Revision: Electromagnetic Induction and Alternating Currents >> Alternating Current Physics (Theory) ISC (Science) ISC Class 12 CISCE

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 a.c. generator is a device which converts the mechanical energy into the electrical energy using the principle of electromagnetic induction.

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.

The root-mean-square (rms) value of an alternating current is defined as the square-root of the average of 12 during a complete cycle, where I is the instantaneous value of the alternating current.

When a charged capacitor is discharged through an inductor of negligible ohmic resistance, electrical oscillations take place in the circuit. These are called L-C oscillations.

The alternating voltage or alternating current produced by a coil rotating in a magnetic field is maximum in two positions of the coil. This maximum value of the alternating voltage or alternating current is called the 'peak value' or the amplitude of voltage or ‘current'.

“The time taken by the alternating current to complete one cycle is called the ‘periodic time’ of the current.”

T = \[\frac {2π}{ω}\]

The number of cycles completed by an alternating current in one second is called the 'frequency' of the current.

f = \[\frac {ω}{2π}\]

A diagram representing alternating current and alternating voltage (of same frequency) as rotating vectors (phasors) with the phase angle between them is called a phasor diagram.

“The current which flows through the circuit, due to this alternating voltage, changes continuously between zero and maximum value, and flows in one direction in the first half rotation and in the opposite direction in the next half rotation. This type of current is called an ‘alternating current’.”

A series resonant circuit is the circuit, in which the frequency of the applied voltage is equal to the natural frequency of the circuit.

The current in an alternating-current circuit may, however, be reduced by means of a device which involves little loss of energy. This device is called ‘choke-coil'.

The rate of dissipation of energy in an electrical circuit is called the 'power'.

A transformer is a device, based on the principle of mutual induction, which is used for converting large alternating current at low voltage into small current at high voltage, and vice-versa. Transformers are used only in AC (not in DC).

The transformers which convert low voltages into higher ones are called 'step-up' transformers.
The transformers which convert high voltages into lower ones are called ‘step-down' transformers.

If the resistance in an AC circuit is zero, although current flows in the circuit, yet the average power remains zero, that is, there is no energy dissipation in the circuit. The current in such a circuit is called 'wattless current'.

P = V_rms × I_rms× cos ф,

where cos Φ = \[\frac{R}{\sqrt{R^2+\omega^2L^2}}\]

f = \[\frac{1}{2\pi}\sqrt{\frac{1}{LC}}=\frac{1}{2\pi}\sqrt{\frac{1/C}{L}}\cdot\]

I_rms = \[\sqrt{I^{2}}=\frac{I_{0}}{\sqrt{2}}=0.707I_{0}\]

The root-mean-square value of an alternating current is 0.707 times, or 70.7%, of the peak value.

• In a pure resistive AC circuit, voltage and current are in phase, and the average power is
  \[\vec P\] = V_rmsI_rms.
• The instantaneous power in a resistive circuit varies as sin⁡²ωt and is always positive.
• In an L–R circuit, the current lags behind the voltage by a phase angle ϕ, where
  tan⁡ϕ = \[\frac {ωL}{R}\].
• The average power in an L–R circuit is
  \[\vec P\] = V_rmsI_rms cos ⁡ϕ,
  which is less than that of a pure resistive circuit for the same V and I.
• Since cos⁡ϕ < 1 in an L–R circuit, power dissipation is reduced, though the average power remains positive.

• In a series L–C–R circuit, the average power is maximum at resonance and becomes negligible at very high or very low frequencies.
• The half-power points are the two frequencies ω₁​ and ω₂​ at which the power is half of the maximum power.
• The bandwidth of a resonant circuit is the difference between the half-power frequencies:
  Δω = ω2 − ω1 = \[\frac {R}{L}\].
• Lower resistance yields a narrower bandwidth and a sharper resonance curve, making the circuit more selective.
• The Q-factor measures the sharpness of resonance and is given by
  Q = \[\frac {ω_0}{Δ_ω}\],
  and it is large for small R and large L.

• An AC generator converts mechanical energy into electrical energy via electromagnetic induction.
• It consists of four main parts: the armature (rotating coil), the field magnet, the slip rings, and the carbon brushes.
• As the coil rotates, the magnetic flux linked with it changes continuously, inducing an emf whose direction is given by Fleming’s right-hand rule.
• The induced emf reverses after every half-rotation, producing an alternating current in the external circuit.
• The emf is maximum when the plane of the coil is parallel to the magnetic field and zero when the coil is perpendicular (vertical) to the field.

• Resonance occurs when X_L=X_C​ and current and voltage are in phase.
• At resonance, impedance is minimum (= R) and current is maximum.
• The resonant frequency is
• f₀ = \[\frac{1}{2\pi\sqrt{LC}}\].
• At resonance, the voltage across L and C can be much larger than the applied voltage.
• At f = f₀​: Z is minimum, and I is maximum, so the circuit is used for radio tuning.

• In an AC circuit, the phase difference between voltage and current depends on the circuit elements present (R, L, C).
• In a pure resistive circuit, current and voltage are in the same phase, and the peak current is I₀ = \[\frac {V_0}{R}\].
• In a pure inductive circuit, the current lags the voltage by 90^∘, and the opposition to AC is called inductive reactance X_L = ωL = 2π f L.
• Inductive reactance increases with frequency and becomes zero for DC.
• In a pure capacitive circuit, the current leads the voltage by 90^∘, and the opposition to AC is called capacitive reactance X_C = \[\frac {1}{ωC}\] = \[\frac {1}{2πfC}\].
• Capacitive reactance decreases with frequency and becomes infinite for DC.
• In an L–R series circuit, the impedance is
  Z = \[\sqrt{R^2+X_L^2}\]​​
  and the current lags behind the voltage.
• In a C–R series circuit, the impedance is
  Z = \[\sqrt{R^2+X_C^2}\]​​
  and the current leads the voltage.
• In an L–C circuit, the net voltage depends on V_L − V_C​; when X_L = X_C​, the circuit is in electrical resonance.
• In an L–C–R series circuit, the impedance is
  Z = \[\sqrt{R^2+(X_L-X_C)^2}\]​
  and at resonance, impedance is minimum, current is maximum, and voltage and current are in phase.

• Over one complete cycle, an alternating current flows equally in opposite directions, so its mean (average) value is zero.
• The mean value of AC is defined over one half-cycle, since averaging over a full cycle gives zero.
• For a sinusoidal AC, the mean value over a half-cycle is
  I_m = \[\frac {2}{π}\]I₀ = 0.637 I₀​
  i.e. 63.7% of the peak value.

• A transformer works only with alternating current (AC) and cannot operate on direct current (DC).
• It operates on the principle of mutual induction, where a changing magnetic flux in the core induces an emf in the secondary coil.
• The primary coil is connected to the AC mains, and the secondary emf has the same frequency as the primary (usually 50 Hz).
• In an ideal transformer, the voltage ratio equals the turns ratio:
  \[\frac{V_s}{V_p}=\frac{N_s}{N_p}.\]
• When voltage is stepped up, current is stepped down by the same ratio, and vice versa; thus, energy is conserved.
• The soft-iron core is laminated to reduce eddy currents and hysteresis losses, thereby improving efficiency.
• Step-up and step-down transformers use different wire thicknesses in coils to reduce copper losses.
• Practical transformers have efficiency below 100% due to copper, eddy-current, hysteresis, magnetic, and dielectric losses, yet they are essential for long-distance power transmission because high voltage reduces current and power loss.