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Alternating Current Formulas

Alternating Current may hold less practical advantage compared to direct current. But it is still used in many applications like generators, motors and works better than DC. To help you with the Concept of AC we have curated the Alternating Current Formulas. Make the most out of them and be familiar with the terms like Impedance, Reactance, Power Factor, etc.

You can seek help regarding any concept using the Physics Formulas of ours and improve your subject knowledge. The List of AC Formulae provided covers everything right from basic to the advanced level regarding the concept.

List of 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:
erms = (< e2 >)1/2 = \(\frac{e_{0}}{\sqrt{2}}\) = 0.707E0 = 70.7% e0
and Irms = (< 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.
Ipp = | + I0 | + | – I0 | = 2I0 = 2.8 Irms

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
XL = ωL = 2πnL

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

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:
ZRL = \(\sqrt{R^{2}+X_{L}^{2}}=\sqrt{R^{2}+\omega L^{2}}\)
and tan Φ = \(\left(\frac{\omega L}{R}\right)\) or Φ = tan-1 \(\left(\frac{\omega L}{R}\right)\)
Alternating Current formulas img 1

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

(d) For L-C-R series circuit:
ZLCR = \(\sqrt{\mathrm{R}^{2}+\left(\mathrm{X}_{\mathrm{L}}-\mathrm{X}_{\mathrm{C}}\right)^{2}}\)
= \(\sqrt{\mathrm{R}^{2}+\left(\omega \mathrm{L}-\frac{1}{\omega \mathrm{C}}\right)^{2}}\)
Alternating Current formulas img 3
Alternating Current formulas img 4
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:
Pinst = einst × Iinst

(c) Average power
pav = \(\frac{1}{2}\) e0I0cos Φ = erms Irms cos Φ

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

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

Alternating Current formulas img 5

13. Wattless current
The component of current differing in phase by π/2 relative to the voltage is called wattles current.
Alternating Current formulas img 6
Rms value of wattles current
= \(\frac{I_{0}}{\sqrt{2}}\) sin Φ
= Irms = 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 (XL) becomes equal to the capacitive reactance (XC) 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.
Alternating Current formulas img 7
Alternating Current formulas img 8

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

(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 = \(\frac{1}{2}\) Pmax
and I = \(\frac{I_{\max }}{\sqrt{2}}\),
f1 and f2 are called half-power frequencies.
Alternating Current formulas img 9
Alternating Current formulas img 10

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
VR = e, VL = Oe and Vc = Qe,
where e is the applied voltage
Alternating Current formulas img 11

19. Parallel LCR Circuit
(a) Admittance of the circuit:
Alternating Current formulas img 12
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
Alternating Current formulas img 13
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 XL becomes equal to the capacitive reactance XC 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
fr = \(\frac{1}{2 \pi \sqrt{L C}}\)
Alternating Current formulas img 14
Alternating Current formulas img 15

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

(d) Band-width Δf = 2πfr2CR

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

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