Alternating Current

1. An alternating current (a.c.) is that current which changes continuously in its magnitude and periodically reverses its direction. In its simplest form an a.c. may be sinusoidal function of time and may be expressed as 

$$I = I_0sin(\omega t) = I_0sin(2\pi f t)$$ where f is the frequency. 

2. Similarly an alternating voltage may be expressed as

$$V = V_0sin(\omega t) = V_0sin(2\pi f t)$$

3. Mean (or voltage) value of an alternating current (or voltage) is zero for a whole (complete) cycle.

4. The root mean square (rms) or effective value of an a.c. is that steady current which, when passed through a resistance, produced exactly the same amount of heat is given time as is produced by actual a.c. when flowing through the same amount of heat in given time as is produced by actual a.c. when flowing through the same resistance for same time. It can be show that 

$$I_{rms} = \frac{I_0}{\sqrt{2}} = 0.707I_0$$ $$V_{rms} = \frac{V_0}{\sqrt{2}} = 0.707V_0$$ It is also sometimes referred as virtual value.

5. In an a.c. circuit, unless otherwise specified, we talk in terms of arms values of current and voltage.

6. When an alternating voltage $V = V_m sin(\omega t)$ is applied to a pure resistor, the current flowing through the resistor is 

$$I = \frac{V_0}{R}sin(\omega t) = I_0sin(\omega t)$$ where $I_0 = \frac{V_0}{R}$. This means current is in the phase with the applied voltage.

7. When an alternating voltage $V = V_0 sin(\omega t)$  is applied to a pure inductor (whose resistance is zero is 0) of inductance L, the current flowing through the inductor is

$$I = \frac{V_0}{X_L}sin(\omega t - \frac{\pi}{2}) = I_0sin(\omega t - \frac{\pi}{2})$$ where $X_L = \omega L = 2\pi f L$ is called the inductive reactance of the given circuit. Unit of inductive reactance $X_L$ is a ohm $\Omega$. Moreover, current in a pure inductor lags behind the voltage by a phase angle $\frac{\pi}{2}$.

8. When an alternating voltage $V = V_0 sin(\omega t)$ is applied to a pure capacitance C, the current flowing through the capacitor is 

$$I = \frac{V_0}{X_C}sin(\omega t + \frac{\pi}{2}) = I_0sin(\omega t + \frac{\pi}{2})$$ Where $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$ is called the capacitive reactance of given capacitor. Unit of capacitive reactance $X_C$ is ohm . Moreover, current in a capacitive circuit is ahead in phase than the voltage by a phase angle $\frac{\pi}{2}$.

9. To facilitate the analysis of an a.c. circuit we use a phasor diagram. A phasor is a vector which rotates about the origin with an angular speed. Amplitudes of phasors V and I represent the peak value $V_0$ and $I_0$ and the vertical components of phasors give the instantaneous values of voltage and current.

10. A pure inductor offers no opposition for flow of d.c. but offers an inductive reactance for flow of a.c. Magnitude of inductive reactance is directly proportional to the frequency of a.c.

11. A pure capacitance does not allow d.c. to pass through it but allows a.c. to pass through it. Value of capacitive reactance is inversely proportional to the frequency of a.c.

12. In an alternating current, circuit containing LCR in series, the potential difference may be added by the rule of phasors. As for a given current I, the voltage $V_L$ is ahead in phase by $\frac{\pi}{2}$, $V_R$ is in phase and $V_C$ lags behind in phase by $\frac{\pi}{2}$, hence resultant voltage will be given by 

$$V = \sqrt{V_R^2 + (V_L-V_C)^2}$$

13. Total opposition offered by an a.c. circuit for flows a current through it is called impedance and is denoted by Z. Its unit is ohm. Impedance plays the same role in an a.c. circuit, which is being played by resistance in a d.c. circuit. Thus, in an a.c. circuit, $$Z = \frac{V}{I} = \frac{V_{rms}}{I_{rms}} = \frac{V_0}{I_0}$$

14. When an alternating voltage $V = V_0 sin (\omega t)$ is applied to a LR series circuit, the current in the circuit is given by $$I = \frac{V_0}{\sqrt{R^2 + X_L^2}}sin(\omega t - \phi)=\frac{V_0}{Z}sin(\omega t - \phi) = I_0sin(\omega t - \phi)$$ Where $Z = \sqrt{R^2 + X_L^2}$ is the impedance of the circuit and current lags behind the voltage by a phase angle $\phi$, given by $tan\phi =\frac{X_L}{R} = \frac{\omega L}{R}$.

15. When an alternating voltage $V = V_0 sin (\omega t)$ is applied to a RC series circuit, the current in the circuit is given by $$I = \frac{V_0}{\sqrt{R^2 + X_C^2}}sin(\omega t + \phi)=\frac{V_0}{Z}sin(\omega t + \phi) = I_0sin(\omega t + \phi)$$ where $Z = \sqrt{R^2 + X_C^2}$ is the impedance of the circuit and voltage lags behind the current by a phase angle $\phi$, given by $tan\phi =\frac{X_C}{R} = \frac{1/(\omega C)}{R}$.

16. When an alternating voltage $V = V_0  sin(\omega t)$ is applied to a LCR series circuit, the current in the circuit is given by

$$I = \frac{V_0}{\sqrt{R^2 + (X_L-X_C)^2}}sin(\omega t - \phi)=\frac{V_0}{Z}sin(\omega t - \phi) = I_0sin(\omega t - \phi)$$ where $Z = \sqrt{R^2 + (X_L-X_C)^2}$ is the impedance of the circuit and current lags behind the voltage by a phase angle $\phi$, given by $tan\phi =\frac{X_L-X_C}{R} = \frac{\omega L - 1/(\omega C)}{R}$.

17. The average power dissipated in an a.c. circuit is given by

$$P_{av}=I_{rms}V_{rms}cos(\phi)$$ Where $\phi$ is the phase angle between voltage and current. The term ‘$cos(\phi)$’ is referred  as the power factor. Here following special cases arise :

(i)    For a pure resistive circuit $P_{ av} = V_{ rms} I_{ rms}$

(ii)   For a pure inductive or a pure capacitance circuit, power factor $cos (\phi)$ has a zero value and hence net power consumed over an entire cycle of a.c. is zero. Such type of electrical circuit is known as a “wattles circuit” and current flowing is known as “wattles current”.

18. In a LCR series circuit, in general, 

$$I_0 = \frac{V_0}{\sqrt{R^2 + (X_L-X_C)^2}}=\frac{V_0}{Z}$$ If as a special case $X_L = X_C$, then Z = R = a minimum and consequently the current amplitude $I_0 = \frac{V_0}{R}$= a maximum and current and voltage are in same phase. Such a situation is called “electrical resonance”. Resonance takes place when $X_L = X_C$ or when angular frequency $$\omega_0 = \frac{1}{\sqrt{LC}} \implies f_0 = \frac{1}{2\pi \sqrt{LC}}$$

19. The quality factor (Q factor) of a resonant circuit is a measure of the “sharpness of resonance” and is defined as the ratio of resonant angular frequency $\omega_0$ to the band width $(2\Delta \omega )$ of the circuit, where band width is the difference in angular frequencies $(\omega_0 - \Delta \omega)$ and $(\omega_0 + \Delta \omega )$ at which power is half the maximum power or current is $(\frac{1}{\sqrt{2}})$ times the maximum current value at resonance. Mathematically,

$$Q = \frac{\omega_0}{2\Delta \omega} = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 CR} = \frac{1}{R}\sqrt{\frac{L}{C}}$$ The quality factor is large if resistance R is low or inductance L is high. High quality factor or high sharpness of resonance means high selectivity and the tuning of the circuit for resonance will be better.

20. When a capacitor (initially charged) is connected to an inductor, the charge on the capacitor and the current in the circuit, exhibit electrical oscillation just like a harmonic oscillator. The angular frequency and the frequency of these oscillations are 

$$\omega_0 = \frac{1}{\sqrt{LC}}$$

21. For an ideal L – C circuit there is no dissipation of energy and amplitude of oscillations remains constant. Energy in the system oscillates between the capacitor and the inductor. Average value of electrostatic energy and of magnetic energy is same and total electromagnetic energy 

$$u = \frac{1}{2}\frac{q^2}{C} + \frac{1}{2}LI^2$$ However, practically oscillations are damped one due to two reasons, namely (i) presence of some resistance in the inductor, and (ii) radiation in energy in the form of electromagnetic waves.

22. A transformer is a device used in a.c. circuits to change the voltages. A transformer which increases the a.c. voltage is called ‘set-up’ transformer, where as the ‘step-down’ transformer reduce the a.c. voltage.

23. A transformer works on the principle of mutual induction and consists of a primary coils and a secondary coil wound on a laminated soft iron core. It is found that for an ideal transformer (in which there is no loss of electrical energy), we have 

$$\frac{V_s}{V_p} = \frac{I_p}{I_s} = \frac{N_s}{N_p}= k$$ Where $N_s$ and $N_p$ are the number of turns in windings of secondary and primary coils and k is called the transformation ratio. In step up transformer, $V_s > V_p, N_s > N_p, I_s < I_p$. while in step down transformer, $V_s < V_p, N_s < N_p, I_s > I_p$.

24. In a set up transformer there is some loss energy and hence output given by transformer is less than the input supplied to it. Four main causes of energy losses in a transformer are (i) magnetic flux leakage, (ii) resistance of the windings, (iii) eddy currents, and (iv) magnetic hysteresis. However, by taking appropriate preventive measures, these energy losses can be minimised and controlled.

25. Generally, a.c. power is transmitted from one station to another at highest possible voltage so that line current is less and consequently power loss during transmission is least possible. It is achieved by use of step up transformers at the generating station. At the consumer station, using step down transformers, the power is again supplied to homes and establishments at comparatively low voltages.

Watch Video Lectures

4.6 A.C. Generator Fully explained watch video

4.7  Mean and Root mean square values of Alternating Current and voltage watch video

4.8 A.C. through Resistance and Inductor coil watch video

4.9 A.C. through Capacitors and RLC in series watch video

4.10 A.C. through RL, RC | LC Oscillations and Resonance condition watch video

4.11 Quality Factor and Power dissipation in Resistance, Inductor and Capacitor watch video

4.12 Power dissipation in RLC circuit and Power Factor watch video

4.13 Transformer and Choke Coil watch video