Alternating Current

 Generation of Alternating Current

It is generated by ac generator / alternator / dynamo.

It is generated through conversion of mechanical energy into electrical energy and is based on principle of electromagnetic induction.

When a rectangular coil of loop area ‘A’ and ‘N’ turns rotates in a region of magnetic strength ‘B’ with angular velocity ‘ω’, the generated emf in the coil is E = NABωsinωt => $E = E_osinωt$

The generated emf is sine wave i.e. alternating in nature. ‘$E_o$’ is called amplitude of generating emf.

Introduction to Alternating Current and its parameters

Alternating current (ac) is the current whose magnitude varies continuously with time (or position) and direction reverses periodically.

When an alternating voltage source is applied to electrical device, alternating current flows through device.

$\textbf{Parameters of alternating quantity}$

i. Instantaneous value: It is value of alternating quantity at any instant. It is denoted by i(t) or e(t).

ii. Time period (T): It is time taken to complete one cycle.

iii. Frequency (F): The number of times that wave cycle occurs in one second.

iv. Peak value ($I_o$ or $E_o$): It is amplitude of alternating quantity.

v. Average or mean value($I_{av}$ or $E_{av}$): This is the value of steady current which sends same amount of charge through a circuit for same time as send by ac.

$\to$ For half cycle of ac; $I_{av}$ = $\frac{2}{π} I_o$ and $E_{av}$ = $\frac{2}{π} E_o$

vi. Root mean square speed (rms) or effective value ($I_{rms}$ or $E_{rms}$): RMS is that value of steady current which develops same amount of heat through a device/circuit for same time as developed by ac.

$\to$ For full cycle of ac, $I_{rms}$ = $I_o/\sqrt{2}$ and $E_{rms}$ = $E_o/\sqrt{2}$

Resistance, Reactance and Impedance

Resistance (R) is opposition to flow of current.

The resistance offered by inductor and capacitor in ac circuit are inductive reactance (XL) and capacitive reactance (XC) respectively. SI unit of reactance is ohm.

$X_L = ωL=2πfL$ and $X_C = \frac{1}{ωC}=\frac{1}{2πfC}$

Total resistance offered by multiple components in ac circuit is impedance (Z). Its SI unit is ohm.

Susceptance (S) = 1/X and Admittance (A) = 1/Z. Their SI unit is mho.

Different AC Circuit and their parameters.

The supply is considered as $E = E_osinωt = E_osin2πft,$ volts.




Note: In above cases of RL, RC and RLC; $I_oZ$ gives peak voltage $(E_o)$ and $I_{rms}Z$ gives rms voltage $(E_{rms}).$

Resonance phenomenon in LCR Circuit

It occurs when $X_L = X_C$

Resonant frequency (𝑹) = $\frac{1}{2\pi\sqrt{LC}}$

Circuit behaves as purely resistive in resonance.

Impedance is minimum and current is maximum at resonance.

The series LCR resonance circuits are known as acceptor circuit and used in radio, television and in voltage amplification purpose while parallel RCL circuit is known as rejector circuit.

Quality factor (Q-factor) at resonance =$\frac{VLorVC}{VR}$ = $\frac{1}{R}\sqrt{\frac{L}{C}} \\$

Note: Q-factor also measures the sharpness of resonance curve.

Average power or power dissipated in AC Circuit

$P_{av}=I_{rms}{E}_{rms}$ cos∅$\\$Hence, average power over a complete cycle in an inductive circuit is the product of virtual emf, virtual current and cosine of the phase angle between the voltage and current. The quantity cos∅ is called power factor.$\\$

For resistance only, cos∅ = 0, so $P_{av} = I_{rms} E_{rms}$ (i.e. maximum power loss)$\\$For capacitor only, cos∅ = $90^0$, so $P_av$ = 0$\\$For inductor only, cos∅ = $90^0$, so $P_av$ = 0

Wattless current and Choke coil

The component of AC which does not contribute to any power loss is called wattless current.

Such a current leads the alternating emf by $90^0$.

In purely reactive circuit, only wattless component of current flows.

Choke coil is based on the principle of wattless current.

Choke coil is approximation towards ideal inductor i.e. it has high inductance and negligible resistance.