Download 2 Steady-State Response of Electrodes Full

4B15 Introduction to Bioengineering
Electrodes and Transducers
Lecture 2: Steady-State Response of Electrodes
2.1
Introduction
In measuring electrical bio-signals, electrodes act as the interface between
the surface of the skin and the input of the recording amplifier. They essentially
perform the task of converting the ionic current associated with electrical
activity in the body into electronic current which is fed into the input of a
recording amplifier. This is facilitated through the medium of an electrolyte in
the form of the coupling gel. As the signal of interest is electrical in nature and
the amplifier and its associated circuitry is electrical, it is essential to know how
the electrode will influence the electrical behaviour and performance of the
measurement set-up. This means that it is important to establish an electrical
equivalent model for the electrode so that the electrical behaviour of the
measurement set-up can be examined. Only in this way can the performance
requirements of the recording amplifier be established to ensure that the signal
presented at the output of the amplifier is an accurate and reliable
representation of the electrical activity in the body which it is intended to
observe.
The typical ECG electrodes shown below are pre-jelled with an electrolyte
so that this electrolyte makes contact with the skin when the electrode is placed
and adheres to it.
Fig. 1 Adhesive, Pre-jelled ECG Electrodes
1
2.2 Electrode Electrical Model
A simplified model of the electrode skin interface is shown in Fig. 2. When
the electrolyte gel comes into contact with the skin, the ionic chemical
interaction establishes a polarisation potential which is represented by the dc
voltage source, VPOL, shown. The resistor RS represents the series contact and
lead resistance, which is usually very small compared to the impedances of the
other components. The capacitor CP represents the capacitance formed by the
contact area of the electrode on the one side and the skin on the other. The
resistor RP represents the conductance of the coupling gel as a poor dielectric
but also the electrical interaction with the surface of the skin. These latter two
elements therefore appear in parallel.
RP
VPOL
RS
CP
Fig. 2. A Simple Electrode Model
2.3 Frequency Response
It is imperative to investigate the effect of the equivalent circuit
impedances on the frequency response of the overall recording system. The
front-end of this system can be treated as a signal source feeding two electrodes,
one on either leg of the inputs to a differential amplifier as shown in Fig 3.
RP1
iin
RS1
VS
CP1
RP2
Vin
Rin
Amplifier
RS2
CP2
Fig. 3. A Model of the Front End of the Recoding System
2
For the present the input impedance of the amplifier is taken as purely resistive.
The polarisation potential is dc and so can be ignored from the point of view of
the ac analysis.
For steady state conditions let s = jω then the transfer function is given as:
𝑽𝒊𝒏 = 𝒊𝒊𝒏 𝑹𝒊𝒏
𝒊𝒊𝒏 =
𝑽𝑺
𝑹𝑷𝟏
𝑹𝑷𝟐
𝑹𝒊𝒏 + 𝑹𝑺𝟏 + 𝑹𝑺𝟐 +
+
𝟏 + 𝒋𝝎𝑪𝑷𝟏 𝑹𝑷𝟏 𝟏 + 𝒋𝝎𝑪𝑷𝟐 𝑹𝑷𝟐
Then the transfer function is given as:
𝑽𝒊𝒏
=
𝑽𝑺 𝑹 + 𝑹 + 𝑹 +
𝒊𝒏
𝑺𝟏
𝑺𝟐
𝑹𝒊𝒏
𝑹𝑷𝟏
𝑹𝑷𝟐
+
𝟏 + 𝒋𝝎𝑪𝑷𝟏 𝑹𝑷𝟏 𝟏 + 𝒋𝝎𝑪𝑷𝟐 𝑹𝑷𝟐
If RS1 = RS2 = RS , RP1 = RP2 = RP and CP1 = CP2 = CP then this becomes:
𝑽𝒊𝒏
𝑹𝒊𝒏
=
𝑹𝑷
𝑽𝑺 𝑹 + 𝟐 (𝑹 +
𝒊𝒏
𝑺
𝟏 + 𝒋𝝎𝑪𝑷 𝑹𝑷 )
𝑽𝒊𝒏
𝑹𝒊𝒏 (𝟏 + 𝒋𝝎𝑪𝑷 𝑹𝑷 )
=
𝑽𝑺 𝑹𝒊𝒏 (𝟏 + 𝒋𝝎𝑪𝑷 𝑹𝑷 ) + 𝟐[𝑹𝑺 (𝟏 + 𝒋𝝎𝑪𝑷 𝑹𝑷 ) + 𝑹𝑷 ]
𝑽𝒊𝒏
𝑹𝒊𝒏 (𝟏 + 𝒋𝝎𝑪𝑷 𝑹𝑷 )
=
𝑽𝑺
𝑹𝒊𝒏 + 𝒋𝝎𝑪𝑷 𝑹𝑷 𝑹𝒊𝒏 + 𝟐𝑹𝑺 + 𝟐𝒋𝝎𝑪𝑷 𝑹𝑺 𝑹𝑷 + 𝟐𝑹𝑷
𝑽𝒊𝒏
𝑹𝒊𝒏 (𝟏 + 𝒋𝝎𝑪𝑷 𝑹𝑷 )
=
𝑽𝑺 𝑹𝒊𝒏 + 𝟐(𝑹𝑺 + 𝑹𝑷 ) + 𝒋𝝎𝑪𝑷 𝑹𝑷 (𝑹𝒊𝒏 + 𝟐𝑹𝑺 )
(𝟏 + 𝒋𝝎𝑪𝑷 𝑹𝑷 )
𝑽𝒊𝒏
𝑹𝒊𝒏
=
𝑹𝒊𝒏 + 𝟐𝑹𝑺
𝑽𝑺 [𝑹𝒊𝒏 + 𝟐(𝑹𝑺 + 𝑹𝑷 )] [𝟏 + 𝒋𝝎𝑪 𝑹 (
)]
𝑷 𝑷
𝑹𝒊𝒏 + 𝟐(𝑹𝑺 + 𝑹𝑷 )
3
If Rin >> 2RS then this can be taken as:
(𝟏 + 𝒋𝝎𝑪𝑷 𝑹𝑷 )
𝑽𝒊𝒏
𝑹𝒊𝒏
=
𝑹𝒊𝒏
𝑽𝑺 [𝑹𝒊𝒏 + 𝟐(𝑹𝑺 + 𝑹𝑷 )] [𝟏 + 𝒋𝝎𝑪 𝑹 (
)]
𝑷 𝑷
𝑹𝒊𝒏 + 𝟐(𝑹𝑺 + 𝑹𝑷 )
The term on the left hand side can be taken as the dc attenuation factor:
𝜶=
𝑹𝒊𝒏
𝑹𝒊𝒏 + 𝟐(𝑹𝑺 + 𝑹𝑷 )
This means that the frequency response is of the form:
𝝎
(𝟏 + 𝒋 𝝎 )
𝑽𝒊𝒏
𝒁
=𝜶
𝝎
𝑽𝑺
(𝟏 + 𝒋 𝝎 )
𝑷
where:
𝝎𝒁 =
𝟏
𝑪𝑷 𝑹𝑷
𝝎𝑷 =
𝟏
𝑹𝒊𝒏
(
)𝑪 𝑹
𝑹𝒊𝒏 + 𝟐(𝑹𝑺 + 𝑹𝑷 ) 𝑷 𝑷
=
𝝎𝒁
𝜶
𝝎𝑷 > 𝝎𝒁
Expressing the response in the form of magnitude and phase gives:
𝝎
𝝎
(𝟏 + 𝒋 𝝎 )
(𝟏 + 𝒋 𝝎 )
𝑽𝒊𝒏
𝒁
𝒁
= 𝜶|
𝝎 | ∠[
𝝎 ]
𝑽𝑺
(𝟏 + 𝒋 𝝎 )
(𝟏 + 𝒋 𝝎 )
𝑷
𝑷
𝝎
𝝎
∠
+
𝒋
)|
(𝟏
𝑽𝒊𝒏
𝝎𝒁
𝝎𝒁 )
=𝜶
𝝎
𝝎
𝑽𝑺
|(𝟏 + 𝒋 𝝎 )| ∠ (𝟏 + 𝒋 𝝎 )
𝑷
𝑷
|(𝟏 + 𝒋
𝑽𝒊𝒏
=𝜶
𝑽𝑺
√𝟏 + (
𝝎 𝟐
𝝎
∠𝑻𝒂𝒏−𝟏 ( )
)
𝝎𝒁
𝝎𝒁
𝝎 𝟐
𝝎
√𝟏 + ( ) ∠𝑻𝒂𝒏−𝟏 ( )
𝝎𝑷
𝝎𝑷
4
𝜶<𝟏
𝝎 𝟐
𝟏
+
(
𝑽𝒊𝒏
𝝎
𝝎
𝝎𝒁 )
−𝟏
−𝟏
= 𝜶√
∠
−
𝑻𝒂𝒏
[𝑻𝒂𝒏
(
)
(
)]
𝝎 𝟐
𝑽𝑺
𝝎𝒁
𝝎𝑷
𝟏 + (𝝎 )
𝑷
𝑽𝒊𝒏
=𝜶
𝑽𝑺
𝝎𝟐𝒁 + 𝝎𝟐
(
)
𝝎𝟐𝒁
𝝎𝟐𝑷 + 𝝎𝟐
(
)
𝝎𝟐𝑷
√
∠ [𝑻𝒂𝒏−𝟏 (
𝝎
𝝎
) − 𝑻𝒂𝒏−𝟏 ( )]
𝝎𝒁
𝝎𝑷
𝑽𝒊𝒏
𝝎𝟐𝑷 (𝝎𝟐𝒁 + 𝝎𝟐 )
𝝎
𝝎
−𝟏
−𝟏
= 𝜶√ 𝟐 𝟐
∠
−
𝑻𝒂𝒏
[𝑻𝒂𝒏
(
)
(
)]
𝑽𝑺
𝝎𝒁
𝝎𝑷
𝝎𝒁 (𝝎𝑷 + 𝝎𝟐 )
This is a universal form of transfer function for such a pole-zero combination.
𝝎
In this case with 𝝎𝑷 = 𝒁 we have:
𝜶
𝑽𝒊𝒏
=𝜶
𝑽𝑺
𝝎𝟐𝒁
( 𝟐 ) (𝝎𝟐𝒁 + 𝝎𝟐 )
𝜶
√
𝟐
𝟐 𝝎𝒁
𝝎𝒁 ( 𝟐
𝜶
∠ [𝑻𝒂𝒏−𝟏 (
+ 𝝎𝟐 )
𝝎
𝜶𝝎
) − 𝑻𝒂𝒏−𝟏 ( )]
𝝎𝒁
𝝎𝒁
𝑽𝒊𝒏
𝝎𝟐𝒁 (𝝎𝟐𝒁 + 𝝎𝟐 )
𝝎
𝜶𝝎
−𝟏
−𝟏
= 𝜶√ 𝟐 𝟐
∠
−
𝑻𝒂𝒏
[𝑻𝒂𝒏
(
)
(
)]
𝑽𝑺
𝝎𝒁
𝝎𝒁
𝝎𝒁 (𝝎𝒁 + 𝜶𝟐 𝝎𝟐 )
𝑽𝒊𝒏
𝝎
𝜶𝝎
(𝝎𝟐𝒁 + 𝝎𝟐 )
−𝟏
−𝟏
= 𝜶√ 𝟐
∠
−
𝑻𝒂𝒏
[𝑻𝒂𝒏
(
)
(
)]
𝑽𝑺
𝝎𝒁
𝝎𝒁
(𝝎𝒁 + 𝜶𝟐 𝝎𝟐 )
When ω → 0 we have:
𝑽𝒊𝒏
𝝎𝟐𝒁
= 𝜶√ 𝟐 ∠[𝑻𝒂𝒏−𝟏 (𝟎) − 𝑻𝒂𝒏−𝟏 (𝜶𝟎)]
𝑽𝑺
𝝎𝒁
𝑽𝒊𝒏
= 𝜶 ∠𝟎
𝑽𝑺
5
When ω → ∞ we have:
𝑽𝒊𝒏
𝝎𝟐
√
= 𝜶 𝟐 𝟐 ∠[𝑻𝒂𝒏−𝟏 (∞) − 𝑻𝒂𝒏−𝟏 (∞)]
𝑽𝑺
𝜶 𝝎
𝑽𝒊𝒏
𝜶
= ( ) ∠[𝑻𝒂𝒏−𝟏 (∞) − 𝑻𝒂𝒏−𝟏 (∞)]
𝑽𝑺
𝜶
𝑽𝒊𝒏
= 𝟏 ∠𝟎
𝑽𝑺
Consider also a frequency of the geometric mean of the pole and zero locations
at 𝝎 = √𝝎𝑷 𝝎𝒁 which gives:
𝑽𝒊𝒏
𝝎 𝝎
𝜶√𝝎𝑷 𝝎𝒁
(𝝎𝟐𝒁 + 𝝎𝑷 𝝎𝒁 )
−𝟏 √ 𝑷 𝒁
−𝟏
= 𝜶√ 𝟐
∠
−
𝑻𝒂𝒏
[𝑻𝒂𝒏
(
)
(
)]
𝑽𝑺
𝝎𝒁
𝝎𝒁
(𝝎𝒁 + 𝜶𝟐 𝝎𝑷 𝝎𝒁 )
But again with 𝝎𝑷 =
𝑽𝒊𝒏
=𝜶
𝑽𝑺
𝝎𝒁
𝜶
(𝝎𝟐𝒁
this gives:
𝝎𝟐𝒁
+
𝜶)
𝟐
𝟐 𝝎𝒁
(𝝎𝟐𝒁 + 𝜶 𝜶 )
√
𝝎𝟐𝒁
√ 𝜶
𝝎𝟐𝒁
∠ 𝑻𝒂𝒏−𝟏
[
(
𝝎𝟐𝒁
𝜶√ 𝜶𝟐
𝝎𝒁
− 𝑻𝒂𝒏−𝟏
)
(
)]
𝟏
(𝜶𝝎𝟐𝒁 + 𝝎𝟐𝒁 )
𝑽𝒊𝒏
𝟏
𝜶
−𝟏 √
= 𝜶√
∠
[𝑻𝒂𝒏
(
) − 𝑻𝒂𝒏−𝟏 (√𝜶)]
𝟐
𝑽𝑺
𝜶
𝝎
(𝝎𝟐𝒁 + 𝜶𝟐 𝜶𝒁 )
which finally becomes:
𝑽𝒊𝒏
𝟏
= √𝜶 ∠ [𝑻𝒂𝒏−𝟏 (√ ) − 𝑻𝒂𝒏−𝟏 (√𝜶)]
𝑽𝑺
𝜶
The asymptotic values and the value at the geometric mean of the response allow
a Bode sketch of the gain and phase to be plotted.
6
mag
1
α1
α2
α3
0
ωPZ
ωP1
ωP2
ωP3
ω (log scale)
π/2
α3
phase
α2
α1
0
ωP1 ωZ ωP3 ωZ
ωP2 ωZ
ω (log scale)
Fig. 4 A Bode Diagram of the Frequency Response of the Electrode Model
7
2.4
Amplifier Input Impedance Requirements
It can be seen that the high frequency magnitude is unity while the low
frequency asymtote depends on the attenuation factor, α. The phase response is
asymtotic to zero at high and low frequencies, but has a peak phase which also
depends on the attenuation factor α which is given as:
𝜶=
𝑹𝒊𝒏
𝑹𝒊𝒏 + 𝟐(𝑹𝑺 + 𝑹𝑷 )
Note also that the location of the zero frequency depends only on the properties
of the electrode, while that of the pole depends in addition on the attenuation
factor α since:
𝝎𝒁 =
𝟏
𝑪𝑷 𝑹𝑷
𝟏
𝝎𝑷 =
(
𝑹𝒊𝒏
)𝑪 𝑹
𝑹𝒊𝒏 + 𝟐(𝑹𝑺 + 𝑹𝑷 ) 𝑷 𝑷
=
𝝎𝒁
𝜶
The attenuation factor itself depends of the input resistance of the amplifier, Rin.
Typical values for the electrode components for modern disposable adhesive
electrodes are RS = 50Ω, RP = 200kΩ and CP = 0.5µF. These values give the
frequency of the zero as:
𝒇𝒁 =
𝟏
𝟏
𝟏𝟎
=
=
= 𝟏. 𝟓𝟗𝑯𝒛
𝟐𝝅𝑪𝑷 𝑹𝑷 𝟐 × 𝟑. 𝟏𝟒 × 𝟎. 𝟓 × 𝟏𝟎−𝟔 × 𝟐 × 𝟏𝟎𝟓 𝟔. 𝟐𝟖
It should be noted that this frequency is well within the spectrum of the
ECG signal at low heart rates. The fundamental component in the spectrum of
the ECG at a heart rate of 40 bpm is 0.67 Hz, which is used in some regulation
standards as the lower end of the spectrum. The frequency of the zero given
above is more than twice this.
The frequency of the pole depends on the degree of attenuation present,
i.e. on the value of α. For a range of values of α the frequency of the pole can be
evaluated as in Table I below. The value of the maximum phase actually occurs
at the frequency of the geometric mean of the pole and the zero and also depends
on the value of α and is shown in Table I also.
Table I Pole Frequency and Phase for Different Values of α
α
0.1
0.25
0.5
0.75
.9
fP
15.9
6.36
3.18
2.12
1.77
Hz
36.9
19.5
8.2
3.0
degrees
∠𝝓𝑴𝑨𝑿 54.9
Older performance specifications for electrocardiographs issued by bodies
such as the American Heart Association (AHA) suggest that the magnitude
response within the bandwidth of the ECG should be flat to within a variation of
8
±0.5dB which corresponds to ±6% and should not have a phase shift exceeding
that introduced by a single-pole, high-pass filter with a pole frequency of 0.05Hz.
Such a filter has a phase shift of 5.6O at a frequency of 0.5Hz. In order to keep
the attenuation less than 6%, the value of α needs to be higher than 0.94. This
gives:
𝜶≥=
𝑹𝒊𝒏
> 𝟎. 𝟗𝟒
𝑹𝒊𝒏 + 𝟐(𝑹𝑺 + 𝑹𝑷 )
which means:
𝑹𝒊𝒏 > 𝟎. 𝟗𝟒𝑹𝒊𝒏 + 𝟏. 𝟖𝟖(𝑹𝑺 + 𝑹𝑷 )
or
𝟎. 𝟎𝟔𝑹𝒊𝒏 > 𝟏. 𝟖𝟖(𝑹𝑺 + 𝑹𝑷 )
so that:
𝑹𝒊𝒏 > 𝟑𝟏. 𝟑(𝑹𝑺 + 𝑹𝑷 )
For the values of these components given above as RS = 50Ω, RP = 200kΩ this
means:
𝑹𝒊𝒏 > 𝟔. 𝟐𝟔 𝑴Ω
This is easily attained with an amplifier input resistance of 10MΩ. It can be seen
from Table I above that an attenuation factor of 0.96 is associated with a
maximum phase shift of less than 3O which will therefore also satisfy the phase
requirement of the performance standards. However, it must be noted that
electrodes having values of the model components which may give much higher
impedances than those above will require an amplifier having a much higher
input resistance than 10MΩ. Dry electrodes which do not require a coupling gel
and which are the subject of current research fall into this category.
Of course the universal solution is to make the amplifier input impedance
as high as possible so that:
𝜶 → 𝟏,
𝝎𝑷 → 𝝎𝒁
𝒂𝒏𝒅
9
𝑽𝒊𝒏
→ 𝟏 ∠𝟎
𝑽𝑺