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Single phase series a.c. circuits

Publisher Summary

This chapter discusses the single phase series alternating current (a.c.) circuits. The voltage magnification at resonance is the ratio measured of the quality of a circuit—as a resonator or tuning device—and is called the Q-factor. The advantages of a high Q-factor in a series high frequency circuit is that such circuits need to be selective to signals at the resonant frequency and must be less selective to other frequencies. The disadvantage of a high Q-factor in a series power circuit is that it can lead to dangerously high voltages across the insulation and may lead to electrical breakdown. For series-connected impedances, the total circuit impedance can be represented as a single L–C–R circuit by combining all values of resistance together, all values of inductance together and all values of capacitance together.

1. In a purely resistive a.c. circuit, the current I_R and applied voltage V_R are in phase, see Figure 13.1(a).

2. 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 13.1(b).

3. 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 13.1(c).

4. In a purely inductive circuit the opposition to the flow of alternating current is called the inductive reactance, X_L.

$X_{L} = \frac{V_{L}}{I_{L}} = 2 π f L Ω,$ $X_{L} = \frac{V_{L}}{I_{L}} = 2 π f L Ω,$

where f is the supply frequency in hertz and L is the inductance in henry’s.

X_L, is proportional to f as shown in Figure 13.2(a).

5. In a purely capacitive circuit the opposition to the flow of alternating current is called the capacitive reactance, X_C.

$X_{C} = \frac{V_{C}}{I_{C}} = \frac{1}{2 π f C} Ω,$ $X_{C} = \frac{V_{C}}{I_{C}} = \frac{1}{2 π f C} Ω,$

where C is the capacitance in farads. X_C varies with f as shown in Figure 13.2(b).

6. In an a.c. series circuit containing inductance L and resistance R, the applied voltage V is the phasor sum of V_R and V_L (see Figure 13.3(a)) 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.

7. 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 13.3(b)) 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.

8. In an a.c. circuit, the ratio

$\frac{applied voltage V}{current I}$ $\frac{applied voltage V}{current I}$

is called the impedance Z,

$i .e . Z= \frac{V}{I} Ω$ $i .e . Z= \frac{V}{I} Ω$

9. From the phasor diagrams of Figure 13.3, the ‘voltage triangles’ are derived.

(a) For the R-L circuits:

$\begin{array}{l} V = \sqrt (V_{R}^{2} + V_{L}^{2}) (by Pythagoras’ theorem)), and \\ tan ϕ = \frac{V_{L}}{V_{R}} (by trigonometric ratios) \end{array}$ $\begin{array}{l} V = \sqrt (V_{R}^{2} + V_{L}^{2}) (by Pythagoras’ theorem)), and \\ tan ϕ = \frac{V_{L}}{V_{R}} (by trigonometric ratios) \end{array}$

(b) For the R-C circuit:

$\begin{array}{l} V = \sqrt (V_{R}^{2} + V_{C}^{2}), and \\ tan α = \frac{V_{C}}{V_{R}} \end{array}$ $\begin{array}{l} V = \sqrt (V_{R}^{2} + V_{C}^{2}), and \\ tan α = \frac{V_{C}}{V_{R}} \end{array}$

10. If each side of the voltage triangles in Figure 13.3 is divided by current I then the ‘impedance triangles’ are derived.

(a)

$\begin{array}{l} For the R-L circuit: Z = \sqrt (R^{2} + X_{L}^{2}) \\ \tan ϕ = \frac{X_{L}}{R}, \sin ϕ = \frac{X_{L}}{Z} a n d \cos ϕ = \frac{R}{Z} \end{array}$ $\begin{array}{l} For the R-L circuit: Z = \sqrt (R^{2} + X_{L}^{2}) \\ \tan ϕ = \frac{X_{L}}{R}, \sin ϕ = \frac{X_{L}}{Z} a n d \cos ϕ = \frac{R}{Z} \end{array}$

(b)

$\begin{array}{l} For the R-C circuit: Z = \sqrt (R^{2} + X_{C}^{2}) \\ \tan α = \frac{X_{C}}{R}, \sin α = \frac{X_{C}}{Z} a n d \cos α = \frac{R}{Z} \end{array}$ $\begin{array}{l} For the R-C circuit: Z = \sqrt (R^{2} + X_{C}^{2}) \\ \tan α = \frac{X_{C}}{R}, \sin α = \frac{X_{C}}{Z} a n d \cos α = \frac{R}{Z} \end{array}$

11. 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 13.4). V_L, and V_C are anti-phase and there are three phasor diagrams possible — each depending on the relative values of V_L and V_C.

12. When X_L>X_C (see Figure 13.4(b)):

$\begin{array}{l} Z = \sqrt [R^{2} + {(X_{L} - X_{C})}^{2}] \\ a n d \tan ϕ = \frac{(X_{L} - X_{C})}{R} \end{array}$ $\begin{array}{l} Z = \sqrt [R^{2} + {(X_{L} - X_{C})}^{2}] \\ a n d \tan ϕ = \frac{(X_{L} - X_{C})}{R} \end{array}$

13. When X_C>X_L (see Figure 13.4(c)):

$\begin{array}{l} Z = \sqrt [R^{2} + {(X_{C} - X_{L})}^{2}] \\ a n d \tan α = \frac{(X_{C} - X_{L})}{R} \end{array}$ $\begin{array}{l} Z = \sqrt [R^{2} + {(X_{C} - X_{L})}^{2}] \\ a n d \tan α = \frac{(X_{C} - X_{L})}{R} \end{array}$

14. When X_L = X_C (see Figure 13.4(d)), the applied voltage V and the current I are in phase. This effect is called series resonance. At resonance:

(i) V_L = V_C

(ii) Z=R (i.e. the minimum circuit impedance possible in an L-C-R circuit).

(iii) I = $\frac{V}{R}$ $\frac{V}{R}$ (i.e. the maximum current possible in an L-C-R circuit).

(iv) Since X_L = X_C, then $2 π f_{r} L = \frac{1}{2 π f_{r} C}$ $2 π f_{r} L = \frac{1}{2 π f_{r} C}$ , from which, $f_{r} = \frac{1}{2 π \sqrt (L C)}$ $f_{r} = \frac{1}{2 π \sqrt (L C)}$ Hz, where f_r, is the resonant frequency.

15. At resonance, if R is small compared with X_L, and X_C, it is possible for V_L and V_C to have voltages many times greater than the supply voltage V (see Figure 13.4(d)).

$Voltage magnification at resonance = \frac{voltage across L (or C)}{supply voltage V}$ $Voltage magnification at resonance = \frac{voltage across L (or C)}{supply voltage V}$

This ratio is a measure of the quality of a circuit (as a resonator or tuning device) and is called the Q-factor.

$\begin{array}{l} Hence Q-factor= \frac{V_{L}}{V} = \frac{I X_{L}}{I R} = \frac{X_{L}}{R} = \frac{2 π f_{r} L}{R} \\ (Alternatively, Q-factor= \frac{V_{C}}{V} = \frac{I X_{C}}{I R} = \frac{X_{C}}{R} = \frac{1}{2 π f_{r} C R}) \\ At resonance f_{r} = \frac{1}{2 π \sqrt (L C)} i . e .2 π f_{r} = \frac{1}{\sqrt (L C)} \end{array}$ $\begin{array}{l} Hence Q-factor= \frac{V_{L}}{V} = \frac{I X_{L}}{I R} = \frac{X_{L}}{R} = \frac{2 π f_{r} L}{R} \\ (Alternatively, Q-factor= \frac{V_{C}}{V} = \frac{I X_{C}}{I R} = \frac{X_{C}}{R} = \frac{1}{2 π f_{r} C R}) \\ At resonance f_{r} = \frac{1}{2 π \sqrt (L C)} i . e .2 π f_{r} = \frac{1}{\sqrt (L C)} \end{array}$

$Hence Qlactor = \frac{2 π f_{r} L}{R} = \frac{1}{\sqrt (L C)} (\frac{L}{R}) = \frac{1}{R} \sqrt (\frac{L}{C})$ $Hence Qlactor = \frac{2 π f_{r} L}{R} = \frac{1}{\sqrt (L C)} (\frac{L}{R}) = \frac{1}{R} \sqrt (\frac{L}{C})$

16.

(a) The advantages of a high Q-factor in a series high frequency circuit is that such circuits need to be ‘selective’ to signals at the resonant frequency and must be less selective to other frequencies.

(b) The disadvantage of a high Q-factor in a series power circuit is that it can lead to dangerously high voltages across the insulation and may lead to electrical breakdown.

17. For series-connected impedances the total circuit impedance can be represented as a single L-C-R circuit by combining all values of resistance together, all values of inductance together and all values of capacitance together (remembering that for series connected capacitors 1/C = 1/C₁ + 1/C₂ + ···). For example, the circuit of Figure 13.5(a) showing three impedances has an equivalent circuit of Figure 13.5(b).

18.

(a) For a purely resistive a.c. circuit, the average power dissipated, P, is given by: P = VI = I²R = V²/R watts (V and I being r.m.s. values). See Figure 13.6(a).

(b) For a purely inductive a.c. circuit, the average power is zero. See Figure 13.6(b).

In Figure 13.6(a)-(c), the value of power at any instant is given by the product of the voltage and current at that instant, i.e. the instantaneous power, p = vi, as shown by the broken lines.

19. Figure 13.7 shows current and voltage waveforms for an R-L circuit where the current lags the voltage by angle ϕ. The waveform for power (where p = vi) is shown by the broken line, and its shape, and hence average power, depends on the value of angle ϕ.

For an R-L, R-C or L-C-R series a.c. circuit, the average power P is given by:

P = VI cos ϕ watts or P = I²R watts (V and I being r.m.s. values).

20. Figure 13.8(a) shows a phasor diagram in which the current I lags the applied voltage V by angle ϕ. The horizontal component of V is V cos ϕ and the vertical component of V is V sin ϕ. If each of the voltage phasors is multiplied by I, Figure 13.8(b) is obtained and is known as the ‘power triangle’.

Apparent power, S = VI voltamperes (VA)

True or active power, P = VI cos ϕ watts (W)

Reactive power, Q = VI sin ϕ reactive voltamperes (VAr)

21. If each of phasors of the power triangle of Figure 13.8(b) is divided by V, Figure 13.8(c) is obtained and is known as the ‘current triangle’. The horizontal component of current, I cos ϕ, is called the active or the in-phase component. The vertical component of current, I sin ϕ, is called the reactive or the quadrature component.

22.

$Power factor= \frac{True power P}{Apparent power S}$ $Power factor= \frac{True power P}{Apparent power S}$

For sinusoidal voltages and currents,

$\begin{array}{l} power factor= \frac{P}{S} = \frac{V I \cos ϕ}{V I} \\ i . e . p . f . = \cos ϕ = \frac{R}{Z} \end{array}$ $\begin{array}{l} power factor= \frac{P}{S} = \frac{V I \cos ϕ}{V I} \\ i . e . p . f . = \cos ϕ = \frac{R}{Z} \end{array}$

(from Figure 13.3).

(The relationships stated in paras 20 to 22 are also true when current I leads voltage V.)

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Table of Contents for Chapter 13: Single phase series a.c. circuits

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Table of Contents for
Chapter 13: Single phase series a.c. circuits