1. Single-phase series a.c. circuits

2. Purely resistive a.c. circuit In a purely resistive a.c. circuit, the current I R and applied voltage V R are in phase. See Figure 1. Figure 1

3. Purely inductive a.c. circuit In a purely inductive a.c. circuit, the current I L lags the applied voltage V L by 90◦ (i.e. π/2 rads). See Figure 2 Figure 2 In a purely inductive circuit the opposition to the flow of alternating current is called the inductive reactance, XL where : f is the supply frequency, in hertz, L is the inductance, in henrys.

4. Purely capacitive a.c. circuit In a purely capacitive a.c. circuit, the current I C leads the applied voltage V C by 90 ◦ (i.e. π/2 rads). See Figure below. In a purely capacitive circuit the opposition to the flow of alternating current is called the capacitive reactance, X C where C is the capacitance in farads.

5. R–L series a.c. circuit In an a.c. circuit containing inductance L and resistance R, the applied voltage V is the phasor sum of V R and V L (see Figure), and thus the current I lags the applied voltage V by an angle lying between 0 ◦ and 90 ◦ (depending on the values of V R and V L ), shown as angle φ. In any a.c. series circuit the current is common to each component and is thus taken as the reference phasor. From the phasor diagram of Figure 15.6, the ‘voltage triangle’ is derived. For the R–L circuit:

6. R–L series a.c. circuit (continued) In an a.c. circuit, the ratio applied voltage V divided by current I is called the impedance Z, i.e. If each side of the voltage triangle in Figure 15.6 is divided by current I then the ‘impedance triangle’ is derived.

7. R–C series a.c. circuit In an a.c. series circuit containing capacitance C and resistance R, the applied voltage V is the phasor sum of V R and V C (see Figure 15.10) and thus the current I leads the applied voltage V by an angle lying between 0 ◦ and 90 ◦ (depending on the values of V R and V C ), shown as angle α. From the phasor diagram of Figure shown, the ‘voltage triangle’ is derived. For the R–C circuit: in an a.c. circuit, the ratio (applied voltage V)/(current I) is called the impedance Z, i.e.

8. R–L–C series a.c. circuit In an a.c. series circuit containing resistance R, inductance L and capacitance C, the applied voltage V is the phasor sum of V R, V L and V C (see Figure). V L and V C are anti-phase, i.e. displaced by 180 ◦, and there are three phasor diagrams possible — each depending on the relative values of V L and V C. When XL > XC (Figure (b)):

9. R–L–C series a.c. circuit (continued) When X C > X L (Figure (c)): When XL =XC (Figure (d)), the applied voltage V and the current I are in phase. This effect is called series resonance