CHAPTER 6 PYTHAGORAS AND TRIGONOMETRICAL FUNCTIONS

6.1       GENERAL

It is necessary here, while dealing with fundamentals, to look again at two important mathematical concepts which many will remember from their schooldays.  Both are widely used in calculations associated with electrical engineering, especially when dealing with alternating currents.  The two concepts are Pythagoras’ Theorem and, associated with it, the basic trigonometrical functions.  At the end of this chapter are some examples showing how these functions are used.

6.2          PYTHAGORAS’ THEOREM

If ABC is a right-angled triangle with the right angle at C, the side AB opposite the right angle is called the ‘hypotenuse’ (Figure 6.1).

FIGURE 6.1

PYTHAGORAS’ THEOREM

Pythagoras’ Theorem states that ‘the square on the hypotenuse is equal to the sum of the squares on the other two sides’.  Put into mathematical form:

                                                            AB²      =    AC² + BC²

                                                                                                                                                 (i)

For example, if AC = 4 inches and BC = 3 inches, then:

                                                            AB²      =    4² + 3²

                                                                        =    16 + 9

                                                                        =    25

                                                      \   AB       =    √ 25

                                                                        =    5 inches

Thus, if the ‘other two’ sides are both known, the length of the hypotenuse can be calculated.

Expression (i) can also be written:

                                                            AC²     = AB² - BC²                                                            ….(ii)

                                                or         BC²     =  AB² - AC²                                              ….(iii)

For example, if AB = 13 inches and BC = 5 inches:

                                                            AC²     =     13² - 5² from equation (ii)

                                                                        =     169 - 25

                                                                        =     144

                                                \           AC     =     √ 144

                                                                        =     12 inches

Similarly, if AB and AC are known, BC can be calculated from equation (iii).

To sum up: in a right-angled triangle, if the lengths of any two of the sides are known, that of the third side can be calculated by using one of the three forms (equations (i), (ii) or (iii)) of Pythagoras’ Theorem:

                                                                  AB²              =          AC² + BC²

6.3       TRIGONOMETRICAL FUNCTIONS

The word ‘trigonometry’ is derived from the Greek, meaning ‘three-cornered’, and, as may be expected, it deals with triangles and the relationships between their sides and angles.  For the sake of this discussion it will be limited to right-angled triangles and so stays within the realm of Pythagoras.

                                                    

FIGURE 6.2

TRIGONOMETRICAL FUNCTIONS

Figure 6.2 shows a right-angled triangle with the right angle at C and with AB as its hypotenuse.  The angle ABC will be called ‘j’; the side BC adjoining j is called the ‘adjacent side’ (symbol ‘A’), and the side AC opposite j is called the ‘opposite side’ (symbol ‘O’).  Keeping to the symbol notation, the hypotenuse AB will be called ‘H’. Then, by Pythagoras:

                                                                  AB²      =    AC² + BC²

or, using the symbol notation,

                                                                  H²        =    A² + O²

This is the Pythagoras relationship between the hypotenuse and the other two sides.

So far only the relationship between the triangle’s three sides have been discussed.  Look now at the angle j and consider three more definitions:

                         

Each of these three is the ratio of a pair of sides, the ratio of different pairs having different names, but all being functions of the angle j.  Tables are available of sines, cosines and tangents from which, if the ratio is known, the angle j can be read off directly.  So, if any two sides of a right-angled triangle are known, not only can the third side be calculated by Pythagoras, but either of the two acute angles can be determined by use of the appropriate table.  (Nowadays ‘scientific’ pocket calculators are used instead of tables.)

Relations (a), (b) and (c) above can also be written:

                                                         

It is in these forms that the relationships are most useful.

Example

In case of the triangle of Figure 6.2, suppose the opposite side ‘O’ had length 10cm and the adjacent side ‘A’ had length 15cm.  Then by Pythagoras:

                                                            H²        =     10² + 15²

                                                                        =     100 - 225

                                                                        =     325

                                                   \      H         =     √ 325

                                                                        =     18.03 cm.

Also:

                                                                

From the tangent tables or pocket

calculator:                                
                                                                     

(Note that this can be calculated without working out H first.)

But if H is known: 

                                           

                                From the sine tables                         From the cosine tables

                                or pocket calculator:                          or pocket calculator:

                 

It will be seen that, whichever of the three functions is used, the calculation leads to the same answer for

When it is stated that , it could equally well be said that j is the angle whose sine is .  This may be written mathematically as:

                                       

                                                    

The expression ‘sin^-1’ or ‘arcsin’ means simply ‘the angle whose sine is ...’, and one or other of these expressions will be found on most scientific pocket calculators.  It is important to distinguish between ‘sin’ (the sine of the keyed-in angle) and ‘sin^-1 (the angle whose sine is the keyed-in figure).

Thus:                                      sin 30° = 0.5, but sin^-1 0.5 = 30°

6.3.1    Use of Trigonometric Functions

In many a.c. electrical power problems much use is made of Pythagoras and of the trigonometrical functions sine, cosine and tangent.  Chapters 8 to 10 show that currents in purely resistive circuits are in phase with the applied voltage, but currents in purely reactive circuits lag or lead 90° on the applied voltage.  We therefore have all the elements of a right-angled triangle to which Pythagoras and the simple trigonometrical functions can be applied.

The principal uses of these methods in electrical circuits are for COMBINING and for RESOLVING.  These two terms are explained below.

6.3.2    Combining

Anticipating for a moment the use of vector notation described more fully in Chapter 7, Figure 6.3 shows an applied voltage V supplying a purely resistive circuit R in parallel with a purely reactive (inductive) circuit X.  The resistive circuit draws a current I_R which is in phase with the voltage, whereas the reactive circuit draws a current I_x which lags 90° on the voltage.  These two currents are so shown in the figure on the right.

The diagram shows that two separate currents are flowing in the two limbs, although of course only one current can come out of the voltage source, and that must clearly be the combination of the two separate currents.  It is therefore required to combine the two currents I_R and I_x to give a single equivalent or ‘resultant’ current I.

FIGURE 6.3 

  COMBINED CURRENTS  

In the diagram on the right of Figure 6.3, OA represents the current I_R in phase with the voltage V, and OB the current I_x lagging 90° on V.  Complete the rectangle OACB and draw the diagonal OC.  Then OC represents the resultant or equivalent current I to the same scale as I_R and I_x

By Pythagoras:

                                                    

 

That is to say, the resultant current is the square root of the sum of the squares of the individual resistive and reactive currents.  Putting numbers to these, if I_R = 40A and I_x = 30A, then:

                                                                     I²  = 40² +   30²

= 1 600 + 900

= 2 500

                                                \                  I    = 50A

Thus the resultant current equivalent to the 40A and 30A with this arrangement is not 70A but 50A.  This is an example of combining.  (Compare Chapter 7, Figure 7.5.)

This combining of currents can also be used to determine the phase angle (j) of the resultant current with respect to the applied voltage.

For, as already defined,

                                                                         

Expressed mathematically:

That is to say, the phase angle j is the angle whose tangent is the ratio of the reactive current (l_x) to the resistive current (l_R).  It will be shown later that this is the same as the ratio of the actual reactance (X) to the resistance (R); these two terms are explained in Chapters 8 to 10.

6.3.3    Resolving

When the actual current is known and also its angle of lag (or lead) j, as shown in Figure 6.4, it is often required to break down, or ‘resolve’, that current into components which are in phase with the applied voltage and at right angles to that voltage - that is the reverse process to ‘combining’ which has just been described.

                                                          FIGURE 6.4

RESOLVED CURRENTS

All that it is necessary to do is to apply the trigonometrical functions cos j and sin j to the current.  In the figure, V is the applied voltage and I the total current which flows.  The angle between I and V is j.  If perpendiculars are dropped from C to the horizontal (CB) and to the vertical (CA), then OA is the in-phase component I_P, of the current I (that is, in the same phase as the voltage), and OB is the right-angled (or ‘quadrature’) component l_Q of the current I (that is, at right angles to the voltage).

Remembering equations (v) and (iv),

                                                        

                                                                       

But OC is the total current I, and OA and OB are the in-phase and quadrature components l_P and I_Q respectively.

                                                                              

                                                        

Thus, if the total current and phase angle are known, the in-phase and quadrature components can be immediately calculated.  (Usually the actual angle j is not given, but the power factor, cos j, is.  I_P then makes direct use of it, but to find sin j it is necessary to use the cosine tables to find j, then the sine tables to find sin j to evaluate I_Q.  A good scientific calculator can do this transition in one operation.)  In finding I_P, and I_Q from I in this manner, the current I is said to be resolved into in-phase and quadrature components.

Example

A circuit has a current of 100A at a lagging power factor of 0.8 on the applied voltage.  Find the in-phase and quadrature components of this current.

	Power factor is 0.8	=	cos j
	From the cosine tables (or calculator) j	=	36°52’
	From the sine tables (or calculator) sin j	=	0.60
\	In-phase current	=	100 cos j = 80A
	Quadrature current	=	100 sin j = 60A

(Note that, if these were recombined by Pythagoras,

	(Total current)2	=	802 + 602
		=	6 400 + 3 600
		=	10 000
\

which is the total current that we started with.  This shows that combining and resolving are merely reverse processes.)