Introduction to Electric Circuits

70 Single-phase a.c. circuits

horizontal position OP the line will have reached position OP 1 after 0~/~o
seconds. After 0z/O2 seconds it will have reached position OP2 and after ~r/2o2 it
will be in position OP 3. Plotting the horizontal projections of the line as it
moves in a circular path results in the sine wave shown. The line OP is called a
phasor which is defined as a line whose length represents the magnitude of a
sinusoidal quantity and whose position represents its phase with respect to
some reference.

Phasor diagrams

Two sinusoidally alternating quantities v = V m sin wt (a voltage say) and
i =Im sin (wt + oh), (a current) may be represented by two phasors as shown in

Figure 4.4

I

V

V

(a) (b)

Fig. 4.4(a). This is called a phasor diagram and it gives the following informa-

tion:

9 the magnitude of the voltage (the length of the line OV); this can be its

maximum value or any constant multiple of it;

~ the magnitude of the current (the length of the line OI); again this could be

its maximum value;

9 the phase difference between the two quantities (the angle ~b).

The phasors by convention rotate in an anticlockwise direction so that in the

example shown the current I leads the voltage V by ~b. Alternatively we could

say that the voltage lags the current by ~b. Note that the diagram of Fig. 4.4(b)

gives exactly the same information as Fig. 4.4(a). The difference between the

two is simply that they have been 'stopped' at different instants of time. Only

quantities having the same frequency can be represented on the same phasor

diagram.

The root mean square (rms) value of a sinusoidai quantity

The rms value of any alternating quantity is found by taking the square root of

the mean of the squares of the values of the quantity. It is often called the

effective value, and the rms value of a current is that value which has the same

heating effect as a steady direct current of the same value. It is represented by a

capital letter.