The power factor of a circuit is defined in one of the following ways:
(i) Power factor = cos Φ= cosine of the angle of lead or leg
(ii) Power factor = \(\frac{R}{Z}\) = \(\frac{Resistance}{Impedance}\)
(iii) Power factor = \(\frac { VI \ cos \ \Phi }{ VI } \) = \(\frac{True \ power}{Apparent \ power}\)

(i) For a purely resistive circuit, the phase angle between voltage and current is zero and cos Φ = 1
ஃ P_av = V_RMSI_RMS
(ii) For a purely inductive or capacitive circuit, the phase angle is ±\(\frac { \pi }{ 2 } \) and cos \(\left( \pm \frac { \pi }{ 2 } \right) \)

The current in the series RLC circuit becomes maximum at resonance. Due to the increase in current, the voltage across L and C are also increased. This magnification of voltages at series resonance is termed as Q-factor. It gives a measure of sharpness of current peak in the resanance condition of LCT circuit

The phenomenon of electrical resonance is possible when the circuit contains both L and C. Only then does the voltage across L and C cancels one another when VL and VC are 1800 out of phase and the circuit becomes purely resistive. This implies that resonance will not occur in RL and RC circuits.

RLC circuits have many applications like filter circuits, oscillators, voltage multipliers, etc. An important use of series RLC resonant circuits is in the tuning circuits of radio and TV systems.

The maximum current at series resonance is limited by the resistance of the circuit. For smaller resistance, a larger current with a sharper curve is obtained and vice versa.

When the frequency of the applied alternating source (ω_r) is equal to the natural frequency \(\left[ \frac { 1 }{ \sqrt { LC } } \right] \) of the RLC circuit, the current in the circuit reaches its maximum value. Then the circuit is said to be in electrical resonance. The frequency at which resonance takes place is called resonant frequency.
Resonant angular frequency, ω_r = \(\frac { 1 }{ \sqrt { LC } } \)

The resistance offered by the inductor, called inductive reactance (X_L). It is measured in ohm.
X_L = ωL
X_L = 2πfL
where f is the frequency of the alternating current. For a steady current, f = 0. Therefore, X_L = 0. Thus an ideal inductor offers no resistance to steady DC current.

If the waveform of alternating voltage is a sine wave, then it is known as sinusoidal alternating voltage which is given by the relation.
v = V_m sin ωt
where v is the instantaneous value of alternating voltage; V_m is the maximum value (amplitude) and w is the angular frequency of the alternating voltage.
i = I_m sin ωt
where I_m is the maximum value (amplitude) of the alternating current. The direction of sinusoidal voltage or current is reversed after every half-cycle and its magnitude is also changing continuously.

An alternating voltage is the voltage which changes polarity at regular intervals of time and the direction of the resulting alternating current also changes accordingly.

At the transmitting point, the voltage is increased and the corresponding current is decreased by using a step-up transformer. Then it is transmitted through transmission lines. This reduced current at high voltage reaches the destination without any appreciable loss. At the receiving point, the voltage is decreased and the current is increased to appropriate values by using a step-down transformer, and then it is given to consumers.

Most of the power stations are located in remote places. Hence the electric power generated is transmitted over long distances through transmission lines to reach towns or cities where it is actually consumed. This process is called power transmission.

The magnetic flux through an area A in a magnetic field is defined as the number of magnetic field lines passing through that area normally and is given by the equation.
\({ \Phi }_{ B }=\int _{ A }^{ }{ \vec { B } .d\vec { A } =BAcos\theta } \)
where the integral is taken over the area A and θ is the angle between the direction of the magnetic field and the outward normal to the area.

For capacitor:

The charge decays exponentially with time. When the capacitor discharge.
For Inductor: