Revision Notes: Alternating Currents | Physics for JEE Main & Advanced PDF Download

Introduction to Alternating Current

An alternating current (a.c.) is a current which continuously, changes in magnitude and periodically reverses in direction.`
i = I₀ sin ωt = I₀ sin (2π/T) t
Here I₀ is the peak value of a.c.

Basic Parameters and Equations

Mean Value of AC or DC Value of AC

Mean value and its calculation for half-cycle
Mean value of a.c. is that value of steady current which sends the same amount of charge, through a circuit, in same time as is done by a.c. in one half-cycle.
(I_av)_{half cycle} = (2/π)I₀
Thus, mean value of alternating current is 2/π times (0.637 times) its peak value.
(V_av)_{half cycle} = (2/π) V₀

Average value of AC over a complete cycle
I_av = 0
The average value of a.c. taken over the complete cycle of a.c.is zero.

Root Mean Square (RMS) value of AC and its calculation
Root mean square value of alternating current is defined as that value of steady current which produces same heating effect, in a resistance, in a certain time as is produced by the alternating current in same resistance in same time. The r.m.s value of a.c.is also called its virtual value.
I_rms = I₀/√2
Root mean square value of alternating current is I/√2 times (or 0.707 times) the peak value of current.
Similarly, V_rms= V₀/√2
Here V₀ is the peak value of e.m.f.

Form Factor
Form Factor = rms value/average value = (V₀/√2)/ (2 V₀/π)  = π/2√2

Current Elements

(b) Capacitative reactance
X_c = 1/ωC
Here C is the capacity of the condenser

AC Circuits with Components

q = CV₀sinωt
I = I₀ sin(ωt +π/2)
V₀ = I₀/ωC
X_c = 1/ωC

Inductor in AC circuit

V_L = L(dI/dt) = LI₀ω cosωt
I = (V₀/ ωL) sinωt
Here, I₀ = V₀/ ωL
X_L = ωL
And the maximum current, I₀ = V₀/XL

R-L circuit

I = ε/R [1-e^-Rt/L]
V = ε e^-Rt/L

Graph between I (amp) and t (sec):-

Graph between potential difference across inductor and time:-

L-C Circuit:-

f = 1/2π√LC
q = q₀ sin (ωt+ϕ)
I = q₀ωsin (ωt+ϕ)
ω = 1/√LC

The total energy of the system remains conserved,
½ CV² + ½ Li² = constant = ½ CV₀² = ½ Li₀²

Series in C-R circuit:-

V = IZ
The modulus of impedance, |Z |= √R²+(1/ωC)²
The potential difference lags the current by an angle, ϕ = tan^-1(1/ωCR)

Series in L-C-R Circuit:

V = IZ
The modulus of impedance, |Z |= √[R²+(ωL-1/ωC)²]
The potential difference lags the current by an angle, ϕ = tan^-1[ωL -1/ωC)/R]

Circuit elements with A.C

Resonance and Related Concepts

Half power frequencies:-

(a) Lower,  f₁ = f_r – R/4πL    or      ω₁ = ω_r – R/2L
(b) Upper,  f₂ = f_r + R/4πL    or      ω₂ = ω_r + R/2L

Band width:- Δf = R/2πL   or   Δf = R/L

Quality Factor
(a) Q = ω_r/Δω = ω_rL/R
(b) As ω = 1/√LC, So Q ∝ √L, Q ∝1/R and Q ∝ 1/√C
(c) Q = 1/ω_rCR
(d) Q = X_L/R   or  Q = X_C/R
(e) Q = f_r/Δf   

At resonance, peak voltages are:-
(a) (V_L)_res = e₀Q
(b) (V_C)_res = e₀Q
(c) (V_R)_res = e₀

Conductance, Susceptance, and Admittance

(a) Conductance, G = 1/R
(b) Susceptance, S = 1/X
(c) S_L = 1/X_L and S_C = 1/X_C = ωC
(d) Admittance, Y = 1/Z
(e) Impedance add in series while add in parallel

Power in AC Circuits

Here E_v and I_v are the virtual values of e.m.f and the current respectively.

Circuit containing impedance (a combination of R,L and C):-
P_av = (E₀/√2)×(I₀/√2) cosϕ = (E_v×I_v) cosϕ
Here cosϕ is the power factor.
(a) Circuit containing pure resistance, P_av = E_vI_v
(b) Circuit containing pure inductance, P_av = 0
(c) Circuit containing pure capacitance, P_av = 0
(d) Circuit containing resistance and inductance,
Z = √R²+(ωL)
cosϕ = R/Z = R/[√{R²+(ωL)²}]
(e) Circuit containing resistance and capacitance:-
Z = √R²+(1/ωC)²
cosϕ = R/Z = R/[√{R²+(1/ωC)²}]
(f) Power factor, cosϕ = Real power/Virtual power = P_av/E_rmsI_rms

Transformer

(a) C_p = N_p (dϕ/dt) and e_s = N_s (dϕ/dt)
(b) e_p/e_s = N_p/N_s
(c) As, e_pI_p = e_sI_s, Thus, I_s/I_p = e_p/e_s = N_p/N_s
(d) Step down:- e_s < e_p, N_s< N_p and I_s> I_p
(e) Step up:- e_s >e_p, N_s>N_p and I_s< I_p
(f) Efficiency, η = e_s I_s/ e_p I_p

AC Generator

e = e₀ sin (2πft)
Here, e₀ = NBAω