Alternating Current Formulas

1. Alternating current (a.c.)
The current whose magnitude changes with time and direction reverses periodically, is called alternating current.
(a) Alternating emf e and current I at any time are given by:
e = e₀ sinωt, where e₀ = NBAω
and I = I₀ sin (ωt – Φ) I₀ = NBAω/R
if frequency of a.c. is n then
ω = 2πn = \(\frac{2 \pi}{\mathrm{T}}\) T → Time period

2. Different values of alternating current and voltage
(a) Instantaneous value: It is the value of alternating current and voltage at an instant t.

(b) Peak value: Maximum values of voltage (e₀) and current (I₀) in a cycle are called peak-values.

(c) Mean value: For complete cycle.
< e > = \(\frac{1}{\mathrm{T}} \int_{0}^{\mathrm{T}} \mathrm{e} \mathrm{dt}=0\)
< I > = \(\frac{1}{T} \int_{0}^{T} I d t=0\)
mean value for half cycle = \(\frac{2 \mathrm{e}_{0}}{\pi}\)
Root-mean-square value:
e_rms = (< e² >)^1/2 = \(\frac{e_{0}}{\sqrt{2}}\) = 0.707E₀ = 70.7% e₀
and I_rms = (< I² >)^1/2 = \(\frac{I_{0}}{\sqrt{2}}\) = 0.707I₀ = 70.7% I₀
RMS values are also called apparent or effective values.

3. Peak to peak value of A.C.
I_pp = | + I₀ | + | – I₀ | = 2I₀ = 2.8 I_rms

4. Form factor
Form factor for sinusoidal voltage or current = \(\frac{I_{\mathrm{rms}}}{<\mathrm{I}>}=\frac{\pi}{2 \sqrt{2}}\)

5. Phase difference between the emf (voltage) and the current in an ac circuit

• For pure resistance: The voltage and the current are in same phase i.e., phase difference Φ = 0.
• For pure inductance: The voltage is ahead of current by π/2 i.e., phase difference Φ = + π/2.
• For pure capacitance: The voltage lags behind the current by π/2, i.e., phase difference Φ = -π/2.

where Φ is the phase difference of the voltage E relative to the current I.

6. Reactance
(a) Reactance X = \(\frac{\mathrm{e}}{\mathrm{I}}=\frac{\mathrm{e}_{0}}{\mathrm{I}_{0}}=\frac{\mathrm{e}_{\mathrm{rms}}}{\mathrm{I}_{\mathrm{rms}}} \pm \pi / 2\)

(b) Inductive reactance
X_L = ωL = 2πnL

(c) Capacitive reactance
X_C = \(\frac{1}{\omega C}=\frac{1}{2 \pi n C}\)

7. Impedance
Impedance Z = \(\frac{\mathrm{e}}{\mathrm{I}}=\frac{\mathrm{e}_{0}}{\mathrm{I}_{0}}=\frac{\mathrm{e}_{\mathrm{ms}}}{\mathrm{I}_{\mathrm{rms}}}\)

(b) For L-R series circuit:
Z_RL = \(\sqrt{R^{2}+X_{L}^{2}}=\sqrt{R^{2}+\omega L^{2}}\)
and tan Φ = \(\left(\frac{\omega L}{R}\right)\) or Φ = tan^-1 \(\left(\frac{\omega L}{R}\right)\)

(c) For R-C series circuit:
Z_RC = \(\sqrt{\mathrm{R}^{2}+\mathrm{X}_{\mathrm{C}}^{2}}=\sqrt{\mathrm{R}^{2}+\left(\frac{1}{\omega \mathrm{C}}\right)^{2}}\)
and tan Φ = \(\left(\frac{1}{\omega C R}\right)\)
or Φ = tan^-1 \(\left(\frac{1}{\omega \mathrm{CR}}\right)\)

(d) For L-C-R series circuit:
Z_LCR = \(\sqrt{\mathrm{R}^{2}+\left(\mathrm{X}_{\mathrm{L}}-\mathrm{X}_{\mathrm{C}}\right)^{2}}\)
= \(\sqrt{\mathrm{R}^{2}+\left(\omega \mathrm{L}-\frac{1}{\omega \mathrm{C}}\right)^{2}}\)


and tan Φ = \(\frac{\left(\omega L \sim \frac{1}{\omega C}\right)}{R}\)
or Φ = tan^-1\(\left(\frac{\omega L \sim \frac{1}{\omega C}}{R}\right)\)

8. Susceptance
Reciprocal of reactance is called susceptance.
∴ Susceptance S = \(\frac{1}{X}\) mho

9. Conductance
Reciprocal of resistance is called conductance.
∴ Conductance G = \(\frac{1}{R}\) mho

10. Admittance
Reciprocal of impedance is called admittance.
∴ Admittance Y = \(\frac{1}{Z}\) mho

11. Power in an ac circuit
(a) Electric power = (current in circuit) × (voltage in circuit)
P = Ie

(b) Instantaneous power:
P_inst = e_inst × I_inst

(c) Average power
p_av = \(\frac{1}{2}\) e₀I₀cos Φ = e_rms I_rms cos Φ

(d) Virtual power (apparent power):
= \(\frac{1}{2}\) e₀I₀ = e_rmsI_rms

12. Power factor

• Factor cos Φ = \(\frac{P_{\mathrm{av}}}{P_{\mathrm{v}}}=\frac{R}{Z}\)
• For pure inductance cos Φ = 1
• For pure capacitance cos Φ = 0
• For LCR circuit
  cos Φ = \(\frac{\mathrm{R}}{\sqrt{\mathrm{R}^{2}+\left(\omega \mathrm{L}-\frac{1}{\omega \mathrm{C}}\right)^{2}}}\)

13. Wattless current
The component of current differing in phase by π/2 relative to the voltage is called wattles current.

Rms value of wattles current
= \(\frac{I_{0}}{\sqrt{2}}\) sin Φ
= I_rms = sin Φ = \(\frac{I_{0}}{\sqrt{2}}\left(\frac{X}{Z}\right)\)

14. Choke coil
An inductive coil used for controlling alternating current whose self inductance is high and resistance is negligible, is called choke coil. The power factor of this coil is approximately zero.

15. Series lcr resonant circuit
(a) When the inductive reactance (X_L) becomes equal to the capacitive reactance (X_C) in the circuit, the total impedance becomes purely resistive (Z = R). In this state voltage and current are in same phase (Φ = 0), the current and power are maximum and impedance is minimum. This state is called resonance.

(b) At resonant frequency:
ω_rL = \(\frac{1}{\omega_{\mathrm{r}} \mathrm{C}}\)
Hence resonant frequency f_r = \(\frac{1}{2 \pi \sqrt{L C}}\)

(c) In resonance the power factor of the circuit is one.

16. Half-power frequencies
Those frequencies f₁ and f₂ at which the power is half of the maximum power (power at resonance), i.e.,
P = \(\frac{1}{2}\) P_max
and I = \(\frac{I_{\max }}{\sqrt{2}}\),
f₁ and f₂ are called half-power frequencies.

17. Band-width
The frequency interval between half-power frequencies is called band-width.
∴ Band-width Δf = f₂ – f₁
For a series LCR resonant circuit.
Δf = \(\frac{1}{2 \pi} \frac{\mathrm{R}}{\mathrm{L}}\)

18. Quality factor (Q)
Maximum energy stored
Q = 2π × \(\frac{\text { Maximum energy stored }}{\text { Energy dissipated per cycle }}\)
\(\frac{2 \pi}{\mathrm{T}} \times \frac{\text { Maximum energy stored }}{\text { Mean power dissipated }}\)
or Q = \(\frac{\omega_{\mathrm{r}} \mathrm{L}}{\mathrm{R}}=\frac{1}{\omega_{\mathrm{r}} \mathrm{CR}}=\frac{\mathrm{f}_{\mathrm{r}}}{\left(\mathrm{f}_{2}-\mathrm{f}_{1}\right)}=\frac{\mathrm{f}_{\mathrm{r}}}{\Delta \mathrm{f}}\)
In the state of resonance
V_R = e, V_L = Oe and V_c = Qe,
where e is the applied voltage

19. Parallel LCR Circuit
(a) Admittance of the circuit:

Y = \(\frac{1}{Z}=\sqrt{\frac{1}{R^{2}}+\left(\omega C-\frac{1}{\omega L}\right)^{2}}\)
= \(\sqrt{\mathrm{G}^{2}+\left(\mathrm{S}_{\mathrm{C}}-\mathrm{S}_{\mathrm{L}}\right)^{2}}\)

(b) The phase difference between the voltage and the current

tan Φ = \(\left(\omega C-\frac{1}{\omega L}\right) R=\frac{S}{G}\)
or Φ = tan ^-1 \(\left(\omega C-\frac{1}{\omega C}\right)\)R
= tan^-1 \(\left(\frac{\mathrm{S}}{\mathrm{G}}\right)\)

20. Parallel resonance or anti-resonance
(a) When the inductive reactance X_L becomes equal to the capacitive reactance X_C the impedance of the circuit is maximum or the admittance and the current are minimum, this state is called parallel resonance or anti-resonance. Parallel resonance is just opposite to the series resonance.

(b) The frequency of anti-resonance when resistance is very small
f_r = \(\frac{1}{2 \pi \sqrt{L C}}\)

(c) The power factor of circuit in the state of parallel resonance is one.

(d) Band-width Δf = 2πf_r²CR

(e) Quality factor Q = \(\frac{1}{\omega_{\mathrm{r}} \mathrm{CR}}=\frac{1}{2 \pi \mathrm{f}_{\mathrm{r}} \mathrm{CR}}\)

Be it any kind of formulas in Physics you can always look up to Physicscalc.Com for all your queries and clear them at the earliest.