and j i are the phase angle of voltage and current, respectively.
While pure resistors offer resistance to a fl ow of current due to friction against the motion of electrons, electrical circuits can also comprise reactance, which can be understood as inertia against the motion of electrons. Taken together, resistance and reactance determine the direction and magni- tude of impedance.
3.2.2 Resistance
Resistance is referred to as the real part of impedance and expressed in Ohm. Assuming uniform current density, resistance ( R ) can be defi ned by the physical geometry and resistivity of the conductor.
Specifi c resistivity is commonly denoted by r ( W ·m).·It·is a measure of the material’s ability to transmit electrical current that is independent of the geometrical constraints. The geometrical constraints are given by the cross-sectional area A (m 2 ) of the conductor and its length l (m) as indicated in
l [ ].
R=rA Ω (3.2)
3.2.3 Reactance
The imaginary part of impedance is termed reactance and also measured in Ohm. Reactance can be traced to capacitive and inductive sources. The latter, however, do presently not play a signifi cant role in most bioimpedance models, and will therefore not be considered in the following (Ward et al.
1999 ; Gluskin 2003 ; Riu 2004 ) .
Capacitors are able to store and release charges, but do not permit a direct, physical fl ow of charges.
However, by changing the polarization, they can virtually “conduct” current in proportion to the rate of voltage change. Hence, they will pass more current for faster-changing voltages (as they charge and discharge to the same voltage peaks in less time), and less current for slower-changing voltages. This frequency-dependent opposition is termed capacitive reactance ( X c ) and defi ned as
= 1 Ω
2 [ ].
Xc
p f C (3.3)
54 A. Stahn et al.
where f is frequency (Hz), and C is capacitance (F). Capacitance depends on the dimensions of the capacitor as well as the composition between the capacitor plates. For simple structures (i.e. parallel- plate capacitors), capacitance can be defi ned as
0 r A[ ].
C F
e e d
= (3.4)
where A (m 2 ) is the cross-sectional area of the plates, d (m) is the distance between the plates, e 0·is permittivity of free space (8.854 e⫺12 F·m -1 ), and e r is the relative permittivity of the permittivity of the material between the plates, i.e. the dielectric (dimensionless). The latter indicates the ease with which localized electrical charge can be polarized by the application of an electrical fi eld, thus affecting C .
3.2.4 Impedance and Phase
Electrical impedance comprises both the opposition to current related to resistance and reactance.
In Cartesian form, electrical impedance can be defi ned as
Z = +R jX[ ].Ω (3.5)
where Z denotes impedance, j is an imaginary unit ( −1) and is used instead of i to avoid confusion with the current symbol. Since, in alternating-current circuits, voltage and current change as a func- tion of frequency, impedance is also dynamic because it changes in direction and amplitude.
Consequently, impedance cannot be represented by a scalar quantity, which only refl ects a single dimension. To represent both direction (i.e. phase shift) and amplitude (i.e. distance or magnitude) complex numbers are used to denote impedance. For these reasons complex impedance is sometimes written in bold to distinguish it from the magnitude of impedance. For simplifi cation, Z is used to denote complex impedance as well as general impedance (magnitude) throughout in this text.
Graphically, complex impedance can be depicted as a vector. It has a defi nite direction and a defi nite length. Thus, it comprises the information of both dimensions. The relationships between Z , R and X c can also be graphically depicted in a complex plane. Figure 3.2 shows an example of such as plane for a resistor-capacitor (RC) circuit.
Due to the representation of R and X c as vectors, the magnitude of Z can be calculated as the vector sum of its individual components. The geometrical relationship between R and X c is also characterized by the phase angle ( j ), which varies between 0 degree and ⫺90 degrees. j represents the delay between voltage and current, and is caused by the ability of the capacitor to store and release charges.
For an ideal capacitor, j is ⫺90 degrees and Z is purely reactive. In contrast, if an alternating-current circuit is purely resistive, i.e. it consists of resistors only, the phase shift between the voltage and current is zero, and hence, Z is equal to R . As a result,
max max max
max max max
sin( )
( ) 0 [ ].
( ) sin( )
v i
V t V V
Z v t
i t I t I I
w j w j
= = + = ∠ = Ω
+ (3.6)
The frequency-dependent response of X c is also refl ected in j . An example for a simple resistor- capacitor (RC) circuit in series is given in Fig. 3.3 .
The behavior of the transfer functions can be explained straightforwardly. In a direct current circuit, once the applied voltage reaches a maximum, no more charges can accumulate on the plates of the
55 3 Use of Bioelectrical Impedance: General Principles and Overview
capacitor. Hence, no further current will pass through the circuit. For these reasons, it is commonly said that capacitors block direct current. Similarly, in an alternating-current circuit, at very low frequencies, capacitors nearly block the current, as the periodic accumulation of charges on the plates of the capaci- tor occurs at a low rate. Hence, as frequency decreases and converges towards 0, the opposition of the capacitor to the fl ow of current increases. In contrast, at high frequencies, the capacitor is continuously charged and discharged with regard to frequency, creating a continuous fl ow of electrons. Thus, it is said that alternating current “passes” capacitors as a function of frequency. Finally, for frequencies converging towards infi nity, the capacitor is short-circuited and the opposition of the capacitor dimin- ishes. Consequently, as indicated in Fig. 3.3 b the magnitude of impedance is equal to 100 W and the phase converges to zero.
Fig. 3.2 Complex impedance plane of an RC circuit. Complex impedance can be graphically visualized as a vector.
The vector indicates both magnitude (or distance) and direction (or phase) of impedance. The magnitude can be com- puted as the sum of the vectors of resistance ( R ) and reactance ( X ). The phase angle is expressed in degrees and can be calculated as the inverse tangent of X and R . Presently, except for a few exceptions X is expected to be purely attributed is to capacitive sources in human body composition models. Reactance is therefore also commonly denoted by the subscript c for capacitance ( X c )
a b
Fig. 3.3 Schematic presentation of serial RC circuit and its corresponding Bode plot. ( a ) Serial RC circuit ( b ) Bode plot for a serial RC circuit with C = 1 m F and R = 100 Ω, depicting magnitude and phase as a function of frequency.
Abscissa and ordinate are both expressed in logarithm to base 10 for magnitude. Bode plot Magnitude Z ( solid line ), phase j ( dashed line )
56 A. Stahn et al.