ABSTRACT
V = VR Y jVL Figure 24.5(a) shows the voltage triangle that is derived from the phasor diagram of Figure 24.4(b) (i.e. triangle Oab). If each side of the voltage triangle is divided by current I then the impedance triangle of
Figure 24.2 (a) Circuit diagram (b) Phasor diagram
Figure 24.3 (a) Circuit diagram (b) Phasor diagram (c) Argand diagram
Figure 24.4 (a) Circuit diagram (b) Phasor diagram (c) Argand diagram
Figure 24.5 (a) Voltage triangle (b) Impedance triangle (c) Argand diagram
Figure 24.5(b) is derived. The impedance triangle may be superimposed on the Argand diagram, as shown in Figure 24.5(c), where it may be seen that in complex form the impedance Z is given by:
Z = R Y jXL Thus, for example, an impedance expressed as 3 C j4 means that the resistance is 3 and the inductive reactance is 4
In polar form, Z D jZj6 where, from the impedance triangle, the modulus of impedance jZj D
√ R2 C X2L and the circuit phase angle
D arctanXL/R lagging
(e) R-C series circuit In an a.c. circuit containing resistance R and capacitance C in series
Figure 24.6 (a) Circuit diagram (b) Phasor diagram (c) Argand diagram
C current I leads the applied voltage V by an angle lying between 0° and 90° —the actual value depending on the values of VR and VC, which depend on the values of R and C. The circuit phase angle is shown as angle in the phasor diagram. The phasor diagram may be superimposed on the Argand diagram as shown in Figure 24.6(c), where it may be seen that in complex form the supply voltage V is given by:
V = VR − jVC
Figure 24.7(a) shows the voltage triangle that is derived from the phasor diagram of Figure 24.6(b). If each side of the voltage triangle is divided by current I, the impedance triangle is derived as shown in Figure 24.7(b). The impedance triangle may be superimposed on the Argand diagram as shown in Figure 24.7(c), where it may be seen that in complex form the impedance Z is given by
Z = R − jXC
Thus, for example, an impedance expressed as 9 j14 means that the resistance is 9 and the capacitive reactance XC is 14
In polar form, Z D jZj6 where, from the impedance triangle, jZj D
√ R2 C X2C and D arctanXC/R leading
Figure 24.7 (a) Voltage triangle (b) Impedance triangle (c) Argand diagram
(f) R-L-C series circuit In an a.c. circuit containing resistance R, inductance L and capacitance C in series (see Figure 24.8(a)), the applied voltage V is the phasor sum of VR, VL and VC as shown in the phasor diagram of Figure 24.8(b) (where the condition VL > VC is shown). The phasor diagram may be
Figure 24.8 (a) Circuit diagram (b) Phasor diagram (c) Argand diagram
given by:
V = VR Y j .VL − VC / From the voltage triangle the impedance triangle is derived and superimposing this on the Argand diagram gives, in complex form,
impedance Z = R Y j .XL − XC / or Z = jZ j6 6 f
where, jZj D √
[R2 C XL XC2] and D arctanXL XC/R When VL D VC, XL D XC and the applied voltage V and the current
I are in phase. This effect is called series resonance and is discussed separately in Chapter 28.
