Overview: AC Circuits

The electric current which has a fixed polarity of voltage (positive and negative terminals remain constant) is called Direct Current (DC).

The electric current for which the polarity of voltage changes periodically is called Alternating Current (AC).

OR

A voltage that varies with time like a sine function is called Alternating Voltage (AC Voltage).

OR

The current produced by an alternating voltage, which changes direction periodically with time, is called Alternating Current (AC).

The emf which varies sinusoidally with time and reverses its direction after every half rotation of the coil is called alternating emf.

The maximum value of an alternating current or emf in either direction is called the peak value.

The average of all instantaneous values of alternating current or voltage over one half cycle is called the average value of AC.

The value of steady (DC) current which produces the same heating effect in a resistance as the given alternating current is called the RMS value (effective value) of AC.

i_av​ = 0.637 i₀​

e_av = 0.637 e₀​

\[i_{rms}=\frac{i_0}{\sqrt{2}}=0.707i_0\]

\[e_{rms}=\frac{e_0}{\sqrt2}=0.707e_0\]

A rotating vector that represents a quantity varying sinusoidally with time is called a phasor. The diagram representing phasors and showing the phase relationship between alternating quantities is called a phasor diagram.

OR

“A phasor is a vector which rotates about the origin with angular speed ω.”

The opposition offered by an inductor to the flow of alternating current is called inductive reactance.

The opposition offered by a capacitor to the flow of alternating current is called capacitive reactance.

The total effective opposition offered by resistance, inductance and capacitance in a series AC circuit is called impedance.

The reciprocal of impedance of an AC circuit is called admittance.

• e = e₀ sin ⁡ωt
• e = iR
• i = i₀ sin⁡ ωt
• i₀ = \[\frac {e_0}{R}\]​​
• Phase difference = 0 (Voltage and current in phase)

• e = e₀​ sin ωt
• e = L\[\frac {di}{dt}\]​
• i = i₀ sin⁡(ωt − \[\frac {π}{2}\])
• i₀ = \[\frac {e_0}{ωL}\]​​
• Inductive reactance:
  X_L = ωL = 2πfL
• Current lags voltage by 90^∘

• e = e₀ sin ⁡ωt
• q = CV
• i = \[\frac {dq}{dt}\]​
• i = i₀ sin⁡(ωt + \[\frac {π}{2}\])
• i₀ = \[\frac {e_0}{ωC}\]​​
• Capacitive reactance:
  X_C = \[\frac {1}{ωC}\] = \[\frac {1}{2πfC}\]​
• Current leads voltage by 90°

• e₀ = i₀\[\sqrt{R^2+(X_L-X_C)^2}\]​
• Impedance:
  Z = \[\sqrt{R^2+(X_L-X_C)^2}\]​
• i₀ = \[\frac {e_0}{Z}\]​​
• Phase angle:
  tan⁡ϕ = \[\frac{X_{L}-X_{C}}{R}\]​​

Power is defined as the rate of doing work.

The power in an AC circuit at a given instant is the product of instantaneous voltage and instantaneous current.

Average power in a purely resistive AC circuit is the average of the instantaneous power over one complete cycle.

P_av ​= e_rms i_rms​

The average power over one complete cycle in an ideal inductor is zero because current lags voltage by 90°.

P_av​ = 0

The average power over one complete cycle in an ideal capacitor is zero because current leads voltage by 90°.

P_av​ = 0

Power factor is the cosine of the phase angle between voltage and current in an AC circuit.

Power Factor = cos ϕ

\[\cos\phi=\frac{\mathrm{True~Power}}{\text{Apparent Power}}\]

OR

The quantity cos φ, where φ is the phase angle between voltage and current.

Current flowing in a pure inductor or capacitor that consumes no average power is called Idle current or Wattless current

When a charged capacitor is allowed to discharge through a non-resistive inductor, electrical oscillations of constant amplitude and frequency are produced. These are called LC oscillations.

If there is no loss of energy in the circuit, the amplitude of oscillations remains constant. Such oscillations are called undamped oscillations.

Energy stored in a Capacitor: E = \[\frac {1}{2}\]\[\frac {Q^2}{C}\]

Energy stored in an Inductor: E = \[\frac {1}{2}\]CV²

• Energy Storage: A capacitor stores energy in an electric field; an inductor stores energy in a magnetic field.
• Energy Transfer: In an ideal LC circuit, energy continuously oscillates between the capacitor and the inductor.
• Condition for LC Oscillations: When a charged capacitor discharges through a non-resistive inductor, electrical oscillations are produced.
• Undamped Oscillations: If there is no energy loss, oscillations have constant amplitude and frequency.
• Damping Causes: Oscillations become damped due to (i) resistance, causing heat loss, and (ii) radiation of electromagnetic waves.

Resonance is the phenomenon in which the amplitude of oscillations becomes maximum when the frequency of the applied (driving) force is equal to the natural frequency of the system.

OR

The phenomenon in which the amplitude of oscillation becomes large when a system is driven at a frequency close to its natural frequency.

A circuit in which inductance (L), capacitance (C), and resistance (R) are connected in series and the circuit admits maximum current at a particular frequency is called a series resonance circuit.

The frequency at which inductive reactance equals capacitive reactance and the current becomes maximum is called the resonant frequency.

• Resonance occurs when X_L = X_e
• Resonant frequency f_r = \[\frac{1}{2\pi\sqrt{LC}}\]
• Impedance is minimum, and the circuit is purely resistive.
• Current has a maximum value.
• When a number of frequencies are fed to it, it accepts only one frequency (f_r) and rejects the other frequencies. The current is maximum for this frequency. Hence, it is called acceptor circuit.

A parallel resonance circuit consists of an inductor (L) and a capacitor (C) connected in parallel to an AC source.

The frequency at which the current drawn from the source is minimum and the impedance is maximum is called the resonant frequency of a parallel resonance circuit.

• Resonance occurs when X_L = X_C.
• Resonant frequency f_r = \[\frac{1}{2\sqrt{LC}}\] 
• Impedance is maximum.
• Clirrent is minimum.
• When alternating current of different frequencies is sent through a parallel resonant circuit, it offers very high impedance to the current at the resonant frequency (fr) and rejects it, but allows the current at the other frequencies to pass through; hence, it is called a rejector circuit.

The Q-factor of a series resonant circuit is defined as the ratio of the resonant frequency to the bandwidth (difference between the two half-power frequencies).

Mathematically, Q = \[\frac{\omega_r}{\omega_1-\omega_2}\] or \[\frac{\text{Resonant frequency}}{\mathrm{Bandwidth}}\]

Bandwidth is the difference between the two half-power angular frequencies on either side of the resonant frequency.

Bandwidth = ω₁ ​− ω_2​ = 2Δω

The frequencies at which the current amplitude becomes \[\frac {1}{\sqrt {2}}\]​ times its maximum value are called half-power frequencies.

A choke coil is an inductor used to reduce the current in an AC circuit without much loss of energy.

P = V_rms ​I_rms​ cos ϕ

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

A device used to change (transform) an alternating voltage from one value to another using the principle of mutual induction.

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

• A transformer works on the principle of mutual induction and is used to change the AC voltage from one value to another.
• It consists of two coils: primary (Np turns) and secondary (Ns turns) wound on a soft-iron core.
• Voltage ratio: V_s / V_p = Ns / Np​; voltage depends on the number of turns in the coils.
• Step-up transformer: Ns > Np​ → voltage increases and current decreases.
• Step-down transformer: Ns < Np → voltage decreases and current increases.