Establishment of a
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Literal calculation |
Application of literal
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Created it, 05/10/15
Update it, 05/12/23
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MATHEMATICS “1st Part”
In this lesson, we will treat few simple concepts of mathematics.
It is not a question of a course of mathematics itself, which is not our goal, but to expose it knowledge necessary to the good comprehension of the various theories treating of the study of the electronic circuits without forgetting data processing.
We consider that the elementary operations of arithmetic are known, i.e. the addition, the subtraction, the multiplication and division.
The spirit with which this mathematics will be treated having been specified, we will present to you the contents of our first theory.
The first paragraph is devoted to the formulas. What a formula, how the lira, how to make use of it, it is what we will learn. That will quite naturally bring us to the literal calculation which is the subject of the second paragraph.
We will supplement our knowledge of arithmetic by the study of a new operation : rise with a power, to which the third paragraph is devoted. The opposite operation, i.e. the extraction of root will be treated in the fourth.
The fifth, as for him, will recall us (or will learn to us) what are the fractions, how to simplify them, add them, withdraw them, multiply them …
The sixth, as for him, we will see that a fraction can be sometimes called a report/ratio. Lastly, always in the same spirit of fractions or reports/ratios, the seventh and last paragraph will learn how to us to handle the proportions.
In your interest, we invite you to carry out the examples given, not only the day of the reading, but as well after this one (several days, weeks or month. In mathematics, like and much of other matters, nothing is worth the practice and the repetition for including/understanding well and retaining.
The use of the abbreviations is very widespread. Many administrations, industrial engineering, commercial or political, are made known by formed initials of some letters that the public keeps easily in memory, even if it often forgets their exact significance (the SNCF, EDF-GDF, CNRS. The practice of the abbreviations entered for a very long time the field of mathematical, chemical and physical sciences. That gives a degree of concision very high to scientific descriptions what is an appreciable advantage for the exactitude of calculations carried out.
Let us see by which letters it was agreed to represent certain electric quantities:
Potential difference, electric tension : V or U
Electric capacity : C
Intensity of the electrical current : I
Electric resistance : R
Quantity of electricity : Q
Resistivity : p (to pronounce “RHÔ”), letter R of the Greek alphabet)
The abbreviations of this type are at the base of all simplifications which it is advisable to carry out to obtain from the mathematical expressions. The method can be extended to the representation of all the sizes appearing in the study of the phenomena, with the proviso always of specifying the significance of each letter, and of then taking care, to use each one of those only to indicate the corresponding size.
Consequently, if we established for example that the letters V, I and R respectively represent the tension, the intensity of the current and the resistance of an electric circuit, we will not be able, thereafter, to allot to these same letters another significance or to express these same sizes by different letters.
Each letter of the mathematical expression represents not only a given type of size, but also all the possible values of this size. For example the letter V, used to indicate the tension, can mean 1 volt, 25 volts, 220 volts… i.e. all the possible values of tension. In a similar way, letter I used to indicate the intensity of the current, can mean 0,5 amp, 2 amps, 5 amps… In the same way, the letter R representing the resistance of the circuit can mean 3 ohms, 100 ohms, 10 000 ohms…
This rule is valid for any other letter used in a mathematical expression. One also finds numbers which remain unchanged when one successively assigns with the letters all the possible values. These numbers are called coefficients.
It is obvious that the absence of any sign cannot be justified when no confusion is possible, (i.e. +, -, x, :, / etc…), it is the case of the literal calculation in question further.
1. 2. - ESTABLISHMENT
OF A FORMULA
Let us see now how one obtains the mathematical simplification of the statement of a physical law while taking as example : the law of Ohm.
The law of Ohm is expressed in the following forms :
The first form of the statement (of red color) is purely descriptive and cannot be translated by a mathematical expression because it does not specify the intensity up to what point increases when the tension applied to the circuit increases. On the contrary, the three following expressions lend themselves to a complete mathematical translation.
Here how one proceeds in each case :
1 - One obtains resistance by dividing the tension by the intensity.
One can thus write : Resistance = Tension / Intensity
That is the first step and as most important as one has just achieved to arrive at the mathematical expression. The following simply consists in replacing the words by letters. If we agree to represent resistance by the letter R, the tension by the letter V and the intensity by letter I, we can express the preceding relation in the following way :
With this result, we obtained the most pushed simplification first part of the statement, by highlighting the operation well that it is necessary to carry out to obtain the value of the resistance of a circuit by knowing the values of the tension and the intensity.
The expressions of this kind, which contain letters (replacing the names of sizes given), of the symbols of operations (to be carried out with the values of the sizes) and the symbol of the equality (which expresses the existing bond between a size and the others) are called mathematical formulas or literal expressions or more simply formulas.
When the formula is obtained, it is enough to remember it as well as exact significance to the letters in order to be able to use it in calculations.
The application of the formula is very easy: it is enough to replace the letters by the known values of the sizes and to carry out calculations.
Let us take an example : A circuit to which one applies a tension (V) of 15 volts is traversed by a current of intensity (I) of 3 amps ; calculate according to these data, the resistance (R) of this circuit.
To carry out this calculation in an ordered way, it is advisable to write the formula initially and then, under the latter, the expression which one obtains by replacing the letters by their respective value :
The number 5 which was obtained by carrying out division 15 / 3, represents the value of resistance R.
In conclusion of the calculation carried out according to the law of Ohm, one can affirm that a circuit, to which one applies a tension of 15 volts and in which passes a current of an intensity of 3 amps, must have a resistance of 5 Ohms.
Now let us try to interpret by the same process, the following statement :
2 - One obtains the tension by multiplying resistance by the intensity.
By taking again simplification, one can write :
And by preserving at the letters the same significance that before, we obtain the following formula :
The application of this formula is quite as simple as the preceding one.
Example : A circuit having a resistance (R) of 5 Ohms is traversed by a current of an intensity (I) of 3 amps ; calculate the value of the tension (V) applied to this circuit :
The tension applied to the circuit is 15 volts.
Now let us translate the last statement of the law of Ohm by putting it in formula :
3 - One obtains the intensity by dividing the tension by resistance.
One can thus write : Intensity = Tension / Resistance and by replacing the words by the corresponding letters :
Example : One applies to a circuit, having a resistance (R) of 5 Ohms and a tension (V) of 15 volts ; calculate the intensity of the current (I) absorptive by this circuit :
The intensity of the current absorptive by the circuit is thus 3 amps.
Now let us compare the three formulas obtained starting from the statements of the law of Ohm :
Since there are three distinct formulas, we could be brought to think that there are three laws of Ohm, this conclusion would be erroneous. Actually, the three formulas are equivalent between them because they represent three aspects of the same bond.
This assertion can be easily shown using examples chosen randomly, which show that all the values which satisfy only one of the three formulas, also satisfy the two others.
In the three examples above, we already found that the values of 15 volts, 3 amps and 5 ohms, answer in a satisfactory way the formulas R = V / I ; V = R x I and I = V / R.
Let us consider another example chosen now randomly.
A circuit having a resistance (R) of 150 ohms is traversed by a current of an intensity (I) of 2 amps ; calculate the value of the tension (V) applied to this circuit.
To calculate the value of the tension, it is necessary to choose between the three formulas, that which presents, on the left sign = the letter V, which represents the tension. It is seen immediately that the second formula is that which is appropriate ; therefore, while following the usual procedure, we will have :
The tension applied to the circuit is thus 300 volts. We can now ensure us that these same values of 150 ohms, 2 amps and 300 volts also answer the two other formulas of the law of Ohm.
By using the first formula, we obtain indeed :
The result of 150 ohms is exactly that until we wait for a tension of 300 volts and an intensity of 2 amps, on the basis of preceding calculation.
By using the third formula, we obtain :
This result of 2 amps is also that obtained previously.
One could take many examples similar to this one.
Each one of them would always show that three values: tension, intensity and resistance of a circuit, answer in a satisfactory way the three formulas. You can then ask you what serve three equivalent formulas, derived from the law of Ohm, binding between them the same values. Wouldn't only one formula be enough ? Not, because each one of them makes it possible to quickly carry out a well defined calculation :
It is interesting to note that the number of equivalent formulas obtained starting from the law of Ohm, is three, just as it appears three different letters in the formulas corresponding to the three sizes characteristic of the circuit (tension, intensity, resistance).
This agreement of numbers is not the fact of the chance.
If the number of the sizes dependant between them were three, four or five…, we would respectively obtain formulas containing three, four or five letters and the same number of equivalent formulas, all also ready to completely express the bond between the sizes considered.
If, for all the laws of electronics, we must follow the same procedure that we adopted for the law of Ohm, we would obtain an incredible quantity of formulas, which would make from there the study excessively long and difficult.
One is thus brought to put a question: is it really necessary to draw all the equivalent formulas from a law, since they mean all the same thing ?
The answer is enough reassuring: it is not necessary to draw directly from the statement of a law all the equivalent formulas : it is enough to draw only one from them, whatever it is, and from this one, one will be able to then reconstitute all the others, by means of the simple rules of mathematical calculation.
From this point of view, we now will establish some of the simple rules which will enable us to pass from a formula given to another.
2. - LITERAL CALCULATION
2. 1. - RULES
To prepare us with the rules which make it possible to pass quickly and without sorrow of a formula to the other, let us consider under an angle different the three formulas from the law of Ohm.
Initially, let us be appropriate to call first member, the part of the formula which is on the left sign =, and second member, the part which is on its line.
This convention is valid for any mathematical expression made up of two parts dependant between them by the relation of equality. When the sign = appears in a mathematical expression, this one is called equality, and the following rule is always valid :
Regulate 1 : If one inverts between them the 2 members of an equality, one obtains a new equality equivalent to the first.
By basing us on this rule, we will be able to write indifferently :
Indeed, if we consider the values corresponding to the letters (15 volts, 3 amps and 5 ohms) which were found during the first three exercises, we do not find any difference between :
When a mathematical expression and in particular a formula contain a multiplication, we will be able to observe the following rule :
Regulate 2 : The value of a multiplication is not changed, if the order of the factors is inverted.
By combining this rule with the preceding one, we can write for example, the second formula of the law of Ohm in the four following ways, all also correct :
|
V = RI |
or |
V = IR |
or |
RI = V |
or |
IR = V |
Indeed, by taking again the preceding numerical example, one finds that the acceptable values in the first of the four formulas indicated here, are it also for the three others :
|
15 = 5 x 3 |
or |
15 = 3 x 5 |
or |
5 x 3 = 15 |
or |
3 x 5 = 15 |
Up to now we considered the aspect of the formulas by changing the letters of place without changing the values them. Now, we will do something moreover: by basing us on some other rules, we will change the values into preserving however the equality between the first and the second member.
Regulate 3 : if one multiplies the two members of an equality by the same number, one preserves the equality.
Let us consider the first form of the law of Ohm.
Let us multiply the first and the second member of the formula R = V / I by the same number, for example 7, and, take again our circuit with the value R is 150 ohms, that of V of 300 volts and that of I of 2 amps ; we will have :
The result indicates in an obvious way to us that in spite of the multiplication carried out, the two members remain always equal between them.
Regulate 4 : If one divides the two members of an equality by the same number, one preserves the equality.
Let us consider the second formula of the law of Ohm : V = RI. Let us take the same values that in the preceding example and divide each term by 50 :
This result shows that in spite of divisions carried out, the two members remain always equal between them.
In the two preceding examples, one chose randomly numbers 7 and 50, but any number could be used to show the exactitude of the rules three and four.
Here now two other rules allowing to pass quickly from a formula given to its equivalent :
Rules 5 : If a member of an equality is consisted a division, the divider can be transferred to the other member like multiplication factor.
Let us take the third form of the law of Ohm, that is to say I = V / R. By basing us on the rule which has been just stated, one obtains :
Example : In a circuit the value of the tension (V) is equal to 60 volts and that of resistance (R) to 15 ohms. The intensity (I) which one obtains by dividing 60 by 15, has as a value :
Let us make now pass R (dividing in the 2nd member) in the first member like multiplier. We have :
These two results show that while passing from the first form at the second, one preserves the equality.
Regulate 6 : If a member of an equality is consisted a multiplication, one of the factors whatever it is can be transferred to the other member like divider.
Example :
|
1) V = R x I |
2) V / I = R |
3) V / R = I |
|
60 = 15 x 4 |
60 / 4 = 15 |
60 / 15 = 4 |
|
60 = 60 |
15 = 15 |
4 = 4 |
The three results show that one preserved the equality of the two members while passing from a form at the other.
The operations with the means of which one can transfer in a mathematical expression a letter from a site to another, by preserving the equality of the two members, take the name of literal calculation (or calculus), by analogy with the name of numerical calculation (or arithmetic calculation) which one gave to the operations carried out with numbers.
We now will specify the concepts which were here exposed, by means of an example of application of literal calculation to the field of electronics.
2. 2. - APPLICATION
OF LITERAL CALCULATION
We will see that the electric resistance of a driver depends its length, on its section and its resistivity. (The resistivity is a characteristic of material constituting the driver).
If we want to translate into mathematical formulas this statement, we are extremely embarrassed. Indeed, it is not specified from which quantity the resistance of the driver varies when one increases or one decreases the length, the section or the resistivity. To surmount this obstacle, it is necessary to refer to a more precise statement, such as : one obtains the resistance of a driver by multiplying his resistivity by his length and by dividing the product obtained by his section.
This statement can be shortened in the following way :
Let us be appropriate to represent resistance by the letter R, the resistivity by the Greek letter r, the length of the driver by the small letter I, and its section by the capital letter S, we obtain the following formula :
R = r x I / S
By means of this formula, we can then calculate the resistance of an unspecified driver, knowing its length, its section and its resistivity.
If the resistivity is expressed by means of the measuring unit ohm-centimetre, the length in centimetres and the section in centimetre squares, the value of resistance will be expressed in ohms.
Example : A cylindrical bar has a length (I) of 60 centimetres, a section (S) of 3 centimetres square and consists of a matter having a resistivity (r) of 2 ohm-centimetre ; calculate its resistance (R).
R = r x I / S
R = 2 x 60 / 3 = 2 x 20 = 40 ohms
The resistance of the bar is of 40 ohms.
Up to now, the procedure is similar to that which we followed while studying the law of Ohm.
However, it remains to be seen how one must proceed to obtain, starting from the single formula which one found, the other equivalent formulas, is :
The equivalent formulas are thus four since there are four sizes. As each size is given according to the three others :
A formula is directly obtained starting from the statement,
The three others are obtained starting from the
latter by means of simple literal arithmetic operations.
Here how one proceeds :
Let us write the known formula :
R = r x I / S
and let us observe (by basing us on rule 6), that we can transfer the letter (r) or the letter (I) in the first member, and while basing us on rule 5, we can carry in the first member, the letter S. Exécutons successively the operations necessary :
1) Let us carry initially the letter S (dividing) in the first member (thus multiplying) :
R x S = r x I
Let us carry now the letter I (multiplying) in the first member (thus dividing).
R x S / I = r
The expression obtained : r = R x S / I is the first of the three equivalent formulas which we seek to establish, i.e. the formula which makes it possible to calculate the resistivity according to the three other sizes.
Example : The length (I) of a driver is 30 cm, its resistance (R) of 10 ohms and its section (S) is 3 cm2. Let us calculate the resistivity (r) of the matter of which the driver is made up.
r = R x S / I
r = 10 x 3 / 30 = 30 / 30 = 1 ohm-centimetre
The resistivity is thus 1 ohm per centimetre.
2) - Let us carry now in the first member of our starting formula, initially the letter (S) and then the letter r.
R x S = r x I
R x S / r = I
The expression obtained : I = R x S / r is the second of the sought equivalent formulas, which is that makes it possible to calculate the length of a driver according to the three other sizes.
Example : The resistivity (r) of a driver is 3 ohm-centimetre, its resistance (R) is 45 ohms, its section (S) is 1 square centimetre ; calculate the length (I) of this driver.
I = R x S / r
The length of the driver is thus 15 centimetres.
3) - It remains to find the last formula equivalent. On the basis of our basic formula, we carry the letter (S) of the second member in the first and then the letter (R) of the first member in the second.
R x S = r x I (basic formula)
S = r x I / R
The expression obtained: S = r x I / R is the third of the sought equivalent formulas, i.e., the formula which makes it possible to calculate the section of a driver according to the three other sizes.
Example : The resistance (R) of a driver is of 50 ohms, its resistivity (r) is 1 ohm-centimetre, its length (I) is 5 cm ; calculate the section (S) of this driver :
S = r x I / R
S = 1 x 5 / 50 = 5 / 50 = 0,1 cm2
The section of the driver is thus 0,1 square centimetre.
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