Revision: Electromagnetic Induction and Alternating Currents >> Alternating Current Physics Science (English Medium) Class 12 CBSE

The square root of the mean of the squares of the instantaneous current values over one complete cycle, equal to i_m\[\sqrt{2}​\], is called the root mean square current (or effective current).

The voltage that varies sinusoidally with time, expressed as v = v_m sin⁡ ωt, is called alternating voltage.

The current whose magnitude changes with time and direction reverses periodically is called alternating current (AC).

A phasor is a rotating vector about the origin with a constant angular speed ω\omegaω. Its vertical component at any instant gives the instantaneous sinusoidally varying value of the AC quantity (voltage v or current i) it represents.

The effective resistance offered by an inductor to the alternating current is called inductive reactance.

The effective resistance offered by the capacitor is called capacitive reactance (ΧC).

`Χ_C = 1/omega_C`

`= 1/(2 pi fC)`

The effective resistance offered by a capacitor to the alternating current is called capacitive reactance.

The effective opposition offered by the inductor, capacitor and resistor connected in series to flow of AC current. is called impedance.

Z = `sqrt("R"^2 + (Χ_"L" - Χ_"C")^2)`

A series LCR circuit consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series to an AC voltage source.

Resonance in a series LCR circuit is the phenomenon that occurs at a particular frequency at which the amplitude of current is maximum. This happens when the inductive reactance equals the capacitive reactance: X_C = X_L.

The current flowing in a purely inductive or purely capacitive circuit for which cos φ = 0 and no power is dissipated even though the current is flowing is called wattless current.

OR

Wattless Current (also called reactive current) is the component of current in a purely inductive or purely capacitive circuit that flows without consuming any power.

In the expression Pav = V_rmsI_rms cos⁡ ϕ, the quantity cos φ is called the power factor.

OR

Power Factor is the cosine of the phase angle (φ) between the voltage and current in an AC circuit. It equals the ratio of true (average) power to apparent power.

The Power Factor is the ratio of True Power (measured in Watts) to Apparent Power (measured in Volt-Amperes) in an AC circuit.

Power factor (cos Φ) = `"True power"/"Apparent power"`

An electrical device which converts low alternating voltage at high current to high alternating voltage at low current (or vice versa) — i.e., a device which reduces or increases the voltage in an AC circuit through mutual induction — is called a transformer.

The transformer is a device used for converting low voltage into high voltage and high voltage into low voltage. It works on the principle of electromagnetic induction.

X_L ​= 2πfL (∝ f)

X_C​ = \[\frac {1}{2πfC}\]​ (∝ 1/f)

Z = \[\sqrt{R^2+(X_L-X_C)^2}\] = \[\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}\]

e = e₀​sin ωt

ϕ = tan⁻¹\[\left(\frac{X_C-X_L}{R}\right)\]

P_av ​= VI cos ϕ = I²Z cos ϕ = \[\frac {V_m​I_m}{2}\]​​cos ϕ = V_rms​I_rms​ cos ϕ

\[\cos\phi=\frac{P_{av}}{P_{apparent}}=\frac{P_{av}}{V_{rms}\cdot I_{rms}}\]

P = VI = \[\frac{V_mI_m}{2}[\cos\phi-\cos(2\omega t+\phi)]\]

Where:

• V_m​ = Peak voltage
• I_m​ = Peak current
• ϕ = Phase angle between voltage and current
• ω = Angular frequency
• t = Time

 P_input = P_output

\[\frac{V_s}{V_p}=\frac{N_s}{N_p}=\frac{I_p}{I_s}=k\]

\[\frac{V_s}{V_p}=\frac{N_s}{N_p}=k\]

Where:

• V_s = Secondary (output) voltage (V)
• V_p = Primary (input) voltage (V)
• N_s = Number of turns in secondary coil
• N_p = Number of turns in primary coil
• k = Transformation ratio (turns ratio)

\[\frac{I_p}{I_s}=\frac{N_s}{N_p}=\frac{V_s}{V_p}\]

A transformer is based on the principle of mutual induction, i.e., whenever the magnetic flux linked with a coil changes, an emf is induced in the neighbouring coil. For an ideal transformer there is no loss of power, so P_input = P_output​. On the basis of winding, transformers are of two types — step-up and step-down.

• A phasor is a rotating vector about the origin with angular speed ω.
• Its vertical component gives the instantaneous value of v or i.
• The length of the phasor equals the peak value (v_m or i_m​).
• For a pure resistor: current and voltage are in phase — phasors point in the same direction, phase angle = 0°.
• For inductors, capacitors, or combinations: the current is NOT in phase with voltage — phasors differ in direction.