Introduction to Electric Circuits

86 Single-phase a.c. circuits

the real and imaginary axes. Note that V- IVIL_~: Vcos~+

j V sin ~b = a + jb say (with a = V cos ~b and b = V sin ~b). From the geometry of

the diagram we see that V = X/(a 2 + b 2) and that 4~ = tana (b/a). Thus

V = IVI/__4, = V/(a 2 + b2)/_tan ~ (b/a) (4.23)

This is called the polar coordinate form of the phasor V. Also

V- IVI cos ~b + jlVI sin ~b = a + jb (4.24)

This is called the rectangular coordinate form of the phasor V. It is a simple

matter to change from one form to the other.

Example 4.11

Express (1) I = (4 + j3) A in polar form, (2) V = 25/_-30 ~ V in rectangular

form.

Solution

1 The magnitude of the current is ~v/(4 2 + 32) = 5 A.
The angle ~b = tan -1 (3/4) = 36.86 ~
Thus in polar form we have I - 5/36.86 ~ The current is shown in both
forms in Fig. 4.25(a).
5A

4A

(a)

Figure 4.25

j3A 21.65V

(reference)

(b)

12.5V

(reference)

2 The component of V along the real axis is 25 cos 30 ~ = 21.65 V.
The component of V along the negative imaginary axis is
25 sin 30 ~ - 12.5 V.
Thus in rectangular form we have V - (21.65 - j12.5) V. The voltage is
shown in both forms in Fig. 4.25(b).

Addition and subtraction of complex quantities

lit is more convenient to do addition and subtraction using the rectangular

coordinates form of the quantities. The real parts of the quantities are added

(or subtracted) to give the real part of the resultant quantity. Similarly the

imaginary parts are added (or subtracted) to give the imaginary part of the

resultant.