Introduction to Electric Circuits

90 Single-phase a.c. circuits

ahead of the reference axis. The total circuit voltage V is the phasor sum of

these two. In complex form we write"

V= VR +J VL

the j in front of V L indicating that it is 90 ~ ahead of the reference (and VR).

Thus

V = IR + jlXL = I(R + jXL) = IZ

where Z is the impedance of the circuit:

Z = R + jXL (4.25)
Similarly, for the series RC circuit of Fig. 4.17 and its phasor diagram of
Fig. 4.18(a)

V=VR-jVc

the -j in front of Vc indicating that it is 90 ~ behind the reference. Thus

V = I(R -jXc) = IZ

where again Z is the impedance of the circuit:

Z = R -jXc (4.26)

For the series RLC circuit of Fig. 4.19 and its associated phasor diagram
(Fig. 4.20(a))

V "-- V R + j(VL- Vc)

Thus

V = I[R + j(XL- Xc)] = IZ

Z = R + j(XL- Xc) (4.27)

If XL > Xc the j term is positive indicating a predominantly inductive circuit.

For XL < Xc the j term is negative, indicating a predominantly capacitive

circuit. When XL = Xc there is no j term and the circuit is purely resistive.

The impedance triangles of Fig. 4.16(b) and 4.18(b) take the forms given in

Fig. 4.27(a) and (b), respectively.

R

jXc

R

(a) (b)

Figure 4.27