Alternating Current Class 12 Notes Physics

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Class 12 Physics notes Chapter 7 Alternating Current

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CBSE Class 12 Physics Revision Notes Chapter 7 Alternating Current

Alternating Current: 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 am given by:

$Alternating Current Class 12 Notes$ Where, $Alternating Current Class 12 Notes$

$I={{I}_{0}}\sin (\omega t-\phi )$ Where, $Alternating Current Class 12 Notes$

$Alternating Current Class 12 Notes$ Where T is the time period.

Values of Alternating Current and Voltage

Instantaneous value: It is the value of alternating current and voltage at an instant t.
Peak value: Maximum values of voltage E₀ and current I₀ in a cycle are called peak values.
Mean or average value of a.c.: It is defined as that value of direct current which sends the same charge in a circuit in the same time as is sent by the given alternating current in its half time period. For complete cycle, $\begin{align} & <E>=\frac{1}{T}\int\limits_{0}^{T}{Edt=0} \\& <I>=\frac{1}{T}\int\limits_{0}^{T}{Idt=0} \\ \end{align}$ Mean value for half cycle: ${{E}_{mean}}=\frac{2{{E}_{0}}}{\pi }$ $Alternating Current Class 12 Notes$
Root – mean- square (rms) or effective or virtual value of a.c.: It is defined as that value of direct current which produces the same heating effect in a given resistor as is produced by the given alternating current when passed for the same time. $Alternating Current Class 12 Notes$ ${I_{rms}} = {( < {I^2} > )^{1/2}} = \frac{{{I_0}}}{{\sqrt 2 }} = 0.707{I_0} = 70.7\% {I_0}$ RMS values are also called apparent or effective values.

Phase difference Between the EMF (Voltage) and the Current in an AC Circuit: A diagram that represents alternating current and voltage of the same frequency as rotating vectors (phasors) alogwith proper phase angle between them is called a phasor diagram.

For pure resistor: The voltage and the current are in same phase i.e. phase difference $\phi =0$
For pure inductor: The voltage is ahead of current by $\frac{\pi }{2}$ i.e. phase difference $\phi =+\frac{\pi }{2}$
For pure capacitor: The voltage lags behind the current by $\frac{\pi }{2}$ i.e. phase difference $\phi =-\frac{\pi }{2}$

Reactance: The non-resistive opposition to the flow of a.c. is called reactance.

${\text{Reactance }}$ ${\text{X = }}\frac{E}{I} = \frac{{{E_0}}}{{{I_0}}} = \frac{{{E_{rms}}}}{{{I_{rms}}}} \pm \pi /2$
${\text{Inductive reactance }}$ ${{\text{X}}_L}{\text{ = }}\omega {\text{L = 2}}\pi {\text{fL}}$
${\text{Capacitive reactance }}$ ${{\text{X}}_C}{\text{ = }}\frac{1}{{\omega C}} = \frac{1}{{2\pi fC}}$

Impedance: It is defined as the effective resistance of the series LCR-circuit which opposes the flow of current through it is called impedance, $Z=\frac{E}{I}=\frac{{{E}_{0}}}{{{I}_{0}}}=\frac{{{E}_{rms}}}{{{I}_{rms}}}\phi$ Where $\phi$ is the phase difference of the voltage E relative to the current I.

For L – R series circuit: ${{Z}_{RL}}=\sqrt{{{R}^{2}}+X_{L}^{2}}=\sqrt{{{R}^{2}}+(\omega {{L})^{2}}}$ $\tan \phi =\left( \frac{\omega L}{R} \right)\ or\ \phi ={{\tan }^{-1}}\left( \frac{\omega L}{R} \right)$
For R – C series circuit: ${{Z}_{RC}}=\sqrt{{{R}^{2}}+X_{C}^{2}}=\sqrt{{{R}^{2}}+{{\left( \frac{1}{\omega C} \right)}^{2}}}$ $\tan \phi =\frac{1}{\omega CR}$ Or $\phi ={{\tan }^{-1}}\left( \frac{1}{\omega CR} \right)$
For L – C series circuit: ${{Z}_{LCR}}=\sqrt{{{R}^{2}}+{{({{X}_{L}}-{{X}_{C}})}^{2}}}$ $=\sqrt{{{R}^{2}}+{{\left( \omega L-\frac{1}{\omega C} \right)}^{2}}}$

$\tan \phi =\frac{\left( \omega L-\frac{1}{\omega C} \right)}{R}$ Or $Alternating Current Class 12 Notes$

Conductance: Reciprocal of resistance is called conductance.

$G=\frac{1}{R}\ mho$

Power in an ac circuit:

Electric power = (current in circuit) x (voltage in circuit)P = IE
Instantaneous power: ${P_{inst}} = {\text{ }}{E_{inst}} \times {\text{ }}{I_{inst}}$
Average power: $Alternating Current Class 12 Notes$
Virtual power (apparent power): $Alternating Current Class 12 Notes$

Power Factor: It is defined as the ratio of the true power to the apparent power of an a.c. circuit.

$Alternating Current Class 12 Notes$

For purely resistive circuit- $\text{Power factor cos}\phi \text{ = 1}$
For purely inductive or capacitive circuit- $\text{Power factor cos}\phi \text{ = 0}$
For LCR circuit- $Alternating Current Class 12 Notes$ $Alternating Current Class 12 Notes$

Wattless Current: The component of current differing in phase by $\frac{\pi }{2}$ relative to the voltage, is called wattles current.
The rms value of wattless current:

${P_{av}} = {\varepsilon _{rms}}\left( {{I_{rms}}\sin \phi } \right).\cos {\pi \over 2} = 0$

The component ${{I_{rms}}\sin \phi }$ is called wattless current because it does not consume any power in a.c. circuit.

Choke Coil:

An inductive coil used for controlling alternating current whose self- inductance is high and resistance negligible, is called choke coil.
The power factor of this coil is approximately zero.
It controls current without consuming any power.

Series Resonant Circuit

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, the voltage and current are in same phase ( $\phi$ = 0), the current and power are maximum and impedance is minimum. This state is called resonance.
At resonance, ${{\omega }_{r}}L=\frac{1}{{{\omega }_{r}}C}$ Hence, resonance frequency is, ${{f}_{r}}=\frac{1}{2\pi \sqrt{LC}}$
In resonance, the power factor of the circuit is one.

Half – Power Frequencies: Those frequencies f₁ and f₂ at which the power is half of the maximum power (power at resonance), i.e., f₁ and f₂ are called half – power frequencies.

$\begin{align} & P=\frac{1}{2}{{P}_{\max }} \\ & I=\frac{{{I}_{\max }}}{\sqrt{2}} \\ & \therefore P=\frac{{{P}_{\max }}}{2} \\ \end{align}$

Band – Width:

The frequency interval between half – power frequencies is called band – width. $\therefore \text{ Bandwidth }\Delta \text{f = }{{f}_{2}}-{{f}_{1}}$
For a series LCR resonant circuit, $\Delta \text{f = }\frac{1}{2\pi }\frac{R}{L}$

Quality Factor (Q): It is defined as the ratio of the resonant frequency to the difference in two frequencies taken on both sides of the resonant frequency such that at each frequency, the current amplitude becomes ${1 \over {\sqrt 2 }}$ times the value at resonant frequency.

$Alternating Current Class 12 Notes$

LC-Oscillations:- When a charged capacitor is allowed to discharge through a non-resistive inductor, electrical oscillations of constant amplitude and frequency are produced. These oscillations are called LC-oscillations.
Transformer:- It is an electrical device for converting an alternating current at low voltage into that at high voltage or vice versa.
If transformer increases the input voltage, it is called step-up transformer.
If transformer decreases the input voltage, it is called step-down transformer.
It works on the principle of mutual induction, i.e. when a changing current is passed through one of the two inductively coupled coils, an induced emf is set up in the other coil.

${{{I_1}} \over {{I_2}}} = {{{\varepsilon _2}} \over {{\varepsilon _1}}} = {{{N_2}} \over {{N_1}}}$

Energy losses in transformers-

Copper losses
Eddy current losses
Hysteresis loss
Flux leakage

A.C. Generator:- It is a device which converts mechanical energy into electrical energy.
Principle: It is based on the principle of electromagnetic induction.

CBSE Class 12 Revision Notes and Key Points

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