Study Notes on Alternating Current

ALTERNATING CURRENT

LEARNING OBJECTIVES

  • Understand and describe time period, frequency, peak and root mean square values of alternating current and voltage.

  • Know and use the relationship for the sinusoidal wave.

  • Understand the flow of alternating current (A.C.) through resistors, capacitors, and inductors.

  • Understand how phase lags and leads in the circuit.

  • Apply knowledge to calculate the reactances of capacitors and inductors.

  • Describe impedance as vector summation of resistances.

  • Know and use formulas of A.C. power to solve problems.

  • Understand the function of resonant circuits.

  • Appreciate the principle of metal detectors used for security checks.

  • Describe the three-phase A.C. supply.

  • Understand the production, transmission, and reception of electromagnetic waves.

DEFINITION OF ALTERNATING CURRENT

Alternating current (A.C.) is defined as current produced by a voltage source whose polarity keeps reversing over time. This means that in any given interval, one terminal (A) is positive while the other terminal (B) is negative, and this relationship switches after a specified time period (T).

Time Period

The time interval (T) during which the voltage source changes its polarity once is known as the period T of alternating current or voltage. The relationship between frequency (f) and time period (T) is given by the equation:

f = \frac{1}{T}

Waveform of Alternating Voltage

The output voltage (V) of an A.C. generator at any instant can be described by the formula:

V = V0 \sin(\omega t) = V0 \sin\left(\frac{2\pi}{T} t\right)
where:

  • V_0 is the maximum value of the voltage,

  • T is the period of rotation of the coil of the generator,

  • t is time,

  • and ( \omega ) is the angular frequency defined as ( \omega = \frac{2\pi}{T} ).

Graphical Representation of Waveform: The graph of V against time showcases the sinusoidal wave, with key points defined:

  • At t = 0, V = 0

  • At t = T/4, V = V_0

  • At t = T/2, V = 0

  • At t = 3T/4, V = -V_0

  • At t = T, V = 0

INSTANTANEOUS VALUE

The instantaneous value of voltage or current at any given time t from a reference point is defined as:

V = V0 \sin(\omega t) = V0 \sin(2\pi ft)
where V varies continuously with time.

PEAK VALUE

The peak value, denoted as V_0, refers to the maximum voltage or current observed in one complete cycle of A.C.

PEAK TO PEAK VALUE

The peak-to-peak value is defined as the sum of the positive and negative peak voltages and is represented as:

V{p-p} = 2V0

ROOT MEAN SQUARE (RMS) VALUE

The root mean square (rms) value is derived from the mean square current and voltage, and is calculated using the formula:

V{rms} = \frac{V0}{\sqrt{2}} = 0.707 V0 I{rms} = \frac{I0}{\sqrt{2}} = 0.707 I0
This value allows for a more practical measurement of A.C. current or voltage. Most A.C. measuring devices (like voltmeters) are calibrated to read rms values.

PHASE OF A.C.

In terms of A.C, the phase refers to the angle θ (in radians or degrees) that corresponds to the instantaneous value of alternating voltage or current:

V = V_0 \sin(\theta)

  • Where θ represents the angle which defines the phase of the sinusoidal wave.

  • For certain points on the waveform A, B, C, D, and E, the respective phases are 0, π/2, π, 3π/2, and 2π.

PHASE LAG AND PHASE LEAD

Phase lag and lead describe the difference in phase between two alternating waveforms. For instance, if we have two waveforms:

  • Waveform 1 at point B has a phase of π/2 and wave 2 at this point has a phase of 0, we say waveform 2 lags behind waveform 1.

  • In contrast, if waveform 2 leads, it would be at a phase greater than that of waveform 1 at each equivalent point in the cycle.

Vector Representation of Alternating Quantities

Vals representing A.C. quantities can be depicted using vectors which rotate counterclockwise where the length of the vector represents its peak or rms value and the angular frequency is consistent with that of the alternating quantity.

A.C. CIRCUITS

In A.C. circuits, resistors, capacitors, and inductors determine the behaviour of voltage and current.

A.C. THROUGH A RESISTOR

Using Ohm's law in purely resistive circuits, we know:

V = I R
In this case, the voltage across the resistor and the alternating current are in phase. This leads to the instantaneous power given as:
P = I^2 R

A.C. THROUGH A CAPACITOR

For capacitors, A.C. can flow because the alternating voltage continuously changes the polarity of the capacitor's plates, charging and discharging it.
The basic relationship between voltage (V) and charge (q) for a capacitor is:
q = C V
The alternating current through a capacitor leads the voltage by 90 degrees or π/2. The current can be determined as:
I = \frac{dq}{dt}
Current and voltage are sinusoidal but out of phase, with the current leading the voltage, represented as:
X_C = \frac{1}{2\pi f C}
This shows that reactance is inversely related to frequency; low frequency leads to high reactance and vice versa.

A.C. THROUGH AN INDUCTOR

For inductors, a changing current generates a back electromotive force (back EMF) opposing the flow, causing the current to lag behind the voltage by 90 degrees or π/2. When current is represented graphically, the respective currents and voltages follow:
X_L = 2\pi f L
The properties of inductors in A.C. circuits result in no power loss.

IMPEDANCE

Impedance (Z) is the combination of resistance and reactance in circuit elements and can be represented mathematically as:
Z = \sqrt{R^2 + (XL - XC)^2}
Where the angle θ shows the phase difference between current and voltage given by:
θ = \tan^{-1}\left(\frac{XL - XC}{R}\right)

R-C AND R-L SERIES CIRCUITS

In an R-C series circuit, reactance causes voltage to lag behind current while in R-L circuits, voltage leads the current.

POWER IN A.C. CIRCUITS

For A.C. circuits, power is given by:
P = V{rms}I{rms} cos θ
Where cos θ is termed as the power factor indicating efficiency.

RESONANCE CIRCUITS

SERIES RESONANCE (R-L-C)

In R-L-C series circuits, resonance occurs at a condition when reactance of inductor equals that of the capacitor, where the total impedance is minimal, and can be mathematically written as:
f_r = \frac{1}{2\pi \sqrt{LC}}

PARALLEL RESONANCE CIRCUITS

These circuits operate in a manner whereby at resonance frequency minimum current is drawn, given by:
f_r = \frac{1}{2\pi \sqrt{LC}}

THREE PHASE A.C. SUPPLY

In a three-phase A.C. generator, three coils inclined at 120 degrees to one another produce three alternating voltages with phase differences of 120 degrees between them. The advantages include more efficient power distribution across multiple load types.

METAL DETECTORS

Metal detectors function by detecting changes in inductance when an object interferes with the oscillating magnetic field generated by coils, allowing identification of metal within proximity.

CHOKE

Chokes are coils with significant inductance and small resistance, limiting current in A.C. circuits with minimal energy wastage compared to resistors.

ELECTROMAGNETIC WAVES

These waves propagate through vacuum, generated through acceleration of electric charges. They consist of mutually perpendicular electric and magnetic fields, with speed given by:
c = fλ

MODULATION

Modulation is a technique used in electronic communication, where information is impressed on a carrier wave. There are two types of modulation:

  1. Amplitude Modulation (AM) - The amplitude of the carrier changes with signal strength.

  2. Frequency Modulation (FM) - The frequency of the carrier changes with signal strength.

SOLVED EXAMPLES

  • Example 16.1: Calculate peak and instantaneous values given Vrms = 250 V, f = 50 Hz.

  • Example 16.2: Find reactance and current of a 100 μF capacitor connected to24 V A.Cof50 Hz`.

  • Further examples cover various circuit scenarios including impedance, current flow, and resonance calculations, demonstrating practical applications of A.C. theory in circuits.