Formulas 61 to 73       1. - Electronics
 Calculation of the intensity of current Calculation quantity of electricity Calculation of time necessary
Calculation of the resistance of a driver  Calculation length of a driver   Section of a driver
 Calculation of the resistivity of material Calculation of the section of a driver Calculation of the diameter of a wire  
Calculation of the conductance of a driver  Calculation of the resistance of a driver  Conductivity of a driver
Calculation of the resistivity of a driver    
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Created it, 05/10/15

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Reception

MATHEMATICAL FORMS 2      “3rd part”

1. - ELECTRONICS

This form is devoted to the electric quantities examined in the first theories. Generally, the statements which precede the fundamental formulas are drawn directly from the theories quoted between brackets after each statement. It can be useful or even necessary to re-examine in the theories the concepts which are at the base of the mathematical formulas and their applications. This form is devoted to the electric quantities examined in the first theories. Generally, the statements which precede the fundamental formulas are drawn directly from the theories quoted between brackets after each statement. It can be useful or even necessary to re-examine in the theories the concepts which are at the base of the mathematical formulas and their applications.

In the forms of electrical engineering, as was already the case in the forms devoted to the geometry and physics, one uses the measuring units of system S.I, or their multiples and submultiples own.

Some of these measuring units were already presented in the theories, when the covered subject required it.

HIGH OF PAGE FORMULATE 61 - Calculation of the intensity of current knowing the quantity of electricity which crosses the section of a driver in a given time.

Statement : The intensity of current expressed in amps, is given by the quantity of electricity, expressed in coulombs, which crosses during each second a section of the driver.

To translate into formula the statement precede, it is enough to observe that the quantity of electricity passing in one second through the section of a driver, is obtained by dividing the quantity of electricity passing in the interval of time considered by last time.

I = Q / t

 

 

 

 

 

 

I = intensity of the current in A (amp)

 

 

 

 

 

 

 

Q = quantity of electricity out of C (Coulomb)

 

 

 

 

 

 

 

t = time in seconds

Example :

Data : Q = 10 C ; t = 4 s

 

 

Intensity of the current : I = 10 / 4 = 2,5 A.

HIGH OF PAGE FORMULATE 62 - Calculation of the quantity of electricity which crosses the section of a driver in a given time, knowing the intensity of the current.

Q = It

 

 

 

 

Q = quantity of electricity out of C (Coulomb)

 

 

 

 

 

 

I = intensity of the current in A (amp)

 

 

 

 

 

 

t = time in seconds

(This formula is drawn from formula 61).

Data : I = 2,5 A ; t = 4 s

 

 

 

 

 

 

Quantity of electricity : Q = 2,5 x 4 = 10 C.

HIGH OF PAGE FORMULATE 63 - Calculation of time necessary so that a quantity of electricity given crosses the section of a driver, intensity of the current being known.

t = Q / I

 

 

 

 

t = time in seconds

 

 

 

 

 

Q = quantity of electricity out of C (Coulomb)

 

 

 

 

 

I = intensity of the current in A (amp)

(This formula is drawn from formula 61).

Data : Q = 10 C ; I = 2,5 A

 

 

 

 

 

Time : t = 10 / 2,5 = 4 s

HIGH OF PAGE FORMULATE 64 - Calculation of the resistance of a driver, knowing the resistivity of material, the length and the section of this driver.

Statement : The resistance of a driver, expressed in ohms, is obtained by multiplying the resistivity expressed out of microhmmeters, by the length expressed in meters, the whole divided by the section expressed into square millimetres.

R = pl / S

 

 

 

 

 

R = resistance in W (ohm)

 

 

 

 

 

 

p = resistivity in µW.m (microhmmeter)

 

 

 

 

 

 

I = length in meters

 

 

 

 

 

 

S = section in square millimetres

Example :

Data relating to a copper driver : pa_peu_ pres_egal 0,0176 µW.m (resistivity of copper ; to see table III, figure 1) ; I = 100 m ; S = 0,7854 mm2.

Resistance of the driver R a_peu_ pres_egal 0,0176 x 100 / 0,7854 = 1,76 / 0,7854 = 2,24 W

OBSERVATION : in the central column of table III (figure 1) the values of resistivity of the principal drivers of electricity are indicated; these values are expressed out of microhmmeters (µW.m), submultiple of the measuring unit ohmmeter (W.m).

1 µW.m = 0,0000 001 W.m

 

 

 

 

 

 

1 W.m = 1 000 000 µW.m

Sometimes one uses in the technical handbooks the symbol W.mm2 / m (square ohm-millimetre per meter) which with the same significance as µW.m.

One uses sometimes, for a greater simplicity of calculation, the submultiple W.cm (ohm-centimetre).

1 W.cm = 0,01 W.m

1 W.m = 0,01 W.cm

 Resistivite_et_coefficient_de_temperature

Moreover, in the same paragraph, one expressed the length of the driver in (cm) and not in (m) and the section in cm2 and not in mm2.

However, that it is in the preceding example or those of the mathematical lessons entitled “the formulas”, the measuring units in such way were chosen that at the end of calculations, the resistance of the driver is expressed in ohms.

The unit of resistivity ohmmeter (W .m) is not used in practice to indicate the resistivity of the drivers.

HIGH OF PAGE FORMULATE 65 - Calculation length of a driver knowing resistance, the section of the driver and the resistivity of material.

I = (RS) / p

 

 

 

 

 

l = length in meter

 

 

 

 

 

R = resistance in W (ohm)

 

 

 

 

 

S = section in mm2

 

 

 

 

 

p = resistivity in µW.m (microhmmeter)

(This formula is drawn from formula 64).

Data relating to a driver of nickel-chromium: R = 10 W ; S = 0,7854 mm2 ; p = 1,04 µW.m (resistivity of nickel-chromium; table III, figure 1).

Length of the driver : I = (10 x 0,7854) / 1,04 = 7,854 / 1,04 = 7,552 m

HIGH OF PAGE FORMULATE 66 - Calculation of the section of a driver knowing the resistivity of material, the length and the resistance of this driver.

S = pl / R

 

 

 

 

 

S = section in mm2

 

 

 

 

 

p = resistivity in µW.m (microhmmeter)

 

 

 

 

 

I = length in m

 

 

 

 

 

R = resistance in W

(This formula is drawn from formula 64).

Data relating to a constantan driver : p = 0,5 µW.m (resistivity of constantan; table III, figure 1); I = 100 m; R = 63,66 W

Section of the driver : S = (0,5 x 100) / 63,66 = 50 / 63,66 = 0,7854 mm2

HIGH OF PAGE FORMULATE 67 - Calculation of the resistivity of material of a driver knowing the section, the resistance and the length of this driver.

p = SR / I

 

 

 

 

 

p = resistivity in µW.m (microhmmeter)

 

 

 

 

 

 

S = section in mm2

 

 

 

 

 

R = resistance in W (ohm)

 

 

 

 

 

 

I = length in meters

(This formula is drawn from formula 64).

Data relating to an electric unknown material driver :

S = 1,76 mm2 ; R = 14 W ; I = 3 m.

 

 

 

 

 

 

Resistivity of material : p = (1,76 x 14) / 3 = 24,64 / 3 = 8,21 µW.m

OBSERVATION : so that the result obtained represents indeed the resistivity of material, it is necessary that its composition is homogeneous.

HIGH OF PAGE FORMULATE 68 - Calculation of the section of a thread-like driver knowing the diameter.

S a_peu_ pres_egal 0,7854 d2

 

 

 

 

 

S = section of the wire in mm2

 

 

 

 

 

d = diameter of the wire in mm

(This formula is an application of formula 19 of form 1, entitled “geometry”).

Data : d = 0,1 mm

 

 

 

 

 

 

Section of the wire: S a_peu_ pres_egal  0,7854 x 0,12 = 0,7854 x 0,01 = 0,007854 mm2

OBSERVATION : In the first two left-hands column of table IV (figure 2), one respectively indicated the values of the diameter and the section of wire of nickel-chromium, constantan and manganin, materials frequently used in the electric installations. If the diameter of a wire is equal to the one of the values indicated in the first column, one can determine the value of the corresponding section by reading it directly in the second column.

The diameters are expressed in millimetres and the sections in square millimetres.

Resistance_par_meter_de_quelques_fils

HIGH OF PAGE FORMULATE 69 - Calculation of the diameter of a wire knowing the value of its section.

Calcul_diametre

OBSERVATION : If the section of the wire corresponds roughly to the one of the values indicated in the second column, one can take as value of the diameter corresponding that deferred in the first column.

For example, for a value of section equal to 0,28 mm2 (or 0,283 mm2 or 0,28274 mm2), one will take the diameter 0,60 mm which gives in table IV (figure 2) a section of 0,2827 mm2

HIGH OF PAGE FORMULATE 70 - Calculation of the conductance of an electric driver, knowing its resistance.

Statement : The conductance, expressed in mho, is the reverse of the resistance expressed in ohms.

To translate into formula the preceding statement, it is enough to remember that one obtains the reverse of a size by dividing number 1 by the value of the size considered.

G = 1 / R

 

 

 

 

 

G = conductance in S (mho)

 

 

 

 

 

R = resistance in W (ohm)

Data : R = 20 W

 

 

 

 

 

 

Conductance: G = 1 / 20 = 0,05 S (mho).

HIGH OF PAGE FORMULATE 71 - Calculation of the resistance of a driver knowing its conductance.

R = 1 / G

 

 

 

 

 

R = resistance W (ohm)

 

 

 

 

 

G = conductance in S (mho)

(This formula is drawn from formula 70).

Data : G = 0,25 S

 

 

 

 

 

Resistance : R = 1 / 0,25 = 4 W

 

HIGH OF PAGE FORMULATE 72 - Calculation of the conductivity of a driver knowing its resistivity.

Statement : The conductivity, expressed in mho per meter, is equal contrary to the resistivity expressed in ohmmeters.

y = 1 / p

 

 

 

 

 

y = conductivity in S / m (mho per meter)

 

 

 

 

 

 

p = resistivity in W.m (ohmmeter)

Data : p = 50 W.m (resistivity of the ground in a wet place)

 

 

 

 

 

 

Conductivity : y = 1 / 50 = 0,02 S / m

OBSERVATION : If the resistivity is expressed out of microhmmeter (see formula 64), by applying formula 72, one obtains the value of the conductivity expressed in méga mho per meter (MS / m).

1 MS / m = 1 000 000 S / m

 

 

 

 

 

 

1 S / m = 0,000 001 MS / m

The méga mho per meter is indicated by the symbol m / W.mm2 (meter per square ohm-millimetre).

HIGH OF PAGE FORMULATE 73 - Calculation of the resistivity of a driver knowing its conductivity.

p = 1 / y                      

 

 

 

 

 

 

p = resistivity in W.m (ohmmeter)

 

 

 

 

 

 

y = conductivity in S / m (mho per meter)

(This formula is drawn from formula 72).

Data : y = 2 000 000 S / m (maximum conductivity of constantan).

 

 

 

 

 

 

Resistivity of constantan : p = 1 / 2 000 000 = 0,000 0005 W.m = 0,5 µW.m

(Value deferred in table III, figure 1). 

 

     

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