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Alternating Current

AC is the current produced by a voltage source that reverses its polarity with time.

• Time period \(T\) of alternating or voltage (that produced it) is the time interval \(T\) in which the source voltage reverses its polarity once.
• The sinusoidal waveform of AC is a graph between voltage and time.
• The voltage of AC generator varies with time and at an instant. $$ V=V_o\sin\omega t $$ $$=V_o\sin\frac{2\pi}{T}\times t $$ $$ =V_o\sin 2\pi ft $$ $$ V=V_o\sin \theta $$ Where \(\theta=\omega t\) is the angle through which coil rotates in time \(t\) & \(\theta\) is the phase which specifies instantaneous value.

Types of AC Waves

• Square Waves
• Saw-tooth Waves
• Sinusoidal Waves

Instantaneous Values

The value of voltage or current that exists at any given instant \(t\) in a circuit is known as instantaneous value. $$ V_{inst} = V = V_o\sin \theta= V_o\sin2\pi ft= V_o \sin\omega t $$ $$ I_{inst} = I = I_o\sin \theta= I_o\sin2\pi ft= I_o \sin\omega t $$

• \(V\) is the instantaneous value between \(+V_o\) & \(-V_o\)
• Sinusoidal graph is the set of all instantaneous values.

Peak Value (\(V_o\) or \(I_o\))

It is the highest value of voltage or current.

Peak to Peak Value

It is the sum of +ve & -ve peak values.

Root Mean Square (rms) Value

It is the square root of the average of squared instantaneous values of voltage or current in a period. $$ V_{rms} = V_o/\sqrt{2} = 0.7V_o $$ $$ I_{rms} = I_o/\sqrt{2} = 0.7I_o $$

• It is also known as effective value.

Info

The average value of sinusoidal symmetrical signal is zero over a cycle but RMS value is not. That is why practically all the calculations are based on RMS value.

Phase

The angle \(\theta\) which specifies the instantaneous value of alternating voltage or current is known as its phase.

Phase Lag & Lead

• Phase lag occurs when the phase of waveform 2 is less than the phase of waveform 1.
• Phase lead occurs when the phase of waveform 2 is greater than the phase of waveform 1.

Vector Representation of Phase & Alternating Quantities

Projection of a counter clockwise rotation vector along represents a sinusoidally alternating voltage or current.

Vector Diagram

A set of counter clockwise rotating vectors whose lengths represents the peak values of RMS value of the signals and the angle between them represents the current of phase in AC signals is collectively called vector diagram.

DC Circuit

• Resistor \(R\) is the basic circuit element in DC circuit.
• \(R\) controls voltage and current.
• Ohm's law can be used.

AC Circuit

• Resistor, Inductor & Capacitor are three main components of a AC circuit.

AC through Resistor

• Through resistor, \(V=V_o \sin\omega t\) and \(I=I_o\sin\omega t\) voltage and current are in phase if one increases or decreases, the other does the same.
• \(P=VI=I^2R=V^2/R\), holds true only when \(V\) & \(I\) are in phase.

AC Through Capacitor

• Due to DC, a capacitor behaves as insulator. It only store charges. Due to AC, a capacitor behaves as conductor.
• Through capacitor, charge on parallel plates of a capacitor. $$ q=CV_o\sin\omega t=q_o\sin\omega t $$
• Current leads voltage through capacitor (seems passing) because of the continuously reversing polarity.
• Reactance of capacitor (measure of opposition of capacitor against flow of AC) is given by: $$ X_C=\frac{V_{rms}}{I_{rms}}= \frac{1}{2\pi fC}= \frac{1}{\omega C} $$
• As, frequency decreases \(X_C\) decreases.
• Voltage lags current by \(90\degree\) in a pure capacitor circuit.

AC Through Inductor

• Through inductor, Ac produce self, inductive effect because it constantly changes.
• When potential difference is applied back emf is produced. $$ \epsilon_L=\frac{L\Delta I}{\Delta t} $$
• Current lags behind voltage by \(90\degree\) or \(\pi/2\)
• Inductive reactance is measure of opposition offered by the inductor coil to to flow of AC through it. $$ X_L=\frac{V_{rms}}{I_{rms}}=2\pi fL = \omega L $$
• As frequency increases \(X_L\) increases.
• In a pure inductor, input energy = output energy.
• No power loss take place in pure inductive or capacitive circuit.
• Current lags voltage by \(90\degree\) in inductive circuit.

Impedance (\(Z\))

Combined effect of resistance \(R\), capacitive reactance \(X_C\) & inductive reactance \(X_L\) in a circuit in which AC is provided is called Impedance. $$ Z=\frac{V_{rms}}{I_{rms}} $$

• Impedance is measured in ohms

Circuit Elements	Impedance (\(\Omega\))	Phase Angle (\(\phi\))
R	\(Z=R\)	\(0\degree\)
L	\(Z=X_L=\omega L\)	\(+90\degree\)
C	\(Z=X_C=1/\omega C\)	\(-90\degree\)
R-L	\(Z=\sqrt{R^2+X_L^2}\)	\(0\degree<\phi<90\degree\)
R-C	\(Z=\sqrt{R^2+X_C^2}\)	\(-90\degree<\phi<0\degree\)
R-L-C	\(Z=\sqrt{R^2+(X_L-X_C)^2}\)	\(\phi>0\degree\) if \(X_L>X_C\) \(\phi=0\degree\) if \(X_L=X_C\) (Resonance) \(\phi<0\degree\) if \(X_L<X_C\)

Info

Current vector is taken along X-axis

RC series circuit

\[ \text{Impedance }Z=\sqrt{R^2+\frac{1}{\omega C}^2} \]

• Phase between \(V\) & \(I\) is \(\theta=tan^{-1}(1/\omega CR)\)
• Current leads voltage.

RL Series Circuit

\[ \text{Imedance }Z=\sqrt{R^2+(\omega L)^2} \]

• Phase between \(V\) & \(I\) is \(\theta=tan^{-1}(\omega L/R)\)
• Current lags voltage.

Power in AC Circuit

• Power dissipation \(= 0\), in purely inductive & capacitive circuit.
• \(P=IV\) only when \(V\) & \(I\) are in phase in a purely resistive circuit.
• \(P=IV\cos \theta\) in real \(L\) & \(C\) circuit where \(\cos\theta\) is power factor & \(V \cos\theta\) is component voltage vector along current.

Series Resonance Circuit

\[ X_L=\omega L\;\;\text{and}\;\; X_C=\frac{1}{\omega C} \]

• At low frequency capacitance dominates and circuit behave as RC series circuit and at high frequency inductance dominates and behaves as RL series circuit.
• The Impedance diagram is shown in figure.
• If \(X_L=X_C\), then it is 'resonance' and frequency at this time called resonance frequency $$ \omega_rL=1/\omega_rC $$ $$ f_r=\frac{1}{2\pi\sqrt{LC}} $$
• At resonance, impedance is minimum and is equal to \(R\).
• At resonance \(V_L\), the voltage across inductance and \(V_C\) the across capacitance may be much larger than the source voltage.

Parallel Resonance Circuit

• The resonance frequency is $$ f=\frac{1}{2\pi\sqrt{LC}} $$
• At resonance the impedance is maximum. Hence current is minimum at resonance and in phase with voltage.
• At resonance branch current may be larger than source currents.
• At resonance power factor is one.

Three Phase AC Supply

• In three phase AC generator, there are three coils inclined at \(120\degree\) angle to each other.
• Each coil is connected to its own pair of slip rings.
• In three phase supply, total load is dived in three parts.
• There will be no drop of voltage supply in three phases.
• Three phase supply can provide \(400V\) between two lines.
• The lines to neutral voltage is \(230V\)
• Star type connector is used to connect three phase wire and one neutral wire.

Principle of Metal Detector

• Coil (\(L\)) Capacitor (\(C\)) are coupled to produce oscillations of currents.
• LC circuit behaves like an oscillating mass spring system.
• Energy oscillations between a capacitor and inductor called an electrical oscillator.
• LC circuit produce beats phenomenon for neutral detection.

Choke

• It is an inductive coil.
• It consumes extremely small power.
• It is used in AC to limit current.

Electromagnetic Waves (EMW)

• Electromagnetic waves were produced by the Max well and experimentally produced by Frack Hertz.
• Electromagnetic waves are oscillating electric and magnetic fields.
• Changing magnetic flux produces electric field given by: $$ E=\frac{I}{2\pi r}\frac{\Delta\Phi_m}{\Delta t} $$
• Changing electric flux produced magnetic field given by: $$ B=\left[\frac{\mu_o \epsilon_o}{2\pi r}\right]\frac{\Delta \phi_m}{\Delta t} $$
• Speed of e.m waves in vacuum is given by $$ v=\frac{1}{\sqrt{\mu_o\epsilon_o}} =3\times10^9ms{-1} $$
• The frequency is about \(10^4Hz\) depending upon wavelengths.

Production of EM Waves

• Electromagnetic waves in laboratory are by ossillations of electrons/
• frequency of osicallatory circuit is $$ f=\frac{1}{2\pi\sqrt{LC}} $$

Reception of EM Waves

The reception of EM waves is based on a \(LC\) circuit in which the value of fixed but varying capacitor gives different frequencies.

Modulation

The process of mixing the low frequency signal (sound) with high frequency radio waves is called modulation. The resultant waves is called modulated wave.

Types

• Amplitude modulation (AM)
• Frequency modulation (FM)

• Phase modulation (PM)

• Modulating waves are demodulated by receiving sets (TV or radio sets), rectified and then detected.

• Detection of modulation waves takes place only when frequency of tuning circuit equals p that of transmitting circuit and hence the electrical resistance is produced.

Info

• Frequency range of AM signals is from \(540KHz\) to \(1600KHz\).
• Frequency range of FM signals is from \(88MHz\) to \(108MHz\).
• FM signal does not contain any noise signal as the AM does.

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