Alternating Current 

Alternating Current

Alternating current is a current which keeps changing in magnitude continuously.  Alternating current also changes its direction periodically. The following formula gives the alternating current.

I = I₀ sin ωt

I = I₀ sin (2π/T) t

where I₀ is the peak value of Alternating Current (AC), Angular frequency, ω= 2πn (n is the frequency of AC)

So, I = I₀ sin 2πnt

Since Alternating Current is always fluctuating we consider its mean, average or root mean squared value for calculations.

Mean Value of AC

Mean value is that value of a steady current which sends the same amount of charge through a circuit as an alternating current in does in one half of the cycle. Thus, peak value multiplied with 2/π = (0.637) gives the mean value of alternating current.

Half cycle (I_av) = (2/π)I₀

Half cycle (V_av) = (2/π) V₀

Average value of AC

In one complete cycle of AC, the magnitude of the current remains the same at the start and the end point. Hence the average of alternating current is

I_av = 0

RMS Value of AC

Root Mean Square value of the steady current is that value which produces a certain amount of heat produced in a resistance in a given time as much produced by the alternating current itself in the same time.

The peak value when multiplied with 1/√2 (0.707) gives the RMS value.

I_rms = I₀/√2

V_rms= V₀/√2

Inductive reactance- Inductive reactance is the opposition that is offered to the current flow in an AC circuit. It is the electrical resistance of the inductor coil and is measured in ohms. Its formula is

X_L = ωL 

where ω = 2πn (n is the frequency of AC), L = coefficient of self-inductance of the coil.

Capacitive reactance- Capacitive reactance is the resistance in the flow of current, offered by a capacitor in an AC circuit. It is given by

X_c = 1/ωC

where C is the capacity of the condenser

Form Factor- The form factor is a ratio of the RMS value of the alternating voltage with that of the average voltage. 

Form Factor = RMS value of voltage/Average value of voltage

Form Factor = 

AC Voltage Applied to a Resistor

Let a resistance (R) is connected to a supplied alternating current source as shown in the figure.

Current flowing through the circuit is, I = I₀ sin(ωt)

Power dissipated in the resistance is, P = I²R = I²₀sin²(ωt)

Phasor diagram for a pure resistance circuit 

For a pure resistance circuit V_R (Voltage across resistance) and I_R are in the same phase.

AC Voltage applied to a Capacitor

Let a capacitor (C) is connected to a supplied alternating current source as shown in the figure. The plates of the capacitor are alternatively charged with each half cycle.

The charge q on the capacitor at a given time (t) is, q = CV₀ sinωt

where V₀ = I₀/ωC

Current flowing in the circuit, I = I₀ sin(ωt +π/2)

The impedance of the capacitor is, X_c = 1/ωC

Phasor diagram for a capacitive circuit 

For a pure capacitor, V_C (Voltage across the capacitor) lags I_C by 90^o

AC Voltage applied to an Inductor

Let an inductor (L) is connected to a supplied alternating current source as shown in the figure. An inductor in an AC circuit offers inductive reactance and limits the flow of current, similar to what a purely resistive circuit does.

Impdeance of the inductor is, X_L = ωL

Voltage across the inductor is, V_L = L(dI/dt) = LI₀ω cosωt

Current flowing through the circuit is, I = (V₀/ ωL) sinωt

And the maximum current in the circuit is, I₀ = V₀/X_L

Phasor diagram for an inductive circuit

For a pure inductor circuit, V_L (Voltage across the inductor) lags I_L by 90^o

AC Voltage applied to an LCR circuit

In an LCR circuit, Resistor, Capacitor, and an Inductor are connected in series with an AC voltage supply.

According to Ohm's law supplied voltage is, V = IZ

Where Z is the impedance of the circuit and I is the current flowing through the circuit.

The impedance of the circuit is, |Z |= √[R²+(ωL-1/ωC)²]

If X_c > X_L, the circuit is considered to be dominantly capacitive.

If X_c < X_L, the circuit is considered to be dominantly inductive.

 Phaser diagram of LCR circuit

The potential difference lags the current by an angle θ, θ = tan^-1[ωL -1/ωC)/R]

The phasor diagram of an RLC circuit also represents the impedance diagram.

Resonance in an RLC circuit

The condition of resonance occurs when the capacitive reactance and the inductive reactance are equal in magnitude but are 180 degrees apart in phase, which is why they cancel out each other. This frequency of the system is called its natural frequency.

Resonance frequency is, f_r = 1/2π√LC

In the resonance condition,

X_L = X_C, ϕ = 0, Z = R(minimum), cosϕ = 1, sinϕ = 0 and the current is maximum (= E₀/R)

Power in AC circuit

Power in purely resistive circuits is, P_av = (E₀/√2)×(I₀/√2)

where E_v and I_v are considered to be the virtual values of e.m.f and the current respectively.

Power in an LCR circuit is given, P_av = (E₀/√2)×(I₀/√2) cosϕ = (E_v×I_v) cosϕ

where cosϕ is the Power Factor

Power factor, cosϕ = Real power/Virtual power = P_av/E_rmsI_rms

For a purely resistive circuit, P_av = E_vI_v

For a purely inductive circuit, P_av = 0

For a purely capacitive circuit, P_av = 0

LC Oscillations

In an LC circuit, when the capacitor has charged and is allowed to discharge through a non-resistance, then electrical oscillations of a constant frequency and amplitude are produced. This phenomenon is termed as LC oscillations. 

The frequency of oscillation, ω₀= 1/√LC

The total energy in LC circuit is, 

Transformers

Transformers are used to change the value of an alternating current from a higher value to a lower one or lower value to higher value. It has primary and secondary windings that help step up or step down voltage.

Where N_p is the number of primary windings and N_s is the number of secondary windings.

_{Induced emf in a primary coil is, ep} = N_p (dϕ/dt)                     .....(1)

Induced emf in secondary coil is, e_s = N_s (dϕ/dt)                  ......(2)

equation (1)/(2), e_p /e_s = N_p/N_s

Also, e_pI_p = e_sI_s  (where I_s is the current flowing through the secondary coil, I_p is the current flowing through the primary coil)

Therefore  I_s/I_p = e_p/e_s = N_p/N_s

Transformer are two types

Step down Transformer- Step down transformer is one which converts high voltage at the primary coil to low voltage at the secondary coil. In step down transformer number of turns in the secondary coil is less than the number of turns in the primary coil ( N_s< N_p ).

In step down transformer, e_s < e_p and I_s > I_p

Step up transformer- Step up transformer is one which converts low voltage at the primary coil to high voltage at the secondary coil. In step up transformer number of turns in the primary coil is less than the number of turns in the secondary coil (N_s > N_p ).

In step-up transformer, e_s > e_p,  and I_s < I_p

The efficiency of the transformer is, η = (e_s I_s)/(e_p I_p)

 If you're thinking of ways to score better in JEE 2021, check out 

Rankup JEE: A 70-Day Score Booster Course for JEE Main April 2021 

Prepare with best-in-class teachers from Kota & Delhi, having mentored 100+ double-digit JEE rankers. Comprehensive day wise study plan for last 70 days to become exam ready.

Short Notes on Alternating Current Download PDF!

All India Free Mock Test for JEE Main | Attempt Now

Subscribe to YouTube Channel for JEE Exam

All the best! 

Team BYJU'S Exam Prep 

Sahi Prep hai toh Life set hai

 Download BYJU'S Exam Prep, the best IIT JEE Preparation App