Practical employment №8


Ternopil state medical university

by I.Ya.Horbachevsky


Department medical informatics with physics





Tutorial for practical lessons

for the students І  course


“ Medical and  biological physics”





Physical bases of electrography


























Ternopol ­– 2010

Theme: Physical bases of electrography.

Amount of hours: 6 hours

Place of leadthrough: physical laboratory.

Purpose: To learn basic descriptions of electric-field, dielectric properties of biological fabrics. To learn a structure and principle of work of електрокардіографа. To learn basic conformities to the law of physical processes which flow in biological fabrics at the action  of electric direct-current. To learn to use a direct electric current for electrotherapy by a vehicle for galvanization.

Professional orientation of students.

The electric field is a variety of matter, with the help of which the power operating is carried out on electric charges which are in this field. Descriptions of electric-field which is generated biological structures are a state information generator organism. Registration of biopotentials of fabrics and organs with a diagnostic purpose got the name of electrocardiography, and registration of biopotentials of cardiac м"язу is at his excitation of electrocardiography of /EKG/. Biopotentials have ionic nature and arise up in connection with the changes of concentration of the proper ions for the even sides of membranes of cages. As biopotentials very thinly represent consisting of organs and fabrics of norm and in pathology, their correct registration and decoding is with the widely widespread reception of medical researches.

Program of independed prepare of students.

1.                 Basic descriptions of electric-field and connection are between them.

2.                 Physical bases of electrography.

3.                 Electrocardiogram.

4.                 Structure and principle of work of electrocardiograph.

5.                 Record and decoding electrocardiogram.

6.                 Sensors and their types.

7.                 Temperature of body of man, methods of its measuring.

8.                 Structure and principle of work of thermopar.


Information generators


1.                 Marcenyuk v.P., Didukh v.D., Ladika r.B., Baranyuk I.O., Sverstyuk a.S., Soroka I.S., Naumova l.V. Medical biophysics and medical apparatus: textbook. it is Ternopil': TDMU, 2008. – 170-177, 179-181, 185-193

2.                 Lopushanskiy ya.Y., Collection of tasks and questions from medical and biological physics.- Lvov, 2006, 163-186.

3.                 Chaliy o.V. Medical and biological physics. Kyiv: «Vipol», 1999. 260-294, 305-320.


1.    Timanyuk v.A. Fizika, Khar'kov, Basis, 1996.С — 285-345с.


Practical work – 9:00-11:15 (3 р)

Method of implementation of practical work:


Theoretical information


Basic descriptions of electric-field

The electric field arises up round any charged body, regardless of it moves or not. Power description of electric-field is a size of tension of electric-field, which equals the relation of force with which the field operates on a point charge, to the size of this charge:



Direction of vector  coincides with direction of force which operates on a positive charge.

Name the relation of potential energy of charge potential of electric-field in the field  to the size of this charge :






Potential is a scalar physical size which characterizes the field ability to execute work. The electrostatic field is potential, that is why work, executed the field, equals diminishing of potential energy:



where  is a difference of potentials, it is named also tension:


The difference of potentials (tension) between two points equals the relation of work which is executed by the field for transferring of charge from an initial point in eventual, to the size of charge.

Differentially connection looks like between the difference of potentials:      


The projection of vector of the field tension on the set direction equals speed of diminishing of potential in this direction.

If the electric field is created the system  of point charges, the field tension is equal to the vectorial sum of the tension fields, created every charge separately:



and potential of electric-field – as a sum of algebra of potentials from every charge:



Principle of superposition of the fields consists herein.


 Electric dipole. Weeds dipole

Name the system electric dipole from two even after a size and opposite after a sign point charges, located in the distance one from another. Description of диполя is a dipole moment – vector, even work of charge  on the shoulder of dipole :



where  is a vector, directed from negative to the positive charge (shoulder of dipole).

Potential of electric-field of point charge is in the distance  from him 



where  is a relative inductivity of environment,  – the electric became.

In obedience to the law of Ohm for a complete circle



If . Consequently, the size of current does not depend on resistance of external environment. Current диполь can be characterized Therefore, by analogy with electric диполем, current dipole moment :



where  is a vector which connects the poles of dipole of “– and “+”

In the homogeneous leading environment of unipole creates the electric field potential of which in the remote point of the field is equal



where  is a current through unipole,  – distance to the point, potential is determined in which,  is conductivity of environment.

Determine potential of current dipole after a formula:


Consequently between current dipole and electric dipole there is a considerable analogy which is based on the general analogy of electric-field in a leading environment and electrostatic field (, ).


Examples of decision of typical tasks

Task 1. Between the inside of cage and external solution there is a difference of potentials of U = 80mV. Counting the electric field into a membrane homogeneous, and thickness of membrane of l = 8nm, find tension of this field.


It is given:

U = 80 mV

l = 8 nm



For homogeneous electric-field into a membrane there is tension

where U— difference of potentials, l is a thickness of membrane



Answer: .

Task 2. Between point charges  distance of r1=40 sm. What work does need to be executed, to draw together them to distance of r2=25 sm?


It is given:

r1=0,4 m

r2=0,25 m



Work against forces of the field on transferring of charge of q2 from distance of r1 on distance r2:



Task 3. Equalization of potentials between the plates of flat condenser is evened 90V. Area of every plate of 60 sm2 and charge of 10-9C. What distance are plates on?


It is given:






The capacity of flat condenser is determined after a formula

Taking into account that

,       will get:

Putting numerical values, find:

Answer: .


Task 4. To define the middle rate of the directed movement of electron along a copper explorer at the closeness of direct-current of J = 11 A/мм2, if to consider that there is one lone electron on every atom of copper. Atomic mass of copper A = 64 kg/moll. Closeness of copper .


It is given:

J = 11 A/mm2

A= 64 kg/moll



Closeness of current

where q and n is a charge and concentration of transmitters, is middle speed.



Task 5. Capacitance of a parallel capacitor metal disks , 5.00 cm in consists of two metal disk, 5.00 cm in radius. The disk are separated by air and are a distance of 4.00 mm apart. A potential of 50.0 V is applied across the plates by a battery . find (a) the capacitance of the capacitor and (b) the change on the plat.


a.     the area of the plate is

A=r=(0.50  m)=7.8510m

The capacitance is


=17.4=17.4F=17.4 pF

note how the conversion factors have  been carried tkrough in the example to show that the capacitance does inded com out to have  the unit of farads.

b.    the charge on the plate is

q= CV=(17,4X10F)(50v)=8.7X10C.


Task 6. Total resistance of an ammeter. Find the total resistance of the ammeter in the previous example.


Since resistors  andare in parallel, their combined resistance, which is the resistance of the ammeter ,is \


Task 7. Power dissipated in a parallel circuit. What is the power dissipated in each resistor in example 20.14 ?




The power dissipated in each resistor is

P =  IR =  (0.300 A)(20.0) = 1.80 W

P =  IR =  (0.200 A)(30.0) =1.20 W

P =  IR = (0.150 A)(40.0) =0.900 W

Again note that the sum of the powers dissipated in each resistor

P + P + P =1.80 W + 1.20 W + 0.900 W= 3.90 W

is the same as the total power supplied to the circuit by the battery, within round-off error.


Task 8. The emf of a battery connected to a circuit. A battery has an emf of 1.50 V and an internal resistance of 3.00. It is connected as shown in figure 20.15 to a resistance of 500. Find the current in the circuit, and the terminal voltage of the battery.


The internal resistance r is in series with the load resistor R  , so the total resistance in the circuit is just the sum of the two of them. Using Ohms law to determine the current we have

I = = =  2.98 x 10 A

The terminal voltage across the battery is

V = - Ir= 1.50V – (2.98 x 10A)(3.00)=1.49 V

III. Modelling Electrical Signals in a Heart


Electrical Signals in a Heart

This model is kindly provided by Prof. Simonetta Filippi and Dr. Christian Cherubini from Universitа Campus Biomedico di Roma, Italy.


Modeling the electrical activity in cardiac tissue is an important step in understanding the patterns of contractions and dilations in the heart. The heart produces rhythmic electrical pulses, initiated from a point known as the sinus node. The electrical pulses, in turn, trigger the mechanical contractions of the muscle. In a healthy heart these electrical pulses are damped out, but a number of heart conditions involve an elevated risk of re-entry of the signals. This means that the normal steady pulse is disturbed, a severe and acute condition which is often referred to as arrhythmia.

This section presents two mathematical models describing different aspects of electrical signal propagation in cardiac tissue: the FitzHugh-Nagumo equations and the Complex Ginzburg-Landau equations, both of which are solved on the same geometry. Interesting patterns emerging from these types of models are, for example, spiral waves, which, in the context of cardiac electrical signals, can produce effects similar to those observed in cardiac arrhythmia.

excitable media and the fitzhugh-nagumo equations

It has been shown that many important characteristics of electrical signal propagation in cardiac tissue can be reproduced by a class of equations which describe excitable media, that is, materials consisting of elementary segments or cells with the following basic characteristics:

  • Well-defined rest state
  • Threshold for excitation
  • A diffusive-type coupling to its nearest neighbors

Excitable media is a rather general concept, which is useful for modeling of (in addition to the electrical signals in cardiac tissue) a number of different phenomena, including nerve pulses, the spreading of forest fires, and certain types of chemical reactions. One of the most important qualitative characteristics displayed by excitable media, and equally a common denominator between the diversity of phenomena mentioned above, is the almost immediate damping out of signals below a certain threshold. On the other hand, signals exceeding this threshold propagate without damping.

The heart works by passing ionic current inside the muscle, thus triggering the rhythmic contractions that pump blood in and out. The ions move through small pores or gates in the cellular membrane, which can be either open (excitation state) or closed (rest state).

In nerve cells and cardiac cells the three abstract characteristics of excitable media are manifested as

  • Rest cell membrane potential
  • Threshold for opening the ionic gates in the cellular membrane
  • The diffusive spreading of the electrical signals

The state of the membrane gates is random on a microscopic scale, but the probability of a given state can be modelled as a continuous function of the voltage, thus allowing an averaged macroscopic continuum description of the current flow.

The FitzHugh-Nagumo equations for excitable media describe the simplest physiological model with two variables, an activator and an inhibitor. In these heart models the activator variable corresponds to the electric potential, and the inhibitor is a variable that describes the voltage-dependent probability of the pores in the membrane being open and ready to transmit ionic current.

chaotic dynamics and the complex landau-ginzburg equations

The complex Landau-Ginzburg equations provide a relatively simple way of modeling some aspects of the transition, displayed by many dynamical systems under the influence of strong external stimulus, from periodic oscillatory behavior into a chaotic state with gradually increasing amplitude of oscillations and decreasing periodicity.

Although their first use was to describe the theory of superconductivity, the complex Landau-Ginzburg equations are also generic in their nature (as are the FitzHugh-Nagumo equations), and examples of dynamical systems which you can model successfully using these equations are:

  • The formation of vortices behind a slender obstacle in transversal fluid flow
  • Oscillating chemical reactions of the Belousov-Zhabotinsky type

In this model the complex Landau-Ginzburg equations simulate the dynamics of the spiral waves in excitable media.

Model Definition

The geometry here is a simplified 3D model of a heart with two chambers, represented with semispherical cavities1.

Figure 5-1: Model geometry.

the FitzHugh-nagumo equations

The equations are the following:

Here u1 is an action potential (the activator variable), and u2 is a gate variable (the inhibitor variable). The parameter α represents the threshold for excitation, ε represents the excitability, and β, γ, and δ are parameters that affect the rest state and dynamics of the system.

The boundary conditions for u1 are insulating, using the assumption that no current is flowing into or out of the heart. The initial condition defines an initial potential distribution u1 where one quadrant of the heart is at a constant, elevated potential V0, while the rest remains at zero. The adjacent quadrant has instead an elevated value ν0 for the inhibitor u2. It is convenient to implement this initial distribution using the following logical expressions, where TRUE evaluates to 1 and FALSE to 0:

Here d is equal to 10-5, and it is added into the expressions to shift the elevated potential slightly off the main axes.

the complex landau-ginzburg equations

The complex Landau-Ginzburg equations are:

The two variables u1 and u2 are the activator and inhibitor, respectively. The constants c1 and c2 are parameters reflecting the properties of the material. These constants also determine the existence and nature of the stable solutions.

As in the previous model, the boundary conditions are kept insulating. The initial condition, which gives a smooth transition step near z = 0, are the following:

Modeling in COMSOL Multiphysics

The simplified geometry is quite straightforward to create using the drawing tools in COMSOL Multiphysics. The FitzHugh-Nagumo and Landau-Ginzburg equations are also readily entered in one of the PDE-based application modes2.

It is important to note that these equations are strongly nonlinear. It is therefore necessary (especially in full 3D models like these) to use a much finer mesh or use higher element order than in these examples to get results with some degree of reliability for the time intervals of interest. This is particularly important in solving the complex Landau-Ginzburg equations, which describe inherently chaotic phenomena. They are highly sensitive to perturbations in the initial value and similarly to numerical errors during the course of the time-dependent solution. We recommend the use of the fourth-order Hermite element for the complex Landau-Ginzburg equation.

For the reasons above, the results presented here are only intended as a first rough estimate of the qualitative behavior that you can expect the system to show under a given stimulus. Consequently, higher-order elements, finer meshing, and smaller relative and absolute time dependent tolerances clearly give quantitatively more correct simulation results. These improvements may require several hours of computational time to solve the equations, while the rough model described here should solve within around 20 minutes on a standard PC. When attempting these types of large models we strongly recommend the use of 64-bit platforms.


fitzhugh-nagumo equations

The plots in Figure 5-2 below show the action potential u1. A quarter of the outside shell of the heart is suppressed in the plot, as well as one of the chamber surfaces, to visualize the solution on the inside.

The parameters used in the model along with the initial pulse lead to a reentrant wave which travels around the tissue without damping in a characteristic spiral pattern.

Figure 5-2: Solution to the FitzHugh-Nagumo equations at times t = 125 (left figure) and t = 500 (right figure).

Landau-Ginzburg equations

Figure 5-3 below shows the species u1 at different times

Figure 5-3: Solution to the Complex Landau-Ginzburg equations at times t=50 (left) and t=75 (right).

The equation parameters and initial condition used here lead the diffusing species (u1) to display characteristic spiral patterns with growing complexity over time.

Modeling Using the Graphical User Interface

The instructions below describe how to create the model containing the FitzHugh-Nagumo equations for excitable media. To create the model containing the complex Landau-Ginzburg equations, follow the instructions from the beginning until you have completed the “Geometry Modeling” section. Then jump to the section called “Modifications.”

Model Library path: COMSOL_Multiphysics/Equation-Based_Models/heart_fhn


Seminar discussion – 11:15-13:15 (2 hour ) (1200-1230 interruption)

Discussion of theoretical questions.


1.                 What biopotentials, nature and mechanism of their origin.

2.                 Physical bases of theory of Eyntkhovena.

3.                 What electrocardiogram

4.                 Structure and principle of work of електрокардіографа.

5.                 Basic descriptions of electric-field and connection are between them.

6.                 Physical bases of electrography.

7.                 Descriptions of impulsive current.

8.                 Methods of electrostimulation.

9.                 Law of electromagnetic induction of Faradeya.

10.            Phenomenon of self-induction.

11.            Electromagnetic vibrations and their descriptions.

12.            Impulsive current and his descriptions.


Physical base electrography


Electrocardiogram (EKG, ECG)

As the heart undergoes depolarization and repolarization, the electrical currents that are generated spread not only within the heart, but also throughout the body.  This electrical activity generated by the heart can be measured by an array of electrodes placed on the body surface. The recorded tracing is called an electrocardiogram (ECG, or EKG).  A "typical" ECG tracing is shown to the right.  The different waves that comprise the ECG represent the sequence of depolarization and repolarization of the atria and ventricles. The ECG is recorded at a speed of 25 mm/sec, and the voltages are calibrated so that 1 mV = 10 mm in the vertical direction. Therefore, each small 1-mm square represents 0.04 sec (40 msec) in time and 0.1 mV in voltage. Because the recording speed is standardized, one can calculate the heart rate from the intervals between different waves.

P wave

The P wave represents the wave of depolarization that spreads from the SA node throughout the atria, and is usually 0.08 to 0.1 seconds (80-100 ms) in duration.  The brief isoelectric (zero voltage) period after the P wave represents the time in which the impulse is traveling within the AV node (where the conduction velocity is greatly retarded) and the bundle of His. Atrial rate can be calculated by determining the time interval between P waves. Click here to see how atrial rate is calculated.

The period of time from the onset of the P wave to the beginning of the QRS complex is termed the P-R interval, which normally ranges from 0.12 to 0.20 seconds in duration.  This interval represents the time between the onset of atrial depolarization and the onset of ventricular depolarization.  If the P-R interval is >0.2 sec, there is an AV conduction block, which is also termed a first-degree heart block if the impulse is still able to be conducted into the ventricles.

QRS complex

The QRS complex represents ventricular depolarization. Ventricular rate can be calculated by determining the time interval between QRS complexes. Click here to see how ventricular rate is calculated.

The duration of the QRS complex is normally 0.06 to 0.1 seconds.  This relatively short duration indicates that ventricular depolarization normally occurs very rapidly.  If the QRS complex is prolonged (> 0.1 sec), conduction is impaired within the ventricles.  This can occur with bundle branch blocks or whenever a ventricular foci (abnormal pacemaker site) becomes the pacemaker driving the ventricle. Such an ectopic foci nearly always results in impulses being conducted over slower pathways within the heart, thereby increasing the time for depolarization and the duration of the QRS complex.

The shape of the QRS complex in the above figure is idealized. In fact, the shape changes depending on which recording electrodes are being used. The shape will also change when there is abnormal conduction of electrical impulses within the ventricles. The figure to the right summarizes the nomenclature used to define the different components of the QRS complex.

ST segment

The isoelectric period (ST segment) following the QRS is the time at which the entire ventricle is depolarized and roughly corresponds to the plateau phase of the ventricular action potential.  The ST segment is important in the diagnosis of ventricular ischemia or hypoxia because under those conditions, the ST segment can become either depressed or elevated.

T wave

The T wave represents ventricular repolarization and is longer in duration than depolarization (i.e., conduction of the repolarization wave is slower than the wave of depolarization). Sometimes a small positive U wave may be seen following the T wave (not shown in figure at top of page). This wave represents the last remnants of ventricular repolarization. Inverted or prominent U waves indicates underlying pathology or conditions affecting repolarization.

Q-T interval

The Q-T interval represents the time for both ventricular depolarization and repolarization to occur, and therefore roughly estimates the duration of an average ventricular action potential.  This interval can range from 0.2 to 0.4 seconds depending upon heart rate.  At high heart rates, ventricular action potentials shorten in duration, which decreases the Q-T interval.  Because prolonged Q-T intervals can be diagnostic for susceptibility to certain types of tachyarrhythmias, it is important to determine if a given Q-T interval is excessively long.  In practice, the Q-T interval is expressed as a "corrected Q-T (QTc)" by taking the Q-T interval and dividing it by the square root of the R-R interval (interval between ventricular depolarizations).  This allows an assessment of the Q-T interval that is independent of heart rate.  Normal corrected Q-Tc intervals are less than 0.44 seconds.

There is no distinctly visible wave representing atrial repolarization in the ECG because it occurs during ventricular depolarization.  Because the wave of atrial repolarization is relatively small in amplitude (i.e., has low voltage), it is masked by the much larger ventricular-generated QRS complex.

ECG tracings recorded simultaneous from different electrodes placed on the body produce different characteristic waveforms.  To learn where ECG electrodes are placed, CLICK HERE.

Chest Leads (Unipolar)

The last ECG leads to consider are the precordial, unipolar chest leads.  These are six positive electrodes placed on the surface of the chest over the heart in order to record electrical activity in a plane perpendicular to the frontal plane (see figure at right).  These six leads are named V1 - V6.  The rules of interpretation are the same as for the limb leads.  For example, a wave of depolarization traveling towards a particular electrode on the chest surface will elicit a positive deflection.  

In summary, the twelve ECG leads provide different views of the same electrical activity within the heart.  Therefore, the waveform recorded will be different for each lead.  To understand how cardiac electrical currents actually generate and ECG tracing and why the different leads display that electrical activity differently, it is necessary to understand volume conductor principles and vectors

The difference of potentials, which are registered at Electrocardiography, turns out at excitation nerves - muscles of device of heart. Nervous or muscles the fibers in a condition of rest is polarized so, that the external surface of its environment has a positive charge, and internal negative. At excitation this difference of potentials sharply decreases, and then changes a mark to opposite. In process of passage of a wave of excitation along a fibers the difference of potentials on its sites comes back to initial state.

The device is included between an external surface of an environment and internal environment of a fiber, will register change of potentials shown on a

The part of a curve (a) answers a phase "depolarization", part (b) - "repolarization" of an environment and part (c) - "remain" to potential. The phenomenon as a whole name as formation" of potential of action ".

Biopotentials, sum on all elements nervously - muscles of the device, form a common difference of potentials, which refers to as electromotive force of heart.

          The size of the loops is determined in mm from a zero line to upwards for positive P, R, and T, and downwards - for negative Q, S and is compared with calibrated by a signal, which the voltage U = 1mV determined. Size greatest loops R: UR=2,5mV. The duration loops and intervals of absence of a signal is determined on a special grid located on electrocardiograms. All intimate cycle lasts approximately 1c, and most short-term loops - 100-th shares of second. Thus, electrocardiograph should register a difference of potentials with frequency from 0,3 up to 120-150 Hz and amplitude about 1mV. It requires amplification biopotentials in tens thousand times.

          There are many different marks electrocardiograph we shall work with Cardio complex. A principle of action electrocardiograph based on direct amplification and registration as a curve (electrocardiograms) of a voltage of signals from electrodes of the body, imposed on the appropriate point, of the patient. The electrodes join to electrocardiographs through a cable of loops, which consists of conductors, which correspond  to number of electrodes, and come to an end by probes with multi-colored cables. The display of the information can be on the monitor of computers or can be printed out on a paper.

Considered us loops are basic. In the further number loops was increased at the expense of electrodes, which are imposed on a surface thorax of a crate in the field of an arrangement of heart. These loops correspond to a projection of a vector electric of force of heart a horizontal plane.







Medical equipment for mesument electrical signals of heart.

High resolution 16-channel ECG system

A CE-certified Reference-Electro-Cardiogram-Device (ECG-device) providing a number of special technical features has been developed and manufactured by the Physikalisch-Technische Bundesanstalt (PTB). The system records and stores bio-electrical potential differences at the body surface caused by the excitation of the heart.
In contrast to common ECG-systems the reference ECG-device stores the potential differences measured with respect to a reference electrode. There is no hardware lead network. The conventional ECG is calculated exactly by software.
This approach enables the simulation of human electrical heart activity. The signal can directly be fed into the patient cables of a commercial ECG-device for testing , -also for measuring and analysing ECG-devices- The system offers the possibility to record simultaneously up to 16 channels. This allows to record the data for the standard ECG-leads, to calculate the Frank-leads as well as to acquire reference signals regarding breathing and the frequency of the main power voltage.

  • Measurement device providing a total 16 channels, 14 channels for ECG's and one channel each for breathing and power voltage
  • Input voltage of ±16 mV with an offset of up to ±300 mV that can be compensated
  • Input impedance: 100 MOhm (DC)
  • Resolution: 16 bit with 0,5 µV/LSB
  • Signal band width: 0...1 kHz (synchronous sampling of all channels)
  • Noise: max. 10 µV (pp) or 3 µV (rms) for short circuit at input
  • Online-measurement of skin impedance before and after data acquisition
  • Noise-measurement during data acquisition



Fig 1. High resolution 16-channel ECG system              Fig 2. Electrocardiogram taken on a patientAmplifier module of the ECG systems


Technique of performance of laboratory work.

1.  On the basis of offered electrocardiogram execute the following tasks:               

2.  Results of amplitudes and intervals of loops electrocardiograms note in the table.

1.     On calibrated of a signal by a voltage 1 mV determine scale of a voltage   where n - height calibrated, signal in mm;

2.     Determine electric of force

3.     Determine a time scale

4.     Determine time intervals loops on a time scale and distance between loops;

8.     Find a rhythm of work of heart - time interval " between loops" R-R ";

9.     Calculate height of intimate reductions Fhb(Frequencyof  heartbeet) under the formula   where t - meanings of a time interval in c.

The table ¹1 Measurement of amplitude loops.



Calibrated signal

Amplitude loops     








Norm, mV







Height loops

 in mm













Electric of force

loops in mV








The table Megements of time intervals






P - Q



Q -R



R - S




     S - T



P - T



R - R


Norm in with













Distans in mm














Duration in with, sec







Task for independent work

1.   Biopotentials. A nature and mechanisms of occurrence.

2.   Biopotentials of action and biopotentials of rest.

3.   Physical bases of the theory Einthoven .

4.   Definition electrocardiograms.

5.   Principle of work electrocardiographs.

6.   How to write down electrocardiograms?

1.        Decipher importance characteristic loops of electrocardiograms.



Analysis of the got practical job performances.


Independent work 13:15-14:00.

1. A 2.00-mH inductor is connected to a110-V, 60-HZ line. find the inductive  reactance and (b) the current through the inductor.

2. A resistor , an inductor L=20.0mH,and a capacitor are connected in series. find the impedance if the source is 110V at 400HZ

3. if the impedance of an RLC series circuit of  L=50.0mH and C=3.00m is,find the phase angle between the current in the circuit and the applied voltage.

4. A240-V,50.0-HZ,AC line is connected in series with a resistor  and a capacitor of 9.50mf find (a) the impedance of the circuit and (b) the phase angle of the circuit.

5. A120-V,50.0 HZ,AC line is connected to an  RL series circuit of and L=5.75mH/find the impedance of the circuit and the  phase angle  between the applied voltage and the current in the circuit.

 6. A2.00-mH inductor is connected in series with a 10.0-mf capacitor to an AC line of variable frequency. At what frequency will resonance occur?

7. At what frequency is the inductive reactance equal to the capacitive reactance if  and C=6.00mF?


ІV. Testing of knowledges of students – 14:15-15:00.



Author:                          prof. V.P. Marzeniuk, Vakylenko D.V.,

as. Bagriy-Zayats O.A.

It is ratified on meeting of department

____________ 2009. Protocol №1

It is revised on meeting of department


                   Protocol №__