AC Circuits and Electromagnetic Waves Notes

AC Circuits

Overview of AC Circuits

An AC circuit consists of:

• A combination of circuit elements.

• An AC generator or source/power supply.

Key Equation

The output voltage of an AC generator is given by:
$\Delta v = \Delta V_{max} \sin(2\pi f t)$
where:

• $\Delta v$ is the instantaneous voltage.

• $\Delta V_{max}$ is the maximum voltage of the generator.

• $f$ is the frequency of the voltage changes (60 Hz in the USA, 50 Hz in the EU).

Characteristics of AC Current

Current and Voltage Behavior

• The instantaneous current in an AC circuit can be defined as:
  $i(t) = I_{max} \sin(2\pi f t)$
  where:

• $I_{max}$ is the maximum current.

• The period of the AC signal is the time taken to complete one cycle.

Power in AC Circuits

Power Dissipation in Resistors

The electrical energy dissipated in a resistor is given by:

• Average Power: $P<em>{avg} = I</em>{rms}^2 R$

  • where $I_{rms}$ is the root mean square current and $R$ is the resistance.

RMS Values

• The rms current is found from:
  $I<em>{rms} = \frac{I</em>{max}}{\sqrt{2}}$

• Similar formula holds for voltage:
  $V<em>{rms} = \frac{V</em>{max}}{\sqrt{2}}$

Power Transmission in AC Circuits

Importance of High Voltage Transmission

• Electric power is transmitted at high voltages and low currents to:

  • Minimize I²R losses in transmission lines.

• Example calculation:

  • A power station delivers 20 MW to a city 1 km away with an AC generator output of 22 kV and wire resistance of 2 Ω:

  • $I = \frac{20 \times 10^6 W}{22 \times 10^3 V} \approx 909 A$

  • Power loss due to resistance $P_{loss} = I^2 R = (909 A)^2 \times 2 \Omega = 1.7 \times 10^6 W$

Transformers

Functionality of Transformers

• Transformers increase magnetic flux and transfer it between coils.

• The voltage relationship is defined as:
  $\frac{V<em>2}{V</em>1} = \frac{N<em>2}{N</em>1}$
  where:

• $N_1$ = turns in primary coil.

• $N_2$ = turns in secondary coil.

Types of Transformers

• Step-up Transformer: When N2 > N1.

• Step-down Transformer: When N2 < N1.

• Efficiency rates from 90% to 99% for real transformers.

Electromagnetic Waves

Maxwell’s Equations

• Gauss's Law for E: Electric field lines originate on positive charges and terminate on negative charges.

• Gauss's Law for B: Magnetic field lines form closed loops.

• Faraday’s Law: A varying magnetic field induces an electric field.

• Ampère’s Law: Magnetic fields generated by moving charges or changing electric fields.

Speed of Electromagnetic Waves

• In empty space, electromagnetic waves propagate at the speed of light $c$.

Properties of Electromagnetic Waves

• Transverse nature:

  • Electric field $E$, magnetic field $B$ are perpendicular to each other and to the direction of motion.

• Energy distributions:

  • Energy is equally shared between electric and magnetic fields.

  • Waves carry energy and linear momentum.

EM Spectrum

• Relationship: $c = f \lambda$ (where $f$ is frequency and $\lambda$ is wavelength).

• Examples of EM spectrum ranges (frequency and corresponding wavelengths):

  • Gamma rays: 10²² Hz, 1 pm

  • X-rays: 10¹⁹ Hz, 1 nm

  • Visible light: 4 × 10¹⁴ Hz, 400 nm (violet) to 700 nm (red)

  • Radio waves: 10⁶ Hz, 1 km.