Costruzione di un elettrocardiogramma (laboratorio di elettronica), Guide, Progetti e Ricerche di Elettronica

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Studying and functioning of an

ECG

Electronic laboratory report

A.A. 2022-

24 May 2023

Di Lorenzo Francesco - 25377A

1 Introduction

The purpose of this report is analyzing the functioning of an instrumental amplifier and designing it to show the heart pace on oscilloscope applying electrodes to the fingers subject.

2 Equipment

• Oscilloscope
• Function generator
• Tension generator
• 4 capacitors 1 μF
• 3 TL081 operational amplifiers (opamp)
• 2 resistors 47 kΩ
• 2 resistors 47 kΩ
• 4 resistors 2.2 kΩ
• 1 resistor 1 kΩ
• 2 resistors 1 MΩ
• Digital multimeter
• Breadboard and cables

3 The circuit design

To capture and distinguish on the oscilloscope the electrical impulses of the heart pace, it is necessary to build a ”difference amplifier” that amplifies the ”differential mode signal components” V dm, that is the electrical potential difference that occurs with each heart contraction. At the same time the circuit must be designed to cancel the ”common mode gain” Gcm, that is the offset signals of the input voltages, since the most important in- formation for the experiment is the voltage difference between the inputs. Furthermore, considering that the perceptible voltage on the fingers is very low, it is necessary to put before an ”instrumental amplifier”, in order to increase the differential gain and to cancel the the perturbative effects of the human body on the circuit. Indeed the input resistance of the instrumental amplifier is the opamp resistance, which, for our purposes, can be considered infinite. In the end, high pass RC filters are installed on the circuit inputs, in order to do not pass the electrostatic voltage stored in the human body through the movement, and so to prevent the opamp saturation. At the same time the filters allow the cardiac signals pace to pass, transmitting it to the circuit output.

4.1 Capacitors C 1 and C 2

The differential amplifier capacitors are meant to eliminate the high frequency sig- nals disturbances of the electrical network. Indeed a capacitor tends to behave like a short circuit for the high frequencies. The low frequency pulses, such as the 1 Hz heart beat, are blocked by capacitors, forcing them through resistors, contributing to Vout.

5 Instrumental Amplifier

The instrumental amplifier input voltages are VinA and VinB , that are the human body electrical signals. Because of the opamp ”virtual ground”, these voltages are transmitted on the nodes at the ends of the resistor R 2. The voltage on R 2 is there- fore VinA - VinB and the current I 2 on R 2 is:

I 2 =

R 2

VinA − VinB

This current flows on R 1 and R 3 resistors since the virtual contact of the opamps can be approximated to an open circuit. Therefore the voltage generated by R 1 , R 2 and R 3 resistors, that is Vo 1 - Vo 2 is:

Vo 1 − Vo 2 =

R 2

VinA − VinB

(R 1 + R 2 + R 3 ) (6)

The mean value between Vo 1 and Vo 2 is:

Vo 1 + Vo 2 2

VinA + I 2 · R 1 + VinB − I 2 · R 3 2

Rewriting equation (1) as a function of VinA and VinB , we have:

Vout = (VinA−VinB )

R 1 + R 3

R 2

R 5

R 4

(VinA + VinB ) 2

R 5

R 4

R 7

R 6

1 + R 5 /R 4

1 + R 7 /R 6

After this modification and setting the condition R 5 /R 4 = R 7 /R 6 , gains become:

Gdm =

R 1 + R 3

R 2

R 5

R 4

Gcm =

R 5

R 4

R 7

R 6

1 + R 5 /R 4

1 + R 7 /R 6

In this way it is possible to see how the Gdm gain can increase with the (R 1 + R 3 )/R 2 ratio.

6 RC filters

The RC high pass filters are used to block the electrostatic charge voltage accumu- lated, but the sizing of its components must be such as to have a cut-off frequency that allows cardiac signals to pass. For these reasons have been chosen R= 1MΩ and C= 1 μF, to perform a cutoff frequency of 1 rad/s or 0.16 Hz.

7 Nominal values

The measurement uncertainties values have been taken referring to the resistor color code.

Differential amplifier Instrumental amplifier R 4 2170 ± 108 Ω R 1 46.80 ± 0.47 kΩ R 5 2170 ± 108 Ω R 2 990 ± 10 Ω R 6 2180 ± 109 Ω R 3 46.46 ± 0.46 kΩ R 7 2160 ± 108 Ω RinA & RinB 990 kΩ C 1 0.96 μF CinB 1 μF C 2 0.95 μF CinB 1 μF

Table 1: Coils values

The differential amplifier resistors have been chosen all the same to cancel the com- mon mode gain and to set the differential mode gain equal to 1. The R 1 and R 3 have been set to the same value for symmetry reasons and to obtain, with R 2 of 1 kohm, a gain of almost two orders of magnitude.

Theoretical gains ECG Gdm -95.20 ± 6. Gcm -0.004 ± 0.

Table 2:

The errors on the gains are obtained with the errors propagation.

8 Operative measures

To measure the gains, the capacitors have been removed to avoid perturbations due to their charging and discharging effects.

8.1 Gcm

To measure the common mode gain it is necessary to short circuit the two inputs so that VinA=VinB. In this way, through equation (8) and sampling Vout for different Vin values it is possible find out the experimental gain.

8.2 Gdm

To measure the differential mode gain it is necessary set one input to the ground (VinB = 0) and sampling the Vout/Vin ratio with fit of the least squares method. The gain is obtained by rewriting equation (8) with the collected data:

Gdm =

Vout Vin

Gcm 2

Figure 2: Common mode gain linear fit

10.2 Differential mode gain fit

Since the uncertainties on the x axis are not neglectable, it was necessary to make a fit that considered the orthogonal distance of the data from the model graph rather than the ordinate distance from the model. In other words, the least squares fit method was not used but the ODR (Orthogonal Distance Regression) method.

Figure 3: Differential mode gain general fit

11 Data analysis common mode gain

The optimal values from the fit are: Gcm = 0.0013 ± 0. q = -44.11 ± 0.29 mV

11.1 χ^2 value

The χ^2 value is: χ^2 = 5. On 8 degrees of freedom makes a p-value of: p − value=31. Which means a probability of 68.92 % of finding data with larger deviations from the model.

11.2 Residuals

The residuals are normalized on the measurement uncertainties and it is possible to see how they are distributed according to a Gaussian curve. Indeed 5 out of 10 data are underestimated and the other 5 are overestimated. Also 80% of data are within one uncertainties from the model. For this reason it is possible to say that there is no systematic error in the data.

Figure 4: Heart pace ECG